Quantum Memory in Atomic Ensembles

Quantum Memory in Atomic

Ensembles

Joshua Nunn

St. John’s College, Oxford

Submitted for the degree of Doctor of PhilosophyHilary Term 2008

Supervised byProf. Ian A. Walmsley

Clarendon LaboratoryUniversity of Oxford

United Kingdom

For glory. I mean Gloria.

Abstract

This thesis is a predominantly theoretical study of light storage in atomic en-

sembles. The efficiency of ensemble quantum memories is analyzed and optimized

using the techniques of linear algebra. Analytic expressions describing the memory

interaction in both EIT and Raman regimes are derived, and numerical methods

provide solutions where the analytic expressions break down. A three dimensional

numerical model of off-axis retrieval is presented. Multimode storage is considered,

and the EIT, Raman, CRIB and AFC protocols are analyzed. It is shown that

inhomogeneous broadening improves the multimode capacity of a memory. Raman

storage in a diamond crystal is shown to be feasible. Finally, experimental progress

toward implementing a Raman quantum memory in cesium vapour is described.

Acknowledgements

I have been very lucky to work with a fantastic set of people. I owe a debt ofgratitude to my supervisor Ian Walmsley, whose implacable good humour alwaystransmutes frustration into comedy, and who has overseen my experimental failureswith only mild panic. Much of the theory was conceived in the course of meetingswith Karl Surmacz, who has been Dr. Surmacz for a year already. I fried my firstlaser in the lab with Felix Waldermann, and his patience and subtle sense of humourare missed — he is also qualified and long-gone! My current postdocs Virginia Lorenzand Ben Sussman have been a continual source of exciting discussion, and NorthAmerican optimism. We’re gonna make it work guys! And no misunderstandingcan survive a keen frisking at the hands of Klaus Reim, who is currently completinghis D.Phil — and mine — within our group. KC Lee has made the transitionfrom theorist to experimentalist, without a blip in his coffee intake, and he remainsan inspiration. The help and encouragement of Dieter Jaksch, and more recentlyChristoph Simon, are greatly appreciated. Thanks must also be due to my officeneighbours: Pete Mosley, who introduced me to the concept of a progress chart, andwith it a reification of inadequacy, and Dave Crosby, who go-karts better than hesings.

The rest of the ultrafast group divides cleanly into those who drink tea and thosewho do not. A great big thank you to the tea drinkers: you understand that a teabreak is more than a bathroom break with a drink. It lies at the heart of whatit means to prevaricate. Adam Wyatt knows this. He is a tea soldier. As for thetea-less philistines (you know who you are), I have nothing to say to you. (dramaticpause). Nothing.

Thanks to my Oxford massif, Andy Scott and Tom Rowlands-Rees, who know agood lunch when they see one. And in that vein, thanks to Matthijs Branderhorst,who along with Ben, introduced me to the burrito. There is a growing ultrafastdiaspora — good people in far-off places — and of these I should like to big-upDaryl Achilles and Jeff Lundeen, who are awethome even when in errr-rr.

I could not have made it this far without the support of Sonia: there can be fewless attractive prospects than the unshaven maniac that is a D.Phil student writingup. Thank you for keeping me sane!

And lastly my parents. Thanks mum and thanks dad! Next up: driving license.

Contents

1 Introduction 1

1.1 Classical Computation . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Quantum Computation . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 No cloning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.4 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Quantum Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Linear Optics Quantum Computing . . . . . . . . . . . . . . . . . . 11

1.5 Quantum Communication . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Quantum Repeaters . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6.1 The Ekert protocol . . . . . . . . . . . . . . . . . . . . . . . . 21

1.6.2 Entanglement Swapping . . . . . . . . . . . . . . . . . . . . . 22

1.6.3 Entanglement Purification . . . . . . . . . . . . . . . . . . . . 24

1.6.4 The DLCZ protocol and number state entanglement . . . . . 24

CONTENTS v

1.7 Modified DLCZ with Quantum Memories . . . . . . . . . . . . . . . 31

2 Quantum Memory: Approaches 35

2.1 Cavity QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2 Free space coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.1 EIT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.3.2 Raman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.3.3 CRIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.3.4 AFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.4 Continuous Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3 Optimization 67

3.1 The Singular Value Decomposition . . . . . . . . . . . . . . . . . . . 70

3.1.1 Unitary invariance . . . . . . . . . . . . . . . . . . . . . . . . 75

3.1.2 Connection with Eigenvalues . . . . . . . . . . . . . . . . . . 75

3.1.3 Hermitian SVD . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.1.4 Persymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.2 Norm maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.3 Continuous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.3.1 Normally and Anti-normally ordered kernels. . . . . . . . . . 81

3.3.2 Memory Optimization. . . . . . . . . . . . . . . . . . . . . . . 81

3.3.3 Unitary invariance . . . . . . . . . . . . . . . . . . . . . . . . 82

CONTENTS vi

3.4 Optimizing storage followed by retrieval . . . . . . . . . . . . . . . . 85

3.5 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4 Equations of motion 92

4.1 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2 Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.3 Dipole Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.5 Linear approximation (1) . . . . . . . . . . . . . . . . . . . . . . . . 102

4.6 Rotating Wave Approximation . . . . . . . . . . . . . . . . . . . . . 103

4.7 Unwanted Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.8 Linear Approximation (2) . . . . . . . . . . . . . . . . . . . . . . . . 106

4.9 Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.10 Paraxial and SVE approximations . . . . . . . . . . . . . . . . . . . 110

4.11 Continuum Approximation . . . . . . . . . . . . . . . . . . . . . . . 112

4.12 Spontaneous Emission and Decoherence . . . . . . . . . . . . . . . . 116

5 Raman & EIT Storage 120

5.1 One Dimensional Approximation . . . . . . . . . . . . . . . . . . . . 120

5.2 Solution in k-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 123

5.2.2 Transformed Equations . . . . . . . . . . . . . . . . . . . . . 124

CONTENTS vii

5.2.3 Optimal efficiency . . . . . . . . . . . . . . . . . . . . . . . . 125

5.2.4 Solution in Wavelength Space . . . . . . . . . . . . . . . . . . 129

5.2.5 Including the Control . . . . . . . . . . . . . . . . . . . . . . 134

5.2.6 An Exact Solution: The Rosen-Zener case . . . . . . . . . . . 137

5.2.7 Adiabatic Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.2.8 Reaching the optimal efficiency . . . . . . . . . . . . . . . . . 152

5.2.9 Adiabatic Approximation . . . . . . . . . . . . . . . . . . . . 155

5.3 Raman Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.3.1 Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.3.2 Matter Biased Limit . . . . . . . . . . . . . . . . . . . . . . . 167

5.3.3 Transmitted Modes. . . . . . . . . . . . . . . . . . . . . . . . 168

5.4 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

5.4.1 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

6 Retrieval 190

6.1 Collinear Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

6.1.1 Forward Retrieval . . . . . . . . . . . . . . . . . . . . . . . . 191

6.2 Backward Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.3 Phasematched Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . 207

6.3.1 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

6.3.2 Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

6.4 Full Propagation Model . . . . . . . . . . . . . . . . . . . . . . . . . 213

CONTENTS viii

6.4.1 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

6.4.2 Control Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

6.4.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 218

6.4.4 Read out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

6.4.5 Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

6.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

6.6 Angular Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

6.6.1 Optimizing the carrier frequencies . . . . . . . . . . . . . . . 229

6.6.2 Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

7 Multimode Storage 234

7.1 Multimode Capacity from the SVD . . . . . . . . . . . . . . . . . . . 235

7.1.1 Schmidt Number . . . . . . . . . . . . . . . . . . . . . . . . . 237

7.1.2 Threshold multimode capacity . . . . . . . . . . . . . . . . . 239

7.2 Multimode scaling for EIT and Raman memories . . . . . . . . . . . 241

7.2.1 A spectral perspective . . . . . . . . . . . . . . . . . . . . . . 242

7.3 CRIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

7.3.1 lCRIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

7.3.2 Simplified Kernel . . . . . . . . . . . . . . . . . . . . . . . . . 249

7.3.3 tCRIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

7.4 Broadened Raman . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

7.5 AFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

CONTENTS ix

8 Optimizing the Control 276

8.1 Adiabatic shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

8.2 Non-adiabatic shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 279

9 Diamond 286

9.1 Diamond Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286

9.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

9.3 Acoustic and Optical Phonons . . . . . . . . . . . . . . . . . . . . . 290

9.3.1 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

9.3.2 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

9.4 Raman interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

9.4.1 Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

9.4.2 Deformation Potential . . . . . . . . . . . . . . . . . . . . . . 296

9.5 Propagation in Diamond . . . . . . . . . . . . . . . . . . . . . . . . . 299

9.5.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

9.5.2 Electron-radiation interaction . . . . . . . . . . . . . . . . . . 301

9.5.3 Electron-lattice interaction . . . . . . . . . . . . . . . . . . . 307

9.5.4 Crystal energy . . . . . . . . . . . . . . . . . . . . . . . . . . 309

9.6 Heisenberg equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

9.6.1 Adiabatic perturbative solution . . . . . . . . . . . . . . . . . 311

9.7 Signal propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

9.8 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

9.9 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

CONTENTS x

9.10 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

10 Experiments 323

10.1 Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

10.2 Thallium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

10.3 Cesium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

10.4 Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

10.4.1 Temperature control . . . . . . . . . . . . . . . . . . . . . . . 329

10.4.2 Magnetic shielding . . . . . . . . . . . . . . . . . . . . . . . . 329

10.5 Buffer gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

10.6 Control pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

10.6.1 Pulse duration . . . . . . . . . . . . . . . . . . . . . . . . . . 333

10.6.2 Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

10.6.3 Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

10.7 Pulse picker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

10.8 Stokes scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

10.9 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

10.9.1 Optical depth . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

10.9.2 Rabi frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 344

10.9.3 Raman memory coupling . . . . . . . . . . . . . . . . . . . . 346

10.9.4 Focussing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

10.10Line shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

10.11Effective depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

CONTENTS xi

10.12Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

10.12.1 Pumping efficiency . . . . . . . . . . . . . . . . . . . . . . . . 355

10.13Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

10.13.1 Polarization filtering . . . . . . . . . . . . . . . . . . . . . . . 358

10.13.2 Lyot filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358

10.13.3 Etalons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

10.13.4 Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

10.13.5 Spatial filtering . . . . . . . . . . . . . . . . . . . . . . . . . . 362

10.14Signal pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

10.15Planned experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

11 Summary 369

11.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

A Linear algebra 374

A.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

A.1.1 Adjoint vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 377

A.1.2 Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

A.1.3 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

A.1.4 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

A.2 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

A.2.1 Outer product . . . . . . . . . . . . . . . . . . . . . . . . . . 384

A.2.2 Tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . 385

CONTENTS xii

A.3 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

A.3.1 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

A.4 Types of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

A.4.1 The identity matrix . . . . . . . . . . . . . . . . . . . . . . . 391

A.4.2 Inverse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 392

A.4.3 Hermitian matrices . . . . . . . . . . . . . . . . . . . . . . . . 393

A.4.4 Diagonal matrices . . . . . . . . . . . . . . . . . . . . . . . . 394

A.4.5 Unitary matrices . . . . . . . . . . . . . . . . . . . . . . . . . 396

B Quantum mechanics 399

B.1 Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

B.1.1 State vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

B.1.2 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

B.1.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 400

B.1.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

B.2 The Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . 403

B.2.1 The Heisenberg interaction picture . . . . . . . . . . . . . . . 405

C Quantum optics 407

C.1 Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

C.2 Quantum states of light . . . . . . . . . . . . . . . . . . . . . . . . . 410

C.2.1 Fock states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410

C.2.2 Creation and Annihilation operators . . . . . . . . . . . . . . 411

CONTENTS xiii

C.3 The electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

C.4 Matter-Light Interaction . . . . . . . . . . . . . . . . . . . . . . . . . 415

C.4.1 The A.p Interaction . . . . . . . . . . . . . . . . . . . . . . . 415

C.4.2 The E.d Interaction . . . . . . . . . . . . . . . . . . . . . . . 418

C.5 Dissipation and Fluctuation . . . . . . . . . . . . . . . . . . . . . . . 422

D Sundry Analytical Techniques 427

D.1 Contour Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

D.1.1 Cauchy’s Integral Formula . . . . . . . . . . . . . . . . . . . . 429

D.1.2 Typical example . . . . . . . . . . . . . . . . . . . . . . . . . 430

D.2 The Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . 432

D.3 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

D.3.1 Bilateral Transform . . . . . . . . . . . . . . . . . . . . . . . 434

D.3.2 Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

D.3.3 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

D.3.4 Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

D.3.5 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

D.3.6 Transform of a Derivative . . . . . . . . . . . . . . . . . . . . 436

D.4 Unilateral Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

D.4.1 Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

D.4.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

D.4.3 Transform of a Derivative . . . . . . . . . . . . . . . . . . . . 439

D.4.4 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . 440

CONTENTS xiv

D.5 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

D.5.1 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

D.5.2 Memory Propagator . . . . . . . . . . . . . . . . . . . . . . . 442

D.5.3 Optimal Eigenvalue Kernel . . . . . . . . . . . . . . . . . . . 445

E Numerics 447

E.1 Spectral Collocation . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

E.1.1 Polynomial Differentiation Matrices . . . . . . . . . . . . . . 452

E.1.2 Chebyshev points . . . . . . . . . . . . . . . . . . . . . . . . . 453

E.2 Time-stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

E.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 457

E.4 Constructing the Solutions . . . . . . . . . . . . . . . . . . . . . . . . 459

E.5 Numerical Construction of a Green’s Function . . . . . . . . . . . . . 462

E.6 Spectral Methods for Two Dimensions . . . . . . . . . . . . . . . . . 465

F Atomic Vapours 471

F.1 Vapour pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

F.2 Oscillator strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

F.3 Line broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476

F.3.1 Doppler broadening . . . . . . . . . . . . . . . . . . . . . . . 476

F.3.2 Pressure broadening . . . . . . . . . . . . . . . . . . . . . . . 477

F.3.3 Power broadening . . . . . . . . . . . . . . . . . . . . . . . . 480

F.4 Raman polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

List of Figures

1.1 The state space of a qubit . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 The BB84 protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3 Entanglement swapping . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.4 Single-rail entanglement swapping . . . . . . . . . . . . . . . . . . . 26

1.5 QKD with single-rail entanglement . . . . . . . . . . . . . . . . . . . 27

1.6 Λ-level structure of atoms for DLCZ . . . . . . . . . . . . . . . . . . 29

1.7 Generation of number state entanglement in DLCZ . . . . . . . . . . 31

1.8 Modification to DLCZ with photon sources and quantum memories . 33

2.1 The simplest quantum memory . . . . . . . . . . . . . . . . . . . . . 36

2.2 Adding a dark state . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3 Cavity QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4 Confocal coupling in free space . . . . . . . . . . . . . . . . . . . . . 40

2.5 Atomic ensemble memory . . . . . . . . . . . . . . . . . . . . . . . . 41

2.6 EIT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.7 Stopping light with EIT. . . . . . . . . . . . . . . . . . . . . . . . . . 45

LIST OF FIGURES xvi

2.8 Raman storage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.9 CRIB storage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.10 tCRIB vs. lCRIB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.11 AFC storage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.12 Wigner distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.13 Atomic quadratures. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.14 QND memory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.15 Level scheme for a QND memory. . . . . . . . . . . . . . . . . . . . . 66

3.1 Storage map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2 Linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.3 Persymmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1 The Λ-system again. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2 Time-ordering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.3 Useful and nuisance couplings. . . . . . . . . . . . . . . . . . . . . . 105

5.1 Quantum memory boundary conditions. . . . . . . . . . . . . . . . . 124

5.2 Bessel zeros. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.3 Optimal storage efficiency. . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4 The Rosen-Zener model. . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.5 Raman efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.6 Raman storage as a beamsplitter. . . . . . . . . . . . . . . . . . . . . 181

5.7 Modified DLCZ protocol with partial storage. . . . . . . . . . . . . . 182

LIST OF FIGURES xvii

5.8 Comparison of predictions for the optimal input modes in the adia-

batic limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

5.9 Comparison of predictions for the optimal input modes outside the

adiabatic limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

5.10 Broadband Raman storage. . . . . . . . . . . . . . . . . . . . . . . . 187

5.11 Broadband EIT storage. . . . . . . . . . . . . . . . . . . . . . . . . . 188

6.1 Forward retrieval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.2 Phasematching considerations for backward retrieval. . . . . . . . . . 200

6.3 Backward Retrieval. . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

6.4 Non-collinear phasematching. . . . . . . . . . . . . . . . . . . . . . . 209

6.5 Efficient, phasematched memory for positive and negative phase mis-

matches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

6.6 Focussed beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

6.7 Effectiveness of our phasematching scheme. . . . . . . . . . . . . . . 224

6.8 Comparing phasematched and collinear efficiencies. . . . . . . . . . . 225

6.9 Angular multiplexing. . . . . . . . . . . . . . . . . . . . . . . . . . . 230

6.10 Minimum momentum mismatch. . . . . . . . . . . . . . . . . . . . . 232

7.1 Bright overlapping modes are distinct. . . . . . . . . . . . . . . . . . 236

7.2 Visualizing the multimode capacity. . . . . . . . . . . . . . . . . . . 237

7.3 The appearance of a multimode Green’s function. . . . . . . . . . . . 239

7.4 Multimode scaling for Raman and EIT memories. . . . . . . . . . . . 244

LIST OF FIGURES xviii

7.5 Scaling of Schmidt number with broadening. . . . . . . . . . . . . . 252

7.6 Comparison of the predictions of the kernels (7.23) and (7.18). . . . 253

7.7 Understanding the linear multimode scaling of lCRIB. . . . . . . . . 254

7.8 Multimode scaling for CRIB memories. . . . . . . . . . . . . . . . . . 261

7.9 The multimode scaling of a broadened Raman protocol. . . . . . . . 268

7.10 The multimode scaling of the AFC memory protocol. . . . . . . . . . 275

8.1 Adiabatic control shaping. . . . . . . . . . . . . . . . . . . . . . . . . 284

8.2 Non-adiabatic control shaping. . . . . . . . . . . . . . . . . . . . . . 285

9.1 The crystal structure of diamond. . . . . . . . . . . . . . . . . . . . . 287

9.2 Phonon aliasing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

9.3 Phonon dispersion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

9.4 Band structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

9.5 An exciton. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

9.6 The Raman interaction in diamond. . . . . . . . . . . . . . . . . . . 298

10.1 Observing Stokes scattering as a first step. . . . . . . . . . . . . . . . 325

10.2 Thallium atomic structure. . . . . . . . . . . . . . . . . . . . . . . . 326

10.3 Cesium atomic structure. . . . . . . . . . . . . . . . . . . . . . . . . 327

10.4 First order autocorrelation. . . . . . . . . . . . . . . . . . . . . . . . 334

10.5 Second order interferometric autocorrelation. . . . . . . . . . . . . . 336

10.6 Stokes scattering efficiency. . . . . . . . . . . . . . . . . . . . . . . . 342

10.7 Cesium optical depth. . . . . . . . . . . . . . . . . . . . . . . . . . . 345

LIST OF FIGURES xix

10.8 Cesium D2 absorption spectrum. . . . . . . . . . . . . . . . . . . . . 350

10.9 Absorption linewidth. . . . . . . . . . . . . . . . . . . . . . . . . . . 352

10.10Equal populations destroy quantum memory. . . . . . . . . . . . . . 354

10.11Optical pumping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

10.12Verifying efficient optical pumping. . . . . . . . . . . . . . . . . . . . 357

10.13Lyot filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

10.14Stokes filtering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

10.15Backward Stokes scattering. . . . . . . . . . . . . . . . . . . . . . . . 364

10.16A possible design for demonstration of a cesium quantum memory. . 368

A.1 A vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

A.2 The inner product of two vectors. . . . . . . . . . . . . . . . . . . . . 380

A.3 A matrix acting on a vector. . . . . . . . . . . . . . . . . . . . . . . . 383

A.4 Eigenvectors and eigenvalues. . . . . . . . . . . . . . . . . . . . . . . 389

A.5 Non-commuting operations. . . . . . . . . . . . . . . . . . . . . . . . 390

A.6 A unitary transformation. . . . . . . . . . . . . . . . . . . . . . . . . 396

C.1 Symmetrized photons. . . . . . . . . . . . . . . . . . . . . . . . . . . 413

D.1 Contour integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

D.2 Upper closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

D.3 Integration limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

D.4 Lower closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

E.1 The method of lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

LIST OF FIGURES xx

E.2 Periodic extension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

E.3 Chebyshev Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

E.4 Example solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

E.5 A numerically constructed Green’s function. . . . . . . . . . . . . . . 464

E.6 Spectral methods in two dimensions. . . . . . . . . . . . . . . . . . . 470

F.1 Vapour pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

F.2 The Doppler shift. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

F.3 Collisions in a vapour. . . . . . . . . . . . . . . . . . . . . . . . . . . 478

F.4 Polarization selection rules. . . . . . . . . . . . . . . . . . . . . . . . 483

F.5 Alternative scattering pathways. . . . . . . . . . . . . . . . . . . . . 486

Chapter 1

Introduction

The prospect of building a quantum computer, with speed and power far outstripping

the best possible classical computers, has motivated an enormous and sustained

research effort over the last two decades. In this thesis we explore a number of

candidates for the ‘memory’ that would be required by such a device. As we will

see, building a quantum memory is considerably harder than fabricating the RAM

chips used by modern computers. For instance, it is not possible to copy quantum

information, nor can quantum information be digitized. These facts make quantum

storage particularly vulnerable to noise, and loss — problems for which solutions

must be found before quantum computation can mature into a viable technology.

The bulk of this thesis is concerned with optimization of the efficiency and storage

capacity of a quantum memory. We focus on optical memories, in which a pulse of

light is ‘stopped’ for a controllable period, before being re-released.

The structure of the thesis is as follows. In this chapter we introduce the concepts

2

of quantum computing and quantum communication, and we discuss the context

and motivation for the present work on quantum memories. In Chapter 2 we survey

the various approaches to quantum memory, and we describe the principles behind

the memory protocols analyzed later. Chapter 3 introduces the mathematical basis

for our approach to analyzing and optimizing ensemble memories — the Green’s

function and its singular value decomposition. In Chapter 4 we derive the equations

of motion describing the quantum memory interaction in an ensemble of Λ-type

atoms. In Chapter 5 we apply the techniques of Chapter 3 to this interaction. Several

new results are derived, and connections are made with previous work. Chapter 6

is concerned with retrieval of the stored excitations from a Λ-ensemble. It is shown

that both forward and backward retrieval are problematic. A numerical model

is presented that confirms the efficacy of an off-axis geometry, which solves these

problems. In Chapter 7 we move on to consider multimode storage. Our formalism

provides a natural way to calculate the multimode capacity of a memory, and we

study the multimode scaling of all the memory protocols introduced in Chapter 2.

Chapter 8 describes how to optimize a Λ-type memory by shaping the ‘control pulse’.

In Chapter 9 we study the Raman interaction in a diamond crystal: we show that a

diamond Raman quantum memory is feasible. Finally in Chapter 10 we review our

attempts to implement a Raman quantum memory in the laboratory, using cesium

vapour.

But let us begin at the beginning.

1.1 Classical Computation 3

1.1 Classical Computation

Classical computers are conventional computers, like the one I am using to typeset

this document. Their importance as enablers of technological progress, as well as

their utility as a technology in their own right, attest to the fantastic potential of

classical computation. They are typified by the use of bit1 strings — sequences of

1’s and 0’s — to encode information. Information is processed by application of

binary logic to the bits. That is, Boolean operations such as OR, AND or not-AND

(NAND). This last operation is a universal gate, because any logic operation can be

constructed using only NAND gates. Such a gate can be implemented electronically

using a pair of transistors, millions of which can be combined on a single silicon

chip. The rest is history.

Computers have progressed in leaps and bounds over the last fifty years. In 1965

computers were developing so fast that Gordon Moore, a founder of the industrial

giant Intel, proposed a ‘law’, stipulating that the number of transistors comprising

a processor would double every year [2]. Incredibly, this exponential improvement in

computing power has persisted for over 40 years. But improvement by miniaturiza-

tion cannot continue indefinitely. The reason for this is that the physics of electronic

components undergoes a qualitative change at small scales: classical physics becomes

quantum physics. In his 1983 lectures on computation [3], Richard Feynman consid-1‘Bit’ first appeared in Claude Shannon’s 1948 paper on the theory of communication as a

contraction of ‘binary digit’ [1]; the name is apposite, since one bit is the smallest ‘piece’ or ‘chunk’of information there can be: one bit of information is one bit of information. Shannon attributesthe term to John Tukey, a creator of the digital Fourier transform, who is also credited with coiningthe word ‘software’.

1.2 Quantum Computation 4

ers the fate of classical computation as shrinking dimensions bring quantum effects

into play. Two years later David Deutsch published the first explicitly quantum

algorithm [4], demonstrating how quantum physics actually permits more powerful

computation than classical physics allows. A quantum computer, capable of har-

nessing this greater power, must process quantum information, encoded not with

ordinary bits, but with quantum bits.

1.2 Quantum Computation

1.2.1 Qubits

A quantum bit — a qubit2 — is an object with two mutually exclusive states, 0

and 1, say. The only difference with a classical bit is that the object is described

by quantum mechanics. Accordingly, we label the two states by the kets |0〉 and |1〉

(see Appendix B). These kets are to be thought of as vectors in a two dimensional

space: the state space of the qubit (see Figure 1.1). The classical property of mutual

exclusivity is manifested in the quantum formalism by requiring that |0〉 and |1〉 are

perpendicular to one another in the state space. In general, the state of the qubit

can be any vector, of length 1, in the state space. Since both the kets |0〉 and |1〉

have length 1, and since they point in perpendicular directions, an arbitrary qubit

state |ψ〉 can always be written as a linear combination of them,

|ψ〉 = α|0〉+ β|1〉, (1.1)2The term ‘qubit’ first appears in a paper by Benjamin Schumacher in 1995 [5]; he credits its

invention to a conversation with William Wootters.


where α and β are two numbers which must satisfy the normalization condition

|α|2 + |β|2 = 1. States like (1.1), which are a combination of the two mutually

exclusive states |0〉 and |1〉, are called superposition states, or just superpositions.

What does it mean to say that a qubit is in a superposition between its two mutually

exclusive states? Somehow it is both 0 and 1 at the same time. Physically, this is

like saying that a switch is both ‘open’ and ‘closed’, or that a lamp is both ‘on’ and

also ‘off’. Already, for the simplest possible system, without any real dynamics —

no interactions, nothing happening — we see that the basic structure of quantum

mechanics does not sit well with our intuition. Despite these interpretational diffi-sta

te space

Figure 1.1 A visual representation of the state space of a qubit

culties, superposition is central to the success of quantum mechanics. Atomic and

molecular physics, nuclear and particle physics, optics and electronics all make use of

superpositions to successfully explain processes and interactions. From the point of

view of computation, the existence of states like (1.1) provides a clue to the greater

capabilities of a quantum computer. Each qubit has two ‘parts’, the |0〉 part and the


|1〉 part; logical operations on qubits act on both parts together, and the output of

a calculation also has these two parts. So there’s some sense in which a qubit plays

the role of two classical bits, stuck ‘on top of eachother’. David Deutsch coined the

term quantum parallelism for this property — he considers it to be the strongest

evidence for the existence of parallel universes. The B-movie-esque connotations

of this ‘many-worlds’ view make it generally unpopular among physicists, but the

appeal of quantum computing remains, independently of how it is understood.

1.2.2 Noise

A potential difficulty associated with quantum computing is also apparent from

(1.1): the numbers α and β can be varied continuously (subject to the normaliza-

tion constraint). So the number of possible states |ψ〉 of a qubit is infinite! This also

hints at their greater information carrying capacity, but it means that they must

be carefully protected from the influence of noise. Classical bits have exactly two

states; if noise introduces some distortions, it is usually possible to correct these

simply by comparing the distorted bit to an ideal one. Only very large fluctuations

can make a 0 look like a 1, so the discrete structure of classical bits makes them

very robust. By contrast, a perturbed qubit state is also a valid qubit state. In

this respect, the difference between bits and qubits can be likened to the difference

between digital and analogue musical recordings: The quality of music reproduced

by a CD does not degrade gradually with time, whereas old cassettes sound progres-

sively worse as distortions creep into the waveform imprinted on the tape. In fact,


it is possible to correct errors by constructing codes involving bunches of qubits.

The invention of these codes in 1996 by Calderbank, Shor and Steane [6,7] was a

major milestone in demonstrating the practical viability of quantum computation.

Nonetheless these error correcting schemes currently require that noise is suppressed

below thresholds of a few percent, which makes techniques for isolating qubits from

noise a technological sine qua non.

1.2.3 No cloning

Another problematic aspect of quantum information is that it cannot be copied.

The proof of this fact is known as the no-cloning theorem [8]. Suppose that we have

a device which can copy a qubit. If we give it a qubit in state |ψ〉, and also a ‘blank’

qubit in some standard initial state |blank〉, this machine spits out our original qubit,

plus a clone, both in the state |ψ〉. In Dirac notation, using kets, the action of our

qubit photocopier is written as

U |blank〉|ψ〉 = |ψ〉|ψ〉. (1.2)

Here U is the unitary transformation implemented by our machine. Unitary trans-

formations are those which preserve the lengths of the kets upon which they act.

Since all physical states have length 1, and any process must produce physical states

from physical states, it follows that all processes are described by length-preserving

— unitary — transformations (see §B.1.4 in Appendix B). Had we fed our machine


a different state, for example |φ〉, we would have

U |blank〉|φ〉 = |φ〉|φ〉. (1.3)

The length of a ket |ϕ〉 is defined by taking the scalar product 〈ϕ|ϕ〉 of |ϕ〉 with

itself (see §A.1.2 in Appendix A). To prove the impossibility of cloning, we take the

scalar product of the first relation (1.2) with the second, (1.3).

〈ψ|〈blank|U †U |blank〉|φ〉 = 〈ψ|〈ψ||φ〉|φ〉. (1.4)

The U acting on |blank〉 on the left hand side does not change its length, which is

just equal to 1, so the result simplifies to

〈ψ|φ〉 = 〈ψ|φ〉2. (1.5)

Clearly, this expression does not hold for arbitrary choices of |ψ〉 and |φ〉, and there-

fore cloning an arbitrary qubit is impossible. In fact, (1.5) is only true when 〈ψ|φ〉 is

either 1 or 0. The first case corresponds to |ψ〉 = |φ〉, which says that it is possible to

build a machine that can make copies of one particular, pre-determined state. The

second case occurs only when |ψ〉 and |φ〉 are perpendicular, as is the case for |0〉 and

|1〉. This says that it is possible to clone mutually exclusive states. Indeed, this is

precisely what classical computers are doing when they copy digitized information.

An immediate consequence of the no-cloning theorem is that a quantum memory


must work in a qualitatively different way to a classical computer memory. To

store quantum information, that information must be transferred to the memory,

rather than simply copied to it. It is never possible to ‘save a back-up’, as we

routinely do with classical computers. To build a quantum memory, we must find

an interaction between information carrier and storage medium which ‘swaps’ their

quantum states, so that the storage medium ends up with all the information, with

nothing left behind. In this thesis we will examine various ways of accomplishing this

optically, by considering collections of atoms which essentially ‘swallow’ a photon

in a controlled way, completely transferring the quantum state of an optical field to

that of the atoms.

1.2.4 Universality

Any classical computation is possible provided that one is able to apply NAND gates

to pairs of bits. What is required to perform arbitrary quantum computations?

This question is not trivial, but the answer is fortuitously simple [9]. Any quantum

computation can be performed, provided that one is able to arbitrarily control the

state of any qubit (single-qubit rotations), and provided that one can make pairs of

qubits interact with one another (two-qubit gates). It is generally sufficient to have

only a single type of interaction, so long as the final state of both interacting qubits

depends in some way on the initial state of both qubits. Such a gate is known as an

entangling gate, and they are notoriously difficult to implement.

1.3 Quantum Memory 10

1.3 Quantum Memory

In the light of the preceding discussion, a quantum memory can be understood as a

physical system that is well protected from noise, and that can be made to interact

with information carriers so that their quantum state is transferred into, or out of,

the memory. Note how we have distinguished the system comprising the memory

from the information carriers. In many cases, this distinction is artificial. For in-

stance, in ion trap quantum computing [10], the hyperfine states of calcium ions are

used as qubits. These ions are isolated from their noisy environment by trapping

them with oscillating electric fields; the quantum states of the qubits therefore re-

main un-distorted for long periods (on the order of seconds), and so there is no need

to transfer these states into a separate memory. But there are other proposals for

quantum computing that make explicit use of quantum memories. An example is

the use of nitrogen-vacancy centres in diamond for quantum computing [11,12]. Here

a single electron from a nitrogen atom, lodged in a diamond crystal and surrounded

by carbon atoms, is used as a qubit. The electron qubit can be controlled with laser

pulses to perform computations, but this very sensitivity to light makes it suscepti-

ble to damage from noise. Therefore a scheme was devised to transfer the quantum

state of the electron to that of a nearby carbon nucleus. The carbon nucleus is

de-coupled from the optical field, and it can be used to store quantum information

for many minutes.

A common theme among such computation schemes is an antagonism between

controllability and noise-resilience. That is, systems which are easily manipulated

1.4 Linear Optics Quantum Computing 11

and controlled with external fields are susceptible to noise from those same fields,

while well-isolated systems that are not badly affected by noise are generally hard

to access and control in order to perform computations. This trade-off leads to a

natural division of labour between systems that are easily manipulated, but short-

lived, and systems that are not easily controlled, but long-lived. Many quantum

computing architectures put both types of system to use, the former as quantum

processor, the latter as quantum memory.

1.4 Linear Optics Quantum Computing

Since James Clerk Maxwell wrote down the equations of electromagnetism in 1873,

physics has undergone profound upheavals at least twice, with the development of

both Relativity and Quantum Mechanics in the early twentieth century. Maxwell’s

equations have weathered these storms with astonishing fortitude, being both rel-

ativistically covariant and directly applicable in quantum field theory. They are

probably the oldest correct equations in physics. Implicit within them is a descrip-

tion of the photon, the quantum of the electromagnetic field. Photons come with

one of two polarizations, and superposition states of these polarizations are readily

prepared in the lab. In addition, they are themselves discrete entities, and it is pos-

sible to generate superpositions of different numbers of photons. Photons therefore

embody the archetypal qubit, and for this reason Maxwell’s equations remain as

central to the emerging discipline of quantum information processing as they were

to the pioneers of telegraphy and radio.

1.4 Linear Optics Quantum Computing 12

Photons occupy a frustrating territory on the balance sheet of usefulness for

quantum computation. They are ideal qubits, and arbitrary manipulation of their

polarization and number states can be accomplished with simple waveplates, beam-

splitters and phase-shifters. That is, single-qubit rotations are ‘cheap’. Unfortu-

nately, entangling gates between photons are much more difficult to realise. This

is unsurprising, since such a gate requires that two photons be made to interact

with one another, and it is well known that light does not generally interact with

light: torch beams do not ‘bounce off’ each other; rather they pass through each

other unaffected. In 2001 Emanuel Knill, Raymond Laflamme and Gerard Mil-

burn showed how to overcome these difficulties by careful use of measurements [13],

making universal quantum computation possible with only ‘linear optics’. Further

developments [14,15] have cemented linear optics quantum computing (LOQC) as an

important paradigm for the future of quantum computation. However the two-qubit

gates proposed are generally non-deterministic. As the number of gates required in

a computational step increases, the probability that all gates are implemented suc-

cessfully decreases exponentially, so that large computations must be repeated many

times for yielding reliable answers. This problem of scalability can be mitigated if

the photons output from successful gates can be stored until all the required gates

succeed. But photons generally have a short lifetime because they travel at the

speed of light: if they are confined in a laboratory they must be trapped by mirrors

(in a cavity) or by a waveguide (optical fibre), and absorption or scattering losses

are inevitable on time scales of milliseconds or greater [16]. Therefore the ability to

1.5 Quantum Communication 13

transfer the quantum state of a photon into a quantum memory would be a boon to

LOQC.

Another possibility for quantum computing with photons is to implement two-

qubit gates inside a quantum memory. Single-qubit operations are easily performed

on the photons directly; when interactions are needed, the photons are transferred

to atomic excitations which can be manipulated with external fields to accomplish

the entangling gates [17–19].

Applications such as these constitute the most ambitious motivation for the study

of optical quantum memories. In the next section we will see that quantum mem-

ories are also required in extending the range of so-called quantum communication

protocols, which provide guaranteed security from eavesdroppers.

1.5 Quantum Communication

Although practical quantum computing remains beyond the reach of current tech-

nology, another application of quantum mechanics has already made the leap into

the commercial sector. Quantum Key Distribution (QKD) is a technique which al-

lows two communicating parties to be absolutely certain, in so far as the laws of

physics are known to be correct, that their messages have not been intercepted [20].

It is possible to purchase QKD systems from two companies: MagiQ based in New

York, and ID Quantique in Geneva; many other businesses are incumbent, and the

market for such guaranteed-secure communication is estimated at around a billion

dollars annually. The idea behind QKD is simple: if Alice sends a message to Bob


in which she has substituted each letter for a different one, in a completely ran-

dom way, neither Bob, nor anyone else, can decode the message, unless Alice tells

them how she did the substitution. This information is known as the key, and only

someone in possession of the full key has access to the contents of Alice’s message.

Encrypting messages in this way is the oldest and simplest method of encryption. It

is absolutely and completely secure, provided that only the intended recipient has

access to the key. Once the key has been used, it should not be used again, since

with repeated use an eavesdropper, conventionally called Eve, might start to see

patterns in the encrypted messages and begin to guess the substitution rule. For

this reason this encryption protocol is known as the one time pad. For each message

sent, a new, completely random key must be used by both Alice and Bob. How does

Alice send the keys to Bob? If these are encrypted, she will need to send another

key beforehand, and our perfect security is swallowed by an infinite regression. If

she sends the keys unencrypted, can she be sure that Eve has not intercepted them?

If she has, then Eve has access to all of Alice’s subsequent messages, and there’s no

way for Alice or Bob to know their code has been cracked until the paparazzi arrive.

These issues are eliminated by public key cryptography. Here, Bob tells Alice

the substitution rule she should use for her message. The encryption is done in

such a way that Alice’s message cannot be decoded using this rule, so it doesn’t

matter if Eve discovers it. Alice then sends her coded message to Bob, who knows

how to decrypt the message. An implementation of this idea using the mathemat-

ics of large prime numbers was developed in 1978 by Ron Rivest, Adi Shamir and


Leonard Adleman [21]. The RSA cryptosystem is the industry standard for secure

communication over the internet; much of modern finance relies on its security. But

unlike the one time pad, no-one has proved that it is secure. The RSA algorithm

relies on the empirical fact that it is computationally very demanding to find the

prime factors of a large number. That is, if two large prime numbers p and q are

multiplied together to give their product n, it is not practically possible to find p

and q, given knowledge of n alone. The best algorithm, the number field sieve, can

find the factors of n in roughly eN1/3 log

2/32 N computational steps [22], where N is

the number of bits needed to represent n. This exponential scaling means that it

is easy to make the calculation intractably long by making n just a little larger.

But this is not the whole story. In 1994 Peter Shor showed how a quantum com-

puter could be used to perform this factorization much faster [23]. Shor’s algorithm

requires just N2(log2N) log2(log2N) steps, an exponential improvement over the

best conventional methods. A quantum computer that can implement this algo-

rithm efficiently does not yet exist, although proof-of-principle experiments using

LOQC have been performed [24–26]. But it is now known that the RSA cryptosystem

is living on borrowed time: if a practical quantum computer is ever made, modern

secure communications will be spectacularly compromised.

Enter QKD. QKD makes use of quantum mechanics to distribute the keys re-

quired for a one time pad protocol in a secure way, avoiding the infinite regress

arising from a classical protocol, and obviating the need to rely on the fatally flawed

RSA system. The goal is to provide both Alice and Bob with an identical string


of completely random bits, which they can use as keys to encrypt and decrypt a

message. The most widely known protocol used to do this is known as BB84, after

the 1984 paper by Bennett and Brassard [27]. Alice sends photons, one at a time, to

Bob. Alice can choose the polarization of each photon to point in one of four possi-

ble directions: horizontal, vertical, anti-diagonal or diagonal (|H〉, |V 〉, |A〉 or |D〉;

see Figure 1.2). These directions form a pair of perpendicular polarizations, with a

quarter-turn between them. Each of these pairs is known as a basis. The unrotated

basis contains the |H〉 and |V 〉 polarizations, and is known as the rectilinear basis.

The rotated basis contains the |A〉 and |D〉 polarizations, and is referred to as the

45-degree basis. Bob can measure the polarization of the photons he receives from

Alice, but to do so he has to line up his detector with either the rectilinear or the

45-degree basis — he has to choose. The measurement gives one of two possible

results, either 0 or 1, but these results mean nothing unless the photon polariza-

tion belonged to the basis that Bob chose to measure. For example, a |D〉 photon

polarized in the 45-degree basis will give a completely random result, either 0 or 1

with equal probability, if Bob aligns his detector with the rectilinear basis. So if he

gets the basis wrong, his measurement results are useless. If he chooses correctly

however, and the photon is polarized in the same basis that he measures in, then the

0 or 1 results tell him to which of the two possible directions in that basis the photon

polarization belonged. So for instance if Bob aligns his detector with the 45-degree

basis, the |D〉 photon will always give a 1 result. An |A〉 photon would give a 0 result

for this measurement, while either |H〉 or |V 〉 photons would give random results.


This strange property of photons, that measurements give useful or uncertain results

depending on the measurement basis, is a manifestation of Heisenberg’s uncertainty

principle [28]. It is uniquely quantum mechanical.

rectilinear

45-degree

Figure 1.2 BB84 protocol. Alice sends photons to Bob with polar-izations chosen randomly from the four possible directions |H〉/|V 〉and |A〉/|D〉, represented here as qubit states. To measure the polar-ization, Bob (or Eve) must choose a basis, rectilinear or 45-degree,for their measurement. Only photons polarized in this basis will yielduseful information.

To proceed with QKD, Alice generates two completely random, unrelated, bit

strings. For each photon, she uses the first bit string to decide in which basis to

polarize her photon. For instance, 0 could signify rectilinear and a 1 would mean 45-

degree. Then she uses the second bit string to decide which of the two perpendicular

polarizations in that basis to use. When Bob receives Alice’s photons, he records

the results of his measurements, and the basis he used for each measurement. Alice

then sends Bob her first bit string. This tells him the bases each photon belonged

to. He knows that his results are useless every time he chose the wrong basis for

his measurement. So he discards these results. The remaining results tell him the


correct polarization of each photon. That is, Bob’s remaining results now tally

exactly with Alice’s second bit string. Alice and Bob now share a cryptographic

key they can use for a one-time pad. But what about Eve? Well, Eve may have

intercepted Alice’s first bit string, but this only contains information about which

results to discard, it tells Eve nothing about what those measurement results were,

so this does not help her in cracking Alice and Bobs’ code. She could also have

intercepted the photons that Alice sent, and tried to measure their polarizations.

But, just like Bob, she has to guess at which basis to measure in. She gets a result,

but she has no idea whether the basis she chose is correct. She has to send photons

on to Bob, otherwise he will receive nothing and get suspicious. But Eve does not

know what polarization to give her photons, because she doesn’t know whether her

measurements are reliable or not. Suppose she just decides to send photons polarized

according to her measurement results in the basis she chose to measure in. Bob

receives these photo ns and measures them, none the wiser. But after Alice sends

her first bit string, and Bob discards his unreliable measurements, Bob’s remaining

results may not tally perfectly with Alice’s anymore. This is because sometimes Eve

will have chosen a different basis to Alice, obtained useless measurement results, and

sent photons to Bob with the wrong polarization. So Bob and Alice can compare

their keys, or a small part of them, and if they do not match up, they know that Eve

has been tampering with their photons. There is no way for Eve to listen in without

Alice and Bob discovering her presence. In quantum mechanics, measurements affect

the system being measured, and the BB84 protocol exploits this fact to guarantee

1.6 Quantum Repeaters 19

secure communication.

Quantum memories are not needed in the above protocol, provided Alice’s pho-

tons survive to reach Bob. As mentioned in Section 1.4 in the context of LOQC,

photons generally do not survive for longer than around 1 ms in an optical fibre, so

Bob should not be further than around 200 miles from Alice, otherwise her photons

will be scattered or absorbed before they reach him. In order to extend the distance

over which QKD is possible, some kind of amplifier is needed, which can give the

photons a ‘boost’, while maintaining their quantum state — that is, their polar-

ization. But the photons are qubits. They cannot simply be copied; we know this

from the no-cloning theorem (see Section 1.2.3). What is required is a modification

to the protocol described above, and a device known as a quantum repeater. Such

a device requires a quantum memory. If quantum memories can be made efficient,

with storage times on the order of 1 s, intercontinental quantum communication be-

comes possible. In the next section we introduce the quantum repeater, and discuss

the usefulness of quantum memories in this context.

1.6 Quantum Repeaters

A quantum repeater is a device designed to extend entanglement. Entanglement is

a purely quantum mechanical property that can be used as a resource to perform

QKD. In this section we will introduce entanglement, examine how it degrades over

long distances, and how quantum repeaters ameliorate this degradation.

Entanglement is a property of composite quantum systems. As an example,


consider two qubits. Classically, a system composed of two parts could be described

by the states of each part. In quantum mechanics this is not always true. Just as

a qubit can exist in a superposition of different states, so a system comprising two

qubits can exist in a superposition of different combined states. Such states arise

from a blend of correlation and indeterminism. To see this, suppose that we have a

machine that produces two photons, each with the same polarization. Now suppose

that the direction of this polarization is not fixed. It might polarize both photons

horizontally, we’ll label this polarization |0〉, or vertically, |1〉. We have no way of

knowing which of these two polarizations the machine uses, we only know that both

photons will have the same polarization. Such a state cannot be described by talking

about each photon in turn, as is clear from the language we used to describe the set

up. Using subscripts to denote the two photons, the state is written as

|ψ〉 =1√2

(|0〉1|0〉2 + |1〉1|1〉2) . (1.6)

The factor of 1/√

2 appears simply to fix the length of the state vector |ψ〉 to 1. This

state is an entangled state, because there is no way to write it as a product of states

of the first photon with states of the second. It expresses the two properties of our

machine: first, that the two photons always have the same polarization, and second,

that it is not certain which of the two polarizations will be produced. In fact, because

the two possible states |0〉 and |1〉 are mutually exclusive, (1.6) is a maximally

entangled state, sometimes known as a Bell state. Bell states represent much of


what is counter-intuitive about quantum mechanics. Their name derives from John

Bell’s famous 1964 paper [29] in which he proves that these states are incompatible

with local realism. A ‘local’ world is one in which no effect can propagate faster

than the speed of light; a ‘real’ world is one in which all properties can be assigned

definite values at all times. That modern physics describes states which do not admit

a local realistic interpretation is intriguing and controversial. Below we will see that

these states are also a resource for quantum communication. If their use becomes

widespread, we will be in the awkward position of deriving practical benefits from

a technology based on a philosophical conundrum!

1.6.1 The Ekert protocol

In 1991 Artur Ekert proposed a modification of the BB84 QKD protocol based on

the use of Bell states like (1.6) [30]. In this protocol, our machine for generating

entangled photon pairs is used. One photon from each pair is sent to Alice, the

other to Bob. Now Alice and Bob both have polarization detectors; they each have

to choose a basis to measure their photons in. Sometimes they will choose the same

basis as eachother, sometimes they will choose different bases. When they choose

differently, their results are meaningless, but when they choose the same basis, their

results are perfectly correlated. This is obvious for the rectilinear basis by inspection

of the form of (1.6). A little algebra shows that the same perfect correlations also

hold if both Alice and Bob measure in the 45-degree basis. The QKD is accomplished

in the same way as for the BB84 protocol: Alice tells Bob the measurement bases


she used; Bob discards the results of all measurements where his basis differed from

Alice’s. Alice and Bob then compare part of the remaining results to check that

they are correlated, as they should be. Poor correlations signify the presence of an

eavesdropper.

The importance of this modified protocol is that entanglement is a transferrable

resource. Below we will see how entanglement can be swapped between photons to

extend the range of quantum communication.

1.6.2 Entanglement Swapping

Entanglement swapping allows one to entangle two photons that have never encoun-

tered eachother. The situation is sketched in Figure 1.3. Two sources each emit

a pair of entangled photons in the state (1.6). One photon from each pair is sent

into a polarizing beam splitter, which transmits horizontally polarized photons, and

reflects vertically polarized photons. Behind the beamsplitter are a pair of photon

detectors. The beamsplitter has the effect of limiting the information we can learn

about the photons from the detectors. For instance, if both photon detectors D1

and D2 fire together, it could be that photons (2) and (3) were both vertically po-

larized, or that they were both horizontally polarized. That is, a ‘coincidence count’

from D1 and D2 only tells us that photons (2) and (3) had the same polarization;

it reveals nothing about what that polarization was. But we know from the state

(1.6) that photon (1) has the same polarization as photon (2), and similarly that

photon (4) has the same polarization as photon (3). So if photons (2) and (3) have


the same polarization, so do photons (1) and (4). Their polarization is unknown,

but correlated. Therefore, after a coincidence count, the two remaining photons,

(1) and (4), are in a Bell state. The entanglement between photons (1)-(2) and

(3)-(4) has been swapped to photons (1)-(4). This procedure was first demonstrated

experimentally by Jian Wei-Pan et al. in 1998 [31], and is now an essential tool for

LOQC.

&

1 2 3 4

PBS

D1 D2

S1 S2

Figure 1.3 Entanglement swapping. Two independent sources, S1and S2, emit pairs (1)-(2) and (3)-(4) of polarization entangled pho-tons. Photons (2) and (3) are directed into a polarizing beam splitter(PBS). When both detectors D1 and D2 fire behind the PBS, photons(1) and (4), which have never met, become entangled.

It’s clear from the above arguments that entanglement swapping is not much

more than a re-assignment of our knowledge regarding correlations, in the light of

a measurement carefully designed to reveal only partial information. Nonetheless,

a real resource — entanglement — has been extended over a larger distance by this

procedure. And QKD can now be performed using photons (1) and (4).


1.6.3 Entanglement Purification

So far we have shown how entanglement can be extended over large distances by

swapping perfect Bell states, each distributed over shorter distances. In practice,

however, propagating even over short distances can distort the polarizations of the

photons. Small distortions do not completely destroy the entanglement; rather there

is a smooth degradation in the usefulness of the photons for QKD as the distortions

become worse. Nevertheless, with each entanglement swap, these deleterious effects

are compounded, so that the entanglement vanishes after only a few swapping op-

erations. However, it is possible to transform several poorly entangled photon pairs

into one photon pair with near-perfect entanglement using an ‘entanglement pu-

rification protocol’. Several of these exist [32–36], generally they involve mixing and

measuring photons in a way that strengthens the correlations between the remaining

photons. Such procedures allow entanglement to be ‘topped up’, at the expense of

having to use more photons. QKD across distances much larger than those over

which distortions affect photons can then be implemented. Below we will introduce

a paradigm for constructing a quantum repeater which does not explicitly make

use of polarization entanglement, but which combines entanglement swapping and

entanglement purification into a single step.

1.6.4 The DLCZ protocol and number state entanglement

As mentioned previously, photons have several different degrees of freedom that can

be used to encode qubits. We have mostly focussed on polarization qubits for QKD,


but in the following protocol qubits are encoded in the number of photons occupying

a given optical mode, so-called single-rail encoding. Consider a machine similar to

the Bell state sources discussed above, which emits one photon either to the left, or

to the right. Using subscripts L and R for these directions, the state produced by

this machine is

|ψ〉 =1√2

(|0〉L|1〉R + |1〉L|0〉R) . (1.7)

Note that the states |0〉 and |1〉 now refer to the number of photons, rather than the

photon polarization as before. This state is also a maximally entangled state. Like

(1.6) it is also a Bell state. It expresses the correlation that a photon in one mode

always signifies the absence of a photon in the other mode. And it expresses the

indeterminacy, built into our device, that there is no way of knowing whether the

emitted photon will be found in the left or the right mode. How does entanglement

swapping work on such a state? A slightly different set-up is used (see Figure 1.4).

The action of the measurement is particularly clear in this example: the beamsplitter

(BS) mixes the two modes (2) and (3), so that a detection at D1 or D2 tells us only

that one of those modes contained a photon, but not which one. A bit of epistemic

book-keeping reveals the entanglement swap: if (2) contained a photon but (3) did

not, that means (1) had no photon while (4) carries a photon. Similarly if (2) was

empty but (3) contained a photon, (1) must carry a photon while (4) does not.

There is no way to distinguish these possibilities, and therefore a single detection

behind the BS puts the modes (1) and (4) into the entangled state (1.7).

For single-rail encoding, the most damaging effect of propagation is the pos-


|

1 2 3 4

BS

D1 D2

S1 S2

Figure 1.4 Single-rail entanglement swapping. Two independentsources, S1 and S2, emit single photons into modes (1)-(2) and (3)-(4)in the state (1.7). Modes (2) and (3) are mixed on a beam splitter(BS). When one of the detectors D1 or D2 (but not both) fires behindthe BS, modes (1) and (4) become entangled.

sibility of photon loss, through absorption or scattering. This has the effect of

introducing a third term into the state (1.7) of the form |0〉L|0〉R, corresponding to

no photons in either mode (they’ve all been lost!). With the addition of this term,

the quality of the entanglement is reduced. But this ‘vacuum’ component can never

cause any detection events. Therefore if either of the detectors D1 or D2 fire, sig-

naling a successful entanglement swap, the vacuum component is removed, since the

detection of a photon renders it counterfactual. For this type of state, entanglement

swapping also accomplishes a degree of entanglement purification.

The ability to perform both swapping and purification using such a simple mea-

surement makes this type of encoding attractive as a means of distributing en-

tanglement over large distances for quantum communication. However, it is not

immediately obvious how to generalize the Ekert QKD protocol to states encoded in

this way. For example, to perform a measurement in the analogue of the 45-degree

basis, one would require detectors that are sensitive to superpositions of photon


number states. These issues are avoided by combining two states of the form (1.7);

the measurement scheme is shown below in Figure 1.5. Two entangled states of the

1 2

3 4

BS

D2

D4D3

D1

BS

&

&

3

Figure 1.5 QKD with single-rail entanglement. Two entangledstates are distributed between Alice — detectors D1 and D3 on theleft — and Bob — detectors D2 and D4 on the right. The photonson each side are mixed on a beamsplitter (BS). A ‘polarization mea-surement’ is made when a single detector fires on each side. Both D1and D2 firing is a ‘0’. D3 and D4 firing is a ‘1’. Adjusting the phasesφA,B allows to select the measurement bases.

form (1.7), are distributed between Alice and Bob. Alice, on the left hand side,

has two detectors, D1 and D3. Bob, on the right, has detectors D2 and D4. If

Alice only records measurements when one of her detectors fires, she knows that one

photon came to her, while the other went to Bob. Similarly a single detection at

Bob’s side tells him that Alice received the other photon. The phases φA and φB

are independently chosen by Alice and Bob from the set 0, π/2. These phases,

in combination with the beamsplitters, allow Alice and Bob to control the basis in

which their detectors measure, in direct analogy with the rectilinear and 45-degree

bases of the Ekert protocol. When they choose the same phases, their measurements


should be correlated, with D3 and D4 firing together for a ‘1’ result, and D1 and D2

firing together for a ‘0’ result. Their is no correlation if they choose different phases.

Alice and Bob publicly announce their basis choices, and then compare some of their

results to check for the presence of Eve.

The above discussion shows how number state entanglement can be used for

quantum communication over long distances. A specific proposal for implement-

ing this protocol using atomic ensembles to generate the entangled states was first

made in 2001 by Lu-Ming Duan, Michael Lukin, Ignacio Cirac and Peter Zoller [37].

The DLCZ protocol is exciting because it is firmly grounded in feasible technol-

ogy. Several improvements have since been suggested [38–42], which make the scheme

more robust to phase instability, photon loss, detector noise and inefficiency. Below

we briefly introduce the principle behind the original protocol, since this will serve

as a useful introduction to the uses of atomic ensembles in quantum information

processing technologies.

We consider two clouds of atoms L and R, each with internal energy levels ar-

ranged as depicted in Figure 1.6. This type of Λ-structure is ubiquitous in quantum

information protocols. We will encounter it many times in our survey of quantum

memory protocols. It also arises in all but the simplest quantum systems: not just

atoms, but crystals, quantum dots and molecules, as we will see. The atomic ensem-

bles L and R will together play the role of the entangled photon source introduced

earlier. Here’s how it works. Both ensembles are pumped simultaneously by a laser

pulse. The pump pulse is tuned out of resonance with the atomic state |2〉, so that


Generation Readout

(a) (b)

Figure 1.6 Λ-level structure of atoms used in the DLCZ protocol.(a): A pump pulse excites an atom out of the ground state |1〉, whichdecays down to a long-lived metastable state |3〉, emitting a ‘Stokesphoton’. (b): A readout pulse brings the excited atom back to theground state, which emits an ‘anti-Stokes photon’ in the process.

most of the time nothing happens. But provided that the number of atoms in each

cloud is large enough, there is a small probability that the pump pulse will cause a

two-photon Raman transition in one of the atoms (see §2.3.2 in Chapter 2), exciting

it to a long-lived metastable state |3〉, as shown in Figure 1.6 (a). Such an excita-

tion is always accompanied by the emission of a Stokes photon. The probability of

two atoms being excited this way is negligibly small. Therefore at most one Stokes

photon is emitted from the ensembles. Due to the extended, pencil-like geometry

of the ensembles, any Stokes photons tend to be emitted in the forward direction,

so that they can be captured and directed as required [43]. Detectors D1 and D2

are placed behind a beamsplitter in front of the ensembles, as shown in Figure 1.7.

The beamsplitter mixes the optical modes from the two ensembles, so that if one

of the detectors fires, we know that a Stokes photon was emitted from one of the

ensembles, but we don’t know which. After a detection then, the state of the atomic


ensembles is of precisely the form (1.7), where |1〉 and |0〉 now refers to the presence

or absence of an excited atom in an ensemble.

We have now generated the number state entanglement required for the repeater

protocol. A difference with the preceding discussion is that the entanglement is

between the atomic ensembles, rather than optical modes. But this is an advan-

tage, because the atomic excitations are stationary, and can last for a long time,

while photons are generally short lived, as mentioned earlier. Since all the steps of

the protocol involve waiting for particular detectors to fire, it is essential that the

entanglement can be preserved until all the required steps are completed. When

the entanglement is ‘needed’, for instance to perform an entanglement swapping

operation, the atomic excitations can be converted back into photons. This is done

by applying a ‘readout’ pulse to the ensembles, which returns any excited atoms to

their ground states via an anti-Stokes transition, as shown in Figure 1.6 (b). The

anti-Stokes modes inherit the entanglement from the ensembles. If the anti-Stokes

mode from ensemble L is sent off in one direction (left), and the anti-Stokes mode

from the R ensemble is sent in the other direction (right), we now have a device

that can emit entangled states on-demand. The sources S1 and S2 appearing in

Figure 1.4 should now be thought of as each comprised of a pair of entangled atomic

ensembles, waiting to be ‘read-out’ when desired.

So far we have seen how atomic ensembles show promise for quantum communi-

cation protocols, but we have not encountered a specific need for quantum optical

memories. To conclude this section, we describe how the DLCZ protocol can be

1.7 Modified DLCZ with Quantum Memories 31

BS

D1

D2

L

R

Figure 1.7 Generation of number state entanglement using atomicensembles in the DLCZ quantum repeater protocol. A click in oneof the detectors D1 or D2 tells us that one atom, in either of theensembles L or R, is excited, but because of the beamsplitter (BS),we don’t know to which ensemble this atom belongs. Therefore theensembles are left in the entangled state (1.7).

improved by using memories in combination with single photon sources [41].

1.7 Modified DLCZ with Quantum Memories

Recall that we claimed the probability for two atoms in the ensembles L and R to be

excited by the pump pulses was so small as to be insignificant. If two atoms could

be excited, it would mean that sometimes one Stokes photon from L and one from

R is emitted simultaneously. Suppose that one of these photons is lost somehow.

Perhaps it is absorbed in an optical fibre. Then only one of the detectors D1 or

D2 behind the beamsplitter will fire, but both ensembles are in fact excited. To

account for this possibility, we should add a term to the state (1.7) of the form

|1〉L|1〉R — that is, the ensembles are no longer in a maximally entangled state. Of

course the probability of a double excitation can always be made arbitrarily small

by making the pump pulses weaker. But this also reduces the probability of even


a single excitation, so that it is necessary to wait a long time before one of the

detectors fires. If the waiting time becomes too long, the entanglement stored in the

ensembles starts to degrade as the atoms drift and collide, so this limits the distance

over which entanglement can be distributed.

This issue can be resolved if the entanglement is generated using a single-photon

source and a beamsplitter. The modified set up is shown in Figure 1.8. The atomic

ensembles L and R are now used as quantum memories, which we label QML and

QMR. We will explore the details of atomic ensemble quantum memories in Chapters

4–8. For now we only need to know that a photon entering a quantum memory is

trapped until it is needed. We have assumed that we have access to single photon

sources (SL and SR) that each emit one and only one photon, on-demand. Sources

like this are actually rather difficult to make, but research into this technology has

advanced greatly over the last few years [44–47], and including such sources into the

design is no less realistic than the inclusion of quantum memories. To generate

number-state entanglement, we start by triggering both sources SL and SR. Each

emits a photon, which then encounters a beamsplitter (BS). At the beamsplitters

the photons can either be reflected into a quantum memory, or transmitted, in which

case they are brought together on a final BS placed in front of detectors D1 and

D2. In the case that just one of the detectors fires, we know that only one photon

was transmitted at the first BS; the other photon must have been reflected, in which

case it is now stored in a quantum memory. But the final BS prevents us knowing

which quantum memory contains the photon. We therefore have a superposition


state, with either QML excited and QMR empty, or vice versa. This is the desired

entangled state (1.7). What if both photons are transmitted, but one of them is

BS

BS

BS

D1

D2

QML

QMR

SL

SR

Figure 1.8 Modification to the DLCZ protocol using single photonsources (SL and SR) and quantum memories (QML and QMR). Thenumber state entanglement is generated by beamsplitters.

somehow lost? In that case only one detector fires, but neither quantum memory

is excited. We therefore have to add a vacuum component of the form |0〉L|0〉R

to the state (1.7). This certainly degrades the entanglement in the same way as

the |1〉L|1〉R component did for the corresponding error in the DLCZ protocol. But

when we perform entanglement swapping to extend the entanglement, we always

wait until the appropriate detector fires (see Section 1.6.2), and this purifies the

entanglement, removing the vacuum term (see Section 1.6.3). Therefore vacuum

errors do not damage the protocol, and we are able to run it faster and further than

the original DLCZ scheme permits.

Hopefully the possible uses for quantum optical memories is now clear. Presum-


ably future proposals will develop further applications for them. In the next chapter,

we will review the various techniques used for quantum storage.

Chapter 2

Quantum Memory: Approaches

The aim of a quantum optical memory is to convert a flying qubit — an incident

photon — into a stationary qubit — an atomic excitation. This conversion should

be reversible, so that the photon may be re-emitted some time later, at the behest of

the user. In this chapter we review a number of approaches for achieving this kind

of quantum storage. More detailed calculations are deferred until the next chapter.

The simplest system one could imagine for storing a photon would consist of

a single atom, coupled to a single optical mode via an electric dipole transition

(see Figure 2.1). An incident photon, resonant with the transition, is absorbed,

promoting the atom from its ground state |1〉 to its excited state |2〉. The photon

is now ‘trapped’ as a stationary excitation of the atom, and the quantum storage

is complete. However, quantum mechanics is always invariant under time-reversal.

That is to say, the electric dipole interaction (see §C.4 in Appendix C), which allows

the photon to be absorbed by the atom, also causes the atom to re-emit the photon,

36

00

0.5

1

time/

Excita

tio

n p

rob

ab

ility

(a) (b) (c)

Figure 2.1 The simplest quantum memory. (a): A single atomis coupled to a single optical mode. (b): An incident photon is ab-sorbed, exciting the atom. (c): Unfortunately, time reversal symme-try requires that the photon is immediately re-emitted.

almost immediately afterwards. In fact, these two processes compete continuously,

so that over time the population of the excited state oscillates back and forth as

the atom absorbs and re-emits the photon; see Figure 2.1 (c). This behaviour is

known as Rabi flopping, after Isidor Isaac Rabi, who first used the phenomenon

in the context of nuclear magnetic resonance [48]. The Rabi frequency Ω of these

oscillations is proportional to the strength of the coupling between the atom and

the electromagnetic field.

The above scheme needs some modification if it is to store a photon in a controlled

way. One solution is to introduce a third state |3〉 — a dark state — that is not

coupled to the photon mode we want to store. We should have some control field

we can apply that transfers the atomic state from |2〉 to |3〉 once the storage is

complete. In this way the Rabi oscillations are ‘frozen’, and the photon remains

trapped as an excitation of the dark state for as long as is desired. Provided the

atom is well isolated, the dark state will persist for as long as is needed. To retrieve

37

the photon, the control field is applied again, transferring the atomic state from

|3〉 back to |2〉. The Rabi flopping of Figure 2.1 (c) continues and the photon is

re-emitted. The scheme is shown in Figure 2.2. This typifies the approach taken in

(a) (b) (c)

input

control

storage retrieval

Figure 2.2 Adding a dark state. (a): We address the atom withan auxiliary control. (b): After the input photon is absorbed, thecontrol transfers the atom to |3〉. (c): When the photon is needed,the control is re-applied.

many quantum optical memories, although the details of the protocols differ widely.

While single atoms have been used, ensembles of many atoms, each with the Λ-type

structure of Figure 2.2, are also commonly employed. The atom(s) can be enclosed

in an optical cavity, trapped at the centre of a confocal microscope, or addressed

by collimated beams. The optical fields may be resonant, or off-resonant with the

atomic transitions; the shape and timing of the control fields can vary, and indeed

non-optical controls, such as magnetic or electric fields, may be used. We will briefly

review this menagerie of memory protocols in the following sections.

2.1 Cavity QED 38

2.1 Cavity QED

In the forerunning discussion we presumed that a single atom could be coupled to a

single optical mode. This is a highly unnatural state of affairs, since the electromag-

netic field pervades all of space, so that atoms are generally surrounded by a bath of

electromagnetic field modes. The most dramatic effect of this ‘reservoir’ of modes

is that any atomic excitations coupled to the field tend to leak away rather quickly:

an emitted photon is very unlikely to couple back to the atom, so the atomic popu-

lation does not exhibit Rabi flopping; rather it decays exponentially. This is known

as spontaneous emission — the stronger the coupling between an atomic state and

the field, the shorter the lifetime of that state. An associated consequence is that

an incident photon is very unlikely to couple to the atom. Therefore a single atom

in free space cannot be used for quantum storage.

In order to recover strong coupling between an atom and an optical mode, it

is necessary to suppress the interaction with all unwanted field modes. This can

be done interferometrically, but introducing a highly reflective cavity around the

atom (see Figure 2.3). Any fields inside the cavity are reflected back on themselves

by the cavity mirrors. Only those fields with a wavelength equal to a half integer

multiple of the cavity length add constructively when folded back on themselves; all

other wavelengths interfere destructively. The optical modes supported by the cavity

are therefore spaced regularly in frequency. If the resonant frequency of the atomic

transition |1〉 ↔ |2〉 is close to one of these supported modes, the atomic coupling will

be confined to this single mode. All other cavity modes being too far from resonance

2.2 Free space coupling 39

to contribute significantly. If the volume of the cavity is sufficiently small, and the

Figure 2.3 Cavity QED. An atom is confined in a high-finesse op-tical cavity, which supports only a discrete set of optical frequencies.Only one optical mode, resonant with the atomic transition, couplesto the atom.

cavity mirrors sufficiently reflective, it is possible to bring an atom into the so-called

strong-coupling regime [49–53], where the light matter interaction behaves broadly as

described in Figures 2.1 and 2.2. This so-called Cavity QED approach to light-matter

interfaces has been widely applied. Both trapped and moving beams of atoms are

used [16,45,54,55], as well as quantum dots [56–59] and molecules [60]. However, cavities

are rather difficult to fabricate; approaches which do away with this requirement

would be easier to scale up.

2.2 Free space coupling

Another possibility is to focus an incident photon onto an atom in a way that

maximizes the probability of it being absorbed. The cavity is replaced with a pair of

microscope objectives, as shown in Figure 2.4. The rationale behind this approach

is that good coupling should be possible if the spatial shape of an incoming photon

matches the spatial pattern of an outgoing spontaneously emitted photon. This

time-reversal argument suggests that an incident photon should be converted into a

2.3 Ensembles 40

collapsing dipole pattern. The dipole pattern of an atom is very far from a narrow

collimated beam — it is nearly isotropic, covering almost 4π steradians. Therefore

the lenses should be as wide as possible: this in itself represents a technical challenge.

Only preliminary experiments have been done [61], but theoretical work [62,63] suggests

that efficient coupling can be achieved if the numerical aperture of the objective

lenses is made large enough.

Figure 2.4 Confocal coupling in free space. An atom is trappedat the focus of a pair of microscope objectives. If the solid anglesubtended by an incident photon is sufficiently large, the couplingefficiency is predicted to approach unity.

2.3 Ensembles

In addition to the practical difficulties of building a high-finesse cavity, or indeed

a high numerical aperture confocal microscope, there are technological hurdles as-

sociated with trapping and placing single atoms. Another possibility for quantum

storage is to use ensembles of atoms, as shown in Figure 2.5. An incident pho-

ton may have a small probability with interacting with any given atom, but as the

number of atoms in the ensemble increases, the probability that the photon fails

to interact with all the atoms decreases exponentially. Therefore it is possible to

asymptotically approach unit interaction probability simply by adding more atoms.

2.3 Ensembles 41

This approach is the main focus of this thesis. As we will see, introducing many

atoms makes it possible to store more than one photon, or more than one optical

mode. Complicated optical arrangements and cavities are not required, and a wide

range of possible storage media — from atomic vapours to Bose-Einstein conden-

sates, quantum dots, crystals and fibres — are well-suited for protocols of this kind.

We are primarily concerned with schemes based on the absorption, and subsequent

Figure 2.5 Atomic ensemble memory. If an incident photon en-counters enough atoms, it is almost certain to be absorbed.

re-emission, of a freely propagating photon by an atomic ensemble. These schemes

divide broadly into four protocols, all closely related, which we will now introduce.

2.3.1 EIT

EIT stands for Electromagnetically Induced Transparency. It was first observed

by Boller et al. in 1991 [64]. In EIT, a strong laser — the control — is shone

into an atomic ensemble with a Λ-structure, as shown below in Figure 2.6 (a).

Ordinarily, a weak probe beam would be absorbed by the atoms, but the interaction

with the control laser causes the ensemble to become transparent to the probe.

This can be understood by considering the dressed states of the atom, under the

influence of the control [65,66]. The control field couples the states |2〉 and |3〉, so the

Hamiltonian for the electric dipole interaction of an atom with the control is of the

2.3 Ensembles 42

(a) (b) (c)

Detuning (arb. units)

Su

sce

ptib

ility

(a

rb. u

nits)

0

Figure 2.6 EIT. (a): a weak probe beam propagates through anensemble of Λ-type atoms, while a strong control field couples theexcited and metastable states. (b): the control mixes states |2〉 and|3〉 to produce an Autler-Townes doublet. (c): The imaginary part ofthe probe susceptibility (solid line) exhibits a double resonance, witha transparency window at the atomic |1〉 ↔ |2〉 transition frequency.The real part (dotted line) changes quickly within this window, caus-ing marked dispersion.

form H = Ω|2〉〈3| + h.c., where Ω is the Rabi frequency of the control laser on the

|2〉 ↔ |3〉 transition. This can be simply re-written as H = Ω (|+〉〈+| − |−〉〈−|),

where the dressed states are defined by |±〉 = (|2〉 ± |3〉) /√

2. These dressed states

are simply rotated versions of the natural atomic states |2〉 and |3〉. In fact, although

we have changed our notation, the relationship between these states is the same as

between the 45-degree and rectilinear bases discussed in the context of QKD in

chapter 1. The Hamiltonian is diagonal in the dressed basis, and we can read off

the energies of the dressed states as ±Ω. That is, the combination of the control

with the atom produces a system with a double-resonance — known as an Autler-

Townes doublet [67] — with a splitting between the dressed states set by Ω. This is

a manifestation of the dynamic Stark effect: the control brings the energy of state

|3〉 up to that of |2〉, and the dipole interaction induces an anti-crossing.

From the above arguments, we might already expect the probe absorption spec-

2.3 Ensembles 43

trum to divide into two peaks, with a transparency window in between. Figure 2.6

(c) shows the results of a steady-state calculation of the linear susceptibility for the

probe field as a function of its detuning from the |2〉 ↔ |3〉 resonance. The probe

absorption is proportional to the imaginary part, which shows the expected doublet

structure. But the depth of the transparency window cannot be explained simply

by superposing two identical resonances. As is clear from the plot, the absorption

actually vanishes completely in the centre of the transparency window. This total

transparency is due to quantum interference: the contributions to the susceptibility

from the two dressed states are of opposite sign, because one is shifted above, and

the other below the original resonance. There is therefore an exact cancellation at

this resonance, and the susceptibility is identically zero at this point.

In addition to propagating without absorption, the probe beam also propagates

extremely slowly. More precisely, the group velocity of the probe is reduced by

the strong dispersion within the transparency window. The refractive index n of

the atomic ensemble is given by n =√

1 + Re(χ), where χ is the probe suscep-

tibility. Inspection of part (c) of Figure 2.6 shows that the real part of χ varies

extremely rapidly across the transparency window; therefore the refractive index is

also changing quickly in this spectral region. To see why this slows the group veloc-

ity, consider a pulsed probe. A pulse is composed of a range of frequencies, covering

a spectral bandwidth δω, all interfering constructively, so that the phase variation

over the pulse bandwidth is roughly zero. Each frequency component of the pulse

accumulates an optical phase kδz − ωδt over spatial and temporal increments δz,

2.3 Ensembles 44

δt, where k = nω/c is the wavevector of each component. The trajectory of the

pulse is the locus of those points at which all the components of the pulse remain

in phase, so that δkδz = δωδt, with δk the range of wavevectors spanned by the

pulse. The group velocity — the velocity of the pulse — is therefore defined by

vg = dz/dt = dω/dk = c/(n+ωn′), where n′ = dn/dω. The steep increase in Re(χ)

across the transparency window makes n′ large, and therefore vg is small. Group

velocities as low as 17 ms−1 have been demonstrated in the laboratory [68].

The use of EIT as a method for storing light is an elegant application of these

effects. It was first described by Misha Lukin and Michael Fleischhauer in 2000 [69],

and has since been demonstrated experimentally many times [70–74]. The protocol

works as follows. An atomic ensemble of Λ-type atoms is illuminated by a control

field, preparing a transparency window. A probe pulse — to be stored — is now

directed into the ensemble, tuned to the centre of the transparency window. As

described above, it propagates slowly, but without loss. Even if the spatial extent of

the pulse is much longer than the ensemble initially, the sudden slowing of the pulse

as it enters the ensemble causes it to bunch up, so that it fits within the ensemble

as it propagates (see Figure 2.7). As soon as the entire pulse is inside the ensemble,

the control beam is attenuated. That is, the Rabi frequency Ω is reduced, so that

the splitting of the Autler-Townes doublet decreases. The transparency window gets

smaller, and the variation in Re(χ) becomes steeper, so the group velocity of the

probe falls. This process continues until the control is switched off entirely, at which

point the transparency window completely collapses; the dispersion diverges, and

2.3 Ensembles 45

the group velocity vanishes: the probe has been brought to a complete stop! In

fact, the quantum state of the optical field has been transferred to the atoms, and

the state can be stored for as long as the coherence of the atoms survives without

distortions. If the control field is switched back on, the probe field is re-accelerated,

and emerges — hopefully — unchanged, as if the ensemble had not been present.

Figure 2.7 Stopping light with EIT. A long probe pulse bunchesup as it enters the EIT medium, due to the slow group velocity atthe centre of the transparency window.

The arguments just given provide a useful physical picture for the mechanism

behind light-stopping by EIT. In isolation they don’t provide a satisfactory expla-

nation for why the probe is not simply absorbed as the transparency window is

made narrower, nor for precisely how the probe light is transferred to an atomic

excitation. A more complete discussion can be found in the paper by Lukin and

Fleischhauer [69]. Nonetheless we can still draw some conclusions about the circum-

stances under which this procedure will work. The probe spectrum should not be

wider than the transparency window, otherwise it will be partially absorbed, so the

control must be sufficiently intense when the probe enters the ensemble. The control

cannot be turned off too quickly, since the dressed states producing the transparency

2.3 Ensembles 46

window are only a good approximation in the steady, or nearly steady state. The

control intensity must be reduced adiabatically ; we’ll examine this condition more

closely in Chapter 5 (see §5.2.9). Finally, the control should be turned off before

the probe escapes from the ensemble, so the initial group velocity should be suf-

ficiently slow — this means the ensemble density should not be too low. These

considerations tell us that the bandwidth of probe pulses that can be stored via

EIT, and the efficiency of the storage, is limited by the density of the ensemble, and

the available control intensity. We now introduce a related protocol, with the aim

of circumventing some of these limitations. As more detailed calculations show, this

attempt is only partially successful, but the flexibility of the new protocol makes it

an attractive alternative.

2.3.2 Raman

In 1928 Chandrasekhara Venkata Raman was the first to observe the weak inelastic

scattering of light from the internal excitations of vapours and liquids. No lasers

were available, and he used a focussed beam of sunlight to generate the required

intensity [75]. His work earned him the Nobel Prize in 1930; the eponymous Ra-

man scattering is now used routinely in spectroscopy, industrial sensing and indeed

quantum optics.

Raman scattering can be understood rather simply in the context of an atomic

Λ-system. It is represented schematically in Figure 1.6 (a) from Chapter 1 — the

interaction used in the DLCZ repeater protocol is precisely Raman scattering. It

2.3 Ensembles 47

is a two-photon process — that is, second-order in the electric dipole interaction

— in which a pump photon (green arrow in the diagram) is absorbed, and at the

same time a Stokes photon (blue wavy arrow) is emitted. Energy conservation is

satisfied if the frequency difference between the pump and Stokes photons is equal

to the frequency splitting between the states |1〉 and |3〉 in the atoms. The optical

fields need not be tuned into resonance with the |1〉 ↔ |3〉 transition: the likelihood

of Raman scattering decreases as the fields are tuned further away from resonance,

but given a sufficiently intense pump field, and sufficiently many atoms, the Raman

interaction can be made rather strong (see §§10.8, 10.9 in Chapter 10).

In a Raman quantum memory, the Raman interaction is turned on its head: a

signal photon is absorbed, and a strong control field stimulates the emission of a

photon into the control beam (see Figure 2.8). The signal and control fields are

tuned into two-photon resonance, that is, the difference in their frequencies is equal

to the frequency splitting between |1〉 and |3〉.

The process is conceptually very similar to that outlined at the beginning of

this chapter (c.f. Figure 2.2). But in a Raman memory, no atoms are ever excited

into the state |2〉. This is because the fields are tuned out of resonance with this

state, with a common detuning ∆. Instead, the control field creates a virtual state

— represented by the dotted lines in Figure 2.8 — and an atom is excited into

this virtual state by the signal photon. The control field then transfers the atom

into the state |3〉 for storage. Of course, all possible time-orderings for this process

contribute to the interaction, but from this perspective it is clear that no storage is

2.3 Ensembles 48

(a)

Storage Retrieval

(b)

Figure 2.8 Raman storage. Both fields are off-resonant, with adetuning ∆ from the excited state. (a): A signal photon is absorbed,and at the same time a photon is emitted into the control beam. Thistwo-photon Raman transition promotes one atom in the ensemblefrom |1〉 to |3〉. (b): The strong control is applied again, and theinverse Raman process returns the excited atom to its ground state|1〉, re-emitting the signal photon.

possible without the presence of the control field. Hence the name.

We might expect the physics of Raman storage to relate very closely to that of

EIT storage, and indeed they are confusingly similar. To connect Raman storage

with our previous discussion of EIT, we can consider the dressed states of the atoms

in a Raman memory, under the influence of the control. In fact, we have already

done so. The virtual state into which the signal photon is absorbed is precisely the

dressed state produced from the atomic state |3〉 when the control is present. Just

as in the case of the |−〉 state of EIT, the virtual state is really a combination of

the states |2〉 and |3〉. The difference is that it contains a very small amount of

the excited state |2〉, because of the large detuning ∆. The other dressed state, the

equivalent of |+〉 for the Raman memory, is not shown in Figure 2.8, since it is so

close to the bare atomic state |2〉 as to be indiscernible. Again, the large detuning

means that this state is made up almost entirely of |2〉, with almost no contribution

2.3 Ensembles 49

from |3〉. We can therefore frame the difference between EIT and Raman storage in

the following way. In an EIT memory, the photon to be stored is tuned between the

dressed states; it is brought to a halt by turning the control field off. In a Raman

memory, the signal photon is tuned into resonance with one of the dressed states,

which is made up almost entirely of the storage state |3〉. Once it has been absorbed,

the control field is turned off, and the state becomes ‘dark’ — decoupled from the

electromagnetic field.

One motivation for studying Raman storage is the possibility of broadband stor-

age. That is, the storage of temporally short photons, which necessarily comprise a

large range of frequencies. EIT storage requires that the spectral bandwidth of the

input photon should fit within the transparency window. Raman storage does not

rely on a transparency effect, so this limitation does not pertain. In addition, the

spectral width of the virtual state into which the signal photon is absorbed is set by

the spectral width of the control field. We might therefore expect that broadband

photons can be stored with a broadband control. We should ensure that the excited

state |2〉 is never populated, since this state will eventually decay and our photon

will be lost. This means that the detuning ∆ must greatly exceed the bandwidth of

the signal photon. Nonetheless, given sufficiently many atoms, the large detuning

need not weaken the interaction, and efficient broadband storage should be possible.

In fact, as discussed in §5.2.9 of Chapter 5, the Raman memory protocol is ideally

suited to broadband storage, although its advantages over the EIT protocol in this

respect are not entirely clear-cut.

2.3 Ensembles 50

A Raman memory was first proposed by Kozhekin et al. in 2000 [76]. They did

not explicitly address the problem of storing a single photon; rather they considered

the transfer of quadrature squeezing — a phenomenon we’ll describe in the next

section on continuous variables memories — from light to atoms. The theoretical

treatment of a Raman memory for photon storage was published in 2007 [77], and

forms part of this thesis. Raman storage is yet to be implemented experimentally,

but we are currently attempting to demonstrate the protocol in cesium vapour;

details are given in Chapter 10.

2.3.3 CRIB

CRIB stands for Controlled Reversible Inhomogeneous Broadening. In this protocol,

a spatially varying electric or magnetic field is applied to the ensemble. This shifts

the resonant frequency of the |1〉 ↔ |2〉 transition in the atoms, with a frequency

shift proportional to the strength of the field. Therefore, depending on their posi-

tion, some atoms experience a large shift; others a smaller shift, or a negative shift.

The net effect is to produce an inhomogeneous broadening of the atomic resonance.

That is, each atom has a narrow resonance, but the ensemble as a whole absorbs

light over a broad range of frequencies, as determined by the applied field (see Figure

2.9). Storage is accomplished via the general procedure sketched in Figure 2.2 (b).

A signal photon is tuned into resonance with the broadened |1〉 ↔ |2〉 transition,

and is completely absorbed. An important feature of the scheme is that broadband

photons can be stored, because the ensemble resonance covers a wider spectral range

2.3 Ensembles 51

than the unbroadened transition. In particular, photons with a temporal duration

much shorter than the spontaneous emission lifetime of the state |2〉 can be stored.

Therefore the absorption process is finished long before the atoms have had time to

decay, and losses due to spontaneous emission are minimal. Note that so far in the

protocol we have not applied any optical control fields, so there are no dressed states

— the signal photon has simply been resonantly absorbed. When the absorption is

complete, a single atom in the ensemble has been excited into state |2〉. The broad-

ening field is now switched off, so that the atoms recover their natural resonance

frequencies. Then a control field — a laser pulse tuned to the |2〉 ↔ |3〉 transition

— transfers the excited atom to the storage state |3〉.

Storage has now been completed, but the stored excitation is rather mixed up.

To understand why, it is easiest to drop our consideration of photons for a moment,

and consider the memory as it would be described classically. The atoms can be

thought of as a collection of dipoles (separated charges) that are set in motion by an

impinging field — the signal field. Due to the applied inhomogeneous broadening,

each dipole is oscillating at a slightly different frequency. Therefore, although they

all begin oscillating together, driven by the signal pulse, they soon drift out of phase

with one another. When the control pulse is applied, the dipoles are essentially

frozen, since their motion is transferred to the dark state |3〉. In being stored, the

signal pulse has had its phase scrambled. That is, information about the time-of-

arrival of the signal pulse has been lost. The crucial step in retrieving the signal is to

recover this information, by reversing the dephasing process. This is the vital ‘R’ in

2.3 Ensembles 52

CRIB. Because the inhomogeneous broadening is man-made, we are able to switch

it around, by changing the polarity of the external field. When the broadening

is flipped in this way, every atom that was shifted to a higher frequency is now

shifted to a correspondingly lower frequency, and vice versa. This provides us with

a way to completely reverse the dynamics of the memory. Retrieval works like this:

the control pulse is sent into the ensemble. This unfreezes the stored excitation,

transferring the atoms from |3〉 to |2〉. The atomic dipoles are still out of phase, but

then the inhomogeneous broadening is re-applied, this time with the reverse polarity;

see Figure 2.9 (b). The atoms that were red-shifted at storage are now blue-shifted,

and those previously blue-shifted are now red-shifted. The atomic dipoles therefore

eventually re-phase. The collective oscillation of the entire ensemble, in phase, then

acts as a source for the electric field, and the signal pulse is re-emitted.

(a)

Storage Retrieval

(b)

Figure 2.9 CRIB storage. (a): A signal photon is absorbed res-onantly by the broadened ensemble, exciting an atom to state |2〉.Then a strong control pulse transfers the excited atom to the stor-age state. (b): To retrieve the stored photon, the control is appliedagain, and the inhomogeneous broadening is switched on, this timewith reversed polarity. The optical dipoles eventually re-phase, andthe signal photon is re-emitted.

CRIB storage was first proposed by Nilsson and Kroll [78], following analyses of

2.3 Ensembles 53

a generalized photon echo by Moiseev [79–81]; the acronym appeared in a proposal

by Kraus et al. [82]. The protocol has been implemented experimentally by several

groups, all using rare-earth ions doped into solids [83–85].

The reasons for choosing these materials as storage media are manifold. First,

use of atoms in the solid state eliminates limitations to the storage time arising from

atomic collisions, or from atoms drifting out of the interaction region (see §10.5 in

Chapter 10). Second, the rare-earth elements all share a rather peculiar feature in

their electronic structure, namely that the radius of the 4f shell is smaller than the

radii of the (filled) 5s and 5p shells [86]. The optically active electrons in the 4f shell

are therefore shielded by the 5s and 5p shells. These filled shells are spherically

symmetric, and can be thought of as metallic spheres that isolate the f electrons

from external fields, much as would a Faraday cage. Therefore, even when doped

into a solid, rare-earths maintain much of their electronic structure. Perturbations

due to the surrounding ‘host’ material may induce frequency shifts, but noise and

fluctuations, which might reduce the possible storage time, are effectively eliminated.

Optical transitions between different states within the f shell are generally forbidden,

since all these states have the same parity (see §4.3.1 in Chapter 4). This means that

spontaneous emission from these states is greatly suppressed, making them ideal

for use in a quantum memory. Spontaneous lifetimes of several hours have been

measured [87], and photon storage times of up to 30 s are realistic [88]. Of course, if

no transitions are allowed at all, no incident photons can ever be absorbed. But

one effect of the host is to alter the atomic potential so that the f shell acquires an

2.3 Ensembles 54

admixture of d orbital states, making electric dipole transitions possible [89].

Finally, the 4f 4I15/2 ↔ 4I13/2 transition in Erbium has a wavelength of 1.5 µm,

which matches the wavelength at which optical fibres used for telecommunications

have minimal absorption. Particular attention has therefore been paid to the pos-

sibilities for building a solid state quantum memory based on Erbium ions, since it

would integrate extremely well with existing telecoms systems.

We can distinguish two categories of CRIB, based on the direction of variation

of the external field with respect to the propagation direction of the signal (see

Figure 2.10). If these directions are perpendicular, so that the atomic frequencies

are broadened across the ensemble, we call this tCRIB (transverse CRIB). If they

are parallel, with the atoms broadened along the ensemble, we call this lCRIB

(longitudinal CRIB). The latter of these is sometimes referred to as GEM (gradient

echo memory [84,90]). As we will see in Chapter 7, the performance of these two

schemes for photon storage is essentially the same.

(b)

(a)

Figure 2.10 tCRIB vs. lCRIB. (a): In tCRIB, an external fieldbroadens the ensemble resonance in a direction transverse to thepropagation direction of the light to be stored. (b) In lCRIB, thebroadening field is applied parallel to the propagation direction.

2.3 Ensembles 55

2.3.4 AFC

The atomic frequency comb memory protocol was proposed recently [91] by Afzelius

et al.. Their research group in Geneva have focussed on the implementation of CRIB,

and AFC takes a number of cues from their experience in this connection. In the

AFC protocol, we suppose that it is possible to prepare an atomic ensemble with

a large number of absorption lines equally spaced in frequency (see Figure 2.11).

This atomic frequency comb plays a similar role to the inhomogeneous broadening

in CRIB. It increases the bandwidth of the absorptive resonance, so that a tempo-

rally short signal is efficiently absorbed. As with CRIB, this means that the entire

absorption process can be completed long before the excited state |2〉 has any time

to decay. Again, for long-term storage, the excitation is mapped to the dark state

|3〉. The ‘trick’ in the design of the AFC protocol becomes clear at retrieval. Recall

that, after transferring the stored excitation back to |2〉, the atomic dipoles must be

brought back into phase with one another before the signal can be re-emitted. In

CRIB this is done by reversing the atomic detunings, and this would certainly work

for AFC. But even if the atomic resonances are not altered in any way, the AFC

memory still re-emits the signal! The reason is that the atomic frequency comb has

a discrete spectral structure. Therefore, the optical polarization undergoes periodic

revivals — re-phasings — at a rate given by the frequency of the beat note associ-

ated with the comb frequencies. If the frequency separation between adjacent comb

teeth is ∆, the time between re-phasings is roughly 1/∆.

Having introduced the principle behind AFC storage, a number of comments

2.3 Ensembles 56

(a)

Storage Retrieval

(b)

Figure 2.11 AFC storage. (a): A broadband signal photon, witha bandwidth covering many comb teeth, is absorbed by the atomicfrequency comb. The excitation is then transferred to state |3〉 bya control pulse. (b) To retrieve the signal, the control is re-appliedto return the excitation to the frequency comb. The atomic dipolesre-phase, due to the discrete structure of the comb, and the signal isre-emitted.

are in order. First, it is not obvious that an ensemble with a comb-like resonance

will smoothly absorb the signal photon. One might expect that only those parts of

the signal spectrum overlapping with the comb teeth would be absorbed, with the

intervening frequencies simply transmitted and lost. In fact, the absorption between

the comb teeth never vanishes completely. Provided the ensemble contains enough

atoms, the combined absorption over the whole ensemble is enough to store all the

frequencies in the signal.

Second, it is not obvious how the control field can transfer all the atoms in the

frequency comb from state |2〉 to state |3〉 and back again. This must be done

in AFC so as to prevent the periodic revivals in polarization from re-emitting the

signal too early. A sufficiently bright and short control pulse will accomplish the

state transfer — the control pulse bandwidth should span the full spectral width of

the comb. So-called coherent control can be used to shape the control pulse so as to

2.3 Ensembles 57

maximize the transfer efficiency [92–98].

Third, it is not obvious how to prepare an atomic frequency comb. The proce-

dure suggested by Afzelius et al. is based around an implementation with rare-earths

doped into solids. In these materials, natural variations in the position within the

host occupied by the rare-earths causes the resonant frequencies to be shifted ran-

domly over a broad spectral range. This natural inhomogeneous broadening is not

useful for CRIB, since it cannot be ‘reversed’. Therefore in the CRIB protocol, it is

necessary to remove all the atoms except those with the desired resonant frequency,

before applying the external field to artificially broaden the resonances of only these

atoms. To ‘remove’ undesired atoms, an optical pump is employed [99] (see §10.12 in

Chapter 10). This is a series of laser pulses with frequencies tuned so as to transfer

unwanted atoms from the ground state |1〉 into a new state |4〉 (not shown in any

diagrams so far), where they are ‘shelved’ for the duration of the memory protocol.

The shelf state can be chosen to have an extremely long lifetime (several hours, as

mentioned above in Section 2.3.3); this is why we can consider the atoms as having

been simply removed from the ensemble. To prepare the frequency comb for the

AFC protocol, a similar optical pumping procedure is used: all atoms with resonant

frequencies in between the comb teeth are shelved. A significant practical advantage

of AFC over CRIB is now clear. CRIB requires that we pump out all but a single

narrow resonance within the ensemble: we ‘throw away’ a lot of atoms. In AFC,

we pump out all but N narrow resonances, where N is the number of comb teeth,

which may be quite large. Therefore we throw away fewer atoms, and this allows for

2.4 Continuous Variables 58

a much stronger absorption — a much more efficient memory — using an ensemble

with the same doping concentration as a less efficient CRIB protocol. In Chapter

7 we will show that AFC is also well-suited to the parallel storage of multiple sig-

nal fields, making it attractive for use in quantum repeaters. AFC storage has not

yet been demonstrated in its entirety, but proof-of-principle experiments have been

performed by de Riedmatten et al. [100].

2.4 Continuous Variables

So far, our discussion of quantum memories has focussed on the storage of single

photons. For the purposes of QKD and computation discussed in Chapter 1, the

quantum information encoded into these photons is of a discrete nature. Either the

number of photons, or their polarization, can be used to represent the two basis

states of a qubit. This is not the only paradigm for quantum information process-

ing, however. Photons possess other degrees of freedom that are not discrete. The

position of a photon, or its momentum, for example. These quantities don’t lend

themselves to representation in terms of qubit states. But it is still possible to use

such continous variables to encode information (as is done in classical analogue com-

puting). And the specifically quantum features of qubits that confer their increased

computational power — superposition; entanglement — all carry over to continuous

variables. Therefore quantum computing protocols [101,102], and indeed QKD pro-

tocols [103–106], that take advantage of the continuous degrees of freedom possessed

by photons, have all been developed. In general their relationship to the equivalent


qubit-based protocols is similar to that between analogue and digital classical com-

putation. On the one hand, the continuous versions are robust to noise, in the sense

that information is not completely erased in the presence of distortions. On the

other hand, below a certain threshold, digital algorithms can be made essentially

impervious to noise, producing ‘perfect’ outputs, whereas analogue computations

are always subject to fluctuations — they never work perfectly. Anyone who has

ever witnessed the catastrophic failure of ‘crisp and clear’ digital television with a

poor signal will appreciate the fuzzy watchability of analogue television in the same

conditions.

In this section we briefly introduce the concepts of continuous variables, so as

to understand a class of memory based on continuous variables storage. The most

commonly used variables are the field quadratures. These are defined in terms of the

electric field E associated with an optical mode,

E = x cos(ωt) + p sin(ωt), (2.1)

where ω is the optical carrier frequency, and t is the time. The coefficients x and

p for the amplitude of the field are known as the in-phase and out-of-phase field

quadratures, respectively [107]. By convention, the same symbols as would normally

be associated with position and momentum are used, and this is motivated by their

formal similarity. Like the position and momentum of a classical pendulum, the

quadratures are continuous variables that describe the oscillation of the field. Even


more strikingly, in quantum mechanics, x and p are complementary in the same

way as are the position and momentum of a harmonic oscillator. That is to say,

a measurement of one quadrature ‘disturbs’ the other, so that it is impossible to

precisely measure both simultaneously. In appropriately scaled units, the Heisenberg

uncertainty principle applies [103]:

∆x∆p ≥ 1, (2.2)

where ∆x, ∆p are the precisions with which the quadratures are simultaneously

known. Just as in classical physics, the quantum state of an optical field can be

completely described in terms of x and p. Due to the uncertainty principle (2.2),

the state is not a single point in (x,p)-space, or phase space as it is more often called,

but rather a kind of ‘blob’, whose spread represents the uncertainties ∆x and ∆p.

This blob is known as the Wigner distribution [108], and it is the representation of

choice for states parameterized by continuous variables. The most ‘normal’ state

of an optical field — the coherent state — is a monochromatic beam, like that

produced by a laser. It is completely classical, in the sense that the formalism

of quantum mechanics is not required to describe it. Classical electromagnetism

is sufficient. Its Wigner distribution is a Gaussian ‘hill’, with equal uncertainties

in x and p (see Figure 2.12 (a)). Clearly this is not the only type of distribution

compatible with 2.2. Figure 2.12 (b) shows an example of a squeezed state, with

a small uncertainty in x. The price for this increased precision in x is that the


uncertainty in p grows, but states like this can be extremely useful for reducing

the noise on measurements associated with just one of the quadratures. Squeezed

states of light would appear to be rather exotic, but they arise naturally in any

process that converts one frequency into another. Such non-linear processes only

require that the potential in which optically active electrons move is not exactly

quadratic1 in the electrons’ displacement. This happens to some extent in nearly all

materials, and the technology for generating efficient squeezing is now rather well

developed [109–112].

(b)

(a)

Figure 2.12 Wigner distributions in quadrature phase space fortwo different states of an optical mode. (a): A classical coherentstate, with equal uncertainties in x and p. (b) A squeezed state, with∆x halved, but ∆p doubled. The lengths of the dotted lines give thebrightness of the states; their angles give their relative phases.

Research into squeezed light is being actively pursued as a means to improve the

sensitivity of gravitational wave detectors [113,114], and to toughen the noise tolerance

of communication systems [115]. As might be expected from the ubiquity of uncer-1If the potential is not quadratic (i.e. anharmonic), the ‘restoring force’ on the electrons is

non-linear.


tainty relations like (2.2) in quantum mechanics, the phenomenon of squeezing is

not confined to light, and their are metrological benefits that accrue if atoms can be

squeezed [116]. Increased sensitivity to magnetic fields [117], enhanced spectroscopic

precision [118], and better atomic clocks [119–121], as well as quantum computation and

communication, are all enabled by the ability to manipulate non-classical states of

light and matter.

One of the most successful demonstrations of quantum memory has grown out

of research in this area. The mechanism by which storage is accomplished is most

transparent when couched in the language of continuous variables. The memory

works by transferring the quadratures of a light beam into an ensemble of atoms,

where the quadratures X, P associated with the atoms are their ‘coronal’ angular

momenta Jx = X, Jy = P (see Figure 2.13). The storage interaction involves two

steps. In the first, a control and signal beam — both tuned far off-resonance —

are directed through an atomic ensemble. A strong magnetic field is applied to the

ensemble, which aligns the atomic spins, so that all the atoms are initially in state

|1〉 (see Figure 2.15). The control is polarized parallel to the field, so that it does not

induce a turning moment. On the other hand, the signal is polarized perpendicular to

the field. Two-photon Raman transitions, involving both the control and the signal,

can therefore change the z-component of the collective atomic angular momentum,

transferring atoms from |1〉 to |3〉 or vice versa. In terms of the atomic quadratures,


the passage of the optical fields through the ensemble induces the transformation

X → X = X + p, (2.3)

where the tilde denotes the quadrature value at the end of the interaction. At the

same time, the x quadrature of the light is also shunted — a manifestation of the

Faraday effect,

x→ x = x+ P. (2.4)

The other quadratures p, P of both the signal and the atoms are unchanged. For this

reason the interaction is known as a quantum non-demolition (QND) measurement,

since information about the p quadrature of the signal is transferred to the atoms,

without altering it.

In the second step of the storage protocol, the x quadrature of the signal is mea-

sured using balanced homodyne detection [107]. As shown in Figure 2.14, a polarizing

beamsplitter, aligned in the 45-degree basis, mixes the control and signal beams.

The difference in the intensities detected at the two output ports of the beamsplit-

ter is directly proportional to x. This measurement result is then used to determine

the strength of a radio frequency pulse that applies a controlled torque to the atoms,

producing a shift

P → P = P − x = P − (x+ P ) = −x. (2.5)

The two maps (2.3) and (2.5) taken together almost constitute an ideal memory.


Figure 2.13 Atomic quadratures. In the presence of a strong mag-netic field, the atomic spins in an ensemble align with the z-axis. Thequadratures X and P for the atoms are then given by small displace-ments of their angular momenta in the coronal plane, normal to thez-axis.

Apart from an unimportant minus sign in (2.5), the only difficulty is the presence

of the initial atomic quadrature X in (2.3). The average value of X can be made to

vanish, but the finite spread of the initial atomic Wigner distribution still introduces

unwanted fluctuations. This spread can be reduced by squeezing the Wigner distri-

bution of the atoms, so that ∆X → 0, before attempting the memory, and schemes

for doing this are in development [116].

The above type of continuous variables memory, based on a QND interaction

followed by measurement and feedback, was first implemented by Julsgaard et al.

at the Niels Bohr intitute in Copenhagen [122] using an ensemble of cesium atoms.

The same research group, led by Eugene Polzik, has since refined and extended

the technology, and have demonstrated quantum teleportation of light onto atoms,

deterministic entanglement generation, efficient spin squeezing of atomic ensembles


Figure 2.14 QND memory. A magnetic field aligns the atomicspins in an ensemble. A strong control, polarized parallel, and aweak signal, polarized perpendicular to the spins, are sent throughthe atoms. The control cannot rotate the atomic spins, but the signaldoes. A homodyne measurement extracts the resulting x quadratureof the signal, and a radio frequency pulse, generated by the coilsshown, applies a final rotation.

and many other continuous variables protocols. An excellent review of their progress

in this area can be found in the recent review article by Hammerer et al [123].

The conceptual shift between the above description in terms of continuous vari-

ables, and the rather intuitive picture of ‘reversible absorption’ we have used for all

the other memory protocols, makes comparisons difficult. Certainly the optimiza-

tion of this protocol does not fit into the general scheme we apply to the optimization

of the other memory protocols in this thesis. The scheme has enjoyed considerable

success, despite its technical complexity, probably due in large part to the experience

and prowess of those working at the Niels Bohr institute. Nonetheless, it does not

perform as well as the other protocols as a component of a DLCZ-type quantum

repeater. This is because, as described in Chapter 1, the entanglement purification


Figure 2.15 Level scheme for a QND memory. The atoms beginwith their spins aligned with the external magnetic field, in state|1〉. State |3〉 cannot be reached by interaction with the control alone(green arrows), since it is π-polarized — parallel to the B-field —and it cannot induce a spin flip (see Figure F.4 in Appendix F).Raman transitions involving both the control and signal (blue wavyarrows) can transfer atoms to state |3〉. The intermediate excitedstates are collectively labelled |2〉. Both types of Raman interactionare involved: Stokes scattering, as in the DLCZ protocol (see Figure1.6 in Chapter 1), and the anti-Stokes scattering used in the Ramanmemory protocol (see Figure 2.8).

in these repeaters is effective against photon loss, but not against photon addition,

and the Raman transitions shown in Figure 2.15 can produce extra photons that

contaminate the number state entanglement. Other applications for which photon

loss is particularly damaging would benefit from a continuous variables memory,

since loss can be essentially eliminated by increasing the power of the RF pulses

applied in the feedback step, at the expense of distorting the stored state. In any

case, we will not discuss continuous variables memory further.

In the next chapter we introduce the optimization scheme relevant for the absorption-

based memories we have discussed.

Chapter 3

Optimization

Here we introduce the optimization scheme that we will apply to the Raman, EIT,

CRIB and AFC ensemble memory protocols. In all these protocols, an input field,

the signal field, is transferred to a stationary excitation inside an atomic ensemble.

The aim of the optimization is to maximize the efficiency of this transfer, which

amounts to maximizing the amount of stored excitation, given a fixed input. Fortu-

nately, a technique borrowed from the mathematical toolbox of linear algebra makes

this optimization extremely easy to perform.

Suppose that the signal field is a pulse, with a time-dependent amplitude A(τ),

where τ is the time. The action of the memory is to absorb this input pulse, and

convert the incident energy into some kind of long-lasting excitation within the

ensemble (see Figure 3.1). We’ll denote the amplitude of this excitation by B(z),

where z is the position along the ensemble. Here the z dependence allows for the

possibility that the excitation may be distributed over the length of the ensemble

68

in a non-uniform way. This kind of collective, de-localized excitation is typically

referred to as a spin wave, since in many cases the atomic states involved are states

of different spin angular momentum (as in the QND memory discussed at the end

of Chapter 2). We’ll use this term indiscriminately, to describe any distributed

excitation relevant to quantum storage, regardless of the nature of the quantum

numbers associated with the atomic states. In the next chapter, we’ll define the spin

wave more precisely. For the purposes of optimization, it is sufficient to understand

the spin wave as the stationary counterpart of the propagating signal field. That

is, the storage process is a mapping A→ B, and the retrieval process is the inverse

map B → A.

Figure 3.1 Storage map. An incident signal field A(τ) is mappedto a stationary spin wave B(z) within an atomic ensemble.

If the signal field initially contains NA = N photons, we would ideally like the

spin wave at the end of the storage process to contain the same number of excited

atoms NB = N , so that all the input light has been stored. In practice, some of

the input light will pass through the ensemble and be lost, and some of the excited

atoms will decay back down to their ground state, or drift out of the interaction

region. The efficiency η of the storage interaction is simply the ratio of the number

69

of excited atoms to the number of input photons, η = NB/NA, where the number of

quanta, either material of optical, is found by integrating the squared norm of the

relevant amplitude,

NA =∫ ∞−∞|A(τ)|2 dτ, NB =

∫ L

0|B(z)|2 dz. (3.1)

Here L denotes the length of the ensemble. We note that in this type of memory

there is no process that can produce extra excitations. At least in principle, there

is no source of background noise. The failure mode of the memory is photon loss:

photons directed into the memory are not stored, and so are not recovered in the

retrieval process. This differs from the QND memory mentioned in the previous

chapter, which need not suffer from photon loss, but which may introduce noise at

retrieval. Therefore the performance of this type of memory is optimal when the

efficiency is maximized. No other figure of merit, involving the suppression of atomic

fluctuations for instance, is relevant.

A key requirement of a quantum memory is linearity. That is, suppose that we

store a signal field A = αA1 + βA2 built from two contributions A1 and A2. This is

a superposition state, like (1.1) in Chapter 1, and to preserve its encoded quantum

information, the coefficients α, β should not be altered by the storage process. The

resulting spin wave should be of the form B = αB1 +βB2, where B1, B2 are the spin

waves generated by storage of A1, A2 only. This property, required to faithfully store

superpositions, restricts the storage map A → B to be a linear map (and similarly

3.1 The Singular Value Decomposition 70

for the retrieval process). Fortunately, as we will see, all the memory protocols we

will analyze are indeed linear. In general, then, we can always write the storage map

in the following way

B(z) =∫ ∞−∞

K(z, τ)A(τ) dτ. (3.2)

The integral kernel K is known as the Green’s function, or the propagator for the

storage interaction. It contains all the information about how the memory behaves.

In the next chapter we will show how to derive expressions for K in some cases.

Generally, it is possible to construct the form of K numerically, as we will show in

Chapter 5 (see §5.4). For the moment, we suppose that we are able to write the

memory interaction in the form (3.2). It is clear that the efficiency of the memory

depends on achieving a good ‘match’ between K and the shape, in time, of A. For

instance, no storage is possible if K ∼ 0 during the arrival time of the signal. K

and A should ‘overlap’. Optimizing a quantum memory involves finding the shape

of A that maximizes this overlap, so that all of A ends up in B. In the next section

we introduce the singular value decomposition, a valuable analytical tool that can

be used to find this optimal shape, and more besides.

3.1 The Singular Value Decomposition

The SVD is most commonly encountered in the context of matrices. It is one of

the most useful results in linear algebra, and it finds applications from face recog-

nition [124] to weather prediction [125]. Under the name Schmidt decomposition it is


of critical importance in quantum information theory as a tool for the analysis of

bi-partite entanglement [126]. It seems to have been discovered independently several

times in the 19th century [127–129]. Erhard Schmidt applied the decomposition to

integral operators (an example of which is (3.2)) in 1907 [130], and Bateman coined

the term ‘singular values’ in 1908 [131]. The proof of the decomposition for arbitrary

matrices was given in 1936 by Eckart and Young [132].

At the heart of the SVD is a geometric interpretation for the action of a linear

operator. A linear operator, or linear map, takes some vector a as input, and spits

out another vector b as output. Representing the linear operator as a matrix M , we

have

b = Ma. (3.3)

Looking at Figure 3.2 (a), it is clear that such a transformation is equivalent to the

following procedure. (i) rotate a until it lies along one of the coordinate axes (ii)

re-scale the axes so that the length of the rotated version of a matches the length of

b (iii) rotate this re-scaled vector until it sits on top of b. This suggests that it should

be possible to decompose an arbitrary linear transformation by combining rotations

and coordinate re-scaling. The SVD is nothing more than this representation of a

general linear map as a rotation, a re-scaling, and a final rotation. An example is

shown in Figure 3.2 (b), where the action of M is shown on the set a of all vectors

with a certain length. The tips of these vectors trace out the surface of a sphere.

The effect of M is to ‘squish’ this sphere into an ellipsoid. This has to be done in

such a way that the black dot on the sphere ends up as the red dot on the ellipsoid.


(a) (b)

Figure 3.2 Linear transformation. (a): a vector a is mapped ontoa vector b by M , which may be seen as implementing a rotation ofa onto the x-axis, followed by a coordinate re-scaling to increase thelength of a until it matches the length of b, followed by a rotationonto b. (b): The action of M on the set of all initial vectors a(the grey sphere) produces a set of vectors b (the red ellipsoid). Asan example, the red dot is the image of the black dot under underM . The final ellipsoid can be generated from the initial sphere by:rotating the sphere, re-scaling the axes and then rotating again.

To produce the ellipsoid from the sphere, it is necessary to first rotate the sphere

until the black dot is placed appropriately. Then the x, y and z axes are re-scaled

to deform the rotated sphere into the ellipsoid ‘shape’. Finally, a second rotation

puts the ellipsoid in the correct orientation, with the black dot sitting on top of the

red dot.

The above discussion is limited to the intuitive case of a real vector in three

dimensions being mapped to another real vector in three dimensions. In fact, the

SVD exists for all matrices M — all linear transformations. That is, M could be a

complex rectangular matrix of any size, that maps a complex n-dimensional vector

to a complex m-dimensional vector. In general, M can always be written in the

form

M = UDV †. (3.4)


Here U and V are both unitary matrices, and D is a real, positive, diagonal matrix.

The properties of these types of matrices are reviewed in Appendix A. In terms

of the discussion above, V † represents a rotation into a new coordinate system,

D represents a re-scaling of this new coordinate system, and U represents a final

coordinate rotation. The elements of D, lying along its diagonal, are the factors

by which the coordinates are re-scaled in the second step. They are known as the

singular values of M , and they contain a great deal of useful information about the

transformation represented by M . Geometrically, the singular values correspond

to the lengths of the semi-axes of the ellipsoid in Figure 3.2 (b). In this thesis,

singular values are generally denoted by the symbol λ, the same as eigenvalues. It

should always be clear from the context what is meant. By convention, the singular

values, which are all positive, real numbers, are ordered in descending magnitude,

so that D11 = λ1 is the largest singular value, D22 = λ2 is smaller, and so on. It is

sometimes useful to visualize the structure of the matrices in the SVD, and so for

reference we provide the following tableau,

M =

|u1〉

. . . |um〉

︸︷︷︸

U

λ1

. . .

λm

︸︷︷︸

D

〈v1|

...

〈vm|

︸︷︷︸

V †

. (3.5)


Performing the matrix multiplications explicitly, we obtain

M =∑j

λjMj , (3.6)

where each of the matrices Mj = |uj〉〈vj | is an outer product of the jth columns

of U and V (see Section A.2.1 in Appendix A). This last representation provides

a natural way to interpret the action of M , as a sum of independent mappings

from the column-space of V to the column-space of U . To understand why this is

useful, recall that the sets of column vectors |vj〉 and |uj〉 of U and V are both

orthonormal bases, because U and V are unitary (see Section A.4.5 in Appendix

A). Therefore any vector |a〉 to which M is applied can be written in the coordinate

system defined by the |vj〉, and the result |b〉 can always be written in terms of

the |uj〉,

a = |a〉 = a1|v1〉+a2|v2〉+. . .+an|vn〉, and b = |b〉 = b1|u1〉+b2|u2〉+. . .+bm|um〉.

(3.7)

Applying (3.6) to |a〉, and making use of the orthonormality of the u’s and v’s,

〈ui|uj〉 = 〈vi|vj〉 = δij , we find that

b1 = λ1a1, b2 = λ2a2, etc . . . . (3.8)

So M can be viewed as a set of mappings between two special coordinate systems,

each with a different ‘fidelity’, given by the singular values. M may seem like


a complicated transformation, but as long as we use the |vj〉 to define our input

coordinate system, and the |uj〉 to define our output coordinate system, the action

of M is always extremely simple. It just maps each coordinate from the input onto

the corresponding output coordinate, re-scaled by the corresponding singular value.

3.1.1 Unitary invariance

If the input coordinate system is rotated, before performing the SVD, this cannot

change the singular values, but only the input basis that we should use. The same

is true if the output coordinate system is rotated. More generally, consider applying

some unitary transformation W to M . The SVD of this compound operator is then

WM = WUDV † = UDV †, (3.9)

where U = WU is a unitary matrix, since both W and U are unitary. The output

basis vectors are modified by W , but the singular values — the elements of D —

are unchanged. By the same token, the product MW also has the same singular

values as M , but V must be replaced by W †V . Sometimes it is only possible to find

rotated versions of a matrix M , in which case this property of the SVD is useful.

3.1.2 Connection with Eigenvalues

The SVD is connected with the eigenvalues of the ‘square’ of M . More precisely, con-

sider the normally and antinormally ordered productsKN = M †M andKA = MM †.

These two products are both Hermitian, since K†N = KN and K†A = KA. Therefore,


they both have real eigenvalues, and their eigenvectors each form orthonormal bases

(see Section A.4.3 in Appendix A for a derivation of this fact). And inserting the

decomposition (3.4), we see that

KN = V D2V †, and KA = UD2U †, (3.10)

where we have used the relations U †U = V †V = I, which must hold for unitary

matrices. Therefore, the eigenvalues of KN and KA are both given by the squares

of the singular values.

3.1.3 Hermitian SVD

Note also that if M is a Hermitian matrix, with M = M †, we must have that U = V .

That is,

M = UDU †, (3.11)

which is precisely the spectral decomposition of M , with the eigenvalues of M lying

along the diagonal of D, and the eigenvectors of M making up the columns of U . So

the SVD and the spectral decomposition of M are identical for Hermitian matrices.

3.1.4 Persymmetry

Another case, which we will encounter in our treatment of Raman storage, is that

of persymmetry. Suppose that M is a real, square matrix, with n = m. Then M is

persymmetric if it is symmetric under reflection in its anti-diagonal — the diagonal


running from bottom left to top right, as shown in Figure 3.3. M = MP, where

(MP)ij = Mm−j+1m−i+1. This is rather an unusual type of symmetry, and it is

Figure 3.3 Persymmetry. The upper left and lower right portionsof a real persymmetric matrix are mirror images of eachother underreflection in the anti-diagonal, represented as a grey stripe.

not often discussed in textbooks. But the action of a persymmetric matrix can be

viewed as very similar to that of a real Hermitian matrix, the only difference being

that the result is ‘flipped around’. To see the implications of persymmetry for the

SVD, let us write M = XH, where H is a real Hermitian matrix (i.e. symmetric

under reflection in its main diagonal), and where X is a ‘flip’ matrix, with ones

along its anti-diagonal, and zeros everywhere else (left blank for clarity below),

X =

1

1

. . .

1

. (3.12)

The action of X when multiplying a matrix is to flip it around a horizontal axis, so

that its last row becomes its first, and vice versa. For every persymmetric matrix M ,

there must always be some Hermitian matrix H such that M = XH. The property

3.2 Norm maximization 78

of persymmetry can also be easily written in terms of X. A little mental acrobatics

will verify thatMP = XM †X, so that persymmetry requiresM = XM †X. Inserting

the spectral decomposition of H, we obtain

M = XM †X = X(XH)†X = XH†X†X = XHX2 = XUDU † = UDU †, (3.13)

where we used the Hermiticity of H and X, along with the fact that X2 = I (two

horizontal flips cancel each other out), and where U = XU is a unitary matrix whose

columns have been flipped round. Therefore the SVD of a persymmetric matrix is

such that the columns of U , the basis for the output coordinate system, are flipped

versions of the columns of V , the input basis. This result will be of use to us in

Chapter 5.

3.2 Norm maximization

Suppose that we would like to know how to choose a in order to maximize the norm

(the length) of b. Since M is a linear transformation we can always increase the

norm of b just by increasing the norm of a: if we double the length of a, the length

of b also doubles. But this is not interesting. Clearly the direction of a matters;

some directions will result in a larger norm for b. If the norm of a is fixed, how

should we choose its direction? This question is easily answered if we are able to

compute the SVD of M . Without losing generality, let ||a|| = a = 1. That is,

3.3 Continuous maps 79

|a1|2 + |a2|2 + . . .+ |an|2=1. Using (3.8), the norm of b is then given by

b2 = |b1|2 + |b2|2 + . . .+ |bm|2

= λ21|a1|2 + λ2

2|a2|2 + . . . λ2n|an|2. (3.14)

But by definition, λ1 is the largest of the singular values, so that

b2 < λ21(|a1|2 + |a2|2 + . . .+ |an|2)

= λ21, (3.15)

That is, the largest possible norm of b is b = λ1. This maximum norm is obtained

by choosing a1 = 1, with all other components vanishing. So the ‘optimal’ vector

a, with regard to maximizing b, is a = v1 (or |a〉 = |v1〉). From the perspective

of Figure 3.2 (b), this amounts to choosing a so that the resulting b lies along the

largest semi-axis of the ellipsoid generated by M .

3.3 Continuous maps

The preceding discussion of matrices can be extended, without essential modifica-

tion, to the continuous map (3.2) describing the storage interaction in a quantum

memory. The Green’s function K(z, τ) has the same basic structure as a matrix,

except that it has two continuous arguments, instead of two discrete indices. And

it is also amenable to the SVD. The expression (3.6) for a matrix becomes, in the


continuous case

K(z, τ) =∑j

λjψj(z)φ∗j (τ). (3.16)

As before, the λ’s are the singular values. The functions ψj and φj are the continuous

analogues of the input and output basis vectors |uj〉 and |vj〉. We will refer to them

as modes; φ1 is the first input mode, ψ2 is the second output mode, and so on.

Alternatively, since the signal field and spin wave both appear in (3.2), it may be

physically more meaningful to talk of the φ’s as the signal modes, and the ψ’s as

spin wave modes. The inner product between two vectors a, b takes the form of an

overlap integral, for continuous functions a(x), b(x),

a†b = 〈a|b〉 =∑i

a∗i bi −→∫a∗(x)b(x) dx. (3.17)

The orthonormality conditions 〈ui|uj〉 = 〈vi|vj〉 = δij for basis vectors are therefore

replaced by the relations

∫ L

0ψ∗i (z)ψj(z) dz =

∫ ∞−∞

φ∗i (τ)φj(τ) dτ = δij . (3.18)

These conditions tend to produce sets of ‘wiggly’ functions, so that the product of

two different modes alternates between positive and negative values, and integrates

to zero. Sets of oscillating modes, satisfying orthonormality conditions like (3.18),

are common in harmonic analysis and acoustics; they are also the bread and butter

of much of quantum physics — quantum memories are no exception. In general, the


first modes, associated with the largest singular value λ1, are slowly varying. They

represent the basic ‘shape’ of the Green’s function, with broad brushstrokes. Higher

modes tend to oscillate faster, and they represent smaller corrections, at increasingly

fine levels of detail.

3.3.1 Normally and Anti-normally ordered kernels.

The continuous analogues of the matrices KN and KA in (3.10) are found by inte-

grating the product of two K kernels over one of their arguments,

KN (τ, τ ′) =∫ L

0K∗(z, τ)K(z, τ ′) dz,

KA(z, z′) =∫ ∞−∞

K(z, τ)K∗(z′, τ) dτ. (3.19)

These two kernels share the same eigenvalues; they satisfy the eigenvalue equations

∫ ∞−∞

KN (τ, τ ′)φj(τ ′) dτ ′ = λjφj(τ ′),∫ L

0KA(z, z′)ψj(z′) dz′ = λjψj(z′). (3.20)

Sometimes these kernels are more convenient to work with than the kernel K.

3.3.2 Memory Optimization.

Now it is clear how to optimize the performance of a quantum memory. We want

the largest efficiency η = NB/NA. But from the expressions in (3.1), we see that

NB is just the continuous analogue of b2, the squared norm of the output spin wave.


Suppose that we fix NA = 1, just as we fixed a2 = 1 in Section 3.2. How should

we choose the signal field A(τ) to maximize η? The answer carries over directly

from our considerations of matrices. We should choose A(τ) = φ1(τ). With this

choice, the resulting spin wave is B(z) = λ1ψ1(z), and the optimal storage efficiency

is η = λ21.

In practice, the SVD of the Green’s function is almost always computed numer-

ically, and to do this, the continuous function K is discretized on a fine grid: it is

converted into a matrix. Therefore our treatment of matrices applies directly when

calculating the singular values, and the mode functions, for a quantum memory.

Note that, since we must of course have η ≤ 1, any physical Green’s function will

have λ1 ≤ 1, and this condition is a useful check that K has been sampled with a

sufficiently fine grid.

3.3.3 Unitary invariance

When deriving a form for the Green’s function K, it will sometimes be easier to

work in terms of a transformed coordinate system. Suppose that we introduce a

new variable y = y(z), where y(z) is some monotonic single-valued (i.e. invertible)

function. The Green’s function, expressed in terms of y, has the same singular values

as in the original coordinates, provided that we make the transformation unitary by

including a Jacobian factor:

K(y, τ) = K [z(y), τ ]× 1√J(y)

, (3.21)


where z(y) is the inverse transformation, relating z to y, and where J(y) = ∂zy is

the Jacobian, relating the line elements dy and dz. Including the factor involving J

ensures that the norm of K is preserved in the new coordinate system,

∫|K(y, τ)|2 dy =

∫|K [z(y), τ ] |2 × ∂z

∂ydy

=∫|K(z, τ)|2 dz.

The output modes transform accordingly,

ψj(y) =ψj [z(y)]√

J(y). (3.22)

An identical procedure is used when transforming the time coordinate.

Instead of re-parameterizing the kernel, we may wish to transform to a frequency

representation, instead of a temporal one. That is, we might be able to work out an

expression for the Fourier transformed Green’s function K

K(z, ω) =1√2π

∫K(z, τ)eiωτ dτ. (3.23)

Inserting the SVD expansion (3.16) into (3.23), we obtain

K(z, ω) =∑j

λjψj(z)φ∗j (ω), (3.24)

where the modes φj are Fourier transforms of the φj . These Fourier transformed


modes also form an orthonormal set,

∫φ∗i (ω)φj(ω) dω =

12π

∫ [∫φ∗i (τ)e−iωτ dτ

∫φj(τ ′)eiωτ ′ dτ ′

]dω

=∫ ∫

δ(τ − τ ′)φ∗i (τ)φj(τ ′) dτ ′dτ

= δij , (3.25)

where in the penultimate step we used the plane-wave expansion of the Dirac delta

function (see §D.2 in Appendix D),

δ(x) =1

2π

∫e±ixy dy. (3.26)

The expression (3.24) is therefore precisely the SVD of the transformed kernel K.

The singular values of K are the same as those of K. In fact the temporal Fourier

transform we applied is just an example of a unitary transformation; unitary because,

by Parseval’s theorem, it is norm-preserving. That is to say,

∫|f(τ)|2 dτ =

∫|f(ω)|2 dω, (3.27)

for any function f and its Fourier transform f . As described in Section 3.1.1, the

singular values of a mapping are never altered by a unitary transformation.

Other useful possibilities include Fourier transforming over the spatial variable,

3.4 Optimizing storage followed by retrieval 85

to find the Green’s function in so-called k-space,

K(k, τ) =1√2π

∫K(z, τ)eikz dz. (3.28)

Again the singular values are the same as those for K, but the output modes are

k-space versions of the ψj .

Generally the integration limits in these Fourier transforms would be [−∞,∞],

but when boundary conditions are needed, it will sometimes be useful instead to

implement a unilateral transform, where the integration runs from 0 to ∞. The

utility of these techniques will become clear when we examine the memory inter-

action more closely in the next chapter. A brief review of both the bilateral and

unilateral Fourier transforms can be found in Appendix D.

3.4 Optimizing storage followed by retrieval

Suppose that we are not interested in the efficiency of storage alone, but rather

the combined efficiency of storage into, followed by retrieval from the memory. In

many situations it is this combined efficiency that is most experimentally relevant.

The same techniques as outlined above are directly applicable. The entire memory

interaction can be characterized as a map between the input and the output signal

fields,

Aout(τ) =∫ ∞−∞

Ktotal(τ, τ ′)Ain(τ ′) dτ ′. (3.29)

3.5 A Simple Example 86

The total efficiency of the memory is given by the ratio of the norms of Aout and

Ain. The input mode that maximizes this efficiency is therefore found from the

SVD of the Green’s function Ktotal, and the resulting optimal efficiency is given by

ηcombined = λ21, where λ1 is the largest singular value of Ktotal.

If we neglect any decoherence of the spin wave (as we do throughout this thesis),

the kernel Ktotal can be constructed from the kernels describing the storage and

retrieval processes individually. Under certain circumstances — when the retrieval

process is precisely the time-reverse of the storage process [133] — the combined

kernel Ktotal is equal to the normally ordered kernel KN defined in (3.19), and

ηcombined = η2. This is the optimal situation, and in general a mismatch between the

storage and retrieval processes reduces the memory efficiency so that ηcombined < η2.

These issues are explored in more detail in Chapter 6.

3.5 A Simple Example

Before embarking on a detailed derivation of the equations of motion for an ensemble

quantum memory, we run through a simple example that contains many of the

features that emerge from a more rigorous analysis.

We consider a classical optical signal pulse propagating through an ensemble of

identical atoms. We use a classical Lorentz model for the atoms [134], in which the

electric field of the light pulse ‘pulls’ on an electron in each atom, while a harmonic

restoring force ‘pulls back’, keeping the electrons bound around their equilibrium

positions. Let the average displacement at time τ , away from equilibrium, of an


electron located at position z, be given by x(z, τ). The average is taken over all

the atoms with the same position coordinate z, all of which behave identically. The

restoring force on each electron is given by Frestore = −kx, for some constant k.

This is a good approximation for any kind of restoring force, provided the typical

displacement x is small enough. The force on an electron due to the electric field

E of the signal pulse is Flight = eE, where e is the electronic charge. The classical

equation of motion for x is then given by Newton’s second law, Ftotal = ma, where

m is the electronic mass and a = ∂2τx is the acceleration of the electron. Putting

this together yields the equation

(m∂2

τ + k)x = eE. (3.30)

We can write the signal field as E = Aeiωsτ , where ωs is the optical carrier frequency

of the signal, and where A is a slow modulation describing the temporal profile of

the pulse. Irradiating the atoms will produce a response with a similar temporal

structure, so we write x = Beiωsτ , where B is some slowly varying envelope. Sub-

stituting this into the equation of motion, and neglecting the term ∂2τB, gives the

equation

∂τB = −iαA, (3.31)

where α = e2mωs

. Here for simplicity we have assumed that the signal field frequency

is tuned perfectly into resonance with the atoms, so that ωs =√k/m. This equation

describes the response of the atoms to the field; to complete the picture, we would


like to understand how the field responds to the atoms. As a first step, we apply a

(bilateral) Fourier transform from τ −→ ω, so that (3.31) becomes

B =α

ωA, (3.32)

where the tildes denote the transformed variables. The electronic displacement B

acts as a source for the signal field. More precisely, each oscillating electron generates

an electric field E(r) = ae/4πε0c2r in proportion to its acceleration1. Here ε0 as

usual denotes the permittivity of free space; r is the distance from the electron.

Summing the contributions from all the atoms in a thin slice of the ensemble with

thickness δz, we find the total field generated by the dipoles is

Etot = −iδzωsen

4πε0cx, (3.33)

where n is the atomic number density [137]. If we consider the propagation of the

signal through a thin slice of the ensemble of thickness δz, we therefore have

A(z + δz, ω) = A(z, ω)− iδzωsen

4πε0cB(z, ω)

=(

1− iβδz

ω

)A(z, ω), (3.34)

1This can be derived rather neatly from relativistic equivalence. The electrostatic potentialenergy V = e2/4πε0r of two electrons separated by r produces a relativistic mass increase M = V/c2

of the pair of electrons. The extra weight 12gM of one of the electrons, in a gravitational field g,

must have its origin in a vertical force eE exerted by the electric field E of the other electron. Theequivalence principle demands that we cannot tell if the field g is swapped for an acceleration a.Putting this all together, we find the field due to an accelerating charge is E ∼ ae/4πε0c2r [135,136].


where β = e2n8πε0mc

. Taking the limit as δz −→ 0, we derive Beer’s law of exponential

absorption for the signal field, with absorption coefficient β,

A(z, ω) = limδz→0

(1− i

βδz

ω

)z/δzAin(ω)

= e−iβz/ωAin(ω), (3.35)

where Ain(ω) = A(z = 0, ω) is the initial spectrum of the signal field, at the start of

the ensemble [138]. Substituting this into (3.32) gives an expression for the average

electronic displacement in the Fourier domain,

B(ω, z) = αe−iβz/ω

ω× Ain(ω). (3.36)

Using the result (D.40) from §D.5 in Appendix D, along with the convolution theo-

rem (D.21), we can take the inverse Fourier transform of this to obtain the map

Bout(z) = −iα∫ T

0J0

[2√βz(T − τ)

]Ain(τ) dτ ′, (3.37)

where Bout(z) = B(τ = T, z) is the electronic displacement at time τ = T , with

T some time chosen to define the end of the interaction, after the signal pulse has

passed through the ensemble. Here J0 is a zero’th order ordinary Bessel function

of the first kind — we will encounter this function, through similar inverse Fourier

transforms, frequently in Chapter 5. We have arranged for the notation in (3.37) to

appear suggestive of the storage map described at the start of this Chapter. If we


identify the electronic displacement Bout as the amplitude of a spin wave excited by

the signal field, then the storage kernel K in (3.2) can be identified with the Bessel

function appearing in the integrand of (3.37). A numerical SVD of the Green’s

function

K(z, τ) = −iαJ0

[2√βz(T − τ)

](3.38)

would reveal the temporal shape of Ain that maximizes the degree of atomic exci-

tation. We have not taken care to normalize the signal field A, or the electronic

displacement B, and so (3.1) is not quite true, using the current definitions. But

up to some constant normalization, the optimal efficiency of a quantum memory

described by the above model would be provided by squaring the largest singular

value of (3.38).

Our purpose in the above exposition has not been to describe a real quantum

memory in any quantitative detail. But this simple example contains many of the

ingredients we will encounter in the following Chapters. We have used a one di-

mensional propagation model, combined with some atomic physics, to obtain two

equations of motion — the first describing the atomic response to the field, and the

second describing the influence on the field due to the induced atomic polarization.

The solution was found using a Fourier transform, and the result took the form of

the storage map (3.2), with the kernel given by a Bessel function. This story applies

equally to the fuller treatment given shortly. Finally, note that we have made no

use of the formalism of quantum mechanics so far. Of course, a correct description

of the atoms at least requires that we quantize the electronic energies. But the


propagation is simply Maxwellian electrodynamics. It may help to keep in mind

that, although we will treat the signal field quantum mechanically for the sake of

completeness, the quantum memories we consider behave essentially classically. Or,

we might say that their efficiencies may be derived classically, since efficiencies do

not depend on correlation functions of the optical fields, and it is only in the photon

statistics revealed by these correlation functions that non-classicality is manifest.

Chapter 4

Equations of motion

In this chapter we derive the dynamical equations describing the interaction of light

with the atoms of an ensemble quantum memory. We focus on EIT and Raman stor-

age, which can be treated together; inhomogeneously broadened memories, such as

CRIB and AFC, are covered in Chapter 7. We borrow techniques from the treatment

of Stokes scattering by Mostowski et al. [139], and some notation from the later treat-

ment of quantum memories by Gorshkov et al. [140]. The derivation divides broadly

into three parts. First, we write down the Hamiltonian describing the interaction

of a single atom with the signal and control fields, and we obtain Heisenberg equa-

tions for the atomic evolution. Next, we introduce Maxwell’s equations, describing

the coupling of the signal field to the macroscopic atomic polarization as it propa-

gates through the ensemble. Finally, we add up the contributions from many atoms

to form the macroscopic variables describing the atomic polarization and the spin

wave. Having derived the dynamical equations in a convenient form, we investigate

4.1 Interaction 93

various methods of solution, in order to optimize the performance of the memory,

in the next chapter.

4.1 Interaction

We consider the propagation of a weak signal field through an ensemble of Λ-type

atoms, in the presence of a bright control field, tuned into two-photon resonance

with the signal, as depicted in Figure (4.1).

Figure 4.1 The signal (blue) and control (green) fields involved ina Λ-type ensemble quantum memory.

As shown in Section C.4 in Appendix C, the interaction of a light beam with an

atom is well described by the electric dipole Hamiltonian HED = −E.d, where E

is the electric field associated with the light at the atomic position, and where d is

the dipole moment associated with an optically active electron in the atom.

4.2 Electric Field 94

4.2 Electric Field

The electric field is composed of two parts, a bright classical control field, and a

weak signal field that we wish to store, both propagating along the z-axis,

E = Ec +Es. (4.1)

The control field is sufficiently intense that it is not affected by its interaction with

the atoms of the memory, so that we do not need to treat it as a dynamical variable

in the Hamiltonian. We therefore represent it as a classical field,

Ec(t, z,ρ) = vcEc(t, z,ρ)eiωc(t−z/c) + c.c., (4.2)

where vc is the control polarization vector1, ωc is the central frequency of the control

field, and Ec(t, z,ρ) is the slowly varying envelope of the control, describing the

spatio-temporal profile of the control pulse. Here ρ = xx + yy is a transverse

position vector, as shown in Figure 4.1.

The signal field is much weaker than the control, and in general it may be in

a non-classical state (for example, a Fock state — see Appendix C). Therefore we

treat the signal field quantum mechanically. The signal field, when well-collimated,1Note that if the polarization is not linear (circular, for instance), the polarization vector is

complex, satisfying v∗.v = v†v = 1.


can be written in the form

Es(z,ρ) = ivs∫g(ω)a(ω,ρ)e−iωz/c dω + h.c., (4.3)

where vs is the signal polarization vector, g(ω) =√

~ω/4πε0c is the mode amplitude,

and a(ω,ρ) is an annihilation operator for a signal photon with frequency ω, and

transverse position ρ, which satisfies the equal-time commutation relation

[a(ω,ρ), a†(ω′,ρ′)] = δ(ω − ω′)δ(ρ− ρ′). (4.4)

The form of (4.3) is very similar to the expression (C.10) in Appendix C, the only

difference being the inclusion of the transverse position ρ, which allows us to treat

diffraction (this is covered in Chapter 6). In the next chapter we will drop the

transverse coordinate and work with a one dimensional propagation model. Note

that all these operators have no time-dependence in the Schrodinger picture (see

Appendix B). In the Heisenberg picture, commutation with the optical free-field

Hamiltonian (see (4.14) below) gives the annihilation operators the simple time-

dependence

a(ω,ρ, t) = a(ω,ρ)eiωt. (4.5)

It will be useful to factorize the signal field into a carrier wave, and a slowly vary-

ing envelope, as we did with the control field in (4.2). To do this, we make use of

the approximation that the bandwidth of the signal field will be very small in com-


parison to its central frequency ωs. Therefore, only terms with frequencies rather

close to ωs will be important in the integral in (4.3). Since the dependence of the

mode amplitude g(ω) on frequency is quite weak (only ‘square-root’), we make the

replacement g(ω) −→ g(ωs). We are then able to perform the frequency integral

explicitly, to obtain

Es(t, z,ρ) = ivsgsA(t, z,ρ)eiωs(t−z/c) + h.c., (4.6)

where gs =√

2πg(ωs), and where we have defined the slowly varying time-domain

annihilation operator A according to the relation

A(t, z,ρ) = e−iωs(t−z/c) × 1√2π

∫a(ω,ρ, t)e−iωz/c dω. (4.7)

That it retains the property of a photon annihilation operator, albeit for spatio-

temporal, rather than spectral modes, can be seen from its commutator with its

Hermitian adjoint. Inserting (4.5) into (4.7), we find

[A(t, z,ρ), A†(t′, z′,ρ′)] = δ(t− t′ − (z − z′)/c)δ(ρ− ρ′). (4.8)

Aside from the leading phase factor, which removes the rapid time-dependence due

to the carrier frequency ωs, the action of A(t, z,ρ) in free space can be understood

as the annihilation of a signal photon in a spatio-temporal mode centred at position

ρ and retarded time τ = t− z/c.

4.3 Dipole Operator 97

4.3 Dipole Operator

Having developed a convenient notation for the signal and control fields, we now

consider the atomic variables. The Coulomb interaction between the atomic nu-

cleus and the electrons is of course rather complex in general, but the formalism

of quantum mechanics comes to the rescue. The energy levels labelled |1〉, |2〉, |3〉

are eigenstates of the atomic Hamiltonian, and therefore they form an orthonormal

basis for the Hilbert space of quantum states of the atom (see Appendices B and C).

That is, any state of the atom can be described in terms of these states. And any

operator acting on the atomic states can be expressed using the coordinate system

defined by these states. In particular, the electric dipole operator d can be written

in the following way,

d =∑j,k

djkσjk, (4.9)

where the coefficients djk = 〈j|d|k〉 are the matrix elements of the dipole opera-

tor, the σjk = |j〉〈k| are ‘flip operators’ (sometimes known as transition projection

operators), and where the double summation runs over the three atomic states.

To preempt possible confusion, we should clarify that d is a three-dimensional

vector in space, whose elements are quantum mechanical operators acting on the

three-dimensional Hilbert space of the Λ-level atom. The Dirac notation (|1〉, |2〉,

etc...) refers to vectors and/or operators in/on this Hilbert space, while the bold

font notation (d, E, v, etc...) refers to vectors in ordinary space (whether their

elements are numbers, or operators).


4.3.1 Parity

So far, we have not used any properties of the dipole operator specifically — (4.9) is

an identity that holds for any atomic operator. But now we can use the parity of the

dipole operator to remove some terms from the sum in (4.9). Parity refers to the way

a quantity transforms under the operation of inversion, when all spatial coordinates

are reflected in the origin. That is, r −→ −r, where r is any position vector. As

discussed in Appendix C, the dipole operator is simply given by d = −er, where now

r is the position, with respect to the atomic centre-of-mass, of the optically active

electron. Therefore under inversion, we have d −→ −d. The atomic dipole operator

has negative parity. This means that all the diagonal dipole matrix elements must

vanish, djj = 0. To see this, we can express the matrix element djj in terms of the

wavefunction ψj of the state |j〉,

djj = 〈j|d|j〉

=∫ψ∗j (r)dψj(r) d3r. (4.10)

The integral runs over all space, but we can divide it into a pair of integrals: the

first over half of all space, with positive coordinates r (+), and the second over the

remaining half of space, with negative coordinates −r (−). Since the dipole operator


changes sign under inversion, the second integral exactly cancels with the first,

djj =∫

+|ψj(r)|2d d3r +

∫−|ψj(r)|2d d3r

=∫

+|ψj(r)|2(d− d) d3r

= 0. (4.11)

Here we used the fact that |ψj |2 has positive parity (i.e. is unaffected by inversion).

This must be true since |ψj |2 describes the electronic charge density associated with

the state |j〉, and this must be spherically symmetric: there is no interaction to

break the spherical symmetry of the bare atom.

The electric dipole interaction is therefore completely off-diagonal, meaning that

it only couples different states together, never the same state to itself. Furthermore,

we require that the state |3〉 is long-lived, in order that it can serve as an effective

storage state. Therefore we neglect any dipole coupling between the states |1〉 and

|3〉, so that no direct transitions between these states are mediated by the dipole

operator: our goal, instead, is to implement an indirect transition, mediated by the

control field. In the light of these arguments, we arrive at the somewhat pared-down

expression

d = d12σ12 + d23σ23 + h.c.. (4.12)

4.4 Hamiltonian 100

4.4 Hamiltonian

We are now in a position to write down the Hamiltonian for the atom-light system,

H = HA +HL +HED. (4.13)

Here HL is the free-field energy of the light field,

HL =∫ω

∫Aa†(ω,ρ)a(ω,ρ) d2ρdω, (4.14)

where the transverse integral runs over the transverse area A of the signal field.

Here we have neglected the zero-point energy, and also the fixed energy associated

with the control field. HA is the Hamiltonian for the bare atom. The atomic states

|1〉, |2〉, |3〉 are by definition eigenstates of HA, and therefore when written in terms

of these states, HA is purely diagonal (See Section A.4.3 in Appendix A),

HA =∑j

ωjσjj , (4.15)

where ωj is the resonant frequency of state |j〉.

We can now use the Heisenberg equation (see B.10 in Appendix B) to find the

time evolution of the atomic flip operators. Since these operators always commute

with the optical free-field Hamiltonian HL, we drop this from the Hamiltonian (it

4.4 Hamiltonian 101

has no effect on the atoms), and we work with the equation

∂tσjk = i[σjk, HA +HED]. (4.16)

The flip operators satisfy the following multiplicative identity,

σijσkl = σilδjk, (4.17)

and under Hermitian conjugation we have σ†jk = σkj . Using these relations we obtain

five independent atomic equations; two for the atomic populations,

∂tσ11 = −iE. (d12σ12 − h.c.) ,

∂tσ33 = iE. (d23σ23 − h.c.) , (4.18)

and three for the atomic coherences,

∂tσ12 = iω21σ12 − iE. [d∗12 (σ11 − σ22) + d23σ13] ,

∂tσ13 = iω31σ13 − iE. [d∗23σ12 − d∗12σ23] ,

∂tσ23 = iω32σ23 − iE. [d∗23 (σ22 − σ33)− d12σ13] , (4.19)

where we have defined ωjk = ωj − ωk as the frequency difference between the states

|j〉 and |k〉. Note that∑

j σjj = I, the identity, so that the sum of the populations

commutes with the Hamiltonian, and therefore has no time-dependence. This simply

4.5 Linear approximation (1) 102

expresses the fact that the atom remains in one of the states |1〉, |2〉, |3〉, at all times.

Therefore ∂tσ22 = −∂t(σ11 + σ33).

The coupled equations (4.18) and (4.19) constitute a rather complex system,

and it is not possible to extract an analytic solution, in general. Fortunately, the

equations simplify considerably in the linear regime.

4.5 Linear approximation (1)

Provided that we store a small number of photons in the quantum memory, such

that most of the atoms remain in their ground states, with only very few atoms

excited, we can ignore the dynamics of the atomic populations. We replace the

operators σjj by their expectation values on the atomic ground state,

σ11 −→ 1, σ22 −→ 0, σ33 −→ 0. (4.20)

This leaves us with just the three equations for the coherences,

∂tσ12 = iω21σ12 − iE. [d∗12 + d23σ13] ,

∂tσ13 = iω31σ13 − iE. [d∗23σ12 − d∗12σ23] ,

∂tσ23 = −iω23σ23 + iE.d12σ13. (4.21)

4.6 Rotating Wave Approximation 103

4.6 Rotating Wave Approximation

The leading terms in each of the equations in (4.21), of the form iωkjσjk, simply de-

scribe rapid oscillations. The electric field E is also oscillating rapidly; the combined

dynamics of the coherences will therefore contain components oscillating at both the

sum and difference frequencies of these oscillations. Physically, these contributions

give rise to different time-orderings of the two-photon Raman transition between

states |1〉 and |3〉, as shown in Figure 4.2. When the detuning is small compared to

optical frequencies, the sum frequencies are many orders of magnitude higher than

the difference frequencies, and so the sum frequencies average out to zero: they can

be neglected. This is the content of the rotating wave approximation (RWA).

rotating

counter-rotating

Figure 4.2 Time-ordering. The rotating wave approximation ne-glects counter-rotating terms, which correspond to strongly sup-pressed time-orderings for the Raman process.

To implement the RWA, we define rotating coherences by the ansatz

σjk = σjkeiωjkτ , (4.22)

4.6 Rotating Wave Approximation 104

where τ = t− z/c is the retarded time (see the text following (4.8)). Including the

dependence on z here will be useful when we consider propagation. Inserting (4.22)

into (4.21) yields the equations

∂tσ12 = −iE.[d∗12e

−iω21τ + d23σ13e−iω23τ

],

∂tσ13 = −iE.[d∗23σ12e

iω23τ − d∗12σ23e−iω21τ

],

∂tσ23 = iE.d12σ13eiω21τ . (4.23)

Inserting the expressions (4.2) and (4.6) for Ec and Es into E, and multiplying out

the resulting expressions, we find terms multiplying rapidly varying exponentials,

like e−i(ω21+ωs)τ , for instance, and also terms multiplying slowly varying exponen-

tials, like e−i(ω21−ωs)τ = e−i∆τ . Neglecting the fast oscillating terms, we obtain the

equations

∂tσ12 = −id∗12.[vcEce

−i∆+τ + ivsgsAe−i∆τ]− id23.

[vcEce

−i∆τ + ivsgsAe−i∆−τ]σ13,

∂tσ13 = −id∗23.[v∗cE

∗c e

i∆τ − iv∗sgsA†ei∆−τ

]σ12 + id∗12.

[vcEce

−i∆+τ + ivsgsAe−i∆τ]σ23,

∂tσ23 = id12.[v∗cE

∗c e

i∆+τ − iv∗sgsA†ei∆τ

]σ13, (4.24)

where we have defined ∆+ = ω21 − ωc and ∆− = ω23 − ωs (see Figure 4.3).

4.7 Unwanted Coupling 105

(a) (b) (c)

Figure 4.3 Useful and nuisance couplings. (a) The desired quan-tum memory coupling. (b) The control field couples to the groundstate, initiating spontaneous Stokes scattering. (c) The signal fieldcouples to the storage state: it is very weak, and there is no significantpopulation in this state, so the effect of this coupling is negligible.

4.7 Unwanted Coupling

We have already succeeded in dramatically simplifying the dynamical equations, but

it is still not obvious how the behaviour described by this system of equations allows

for the implementation of a quantum memory. Other physical processes obscure

the useful features of the system. For instance, the term involving d∗12.vc represents

the coupling of the control field to the |1〉 ↔ |2〉 transition. If this term is strong,

the control field can initiate spontaneous Stokes scattering, as shown in Figure 4.3

(b). Aside from complicating the equations, this process can be problematic, since it

generates excited atoms in the state |3〉 that are not correlated with the signal field.

When we attempt to retrieve the signal field, some of these uncorrelated atoms may

contribute a noisy background emission. In practice, it is possible to distinguish

the signal from the noise using phasematched retrieval, which we discuss in §6.3.2

in Chapter 6. In any case, a description of the memory interaction requires that we

can eliminate this unwanted coupling. As can be seen from Figure 4.3, the Stokes

scattering process is detuned further from resonance, with detuning ∆+ > ∆. If

4.8 Linear Approximation (2) 106

∆+ is sufficiently large (this requires that the splitting ω31 is large compared to ∆),

the term representing Stokes scattering will oscillate quickly enough to be neglected.

Alternatively, it may be possible to choose the polarization of the control such that

the product d12.vc vanishes due to a selection rule, although strict selection rules

usually require the application of an external magnetic field. Polarization selection

rules are discussed in the context of cesium in §F.4 of Appendix F. Regardless of

the justification, in the following analysis we set d12.vc = 0. By the same token, we

set d23.vs = 0, thereby disregarding any coupling of the signal field to the |2〉 ↔ |3〉

transition, as depicted in Figure 4.3 (c). The equations of motion are now given by

∂tσ12 = d∗12.vsgsAe−i∆τ − id23.vcEce

−i∆τ σ13,

∂tσ13 = −id∗23.v∗cE∗c e

i∆τ σ12 − d∗12.vsgsAe−i∆τ σ23,

∂tσ23 = d12.v∗sgsA

†ei∆τ σ13. (4.25)

4.8 Linear Approximation (2)

Now that we have arrived at the more transparent set of equations (4.25), we can

identify some terms in these equations that are ‘small’, in the sense that ‘weakly

excited’ operators are involved. To be more concrete, we analyse the size of each

term perurbatively. We attach a parameter ε to each of the coherences σjk, and also

to the signal field A. This parameter just labels these quantities as ‘small’; in the

case of the coherences, they are initially vanishing, and in the case of the signal field,

4.8 Linear Approximation (2) 107

only a very few signal photons are sent into the memory. The equations become

ε∂tσ12 = d∗12.vsgsεAe−i∆τ − id23.vcEce

−i∆τ εσ13,

ε∂tσ13 = −id∗23.v∗cE∗c e

i∆τ εσ12 − d∗12.vsgsε2Ae−i∆τ σ23,

ε∂tσ23 = d12.v∗sgsε

2A†ei∆τ σ13. (4.26)

There are two terms proportional to ε2. They correspond to ‘second-order’ pertur-

bative corrections to the dynamics, and they can be neglected. Dropping the label

ε, the linearized equations of motion for the atomic coherences are

∂tσ12 = d∗12.vsgsAe−i∆τ − id23.vcEce

−i∆τ σ13,

∂tσ13 = −id∗23.v∗cE∗c e

i∆τ σ12. (4.27)

These equations contain the important physics of the quantum memory interaction.

They describe how the coherence σ12 is directly excited by the signal field A, and

how this excitation is then coupled to the coherence σ13 through the control field

Ec. Macroscopically, when many identical atoms are involved, we can identify σ12

with the atomic polarization, and σ13 with the spin-wave. Before considering the

collective dynamics of the ensemble as a whole, we first consider the propagation of

the signal field.

4.9 Propagation 108

4.9 Propagation

Heisenberg’s equations describe evolution in time, but it is not obvious how to treat

the spatial propagation of the signal field in this formalism. In fact, it is possible to

do this, but a more transparent derivation utilizes Maxwell’s equations. The atomic

ensemble behaves as a dielectric medium in the presence of the signal field, for which

the appropriate formulation is as follows,

∇.D = ρfree, ∇.B = 0,

∇×E = −∂tB, ∇×H = Jfree + ∂tD. (4.28)

Here D is the displacement field, H is the magnetic field, E is the electric field and

B is the magnetic induction. The distinction between B and H will not be very

important for us, since all the materials we are concerned with are non-magnetic.

We simply take B = µ0H, where µ0 is the permeability of free space, and refer to

B as the magnetic field. The quantities ρfree and Jfree are the free charge density

and current, respectively. The designation ‘free’ refers to the fact that they are not

induced by the fields: they are charges and currents that are not associated with

any material dielectric properties. In any case we only deal with materials in which

there are no free charges or currents, and therefore we set ρfree = Jfree = 0. A wave

equation for the electric field is found by differentiating the equation for H, and

4.9 Propagation 109

substituting in the equation for E,

∇× ∂tH = ∂2tD,

⇒ −∇× (∇×E) = µ0∂2tD. (4.29)

The double curl derivative on the left can be simplified using the vector calculus

identity

∇× (∇×E) = ∇ (∇.E)−∇2E. (4.30)

The signal field is a transverse propagating wave, which is divergence free2. This

is evident from the definition of the Coulomb gauge (see Appendix C), in which

∇.A = 0, with E = ∂tA, where A is the magnetic vector potential. We therefore

drop the first term in (4.30), to obtain

∇2E = µ0∂2tD. (4.31)

The displacement field is formed from the sum of the electric field, and the material

polarization P ,

D = ε0E + P , (4.32)

where P is the polarization density, defined as the dipole moment per unit volume.

Substituting (4.32) into (4.31), and using the relation ε0µ0 = 1/c2, we arrive at the2These arguments apply to a homogeneous and isotropic dielectric containing no source

charges [141]

4.10 Paraxial and SVE approximations 110

wave equation [∇2 − 1

c2∂2t

]E = µ0∂

2tP . (4.33)

This equation relates the propagation of the optical fields to the atomic polarization

in the memory. Since the control field is so intense, it is not significantly affected by

its interaction with the ensemble, and so we do not consider its propagation further.

For the signal field, we use the equation

[∇2 − 1

c2∂2t

]Es = µ0∂

2tPs, (4.34)

where Ps is the component of the atomic polarization which acts as a source for the

signal field. That is, Ps is the component of the polarization oscillating at the signal

carrier frequency ωs. Further simplification is accomplished by making use of the

fact that the signal and control fields are collimated beams.

4.10 Paraxial and SVE approximations

The slowly varying envelope approximation (SVE), and the paraxial approximation,

are both implicit in the decomposition (4.6). We assume that the amplitude A (its

status as an operator is not important for this discussion) is a smooth, slowly varying

function of time and space. The exponential factor then represents the optical carrier

wave, oscillating in time with frequency ωs, and oscillatiing along the z-axis with

wavevector ks = ωs/c.

The paraxial approximation allows us to treat the signal field as a beam traveling

4.10 Paraxial and SVE approximations 111

along the z-axis, with negligible divergence. This approximation is satisfied as long

as the transverse spatial profile of A is much larger than the signal wavelength

λs = 2π/ks.

The SVE approximation allows us to treat the propagation of the signal field

purely in terms of the envelope of the signal pulse, represented by the temporal

shape of A, without having to explicitly model the very fast time-dependence of the

carrier wave, which oscillates much faster. To make this approximation successfully,

we should have that the temporal duration of the signal field is much longer than

the optical period 2π/ωs.

To see how these approximations simplify the theoretical description, we insert

the signal field (4.6) into the wave equation (4.34). We consider only the positive

frequency component of the signal field, since only this component is coupled to

the atoms through the system (4.27). We also define the slowly varying atomic

polarization Ps by factorizing out the signal frequency,

Ps = Pseiωsτ . (4.35)

The resulting wave equation is

[∇2 − 1

c2∂2t

] [ivsgsAeiωs(t−z/c)

]= µ0∂

2t

[Pse

iωs(t−z/c)]. (4.36)

We take the scalar product of both sides with the polarization vector v∗s (this is the

same as taking the inner product with v†s, see Appendix A), and apply the chain

4.11 Continuum Approximation 112

rule for the derivatives to obtain

∇2⊥ +

(∂2z −

1c2∂2t

)− 2i

ωsc

(∂z +

1c∂t

)−[(ωs

c

)2− 1c2ω2s

]A = −i

µ0

gsv∗s .(∂2t + 2iωs∂t − ω2

s

)Ps,

(4.37)

where we have divided out the exponential factor eiωsτ . The transverse Laplacian

∇2⊥ = ∂2

x+∂2y describes diffraction of the signal field as it propagates. Note that the

term in square brackets on the left hand side vanishes. Concerning the remaining

terms, we observe that according to the SVE approximation

∣∣∣∣ks(∂z +1c∂t

)A

∣∣∣∣ ∣∣∣∣(∂2z −

1c2∂2t

)A

∣∣∣∣ . (4.38)

We therefore drop the smaller term. For the same reason, we drop all but the last

term on the right hand side, and we arrive at the equation

(i

2ks∇2⊥ + ∂z +

1c∂t

)A = − µ0ω

2s

2gsksv∗s .Ps. (4.39)

This equation describes the coupling of the atoms to the signal field amplitude.

It now remains for us to connect the atomic evolution equations (4.27) with the

macroscopic polarization Ps.

4.11 Continuum Approximation

We have in mind an ensemble of sufficient density that the atoms form an effective

continuum. Consider a small region within the ensemble at position r = (z,ρ),


with volume δV . We’ll call this a voxel. To make the continuum approximation,

we should have many atoms in each voxel; nδV 1, where n is the number den-

sity of atoms in the ensemble. Each voxel should be ‘pancake-shaped’, so that its

thickness δz along the z-axis satisfies δz λs, while its transverse area δA satis-

fies δA λ2s. The condition on δz ensures that the longitudinal optical phase ksz

is roughly constant throughout the voxel. The condition on δA ensures that the

typical interatomic separation is much larger than the signal wavelength λs, so that

dipole-dipole interactions between the atoms can be neglected, and we can treat the

atoms as isolated from one another. The macroscopic polarization at position r is

then found by adding up the dipole moments in the voxel located at r,

P =1δV

∑β(r)

dβ, (4.40)

where the index β runs over all the atoms in the voxel at position r. Using the

expression (4.12) for each atom, and recalling that the coherence σ23 is negligible

(see Section 4.8), we have

P =1δV

∑β(r)

(d12σ

β12 + h.c.

). (4.41)

We now define macroscopic variables involving sums over the coherences σjk, in order

to ‘tie up’ the system of equations (4.27) with the propagation equation (4.39). For


the macroscopic polarization, we define the operator

P =1√nδV

∑β(r)

σβ12ei∆τ . (4.42)

Note that P is not simply the magnitude of the vector P . They are closely related

(see (4.47) below), but some book-keeping is required to ensure that we keep track of

all the relevant constants. In the same vein, for the spin wave we define the operator

B =1√nδV

∑β(r)

σβ13. (4.43)

These definitions are motivated by analogy with the slowly varying photon annihi-

lation operator A. For instance, the equal-time commutator of B with its Hermitian

adjoint is given by

[B(z,ρ, t), B†(z′,ρ′, t)

]=

1(δV )2n

∑β(r)

∑γ(r′)

[σβ13, σ

γ31

]

=

1

(δV )2n× nδV (σ11 − σ33) if r and r′ label the same voxel,

0 otherwise.

(4.44)

Using the linear approximation (4.20), we find, in the continuum limit δV −→ 0,

[B(z,ρ, t), B†(z′,ρ′, t)

]= δ(z − z′)δ(ρ− ρ′). (4.45)


Identical arguments yield the commutator

[P (z,ρ, t), P †(z′,ρ′, t)

]= δ(z − z′)δ(ρ− ρ′), (4.46)

for the polarization. Therefore both the operators P and B satisfy bosonic commuta-

tion relations. We interpret them as annihilation operators for an atomic excitation

at position r. That is, P (z,ρ) annihilates a distributed excitation of the excited

state |2〉 at position (z,ρ). And B(z,ρ) annihilates a distributed excitation of the

storage state |3〉.

Substituting the definition (4.42) into (4.41), and taking the positive frequency

component (i.e. the component oscillating at a frequency +ωs), we find

Ps =√nd12P, (4.47)

for the slowly varying macroscopic polarization. We can now write down equations

governing both the propagation of the signal field, and the atomic dynamics. These

equations are,

(i

2ks∇2⊥ + ∂z +

1c∂t

)A = −κ∗P,

∂tP = i∆P + κA− iΩB,

∂tB = −iΩ∗P, (4.48)

4.12 Spontaneous Emission and Decoherence 116

where we have defined the control field Rabi frequency

Ω =d23.vc

~Ec, (4.49)

and the coupling constant

κ =d∗12.vs

~×√ngs =

d∗12.vs~

√~ωsn2ε0c

. (4.50)

The equations (4.48) describe the propagation and diffraction of a weak signal field

through an ensemble of ideal atomic Λ-systems, prepared initially in their ground

states. We have not yet included any description of decay processes, such as spon-

taneous emission from the excited state, or collisional de-phasing of the spin wave.

These two processes are expected to happen on very different time-scales, but they

can be treated in exactly the same way, which we now introduce.

4.12 Spontaneous Emission and Decoherence

In Section (C.5) in Appendix (C), we discuss the description of Markovian decay

in the Heisenberg picture using Langevin equations. We used a model in which a

bosonic system was coupled to a large reservoir, also composed of bosons. In the

present discussion, all the operators we consider — A, B and P — are bosonic in

character (see Eqs. (4.8), (4.45) and (4.46) for their commutators). In the case of

spontaneous emission, these operators are coupled to the electromagnetic field, which

is a large reservoir of bosons, and so the model of Appendix C is applicable. Using


this model, spontaneous emission at the rate γ is described by incorporating an ap-

propriate decay term into the dynamical equation for P , along with a Langevin noise

operator FP , which introduces fluctuations that preserve the bosonic commutation

relations of P . The noise operator is delta-correlated, meaning that no correlations

exist between its values at different times. And since all the atoms in the ensemble

are subject to independent fluctuations, no correlations exist between the noise at

different positions. These properties are summarized by the relations

〈F †P (t, z,ρ)FP (t′, z′,ρ′)〉 = 2γnP × δ(t− t′)δ(z − z′)δ(ρ− ρ′),

〈FP (t, z,ρ)F †P (t′, z′,ρ′)〉 = 2γ(nP + 1)× δ(t− t′)δ(z − z′)δ(ρ− ρ′), (4.51)

where the expectation value is taken on the initial state of both the ensemble and

the reservoir (i.e. the electromagnetic field to which the atoms are coupled). Here

the number nP is the initial number of atoms thermally excited into state |2〉, on

average. When dealing with optical transitions, we typically have nP = 0, of course.

Similarly, to treat decoherence of the spin wave B at a rate γB, we add a decay

term and a noise operator FB, which satisfies identical relations to (4.51), again with

nB = 0. The dynamical equations including these dissipative processes are then

(i

2ks∇2⊥ + ∂z +

1c∂t

)A = −κ∗P,

∂tP = −γP + i∆P + κA− iΩB + FP ,

∂tB = −γBB − iΩ∗P + FB. (4.52)


Note that these decay rates are for the atomic coherences. The atomic populations

decay at twice these rates. For instance, the number of spin wave excitations is

given by

NB =∫ L

0

∫AB†(z,ρ)B(z,ρ) d2ρdz. (4.53)

In the absence of any optical fields (with Ω = 0), we find that

∂t〈NB〉 = −2γB〈NB〉, (4.54)

where the expectation value is taken on an arbitrary state. Any terms involving the

noise operator FB vanish when taking the expectation value because its fluctuations

average to zero. The relation (4.54) shows that the number of spin wave excitations

decays at the rate 2γB. The same argument applied to P shows that the number

of excited atoms, in the state |2〉, decays at the rate 2γ, so that the spontaneous

lifetime of the state |2〉 is 1/2γ.

Now that we have introduced losses and decoherence with a degree of formal

rigour, we see that in fact the noise operators FB, FP may be neglected when

optimizing the performance of a quantum memory. The reason is that the efficiency

of a quantum memory depends only on the ratio of stored to input excitations.

That is, only the number operators for the signal field and spin wave are involved.

These number operators involve only normally ordered products, of the form A†A

or B†B, and therefore only normally ordered products of the F operators enter into

the efficiency. Since we have n ∼ 0, any products of the form F †F vanish when an


expectation value is taken, as is clear from (4.51). Therefore the Langevin operators

can be safely dropped from the system of equations (4.52). Of course the decay

terms are important!

Chapter 5

Raman & EIT Storage

Here we use the equations of motion derived in the last chapter to study the opti-

mization of the EIT and Raman quantum memory protocols.

5.1 One Dimensional Approximation

The analysis is greatly simplified if we use a one dimensional model, so that we

only consider propagation along the z-axis. This can always be made a good ap-

proximation by using laser-beams with low divergence. The effects of diffraction

are considered in Chapter 6. In the following, we will average over the transverse

coordinate ρ, to produce a one dimensional propagation model. We re-define the

5.1 One Dimensional Approximation 121

variables A, P and B by integrating over the transverse area A of the signal field,

A(t, z,ρ) −→ A(t, z) =1√A

∫AA(t, z,ρ) d2ρ,

P (t, z,ρ) −→ P (t, z) =1√A

∫AP (t, z,ρ) d2ρ,

B(t, z,ρ) −→ B(t, z) =1√A

∫AB(t, z,ρ) d2ρ. (5.1)

Having averaged the variables in this way, we now drop the transverse Laplacian

∇2⊥ from the propagation equation for A in (4.52). Further simplifications follow.

For instance, in the absence of any transverse structure, the control field envelope

Ec, propagating undisturbed at the speed of light along the z-axis, can be written

as a function of the retarded time τ = t − z/c only. We therefore make a change

of variables from (t, z) to (τ, z), which enables us to write the control field Rabi

frequency as Ω = Ω(τ). Furthermore, the mixed derivative in the propagation

equation for A is simplified, since

∂z

)t+

1c∂t

)z

= ∂z

)τ, ∂t

)z

= ∂τ

)z, (5.2)

where the subscripted parentheses indicate the variables held constant. In this new

coordinate system, the one dimensional equations of motion for a Λ-type quantum

5.1 One Dimensional Approximation 122

memory are therefore given by

∂zA(z, τ) = −κ∗P (z, τ),

∂τP (z, τ) = −ΓP (z, τ) + κA(z, τ)− iΩ(τ)B(z, τ),

∂τB(z, τ) = −iΩ∗(τ)P (z, τ), (5.3)

where Γ = γ−i∆ is the complex detuning. Note that we have dropped the decay term

associated with the decoherence of the spin wave B: by assumption this is negligible

on the time-scale of the storage process, which is what we seek to optimize. We have

also dropped the Langevin noise operator FP associated with spontaneous decay of

the polarization P , since this noise does not affect the efficiency (see the end of

Section 4.12 in the previous chapter). The equations (5.3) are mercifully rather

simple. Certainly they are easier on the eye than any of their previous incarnations

in Chapter 4! The elimination of the Langevin noise operators means that these

equations are now entirely classical in nature: we can treat (5.3) as a system of

coupled partial differential equations in three complex-valued functions A, P and

B. Since the equations are linear, the solutions will be linear, and we need not

worry about issues involving commutators or operator ordering. Our aim is to

find an expression for the Green’s function K(z, τ), relating the input signal field

Ain(τ) = A(z = 0, τ) to the final spin wave Bout(z) = B(z, τ −→ ∞). In the

absence of decoherence of the spin wave, the limit τ −→∞ is simply a mathematical

shorthand for “the end of the storage interaction”, when the control and signal fields

5.2 Solution in k-space 123

have fallen away to zero. As described in (3.2) in Chapter 3, taking the SVD of K

will tell us how to optimize the memory efficiency. We now attempt a solution of

the system (5.3).

5.2 Solution in k-space

5.2.1 Boundary Conditions

We must solve three first order partial differential equations in three functions, and

therefore there must be three boundary conditions. For the storage process, we

begin with no excitations of the atomic polarization, and no spin wave excitations,

so the boundary conditions for the functions P and B are simply

Pin(z) = P (z, τ −→ −∞) = 0, Bin(z) = B(z, τ −→ −∞) = 0. (5.4)

As mentioned above, the boundary condition for the signal field is set by the initial

temporal profile of the signal envelope A, as it impinges on the front face of the

ensemble at z = 0,

Ain(τ) = A(z = 0, τ). (5.5)

Our analysis will tell us the shape for Ain that maximizes the memory efficiency.

These boundary conditions are represented by the tableau in Figure 5.1.


Figure 5.1 Quantum memory boundary conditions. Example solu-tions for the functions A, P and B are shown in each panel, with the zcoordinate running from top to bottom and the τ coordinate runningfrom left to right. The red lines indicate the boundary conditionsthat must be specified to generate the solutions.

5.2.2 Transformed Equations

To proceed with solving the equations of motion, it will be useful to reduce them to

a system of coupled ordinary differential equations. This can be done by applying

a unilateral Fourier transform over the z coordinate (see Appendix D). We define

Fourier transformed variables accordingly,

A(k, τ) =1√2π

∫ ∞0

A(z, τ)eikz dz,

P (k, τ) =1√2π

∫ ∞0

P (z, τ)eikz dz,

B(k, τ) =1√2π

∫ ∞0

B(z, τ)eikz dz.

(5.6)


Using the result (D.27) for the transform of the spatial derivative ∂z, we obtain

−ikA− 1√2πAin = −κ∗P ,

∂τ P = −ΓP + κA− iΩB,

∂τ B = −iΩ∗P . (5.7)

We remark that the independence of Ω from z is critical to the usefulness of this

transformation. The spatial propagation has now been reduced to an algebraic

equation, which we can solve for A.

5.2.3 Optimal efficiency

Even given unlimited energy for the control pulse, the storage efficiency is limited

by spontaneous emission from the excited state |2〉. The storage into the dark state

|3〉, which is not affected by spontaneous emission, is always mediated via coupling

to |2〉. Even with perfect transfer between states |2〉 and |3〉, we can never store

more efficiently into |3〉 than we can couple to |2〉. Therefore the storage efficiency

is bounded by the efficiency with which we can transfer population into |2〉. To

evaluate this upper bound, we simply neglect the spin wave, and solve the equation

for P with Ω = 0. Including the solution for A, we obtain

∂τ P = −(

Γ + i|κ|2

k

)P +

iκ√2πk

Ain. (5.8)


This equation can be integrated directly to give

P (k, τ) = Pin(k)e−(Γ+i|κ|2/k)τ +iκ√2πk

∫ τ

−∞e−(Γ+i|κ|2/k)(τ−τ ′)Ain(τ ′) dτ ′. (5.9)

Since we are concerned with storage, we can set the initial polarization to zero.

We next assume that we are somehow able to transfer all the excitations from P

to the spin wave B, with no loss, at some time τ = T , which marks the end of

the storage interaction. We can then make the substitution Pout → Bout, and the

optimal storage process is described by the map

Bout(k) =∫ ∞−∞

K(k, T − τ)Ain(τ) dτ, (5.10)

where the k-space storage kernel is given by

K(k, τ) =iκ√2πk

e−(Γ+i|κ|2/k)τ . (5.11)

Note that for times τ > T , we set K = 0, so that no storage takes place after τ = T .

Now, some comments are warranted. First, the optimal storage efficiency does not

depend on the detuning ∆. To see this, note that e−Γτ = e−γτ × ei∆τ . The latter

factor, involving the detuning, represents a pure phase rotation. We could absorb it

into the definition of Ain without altering its norm. Therefore we can drop it from

the kernel — it has no effect on its singular values, and no effect on the optimal

efficiency. Second, the optimal efficiency only depends on the resonant optical depth,


defined by

d =|κ|2Lγ

. (5.12)

To see this, we normalize the time and space coordinates, along with the spin wave

and signal field amplitudes, to make them dimensionless. The spontaneous decay

rate γ and the ensemble length L provide natural time and distance scales for this

normalization. We denote the normalized variables by an overbar,

τ = γτ, k = kL,

A =A√γ, B =

B√L. (5.13)

This notation is rather clumsy, but it serves to clarify the re-scaling. With these

definitions, the storage map becomes

Bout

(k)

=∫ ∞−∞

K(k, T − τ

)Ain (τ) dτ , (5.14)

where the kernel has been converted into the dimensionless form

K(k, τ)

=i√d

k√

2πe−(1+id/k)τ . (5.15)

Here we have assumed for simplicity that κ is real, (i.e. κ = κ∗), and we have

dropped the detuning for the reason mentioned above. It is now clear that the

optical depth d is the only parameter associated with the interaction that plays any

role in determining the efficiency of the storage process. This was first shown by


Gorshkov et al [133]. Their explanation is that, regardless of the memory protocol

used, the branching ratio between loss — spontaneous emission — and storage —

absorption — is fixed by the optical depth. We have derived the result by arguing

that we cannot do better than is possible through direct, linear absorption into the

state |2〉. Clearly detuning from resonance cannot improve matters, and so from

this perspective it is unsurprising that the best possible efficiency is only limited

by the resonant coupling, parameterized by d. Below we examine the quantitative

behaviour of the optimal efficiency in more detail.

The optimal efficiency is given by the square of the largest singular value of K.

Note that it makes no difference whether the argument τ or the ‘flipped’ argument

T − τ is used; this just flips the input modes without affecting the singular values.

Unfortunately, K has a singularity at k = 0, and this means we cannot approxi-

mate it as a finite matrix, in order to compute the singular values numerically. To

proceed further, we need to transform to a different coordinate system to remove

the singularity. One possibility is to apply an inverse Fourier transform, and this

works well. Before doing this, however, we first introduce another way to analyse

the kernel (5.15), which gives a degree of insight into its structure. In what follows,

we will drop the overbar notation for the normalized parameters, as a concession to

legibility.


5.2.4 Solution in Wavelength Space

Recall that the singular values of K are unaffected by a unitary transformation.

Consider the coordinate transformation k −→ λ = 2π/k, from k-space to ‘wave-

length space’. The kernel (5.15) is no longer singular in this coordinate system.

To guarantee unitarity, the transformed kernel must include a Jacobian factor of

1/√∂kλ = −i

√2π/λ (see §3.3.3 in Chapter 3). We obtain the result

K(λ, τ) =

√d

2πe−(1+idλ/2π)τ . (5.16)

We now form the anti-normally ordered product KA, as described in (3.19) in Chap-

ter 3, which takes the form

KA(λ, λ′) =1

2πi× 1λ− 4π

d i− λ′. (5.17)

The eigenvalues of KA give the singular values of K, so we should try to solve the

following eigenvalue problem,

12πi

∫ ∞−∞

ψj(λ′)λ− 4π

d i− λ′dλ′ = ηjψj(λ). (5.18)

The integrand has a singularity at λ′ = λ− 4πd i. We consider extending the integral

into the complex plane, and integrating along a semicircle-shaped contour, closed

in the lower half of the complex plane (as depicted in Figure D.4 in Appendix D),

so that the contour encloses the singularity. We assume that the mode functions


ψj fall away to zero along the curved portion of the contour, in the limit that its

radius is made infinitely large. Under this assumption, the only contribution to the

integral comes from the straight portion of the contour, along the real line, which

is precisely the integral in (5.18). We can now use Cauchy’s integral theorem (see

§D.1.1 in Appendix D) to evaluate the left hand side of (5.18),

ψj(λ− 4π

d i)

= ηjψj(λ). (5.19)

By inspection, a possible form for the modefunctions is

ψj(λ) ∝ e±iαjλ. (5.20)

That is, plane waves in λ-space, each with some eigenfrequency αj . We should choose

the minus sign in (5.20), so that the modefunctions are exponentially damped in the

lower half of the complex plane, as we assumed above. The storage efficiencies ηj

are then given by

ηj = e−4παj/d. (5.21)

This form is reassuring, since the ηj −→ 1 in the limit d −→∞, which makes sense.

However, it is not obvious that there is any constraint on the eigenfrequencies αj .

We need a ‘quantization condition’ on the modes. One possibility is to transform

back into ordinary z-space, and look for a physically reasonable condition. First, we


transform back in to k-space. Remembering to include the Jacobian factor, we get

ψj(k) ∝ 1k× e−2πiαj/k. (5.22)

The proportionality symbol reflects the fact that the modes should be properly

normalized; at the moment we are simply concerned with their functional form.

Transforming back into z-space requires taking the inverse Fourier transform of

(5.22). The method is described in §D.5.2 of Appendix D. The result is

ψj(z) ∝ J0

(2√

2παjz), (5.23)

where J0 is a zero’th order ordinary Bessel function of the first kind. In the limit of

large d, we expect that the signal field is completely absorbed, so that there is no

signal left at the end of the ensemble. We are working in normalized coordinates, so

the end of the ensemble is located at z = 1. With no remaining signal field, no spin

wave excitations can be excited, so we should expect ψj(1) ∼ 0. This is satisfied if

we choose αj = .23, 1.21, 2.98, 5.53, etc..., as shown in Figure 5.2. This choice of

quantization is consistent with the orthogonality requirement on the modes,

∫ 1

0ψi(z)ψ∗j (z) dz = 0, if i 6= j, (5.24)

which follows from the orthogonality condition (D.34) of the Bessel functions (see

§D.5 in Appendix D).


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

1

Figure 5.2 The first three zeros of the function J0(2√

2πx). Thecondition that the spin wave should vanish at z = 1 picks out thesezeros as the eigenfrequencies αj .

The optimal storage efficiency is then given by

η1 = e−4πα1/d ≈ 1− 2.9/d, (5.25)

where the approximation holds in the limit of large d.

The above quantization procedure used to derive the optimal efficiency (5.25)

was rather ad hoc, and we should check it against a numerical SVD. We therefore

return to the k-space kernel (5.15). As noted previously the singularity at k = 0

makes this form of the kernel inconvenient. Fortunately it is easy to take the inverse

Fourier transform from k-space back into ordinary space (again using the method

described in §D.5.2 of Appendix D). The result is

K (z, τ) =√de−τJ0

(2√dτz). (5.26)

We could perform a numerical SVD on this kernel directly. The numerical problem


converges better, however, if we form the anti-normally ordered product kernel KA,

which is given by

KA

(z, z′

)=d

2e−d(z+z′)/2I0

(d√zz′). (5.27)

Here I0 is a zero’th order modified Bessel function of the first kind (see §D.5.3 in

Appendix D for a clue as to how to perform the required integral). The optimal

efficiency η1 is the largest eigenvalue of this kernel. In Figure 5.3, we plot the analytic

prediction (5.25) alongside the numerical result, over a range of optical depths. The

analytic formula is an excellent approximation for optical depths larger than ∼ 50.

This scaling of the optimal storage efficiency was first noted by Gorshkov et al. [133].

They also provided an elegant proof of the optimality of the kernel (5.27), that does

not rely on the heuristic assertion that ‘we cannot map more efficiently to B than

we can to P ’. Nonetheless, the result is the same.

0 400 800 1200 1600 200010

−3

10−2

10−1

100

101

Figure 5.3 Optimal storage efficiency. The plot shows the differ-ence between the optimal efficiency η1 and unity, on a logarithmicscale, as a function of the optical depth d. The analytic formula1− η1 ≈ 2.9/d derived above (green), is in excellent agreement withthe numerical result (blue), found by diagonalizing the kernel (5.27).

Having identified the best possible storage efficiency, we now continue with our


analysis of the equations of motion, including the control field. This analysis will

reveal how the temporal profile of the optimal input mode depends on the control

pulse, and in what regimes it is possible to effectively shape the optimal input mode

by shaping the control.

5.2.5 Including the Control

We return to the system (5.7). Solving the first equation for A, and substituting

the result into the second equation for P , yields a pair of coupled linear differential

equations in time only. For each spatial frequency k, we need to solve for the

temporal dynamics of the system (P , B). To do this, we define a vector |ψ〉 whose

two elements are the functions P and B,

|ψ(τ)〉 = P (τ)| ↑〉+ B(τ)| ↓〉, (5.28)

where the basis kets | ↑〉 and | ↓〉 are given by

| ↑〉 =

1

0

, and | ↓〉 =

0

1

. (5.29)

We have suppressed the dependence on the spatial frequency k, which is no longer

a dynamical variable, but which of course must not be forgotten! The equation of

motion for |ψ〉 is found to be

∂τ |ψ〉 = −iM |ψ〉+ |α0〉, (5.30)


where the time-dependent matrix M is given by

M(τ) =

|κ|2k − iΓ Ω(τ)

Ω∗(τ) 0

, (5.31)

and where the time-dependent vector |α0〉 includes the signal field boundary cond-

tion,

|α0(τ)〉 = iκ√2πk

Ain(τ)| ↑〉. (5.32)

Using the normalized variables introduced in the previous section, in which all

lengths are scaled by L and all frequencies by γ, we simply replace κ with√d. The

structure of (5.30) is very similar to Schrodinger’s equation (B.6) for a two-level

system, except that the evolution is not unitary, because M is not quite Hermi-

tian, and because of the ‘driving term’ |α0〉. Nonetheless, techniques applied to the

solution of Schrodinger’s equation remain useful. First, we write down the formal

solution. Suppose that we are able to construct a propagation matrix V (τ) such

that ∂τV = iVM . We then have that

∂τ (V |ψ〉) = (∂τV ) |ψ〉+ V ∂τ |ψ〉

= iVM |ψ〉+ V (−iM |ψ〉+ |α0〉)

= V |α0〉. (5.33)


Integrating this gives

|ψ(τ)〉 = V −1(τ)Vin|ψin〉+ V −1(τ)∫ τ

−∞V (τ ′)|α0(τ ′)〉 dτ ′, (5.34)

where |ψin〉 contains the boundary conditions for P and B, and where Vin is the

propagation matrix at the start of the interaction. Clearly we must have Vin = I,

the identity operator. To find the spin wave at the end of the storage process, for

which |ψin〉 = 0, we take the limit τ −→∞ to obtain

Bout = 〈↓ |V −1out

∫ ∞−∞

V (τ)|α0(τ)〉 dτ, (5.35)

where Vout = V (τ −→∞). The k-space storage kernel is then given by

K(k, τ) =ik

√d

2π〈↓ |V −1

out (k)V (k, τ)| ↑〉, (5.36)

where we have now included the dependence of the V matrices on k explicitly, lest

we forget it. If we can find an expression for V , we can construct the storage kernel,

and so find the optimal input mode by means of its SVD. Here we comment that if

the matrix M(τ) were replaced by an ordinary function, we would simply have

V (τ) = exp[i∫ τ

−∞M(τ ′) dτ ′

]. (5.37)

And in fact this is still true wheneverM is a diagonal matrix, since then [M(τ),M(τ ′)] =

0 (diagonal matrices always commute). Alternatively, if M is constant in time, it


can be pulled out of the integral in (5.37), and there is no issue with commutation

at different times. However, in general, when M is non-diagonal and time-varying,

as we have whenever Ω is not simply a constant, the solution (5.37) is not correct1.

This is an example of what is sometimes known as the great matrix tragedy : the sim-

ple fact that eAeB 6= eA+B when [A,B] 6= 0 is responsible for most of the difficulty

arising in quantum mechanical calculations!

5.2.6 An Exact Solution: The Rosen-Zener case

It is possible to find an exact solution for V , in the particular case that the shape

of the control field envelope is given by a hyperbolic secant,

Ω(τ) =Ω0

Tcsech

(τ

Tc

), (5.38)

where Tc sets the duration of the control pulse, and Ω0 is a dimensionless constant

that sets the strength of the pulse (see Figure 5.4 (a)). The method of solution

is due to Rosen and Zener [142–144]. We include details of the derivation here for

completeness, but in §5.4 below we introduce a numerical approach that is faster,

more accurate and more general. The analytic solutions presented here do provide

useful points of comparison, of course.

We first transform to the interaction picture, to remove the rapid oscillations

generated by the diagonal elements of M . To do this, we define the diagonal and1Sometimes the formal solution is written like this, but in general it is understood that the

exponential must be time ordered


off-diagonal matrices

M0 =

2βTc

0

0 0

, MX =

0 Ω

Ω∗ 0

, (5.39)

where for later convenience we have defined 2β/Tc = d/k− iΓ. Clearly we have that

M = M0 + MX . The interaction picture evolution operator VI is then defined by

VI = V V −10 , where V0 satisfies ∂τV0 = iV0M0. We can write out V0 explicitly as

V0(τ) = eiM0τ since M0 is a constant. The equation of motion for VI is found to be

∂τVI = (∂τV )V −10 + V

(∂τV

−10

)= (iVIV0M)V −1

0 + VIV0

(−iM0V

−10

)= iVIMI , (5.40)

where MI = V0MXV−1

0 is the operator that generates time evolution in the interac-

tion picture, given by

MI =

0 Ωe2iβτ/Tc

Ω∗e−2iβτ/Tc 0

. (5.41)

As described by Pechukas and Light [143], the matrix elements of VI each satisfy a

second order differential equation, which we find by differentiating the equation of


motion,

∂τ (∂τVI) = i (∂τVI)MI + iVI (∂τMI)

= −VIM2I + iVI (∂τMI) . (5.42)

The square of MI is simply given by M2I = |Ω|2I, where I is the identity matrix. We

can express the derivative of MI as ∂τMI = MIG, where G is the diagonal matrix

G =

∂τΩ∗

Ω∗ − 2i βTc 0

0 ∂τΩΩ + 2i βTc

. (5.43)

Inserting this into (5.42) gives

∂2τVI − (∂τVI)G+ |Ω|2VI = 0. (5.44)

The boundary conditions are

VI∣∣τ→−∞ = I, ∂τVI

∣∣τ→−∞ = iMI

∣∣τ→−∞. (5.45)

The equation (5.44) is solved by making a temporal coordinate transformation.

We define the normalized integrated Rabi frequency by

ω(τ) =1W

∫ τ

−∞|Ω(τ ′)|2 dτ ′, (5.46)


where W is a normalization related to the total energy in the control pulse,

W =∫ ∞−∞|Ω(τ)|2 dτ. (5.47)

The coordinate ω runs from 0 to 1, as τ runs from −∞ to ∞ (see Figure 5.4 (b)). ω

can be thought of as the time coordinate marked out by a clock that is powered by

the control field. Using the control field profile (5.38), we can evaluate the integral

in (5.46) explicitly, to get

ω(τ) = 12tanh

( τT

)+ 1

2 . (5.48)

0-6 -4 -2 0 2 4 6

1

-6 -4 -2 0 2 4 6

(a) (b)

Figure 5.4 The Rosen-Zener model. The control field, shown in(a) with Ω0 = Tc = 1, takes the form of hyperbolic secant. (b): theintegrated Rabi frequency ω marks out time at a rate given by |Ω|2.

Under the transformation τ −→ ω, the temporal derivative ∂τ transforms as


follows,

∂τ = (∂τω) ∂ω

=1

2Tcsech2

(τ

Tc

)∂ω

=1

2Tc

(ΩTcΩ0

)2

∂ω, (5.49)

where we used the control shape (5.38). The second derivative is then given by

∂2τ =

(∂2τω)∂ω + (∂τω)2 ∂2

ω

=1T 2c

[(1− 2ω)

(ΩTcΩ0

)2

∂ω +14

(ΩTcΩ0

)4

∂2ω

]. (5.50)

Putting all this together, and using

(ΩTcΩ0

)2

= 4ω(1− ω), (5.51)

we find the equation

ω(1− ω)∂2ωVI + (∂ωVI)

[12

+ iβZ − ω]

+ Ω20VI = 0, (5.52)

where Z is the Pauli matrix

Z =

1 0

0 −1

. (5.53)


The boundary conditions, in terms of the new variable ω, are given by

VI∣∣ω→0

= I, ∂ωVI∣∣ω→0

= iΩ0

0 ω−θ−

ω−θ+ 0

, (5.54)

where θ± = 12 ± iβ. The equation (5.52) is known as a hypergeometric differential

equation. The solutions are known as hypergeometric functions, denoted2 by the

symbol F , and parameterized by the coefficients appearing in the equation. The

hypergeometric functions are special functions that can be evaluated using a mathe-

matics application such as Matlab or Mathematica. Matching the general solutions

to the boundary conditions (5.54), the solution for VI is given by

VI(ω) =

F (Ω0,−Ω0, θ+, ω) iΩ0θ+ωθ+F (θ+ + Ω0, θ+ − Ω0, 1 + θ+, ω)

iΩ0θ−ωθ−F (θ− + Ω0, θ− − Ω0, 1 + θ−, ω) F (Ω0,−Ω0, θ−, ω)

.

(5.55)

The properties of these special functions can be used to show that at the end of the

storage process we have

VI∣∣ω→1

=

Γ2(θ+)

Γ(θ++Ω0)Γ(θ+−Ω0)i sin(πΩ0)cosh(πβ)

i sin(πΩ0)cosh(πβ)

Γ2(θ−)Γ(θ−+Ω0)Γ(θ−−Ω0)

, (5.56)

where Γ(x) is the Euler Gamma function. We substitute these matrices into (5.36) to

obtain an expression for the storage kernel in terms of k and ω. Some manipulations2Sometimes the symbol 2F1 is used, and the designation Gauss hypergeometric function then

distinguishes this from the generalized hypergeometric functions pFq.


reveal that the determinant of the matrix (5.56) is 1, which simplifies forming the

inverse V −1out . The exponential factor ei2βτ/Tc that enters when transforming from VI

back to V = VIV0 translates into the factor

ei2βτ/Tc =(

ω

1− ω

)iβ

. (5.57)

After some algebra, we arrive at the result

K(k, ω) =i√d

k√

2π

(ω

1− ω

)iβ

× iQ(k, ω)√2ω(1− ω)/Tc

, (5.58)

where we’ve defined the function Q in the following way

Q(k, ω) =Γ2(θ+)

Γ(θ+ + Ω0)Γ(θ+ − Ω0)Ω0

θ−ωθ−F (θ− + Ω0, θ− − Ω0, 1 + θ−, ω)

− sin(πΩ0)cosh(πβ)

F (Ω0,−Ω0, θ+, ω). (5.59)

It helps to keep in mind that θ± is a function of k through β. Note we have included

the Jacobian factor√

2ω(1− ω)/Tc =√∂τω in the denominator of (5.58), to make

the transformation from τ to ω a unitary one (see §3.3.3 in Chapter 3). We would

now like to extract the optimal input mode and its associated optimal efficiency by

applying the SVD. However, just as in the case of (5.15), the kernel in (5.58) has a

singularity at k = 0. As before, we remove this by transforming form k-space into

λ-space, where λ = 2π/k is the wavelength of the spin wave excitation. Including


the Jacobian factor, the expression for K in terms of λ and ω is,

K(λ, ω) =i

2π

√Tcd

2ω−θ−(1− ω)−θ+Q(λ, ω), (5.60)

where Q is given, as before, by (5.59), the only difference being that in λ-space, the

parameter β takes the form 2β/Tc = dλ2π − iΓ. There is a minor pathology associated

with the points ω = 0 and ω = 1, which blow up, but in practice this is easily

addressed by introducing a small regularization ε that shifts the singularities into

the complex plane,

ω → ω + iε. (5.61)

After these steps, we have a non-singular kernel that is amenable to a numerical SVD.

From this we obtain the singular values λj, and a set of input modes φj(ω). The

optimal storage efficiency is already given by η1 = λ21. To find the temporal mode

of the signal field that is stored with this optimal efficiency, we need to transform

the mode φ1(ω) back into the temporal domain. Including the Jacobian factor, we

have

φ1(τ) =

2Tcω(τ) [1− ω(τ)]

1/2

φ1 [ω(τ)]

=sech

(τT

)√

2Tcφ1

[12tanh

(τ

Tc

)+ 1

2

]. (5.62)

The analytic solution for K in (5.60) provides a check on the numerical optimiza-

tions we present in §5.4 (see Figures 5.8 and 5.9). In fact, evaluating the function


F (a, b, c, ω) with complex a, b or c can be time-consuming, since these values are

generated by analytic continuation of F into the complex plane. This procedure

is not always accurate, and so the direct numerical optimizations presented at the

end of this chapter are both faster and more reliable. Finally, the analytic solution

(5.60) is, of course, only valid for the particular control (5.38). It would be more

convenient if we could derive an expression that holds for a range of control field

profiles. We now show how to construct an approximation to V that holds in the

adiabatic limit, which is essentially the limit of a slowly varying control.

5.2.7 Adiabatic Limit

The idea behind adiabatic evolution is to adjust Ω sufficiently slowly that at each

moment we can neglect the time dependence of M , and treat the problem as if

it were time-stationary. In this limit, the state |ψ〉 remains in an instantaneous

eigenstate of M at all times. As M changes, the eigenstates of M slowly evolve,

and we arrange for the populated eigenstate at the end of the storage interaction

to overlap with the | ↓〉 state; in this way excitations are transferred into the spin

wave. To see how this works, we re-cast the equation of motion for V in terms of

the adiabatic basis, which is the basis formed by the instantaneous eigenstates of M .

Suppose M has the following eigenvalue decomposition (see §A.4.4 in Appendix A),

M = RDR−1. (5.63)


We define the operator Vad = V R, and differentiate it to obtain its equation of

motion,

∂τVad = (∂τV )R+ V ∂τR

= iVMR+ V ∂τR

= iV RDR−1R+ V R(R−1∂τR

)= iVadMad, (5.64)

where Mad = D− iR−1∂τR generates the time evolution in the adiabatic basis. The

content of the adiabatic approximation is to neglect the term R−1∂τR in Mad, so

that Mad is a purely diagonal matrix. This allows us to solve the equation of motion

for Vad, using the result (5.37). That is,

Vad(τ) = exp[i∫ τ

−∞Mad(τ ′) dτ ′

]. (5.65)

Armed with this solution, we can construct the propagation matrix V = VadR−1, and

therefore the storage kernel (5.36). We now implement this programme explicitly,

after which the conditions under which the adiabatic approximation is justifiable

will become clearer.

Here we remark that corrections to the adiabatic approximation can be gen-

erated, in the current formalism, by making use of the Magnus expansion [143,145],

or Salzman’s expansion [146], which provide approximations to the propagator V


when Mad is non-diagonal and time-dependent. These corrections quickly become

unwieldy however, and so we do not present them here: the numerical approach

presented in §5.4 obviates the need for them.

To find the adiabatic kernel, we start by finding the instantaneous eigenvalues

of M(τ), by solving the equation |M − λI| = 0, where the vertical bars denote the

determinant (see §A.4.2 in Appendix A). The resulting eigenvalues are

λ± = b±√b2 + |Ω|2, (5.66)

where we have defined 2b = d/k − iΓ (thus replacing the notation 2β/Tc defined in

the previous section). Solving the equation M |±〉 = λ±|±〉 for the elements of the

eigenvectors |±〉, we find

|+〉 ∝

λ+

Ω∗

, |−〉 ∝

λ−

Ω∗

. (5.67)

The diagonalizing transformation R is the matrix with |+〉 as its first column and |−〉

as its second. But there is some freedom as to the normalization of the vectors |±〉.

This is fixed by requiring that limτ→−∞R = I, the identity. This just codifies our

knowledge that Ω = 0 at the start of the storage interaction, so that M is initially

diagonal, which means that no transformation need be applied to diagonalize it at


τ −→ −∞. A suitable form for R, that satisfies this boundary condition, is

R =

1 λ−Ω∗

Ω∗

λ+1

. (5.68)

Note that limτ→−∞ λ− = 0. The inverse transformation is then given by

R−1 =1

1− λ−λ+

1 −λ−Ω∗

−Ω∗

λ+1

. (5.69)

We have Mad = diag(λ+, λ−), so that Vad is given by

Vad(τ) =

exp[i∫ τ−∞ λ+(τ ′) dτ ′

]0

0 exp[i∫ τ−∞ λ−(τ ′) dτ ′

] . (5.70)

Combining these results together and substituting them into (5.36), we find there is

only a single non-vanishing term contributing to the storage kernel,

K(k, τ) = − ik

√d

2π×

Ω∗(τ) exp[−i∫∞τ λ−(τ ′) dτ ′

]λ+(τ)− λ−(τ)

. (5.71)

We can further simplify this result if we make the assumption that

|b| |Ω| (5.72)

at all times. This does not have anything to do with the rate at which we change

the control field, but it is usually considered as part of the adiabatic approximation,


as discussed in §5.2.9 below. With this approximation, we can write

λ+ ≈ 2b+|Ω|2

2b, and λ− ≈ −

|Ω|2

2b. (5.73)

Inserting these expressions into (5.71), and making the replacement 2b = d/k − iΓ,

we find, after a little algebra,

K(k, τ) =1√2π×√d

ΓΩ∗(τ) exp

[− 1

Γ

∫ ∞τ|Ω(τ ′)|2 dτ ′

]× 1k + i dΓ

exp

[idΓ2

∫∞τ |Ω(τ ′)|2 dτ ′

k + i dΓ

].

(5.74)

The first exponential factor represents the accumulation of phase due to the dynamic

Stark effect, in which the strong control field ‘dresses’ the atoms and causes a time

dependent shift in the |2〉 ↔ |3〉 transition frequency. The second exponential factor

represents the ‘meat’ of the interaction: the response of the atoms to the incident

signal field. The kernel (5.74) is already amenable to a numerical SVD. However, to

make a connection with previous work, we take the inverse Fourier transform. We

use the method detailed in §D.5.2 in Appendix D, and apply the shift theorem, to

get the adiabatic z-space kernel

K(z, τ) = −i

√d

ΓΩ∗(τ) exp

− 1

Γ

[∫ ∞τ|Ω(τ ′)|2 dτ ′ + dz

]×J0

(2i

√d

Γ

√∫ ∞τ|Ω(τ ′)|2 dτ ′z

).

(5.75)

Note the additional contribution dz to the exponential factor. This contribution

represents the change in refractive index experienced by the signal field as it prop-

agates through the ensemble. On resonance (with Γ = 1 in normalized units), this


term describes exponential attenuation i.e. absorption, with an absorption coeffi-

cient of d. This explains why the quantity d is known as the optical depth: it directly

quantifies the optical thickness of the ensemble on resonance.

A numerical SVD could also be applied to the kernel (5.75) to obtain the optimal

temporal input mode φ1(τ), and its associated optimal efficiency η1 = λ21. This

adiabatic solution appears in the work of Gorshkov et al. [133], although the method

of derivation differs slightly. They provided a method to find the optimal input

mode, in the limit of large control field energy, but they did not use the SVD: The

SVD is a more direct method, and it does not require the assumption of large control

energy. We will discuss the differences between the SVD method and the method of

Gorshkov et al. shortly. First, we make a coordinate transformation that removes

the explicit dependence of K on the shape of the control field. As in the Rosen-Zener

solution above, we transform from τ to ω, where ω is the normalized integrated Rabi

frequency, defined by (5.46). We do not assume a particular shape for the control:

its profile can be arbitrary in the present case (within the limits of the adiabatic

approximation, to be discussed below). By removing the dependence on the control

field with this coordinate transformation, we only need to perform the SVD once, in

the transformed coordinate system, in order to obtain the optimal input mode for

any control field shape. To make the coordinate transformation unitary, we include

a Jacobian factor of√W/Ω∗(τ) in the transformed kernel (see §3.3.3). This rather

conveniently cancels with the factor of Ω∗(τ) in (5.75), so the transformed kernel


can be written as

K(z, ω) = −i

√dW

Γe−(W−Wω+dz)/ΓJ0

(2i√d

Γ

√(W −Wω)z

), (5.76)

A final, cosmetic simplification is achieved by flipping the ω coordinate, ω −→

1 − ω. This has no effect on the singular values of the kernel, but simply flips the

input modes around. The optimal mode for adiabatic storage in a Λ-type quantum

memory, with an arbitrary control field profile, is now found by taking the SVD of

K(z, ω) = −i

√dW

Γe−(Wω+dz)/ΓJ0

(2i√dWΓ

√ωz). (5.77)

We note that both ω and z run from 0 to 1. The optimal temporal input mode, is

then found from the mode φ1(ω) by the relation

φ1(τ) =Ω(τ)√Wφ1 [1− ω(τ)] . (5.78)

The factor of Ω/√W arises from the Jacobian relating the coordinates ω and τ . We

now have a simple prescription for finding the optimal input mode for a quantum

memory. Given a fixed value for W , which essentially quantifies the total energy in

the control pulse, and given values for the detuning and the optical depth, we form

the kernel (5.77) and take the SVD. Then, for any arbitrary shape of the control,

we can construct the optimal input mode φ1, using the transformation (5.78) above.

Some examples of the optimal input modes predicted using this approach can be


found in Figures 5.8, 5.9, 5.10 and 5.11 in §5.4 at the end of this chapter.

In general, the storage efficiency depends on the optical depth, the detuning

(through Γ), and the control pulse energy, through W , since the kernel (5.77) de-

pends on all of these quantities. We also know however, from the discussion in

§5.2.3, that the best possible storage efficiency only depends on the optical depth.

Below we connect these two results together.

5.2.8 Reaching the optimal efficiency

In §5.2.3 we derived an expression for the optimal storage efficiency possible in a

Λ-type ensemble memory. In fact, it is possible to reach this optimal efficiency in the

adiabatic limit. As was first shown by Gorshkov et al. [133], the anti-normally ordered

kernel formed from the adiabatic storage kernel (5.77) is equal to the optimal kernel

(5.27), in the limit of large control pulse energy. To see this, we substitute (5.77) into

the expression (3.19) for the anti-normally ordered kernel, and perform the integral

over ω. In the limit W −→∞, we can evaluate the integral analytically. After some

leg work — see §D.5.3 in Appendix D — we find that the result is exactly (5.27).

That is,

limW→∞

∫ 1

0K(z, ω)K∗(z′, ω) dω =

∫ ∞0

K(z, ω)K∗(z′, ω) dω =d

2e−d(z+z′)/2I0(d

√zz′).

(5.79)

This shows that it is possible to saturate the upper bound on the storage efficiency,

even in the adiabatic limit. The adiabatic limit is not only useful because we can


construct the optimal input mode explicitly. It is also useful because it is possible

to shape the input mode by shaping the control. From the form of (5.78), it is

clear that, by an appropriate choice of Ω(τ), we can choose the shape of φ1(τ). In

particular, we can choose the control so that φ1(τ) matches the temporal profile of

some ‘given’ input field. This is of considerable practical importance, since it may

be experimentally much easier to shape the bright control field, than to shape the

weak input field (this is discussed in Chapter 8). The combination of these two facts

— that adiabatic storage can be optimal, and also that it enables one to shape the

input mode — makes the adiabatic limit an important regime for the operation of

a quantum memory.

Under what circumstances is the equality in (5.79) achieved? The limit of W −→

∞ is not really required. Examining the form of (5.77), it is clear that we just

need to make W large enough that the exponential factor e−Wω/Γ, evaluated at the

limit ω = 1, is sufficiently small that extending the integral further would make no

difference. We should certainly have that W |Γ| then. In addition, we should

ensure that any contribution from the Bessel function is negligible at ω = 1. Using

the approximation J0(ix) = I0(x) ∼ ex/√x for x 1, we see that we should also

have that W d. To summarize, adiabatic storage is optimal if the control pulse

is sufficiently energetic that the conditions

W max (|∆|, d) (5.80)


are satisfied. Recall that ∆ is the common detuning of the signal and control fields

from resonance. We can connect W with the total energy Ec in the control pulse.

The total energy is

Ec = A∫ ∞−∞

Ic(τ) dτ, (5.81)

where Ic = 2ε0c|Ec|2 is the cycle-averaged control pulse intensity. From the definition

of the Rabi frequency (4.49), we then find

Ec =2ε0cA∣∣∣d23.vc

~

∣∣∣2γW. (5.82)

Here the presence of the factor of γ indicates that we have converted back into

ordinary units (rather than normalized units). From the definition of W (5.47), we

find it has the dimensions of frequency, and so in ordinary units it is accompanied

by the factor γ. Let us define the number of photons in the control pulse by Nc =

Ec/~ωc. We also define Na as the number of atoms in the ensemble addressed by

the optical fields, Na = nLA. Using (4.50) and (5.12), we can express the optical

depth as

d =∣∣∣∣d∗12.vs

~

∣∣∣∣2 ~ωs2ε0cAγ

×Na. (5.83)

With these definitions, we see that the condition W d in (5.80) amounts, essen-

tially, to the condition Nc Na (we have used ωs ≈ ωc and |d∗12.vs| ≈ |d23.vc|).

We might therefore describe this condition as describing a ‘light-biased’ interaction,

where the number of control photons dominates over the number of atoms; this is


why it is the latter quantity that limits the efficiency.

In the work of Gorshkov et al. [133], a method is presented for optimizing the

adiabatic storage efficiency in this light-biased limit. The method works by combin-

ing the optimal anti-normally ordered kernel (5.27), whose eigenfunctions are the

optimal spin waves, with the adiabatic storage kernel (5.75), which connects the

control field profile to the signal field profile. Combining these two kernels together

is possible only when Nc Na, and also W ∆. In this limit their method works

extremely well. One advantage of the SVD method however, is that we can apply

it directly to the kernel (5.77), without making this approximation, and we can

therefore find the optimal input modes for arbitrary values of W .

5.2.9 Adiabatic Approximation

We have employed several approximations in the name of adiabaticity. We now

examine the physical content of these approximations. The first assumption we

made, in the text following (5.64), was to neglect the term R−1∂τR in the adiabatic

generator Mad. Clearly this term vanishes if the control field is held constant, and

so the size of this term is set by the rate of variation of the control field. To make

the adiabatic approximation, we must therefore limit the bandwidth of the control

pulse. To find this limit, we introduce the second approximation we made in (5.72),

namely that

|b| |Ω|. (5.84)


With this approximation, we can write the diagonalizing transformations R, R−1 in

the form

R =

1 − Ω2b

Ω∗

2b 1

; R−1 =

1 Ω2b

−Ω∗

2b 1

. (5.85)

For adiabatic evolution, we should have that ||R−1∂τR|| ||D||, where D =

diag(2b + |Ω|2/2b,−|Ω|2/2b). Here the double bars represent the Frobenius norm,

which is found by adding in quadrature the magnitudes of all the elements in a

matrix. We neglect terms of order |Ω/b|, which are small by assumption. We then

arrive at the condition ∣∣∣∣∂τΩ2b

∣∣∣∣2 |Ω|2. (5.86)

Recall that b = 12(d/k − iΓ) depends on the wavevector k, and that Ω varies with

time, so we should be careful to satisfy this condition for all the values of these two

parameters that play a role in the storage process.

We can distinguish two regimes. First, the EIT regime (see §2.3.1 in Chapter

2), in which the optical fields are tuned into resonance with the excited state. In

this case, ∆ = 0 and so Γ = 1 (in normalized units). Since d 1 for reasonably

efficient storage, the contribution to b from the complex detuning Γ is small, and we

have |b| ∼ d/2k. Therefore the adiabatic conditions (5.84) and (5.86) vary strongly

with k, and we must take some care to identify the range of wavevectors that are

important for the storage process.

The second regime is the Raman regime (see §2.3.2 in Chapter 2), in which the

detuning is large compared to the excited state linewidth; ∆ 1 in normalized


units. In this regime |b| ∼ 12(d/k−∆). In this case the large contribution to b from

the detuning makes the adiabatic conditions less dependent on k.

What range of wavevectors are important in the storage process? One answer is

provided by inspection of the k-space kernel (5.74). Using the expansion (k+i dΓ)−1 =

−iΓd + Γ2

d2 k + iΓ3

d3 k2 + . . ., we can re-write the k-space map in the form

K(k, ω) ≈ − i√2π

√W

d× ei

Wd ωk × e−

ΓWd2 ωk2

, (5.87)

where we have introduced the integrated Rabi frequency ω, and ‘flipped’ the kernel,

as we did in deriving (5.77). This expression is a good approximation if |d/Γ| is

large, which is not guaranteed in the Raman regime, but which is generally true in

the EIT regime. The final exponential factor describes a Gaussian profile in k-space,

with a characteristic width given by

δk = δk(ω) =d√Wω

, (5.88)

in the EIT regime with Γ = 1. In this case the most restrictive form of the adiabatic

condition (5.84) can be written as (dropping an unimportant factor of two)

Ωmax d

δk(ωmax), (5.89)

where Ωmax = max(Ω) is the peak Rabi frequency of the control, and where ωmax is

the value of the integrated Rabi frequency when this peak occurs. For a symmetric


control pulse, we would have ωmax = 12 ; in general ωmax will be some fraction that

for our purposes we may approximate as ∼ 1. Substituting (5.88) into (5.89), we

find that this condition in fact restricts the bandwidth δc of the control pulse:

δc 1. (5.90)

Here we made the approximation W ∼ Ω2max/δc. The condition (5.90), in ordinary

units, is δc γ. That is, the bandwidth of the control should not exceed the natural

linewidth of the |2〉 ↔ |3〉 transition in the ensemble, to achieve adiabatic storage on

resonance. This suggests that EIT is a memory protocol best suited for the storage of

narrowband fields. However, the analysis by Gorshkov et al. [133] reaches a different

conclusion: that the adiabatic restriction on the bandwidth of the control is δc dγ

(in ordinary units). This much less stringent condition is derived by evaluating the

adiabatic condition (5.86) using b ∼ d/max(δk) with max(δk) ∼ 1. Identifying the

maximum of the quantity |∂τΩ/Ω| with the bandwidth δc, one arrives at their result.

One justification for using δk ∼ 1 is that for optimal storage, we only need to access

the optimal spin wave mode; higher modes are irrelevant. Since the optimal mode

is that mode which is most slowly varying in space, its width in k-space is limited

to a relatively small region, and the adiabatic approximation need only be satisfied

within this range. It is difficult to argue rigorously about these approximations, but

as we will see, numerics reveal that the reality lies somewhere between these two

cases: the adiabatic approximation breaks down rather quickly in the EIT regime

5.3 Raman Storage 159

as the bandwidth approaches the natural linewidth, although it is true that this is

mitigated somewhat by increasing the optical depth.

In the Raman regime, the analysis is simpler. The adiabatic condition is simply

that

Ωmax |∆|, (5.91)

independent of k. We might comment that there may be some particular value of k

such that there is a cancellation, and b becomes small, but the effect of this isolated

point is generally negligible. The limitation on the control bandwidth comes from

(5.86), which yields the condition

δc |∆|. (5.92)

Therefore adiabatic evolution is guaranteed in the Raman case whenever the detun-

ing ∆ is the dominant frequency involved in the interaction, with both the Rabi

frequency and the bandwidth of the control field small by comparison.

5.3 Raman Storage

So far we have studied the properties of storage in a Λ-type ensemble for arbitrary

values of the detuning. We now specialize to the case of large detunings, ∆ γ

(or ∆ 1 in normalized units). In this Raman regime the storage kernel simpli-

fies further, and an interesting connection between the input and spin wave modes

emerges. The following treatment forms the basis of our Rapid Communication on


Raman storage [77].

Taking the limit ∆ 1, we can write Γ ≈ −i∆, and the storage kernel (5.77)

becomes

K(z, ω) = C × e−i(Wω+dz)/∆ × J0(2C√ωz), (5.93)

where we have defined the Raman memory coupling by

C =

√Wd

|Γ|≈√Wd

∆. (5.94)

The exponential factor in (5.93) represents only phase rotations applied to the ω

and z coordinates. We can absorb these phases into the signal field and spin wave,

and therefore we can drop them from the kernel (we will be careful to ‘put them

back’ when we write down the optimal modes). We can now write down a very

simple recipe for constructing the optimal input mode in the Raman regime. First,

we form the kernel

K(z, ω) = CJ0(2C√ωz). (5.95)

This depends only on the memory coupling C, therefore this parameter uniquely

determines the efficiency of the memory in the Raman limit [77]. We note that

the kernel K is real, and symmetric under the exchange of ω and z. That is, K

is Hermitian (see §A.4.3 in Appendix A). Therefore, its SVD is the same as its

spectral decomposition (see §3.1.3 in Chapter 3). The singular values of K are also

its eigenvalues, and the input modes φj, as functions of ω, have the same form as


the spin wave modes ψj, as functions of z. Their phases differ though, because of

the phase rotations we ‘absorbed’. To be precise, let us define the functions ϕj as

the eigenfunctions of the kernel (5.95). That is to say, ϕj satisfies

∫ 1

0CJ0(2C

√xy)ϕj(x) dx = λjϕj(y). (5.96)

The optimal input mode for the signal field, including the correct phase rotation

and transforming back from ω to τ , is given by

φ1(τ) =1√W

Ω(τ)× exp

iW [1− ω(τ)]∆

× ϕ1[1− ω(τ)]. (5.97)

The optimal output mode for the spin wave, to which this optimal input mode is

mapped by the storage process, is given by

ψ1(z) = e−idz/∆ϕ1(z). (5.98)

The optimal storage efficiency is given by the square of the largest eigenvalue in

(5.96); η1 = λ21. Figure 5.5 shows the variation of this optimal efficiency with C.

In taking the Raman limit, we have neglected spontaneous emission. This means

that the predicted efficiency can approach unity, as long as C is large enough, even

if the optical depth is low. That is, a smaller d can be ‘compensated’ by a larger

W — a more energetic control. Of course the optimal efficiency derived in §5.2.3

remains correct, even in the Raman limit: the best achievable efficiency is always


limited, through spontaneous emission, by the optical depth. If the efficiency pre-

dicted by (5.96) is larger than the upper limit (5.25), then we have reached a regime

where spontaneous emission dominates over other losses. However, generally a Ra-

man memory requires a large optical depth to operate efficiently, as we discuss

below. Therefore the dominant loss mechanism in a Raman memory is not sponta-

neous emission, but insufficient coupling. That is, the large detuning from resonance

makes the interaction weak, so that the biggest problem in getting a Raman memory

to work is to make the coupling strong enough that the signal field is completely

absorbed. The utility of the kernel (5.95), is that it provides a simple way to analyze

the Raman limit before spontaneous emission becomes a limitation. Examples of

the optimal modes predicted by the Raman kernel are shown in Figures 5.8 and

5.10 in §5.4 at the end of this chapter. What is notable about the behaviour of the

0 0.5 1 1.5 2 2.5 3 3.5 410

−5

10−4

10−3

10−2

10−1

100

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

(a) (b)

Figure 5.5 Raman efficiency. (a) the optimal storage efficiencyη1 = λ2

1 predicted by the kernel (5.95) in the Raman limit ∆ 1,versus the memory coupling C =

√Wd/|Γ|. The efficiency ‘saturates’

at around C ∼ 2. (b) A logarithmic plot of the difference 1 − η1between the predicted efficiency and unity.

Raman efficiency is that it rises steeply for small values of C, before ‘saturating’


at C ∼ 2. Physically, this saturation point coincides with the stimulated scattering

regime. For ordinary Stokes scattering, the scattering process becomes efficient as

the coupling is increased beyond this point (see Figure 10.6 in Chapter 10). In the

case of a Raman quantum memory, the transmission of the signal field through the

ensemble drops sharply, and the efficiency of the memory becomes limited by spon-

taneous emission, rather than insufficient coupling. Therefore in designing a Raman

memory, one need only ensure that C & 2, in order that the scheme is viable (see

§9.8 in Chapter 9 and §10.9 in Chapter 10).

5.3.1 Validity

As mentioned above, in deriving the Raman kernel we neglected spontaneous emis-

sion. We did this tacitly when we dropped the real part of Γ in the exponential

factor appearing in the storage kernel. This is valid so long as neither W nor d is

too large. To see this, we define the balance R according to the relation

R =

√W

d. (5.99)

Note that the balance has no relation to the matrix R introduced in §5.2.7 above.

Using the arguments employed in §5.2.8, we see that the balance really expresses

the extent to which the interaction is dominated by light, or matter:

R2 ∼ Nc

Na. (5.100)


That is, the case R 1 corresponds to a light-biased interaction — as described ear-

lier, this is the limit in which the adiabatic kernel saturates the upper bound on the

storage efficiency — and the case R 1 describes a matter-biased interaction, with

a weak control, but a large/dense ensemble. Using the balance, and the definition

C =√Wd/|Γ|, we can re-express the adiabatic storage kernel (5.77) as follows,

K(z, ω) = Ce−iθ × e−iC cos θ(Rω+z/R) × eC sin θ(Rω+z/R) × J0(2Ce−iθ√ωz), (5.101)

where we have defined the phase angle θ such that tan θ = 1/∆; the complex detun-

ing being given by

Γ = −i|Γ|eiθ. (5.102)

The Raman limit is the limit of large detuning, which corresponds to the limit θ 1.

The exponential factor involving cos θ in (5.101) corresponds to the phase rotations

we dealt with previously. On the other hand, the exponential factor involving sin θ

comes from the real part of Γ, and represents spontaneous emission. It cannot

be removed by a unitary transformation of the modes, and it reduces the storage

efficiency. This is the term we neglect in the Raman limit. For this approximation to

hold for both the optical and spin wave modes together, both terms in the exponent

should be small. This is true provided we satisfy the conditions

C sin θR 1, and C sin θ × 1R 1. (5.103)


Using sin θ ≈ tan θ, setting C ∼ 1 (as we should have for a reasonably efficient

memory), and using (5.100), we find that the Raman kernel is a good approximation

whenever

∆ γ, andγ2

∆2 Nc

Na ∆2

γ2, (5.104)

using ordinary units. That is to say, the interaction should be roughly ‘balanced’,

with broadly equal contributions to the coupling originating from atoms and light.

The larger the detuning, the more ‘leeway’ there is to bias the interaction one way

or the other.

Suppose that we set R ∼ 1, so that d ∼W . For reasonable efficiency, we should

have C ∼ 1, or thereabouts. Squaring this, and using R ∼ 1 then, we find d ∼ ∆. In

a Raman memory, ∆ 1, and therefore d 1. That is, a Raman memory described

by (5.95) generally requires a large optical depth. This explains why spontaneous

emission is not as important a limiting factor as is the issue of sufficiently strong

coupling.

Finally, we comment that since ∆ ∼ d, the adiabatic condition on the bandwidth

of the control field can be written as δc d, or in ordinary units δc dγ. This

is the same as the condition derived by Gorshkov et al. for the limitation on the

control bandwidth for resonant EIT storage. It is therefore arguable that a Raman

memory does not allow for more broadband storage than an EIT memory does.

But the adiabatic condition in the Raman case is rather more robust that it is in

the EIT case, since it does not depend on identifying an ‘important’ region in k-

space. Numerics show that the adiabatic storage kernel is a better approximation


in the Raman case, for more broadband control pulses, than it is in the EIT case.

Broadband storage provided the motivation for studying the Raman memory, but

its advantages in this respect are not clear cut.

Other considerations that may favour detuning from resonance include the possi-

bility of dealing with a more complex excited state manifold. Suppose that, instead

of a single excited state |2〉, there are a host of states, perhaps resulting from spin-

orbit or hyperfine splitting (see, for example, Figure 10.3 in Chapter 10). Tuning

into resonance with one of these states could make the dynamics rather complicated.

There may be some direct absorption of the signal field into the nearby states, fol-

lowed by spontaneous emission and loss. Detuning away from all of the states puts

the contribution from each state on an equal footing, so that the dynamics can be

treated just as we did the simple three level system, where we need only swap the

coupling C to a single state for an equivalent coupling that includes the scattering

amplitudes for all the states (see §F.4 in Appendix F). And by detuning we eliminate

the possibility of absorption losses. In addition, a Raman memory is tunable, since

if we tune further from resonance, we can maintain strong coupling by increasing

the control pulse energy. The Raman memory is also affected less adversely by inho-

mogeneous broadening of the excited state than an EIT memory might be. Again,

this is because the coupling to the ensemble is not dominated by resonance with a

single frequency.


5.3.2 Matter Biased Limit

We have already shown how the anti-normally ordered product formed from the

adiabatic storage kernel tends to the optimal kernel (5.27) in the limit of large

control pulse energy. This is the light-biased limit, with R 1. To reach this

limit, the balance R should exceed the upper limit in (5.104), so we should have

R > ∆ (in normalized units). Now, there is a degree of symmetry to the structure

of (5.101): The kernel is unchanged when we swap z and ω, if at the same time we

send R −→ 1/R. It therefore follows that in the matter-biased limit R 1 — that

is Na > ∆Nc/γ in ordinary units — the normally ordered kernel tends to a limit

defined not by the optical depth, but by the control pulse energy:

limd→∞

∫ 1

0K∗(z, ω)K(z, ω′) dz =

W

2e−W (ω+ω′)/2I0(W

√ωω′). (5.105)

The kernel on the right hand side has precisely the same form as (5.27), except that

the efficiency is limited by W rather than by d. Therefore, if we are limited by the

energy of the control, the optimal efficiency achievable is given by η1 = 1− 2.9/W .

Of course, extremely high energy lasers are readily available, whereas the size of the

ensemble is generally not easily varied. Nonetheless, in cases where the ensemble

may be damaged by a high energy laser pulse — as may be the case for a solid state

memory — it may be that the control energy becomes a limitation.


5.3.3 Transmitted Modes.

In this section we describe a connection between the modes that are stored in the

memory, and the modes that are transmitted through it when the efficiency is not

perfect. This connection holds in the Raman limit, and it is clearest when we

form the equations of motion for the signal field and the spin wave in the adiabatic

approximation. This is the way the adiabatic approximation is most commonly

introduced, and so it is informative to run through the procedure. We start with

the equations of motion (5.3), which we reproduce below in normalized units,

∂zA(z, τ) = −√dP (z, τ),

∂τP (z, τ) = −ΓP (z, τ) +√dA(z, τ)− iΩ(τ)B(z, τ),

∂τB(z, τ) = −iΩ∗(τ)P (z, τ). (5.106)

The adiabatic approximation is made by setting ∂τP = 0 on the left hand side of

the second equation. This is reasonable when the natural dynamics of the optical

polarization P are overwhelmed by the motion driven by the signal and control fields

— known as adiabatic following. In this situation, 1/|Γ| is the shortest timescale

in the problem, so that when far-detuned we must have (δc,Ωmax) |∆|, precisely

the adiabatic conditions (5.91), (5.92) derived above. We can then solve the sec-

ond equation for P algebraically, and substitute the result into the first and third


equations. The result is

(∂z +

d

Γ

)A = i

Ω√d

ΓB,(

∂τ +|Ω|2

Γ

)B = −i

Ω∗√d

ΓA. (5.107)

We then switch coordinates from (τ, z) to (ω, z), and we define new variables α and

β for the signal field and spin wave as follows

α(z, ω)e−(Wω+dz)/Γ =√WA(z, τ)Ω(τ)

,

β(z, ω)e−(Wω+dz)/Γ = B(z, τ). (5.108)

These new variables incorporate the Jacobian factor associated with the coordinate

transformation, and also the phase rotations associated with the dynamic Stark shift

and the ensemble refractive index. In general, the transformation linking A and B to

α and β is not quite unitary, because Γ is not strictly imaginary, so the exponential

factors on the left hand side of (5.108) make the norms of A, B different to the

norms of α, β — they are not pure phase rotations. However, taking the Raman

limit ∆ 1, we find Γ ≈ −i∆, and in this case, the transformation is unitary. This

will be important shortly. For now, observe that the equations of motion simplify

greatly when they are cast in terms of the transformed variables α, β. We have

removed the control field, and also the homogeneous terms on the left hand side, so


that we obtain the system

∂zα = −Cβ,

∂ωβ = Cα. (5.109)

Note the symmetry of this system of equations. If we swap α for β, and β for −α,

and then swap z and ω, the system of equations is unchanged. As we will see below,

this symmetry simplifies the form of the solution, so that there are only two different

Green’s functions, instead of a potential four.

We solve these equations by applying a unilateral Fourier transform. Having

eliminated the control field, we can choose either to apply the transform over the ω

coordinate or the z coordinate. We will apply the transform over the z coordinate,

as we have done previously, and solve the equations in k-space. Using tildes to

denote the transformed variables, we have

−ikα− α0√2π

= −Cβ,

∂ωβ = Cα, (5.110)

where α0 = α(z = 0, ω) represents the boundary condition for the signal field.

Solving the first equation for α yields

α = −iC

kβ +

i√2πk

α0. (5.111)


Substituting this result into the second equation, and integrating, we obtain the

solution for β,

β(k, ω) = e−iC2ω/kβ0 + iC√2πk

∫ ω

0e−iC2(ω−ω′)/kα0(ω′) dω′, (5.112)

where β0 is the Fourier transform of the spin wave boundary condition β0 = β(z, ω =

0). Finally, substituting this back into (5.111) gives the solution for α,

α(k, ω) = −iC

ke−iC2ω/kβ0 +

i√2πk

α0(ω) +C2

√2πk2

∫ ω

0e−iC2(ω−ω′)/kα0(ω′) dω′.

(5.113)

These k-space solutions are singular at k = 0, but fortunately it is possible to take

the inverse Fourier transform analytically. We use the results described in §D.5 in

Appendix D, along with the convolution theorem for the unilateral Fourier transform

(see §D.4.2 in Appendix D), to obtain the rather formidable-looking solution

α(z, ω) = α0(ω)− C∫ ω

0

√z

ω − ω′J1

[2C√z(ω − ω′)

]α0(ω′) dω′

−C∫ z

0J0

[2C√

(z − z′)ω]β0(z′) dz′,

β(z, ω) = β0(z)− C∫ z

0

√ω

z − z′J1

[2C√

(z − z′)ω]β0(z′) dz′

+C∫ ω

0J0

[2C√z(ω − ω′)

]α0(ω′) dω′. (5.114)

We re-write this in terms of Green’s functions, or propagators, in order to bring out


its structure.

αout(ω) =∫ 1

0L(ω, ω′)αin(ω′) dω′ −

∫ 1

0K(ω, z)βin(z) dz,

βout(z) =∫ 1

0L(z, z′)βin(z′) dz′ +

∫ 1

0K(z, ω)αin(ω) dω, (5.115)

where αout(ω) = α(z = 1, ω) is the transmitted signal field, and βout = β(z, ω = 1)

is the spin wave at the end of the interaction. We have introduced the Green’s

functions L and K, defined as follows,

L(x, y) = δ(x− y)− CΘ(x− y)× 1√x− y

J1(2C√x− y),

K(x, y) = CJ0

[2C√x(1− y)

]. (5.116)

Note that only two distinct Green’s functions are required, because of the symmetry

between α and β in the adiabatic equations of motion (5.109). The Green’s function

K(x, 1−y) is precisely the adiabatic Raman storage kernel (5.95), and this is why we

have used the same notation. The kernel L describes the relation of the transmitted

signal field to the input field, or equivalently it relates the final to the initial spin

wave. We are considering storage, so that βin = 0, but note that the same solutions

can be used to describe retrieval from the memory; retrieval is dicussed in Chapter

6. The Heaviside step function Θ(x) in L makes the kernel causal, so that the

transmitted signal field is never influenced by future values of the input field.

We now show that there is a connection between the SVD of K, which tells

us about the optimal input mode and its associated efficiency, and the SVD of L,


which tells us about the transmitted fields. Note that it is only correct to associate

the singular values of the Green’s function K with storage efficiencies when the

transformation connecting A, B to α, β is unitary, and this is only true in the

Raman limit ∆ 1. Therefore the following analysis applies only in this limit, and

we will assume that we are tuned far from resonance in the remainder of this section.

To see the connection between the SVDs of K and L, it will help us to use matrix

notation, since it is much more compact. As discussed in Chapter 3, the solutions

5.115 may be considered as the infinite-dimensional limit of the following matrix

equations,

|αout〉 = L|αin〉 −K|βin〉,

|βout〉 = L|βin〉+K|αin〉. (5.117)

To be more precise, we define the vector |αout〉 as a discretized version of the con-

tinuous function αout,

|αout〉 =

αout(0)

αout( 1N−1)

αout( 2N−1)

...

αout(1)

, (5.118)

where N is the number of discretization points. The other vectors are defined


similarly. The matrices K and L are given in terms of the continuous kernels by

Kjk = K

(j − 1N − 1

,k − 1N − 1

)× 1N − 1

, Ljk = L

(j − 1N − 1

,k − 1N − 1

)× 1N − 1

.

(5.119)

The continuous equations are recovered by taking the limit N −→∞, when matrix

multiplication is replaced by integration, and the factors 1/(N − 1) become the

increments dz, dω. It is useful to be able to swap between the continuous and

discretized representations, in order to use both the machinery of calculus and linear

algebra in their ‘natural habitats’.

The relationship between the SVDs of the matrices K and L is fixed by the

equations of motion (5.109), which are unitary. To see this, observe that these

equations imply the identity

∂z(α†α) + ∂ω(β†β) = 0. (5.120)

We have used the † notation for Hermitian conjugation, rather than the ∗ notation

for complex conjugation, since this equation holds whether or not we treat α, β as

ordinary functions, or as quantum mechanical annihilation operators3. Integrating

this identity over all z and ω gives the condition

Nα,out +Nβ,out = Nα,in +Nβ,in, (5.121)

3Strictly, the Langevin noise operators introduced in §4.12 of Chapter 4 contribute unless anexpectation value is taken, but in the Raman limit we neglect spontaneous emission, and theseoperators along with it.


where Nα,out, Nα,in are the numbers of transmitted and incident signal photons,

respectively, and where Nβ,out, Nβ,in are the numbers of final and initial spin wave

excitations:

Nα,out =∫ 1

0α†out(ω)αout(ω) dω, Nα,in =

∫ 1

0α†in(ω)αin(ω) dω,

Nβ,out =∫ 1

0β†out(z)βout(z) dz, Nβ,in =

∫ 1

0β†in(z)βin(z) dz. (5.122)

The condition (5.121) fixes the combined transformation of signal field and spin

wave, implemented by the memory interaction, as unitary, meaning that the total

number of ‘particles’ is conserved. This unitarity holds in the Raman limit approx-

imately, since we have neglected spontaneous emission, which process would scatter

particles into modes other than the signal field or the spin wave, thus violating the

conservation law (5.121). As discussed in §5.3.1, this approximation is generally a

good one for a Raman memory, since large optical depths are required, making spon-

taneous emission losses negligible. The conservation condition (5.121) has the same

form as that for a beamsplitter, which mixes a pair of input modes without loss, to

produce a pair of output modes. Indeed, this is a helpful perspective from which to

view the action of the Raman memory (see Figure 5.6 below). The difference with

a conventional beamsplitter, as we will see, is that the Raman interaction couples

multiple modes together in a pairwise fashion, with each pair of modes ‘seeing’ a

different reflectivity. We derive this fact by combining the solutions for α and β into


the left, to obtain the inverse transformation

|xin〉 = U †|xout〉. (5.127)

Substituting this into the conservation condition, we then find UU † = I. So we see

that U is indeed a unitary transformation (see §A.4.5 in Appendix A). To see the

implications of this for the matrices K and L, we substitute the form (5.125) for U

into the conditions U †U = UU † = I, and perform the matrix multiplications. This

yields the conditions

L†L+K†K = LL† +KK† = I. (5.128)

In terms of the normally and antinormally ordered products, these conditions are

written as

LN +KN = I; LA +KA = I. (5.129)

Therefore we must have that LN and KN commute:

[LN ,KN ] = [LN , I − LN ] = [LN , I]− [LN , LN ] = 0. (5.130)

And similarly for the antinormally ordered products, [LA,KA] = 0. As discussed

in §3.1.2 of Chapter 3, the eigenvectors of KA are the output modes of K, and

the eigenvectors of KN are its input modes. That is, if we write the SVD of K as

K = UKDKV†K , then KA = UKD

2KU†K , and KN = VKD

2KV

†K . The fact that KA

commutes with LA implies that the eigenvectors of LA are the same as those of KA


(see §A.3.1 in A). So we can write LA as

LA = UKD2LU†K , (5.131)

where D2L is a diagonal matrix of positive eigenvalues. The same argument applies

for the normally ordered products, so that we have

LN = VKD2LV†K . (5.132)

Note that the eigenvalues of LN are necessarily the same as those of LA. To sum-

marize, the unitarity of the memory interaction constrains the matrices K and L

such that their SVDs are built from a common set of input and output modes:

L = UKDLV†K K = UKDKV

†K . (5.133)

And substituting these expressions into the conditions 5.129, we find the following

relationship between the singular values,

D2L +D2

K = I. (5.134)

Our analysis is nearly complete. Let us review what we have discovered. A general

transformation from |xin〉 to |xout〉 could involve four Green’s functions — that is,

there could have been four matrices to deal with, one for each of the elements of

U in (5.125). But the symmetry of the equations of motion (5.109) allows us to


express the transformation in terms of just two Green’s functions, L and K. Each

of these has an SVD, so there are potentially four sets of orthogonal modes to deal

with, associated with the input and output modes of the two Green’s functions L

and K. The further condition of unitarity on U , however, allows us to reduce the

number of orthogonal sets from four down to two, since L and K must share the

same input modes, and also the same output modes, as one another. We now show

that in fact the output modes are just flipped versions of the input modes, so that

the entire interaction can be fully described using just a single set of orthogonal

modes. The final step is accomplished by noticing that the matrices L and K are

persymmetric. That is, they are symmetric under reflection in their anti-diagonal

(see §3.1.4 in Chapter 3). This can be seen by examining the functional forms

(5.116) of the Green’s functions. For example, the value of L(x, y) only depends on

the difference x − y. Therefore the contours of L all lie parallel to the line y = x,

which corresponds to the main diagonal of the matrix L. L is accordingly unchanged

by a reflection in the main anti-diagonal, that is to say, L is persymmetric. That

K is also persymmetric follows from the fact that K(x, 1 − y) is Hermitian. As

shown in §3.1.4 in Chapter 3, the input and output modes associated with the SVD

of a persymmetric matrix are simply ‘flipped’ versions of one another. Putting this

last result together with our previous analysis, we can express the Green’s functions


entirely in terms of a single set of modefunctions ϕj,

K(x, y) =∑j

ϕj(x)λjϕj(1− y),

L(x, y) =∑j

ϕj(x)µjϕj(1− y), (5.135)

where the singular values add in quadrature to unity,

µ2j + λ2

j = 1. (5.136)

The modes ϕj can be found from a numerical SVD of the kernel K. Or, equivalently,

they are given by the eigenvalue equation (5.96) introduced previously in §5.3.

The procedure used to connect the SVDs of K and L through the unitarity of

U is known as the Bloch-Messiah Reduction [147–149]. The resulting decomposition

makes the assertion that Raman storage may be understood by analogy with a

beamsplitter into a rigorous correspondence. If we define aj , (bj) as the annihilation

operator for a photon (spin wave excitation) in the jth input mode ϕj , and if Aj

(Bj) annihilates a photon (spin wave excitation) in the jth output mode, then for

each mode, the memory interaction can be written as

Aj = µjaj − λjbj ,

Bj = µjbj + λjaj . (5.137)

These relations are precisely those arising in the quantum mechanical description of


an optical beamsplitter, coupling input modes aj , bj , to output modes Aj , Bj , with

reflectivity R = ηj = λ2j (see Figure 5.6). Optimal storage corresponds to the case

R = 1, so that the incident signal field is entirely ‘reflected’ into a spin wave mode.

Figure 5.6 Raman storage as a beamsplitter. Optical and spinwave modes are mixed pairwise by the storage interaction, as lightbeams are on a beamsplitter. The ideal quantum memory, with unitstorage efficiency, would see the beamsplitter replaced by a perfectmirror.

A possible use of a Raman quantum memory with non-unit storage efficiency

is as part of the modified DLCZ quantum repeater protocol described in §1.7 in

Chapter 1. Instead of using a 50 : 50 beamsplitter in combination with an ideal

quantum memory to generate number state entanglement, a single Raman quantum

memory with η1 = 50% can be used, as shown in Figure 5.7. The quality of the

entanglement generated relies on good overlap of the transmitted field modes on

the final beamsplitter, so the preceding theoretical characterization of the temporal

structure of these modes simplifies the analysis of this type of protocol.

5.4 Numerical Solution 182

BS

D1

D2

QML

QMR

SL

SR

Figure 5.7 Modified DLCZ protocol with partial storage. Whena single detector fires behind the final beamsplitter, number stateentanglement is generated between the memories. The protocol isexplained in §1.7 in Chapter 1. The only difference is that here,a single memory with 50% efficiency replaces the combination of abeamsplitter and an ideal memory.

5.4 Numerical Solution

So far we have succeeded in deriving a form for the storage kernel in the adiabatic

limit, or for arbitrary bandwidths in the case of a hyperbolic secant control. The

general problem of finding the storage kernel for both arbitrary bandwidths and

arbitrary control pulse profiles has not been solved analytically. But it is possible

to construct the kernel numerically. This can be done by integrating the system

of coupled equations (5.106) multiple times, each time with a different boundary

condition. Provided the boundary conditions form a set of orthogonal functions, the

Green’s function can be reconstructed. The easiest set of boundary conditions to

implement is the set of ‘impulses’ — delta functions. To see why this works, recall


the definition of the storage kernel,

Bout(z) =∫ ∞−∞

K(z, τ)Ain(τ) dτ, (5.138)

where we used normalized units for z. Now, if we insert a delta function δ(τ − τj)

as the signal field boundary condition Ain(τ), where τj is some particular time slot,

the resulting spin wave Bout,j is

Bout,j(z) = K(z, τj). (5.139)

We can therefore reconstruct an approximation to the entire Green’s function by

numerically solving for Bout,j repeatedly, with the times τj chosen from a finite

grid. The grid should range over a sufficient range of times that all of the Green’s

function is sampled, and we should make the grid sufficiently fine that no features

of the Green’s function are missed. So long as these requirements can be met while

keeping the computation reasonably fast, this is a convenient way to find the storage

kernel without requiring the adiabatic approximation, and without imposing any

restriction on the temporal profile of the control field. This method was previously

used to reconstruct the adiabatic Green’s functions describing stimulated Stokes

scattering in a dispersive ensemble by Wasilewski and Raymer [148]. In this thesis

we solve the equations of motion numerically using Chebyshev spectral collocation

for the spatial derivatives, and a second order Runge-Kutta (RK2) method for the

time stepping. The method of solution is explained in detail in Appendix E.


In the figures below we compare the optimal input modes predicted by the various

methods presented in this chapter. In Figures 5.8 and 5.9 we plot, side by side, the

optimal input modes found from numerically constructed Green’s functions, the

Rosen-Zener kernel (5.60), the adiabatic kernel (5.77) and the Raman kernel (5.95).

In Figure 5.8 the adiabatic approximation is well satisfied; the Raman approxi-

mation only poorly so, and therefore there is good agreement between all predictions,

save for the Raman prediction, which deviates slightly. But note that the adiabatic

and Raman kernels do slightly overestimate the phase due to the dynamic Stark-

shift.

−5 0 50

0.5

1Rosen Zener

−5 0 5

Numerical

−5 0 5

Adiabatic

−5 0 5

Raman

0

2

4

6

Inte

nsity

(arb

. u

nits)

Ph

ase

Time

Figure 5.8 Comparison of predictions for the optimal input modesin the adiabatic limit. Here we used a hyperbolic secant control, givenby (5.38) with Tc = 1 and Ω0 = Ωmax = 3, an optical depth of d = 300and a detuning ∆ = 15, in normalized units. The control intensityprofile |Ω(τ)|2 is indicated by the dotted lines, scaled for clarity. Theblue lines show the predicted optimal intensity profiles |Ain(τ)|2 =|φ1(τ)|2, and the red lines show the variation of the temporal phaseof the mode φ1 in radians, referred to the axes on the right handside. The storage efficiency is ∼ 90% in all cases, and the predictionsare in good agreement generally. But both the adiabatic and Ramankernels overestimate the phase shift due to the dynamic Stark shift.And the detuning is not quite large enough to render the Ramankernel correct.

In Figure 5.9 the fields are tuned into resonance, the optical depth is reduced,


and the control intensity is increased, so that the adiabatic approximation is no

longer satisfied. There is rough agreement between the numerical and Rosen-Zener

predictions, which do not rely on the adiabatic approximation, although there is

a slight discrepancy that we attribute to the accumulation of numerical errors in

the evaluation of the Hypergeometric functions in the Rosen-Zener kernel (5.60).

The optimal modes predicted by these two methods exhibit oscillations, as can be

seen from the phase jumps indicating sign changes. These oscillations are missing

from the predictions of the adiabatic kernel; the prediction of the Raman kernel is

catastrophically wrong, as we would expect on resonance.

−5 0 50

0.5

1Rosen Zener

−5 0 5

Numerical

Time

−5 0 5

Adiabatic

−5 0 5

Raman

0

2

4

6

Inte

nsity

(arb

. u

nits)

Ph

ase

Figure 5.9 Comparison of predictions for the optimal input modesoutside the adiabatic limit. Again, the control takes the form of ahyperbolic secant, this time with Tc = 1 and Ω0 = Ωmax = 5. Wereduce the optical depth down to d = 10, and we tune into resonance,putting ∆ = 0. The numerical and Rosen-Zener predictions roughlycoincide, and both the predictions exhibit oscillations characteristic ofnon-adiabatic ‘ringing’. Numerical errors in the Rosen-Zener kernelamplify the size of these oscillations slightly. The adiabatic kerneldoes not correctly reproduce the oscillations, while the Raman kernelis totally inappropriate for modelling this resonant interaction. Theoptimized storage efficiency predicted by the numerics is around 82%.

The numerically constructed Green’s functions yield the most reliable predic-

tions. In addition, they take less time to calculate then the Rosen-Zener predictions


— the Matlab code for the numerical method runs in ∼ 40 s on a 3 GHz machine;

the Rosen-Zener kernel takes several hours to construct, simply because the hyperge-

ometric functions are so difficult to evaluate efficiently. Also, the numerical method

is more flexible, since arbitrary control profiles can be used. In general, then, the

direct numerical approach is the method of choice for optimizing a Λ-type quantum

memory.

In the adiabatic and Raman limits, however, it is faster to use the analytic

kernels derived using these approximations. And of course, the optimization can

then be trivially generalized to arbitrary control pulse shapes using the transforma-

tions (5.78) and (5.97). In Figures 5.10 and 5.11 below we compare the numerical

predictions with those of the adiabatic and Raman kernels in the resonant and

Raman limits, using much more broadband control pulses. As expected, there is

good agreement between all three predictions in the Raman limit. In the resonant

case, describing EIT storage, the Raman kernel of course fails completely, but the

adiabatic kernel remains reasonably reliable. That said, its agreement with the nu-

merical prediction is not as good as in the Raman limit, which is symptomatic of

the fragility of the adiabatic approximation on resonance.

When far into the Raman limit, the numerical, adiabatic and Raman methods

all predict a significant temporal phase variation due to the dynamic Stark shift.

Whereas in the EIT case, the optimal input mode has a flat phase. Depending on

the flexibility of the technology used to shape the optical pulses before storage, it

may be easier to implement EIT storage for this reason. On the other hand, Raman


−0.2 0 0.20

0.5

1Numerical

−0.2 0 0.2

Raman

0

2

4

6

−0.2 0 0.2

Adiabatic

Time

Inte

nsity

(arb

. u

nits)

Ph

ase

Figure 5.10 Broadband Raman storage. Here the control is aGaussian pulse Ω(τ) = Ωmaxe

−(τ/Tc)2, where Tc = 0.1 and W =

302.5, so that Ωmax = (2/π)1/4√W/Tc = 49.1. The optical depth is

d = 300, and the detuning is ∆ = 150. As usual all these quantitiesare in normalized units. These parameters give a Raman memorycoupling of C = 2.0 and a balanced interaction with R = 1.0. Notethat the control pulse duration is roughly one tenth of the sponta-neous emission lifetime of the excited state |2〉. Nonetheless, due tothe large detuning, the adiabatic and Raman approximations are wellsatisfied, and the agreement between the numerical and analytic pre-dictions is clear. The optimized storage efficiency is predicted by allthe methods shown to be ∼ 98%.

storage allows for more freedom in the carrier frequencies of the signal and control

fields.

5.4.1 Dispersion

All of the optimal modes shown in the figures above are slightly delayed in time

with respect to the control pulse. This is due to the characteristic dispersion associ-

ated with an absorption process, which produces a superluminal group velocity. Of

course, there is no question of violating causality, it is simply that as the trailing

edge of the signal field is absorbed, the ‘centre of mass’ of the signal pulse advances,

giving the appearance of superluminal propagation. The efficiency of the memory

5.5 Summary 188

−0.2 0 0.20

0.5

1Numerical

−0.2 0 0.2

Adiabatic

−0.2 0 0.2

Raman

Time

Inte

nsity

(arb

. u

nits)

Ph

ase

0

2

4

6

Figure 5.11 Broadband EIT storage. All parameters are the sameas above in Figure 5.10, except that the storage is performed onresonance, with ∆ = 0. The Raman kernel fails utterly, but theadiabatic prediction fairs better, comparing well with the numericalprediction. That the agreement is poorer on resonance is a signaturethat the adiabatic approximation is less robust on resonance. Theoptimized storage efficiency predicted by the numerical method is∼ 99%.

is maximized by ‘pre-compensating’ for this effect, and this explains the time shift

common to the optimal input modes. In the case of resonant storage, this view con-

flicts with our characterization of EIT in §2.3.1 of Chapter 1 as working by slowing

the group velocity of the signal. Perhaps a better perspective is therefore that on

resonance the control must precede the signal in order to ‘prepare’ the transparency

window.

5.5 Summary

We have covered rather a lot of material in this chapter. Here we review the main

results.

1. The analysis of the storage process is greatly simplified if we use a one di-

5.5 Summary 189

mensional propagation model. All the results listed below make use of this

model.

2. The best possible storage efficiency is limited by the optical depth, being given

by η = 1− 2.9/d.

3. The method of Rosen and Zener yields an analytic expression for the storage

kernel for a hyperbolic secant control. It is, however, difficult to evaluate this

efficiently and accurately (at least using Matlab).

4. In the adiabatic approximation, an analytic expression for the storage kernel

can be derived that holds for all control pulse profiles. This is quick to evaluate.

5. When far detuned, the expression further simplifies, yielding the Raman kernel,

which only depends on the Raman memory coupling C. The Raman memory

can be decomposed as a set of beamsplitter transformations between light and

matter, using just a single set of modefunctions.

6. If none of the above methods are appropriate, the storage kernel can be directly

constructed by repeated numerical integration of the equations of motion for

the memory. This method yields the correct optimal input mode for arbitrary

control profiles and detunings, for all values of the optical depth.

In the next chapter, we consider retrieval from a Λ-type memory.

Chapter 6

Retrieval

So far we have considered the optimization of storage in a Λ-type quantum mem-

ory. In some circumstances this optimization is sufficient to maximize the combined

efficiency of storage into, followed by retrieval from the memory. Specifically, this is

true when the retrieval process is the time reverse of the storage process [133]. When

this is not the case, the optimization of this combined efficiency is a distinct prob-

lem. In this Chapter, we discuss various strategies for retrieval, and we present the

results of a numerical analysis of their effectiveness.

6.1 Collinear Retrieval

In order to convert the stored excitation back into a propagating optical signal, we

send a second control pulse — a read out pulse — into the ensemble. Since the

control mediates coupling between the signal field and the spin wave, the rationale

is that the spin wave excitations will transfer back to the optical field, emerging

6.1 Collinear Retrieval 191

as a collimated pulse traveling in a well-defined direction. We have met with some

success in analyzing the storage process using a one dimensional propagation model,

and so it is natural to consider read out from this perspective. We will see later

that certain advantages accrue if a small angle is introduced between the signal

and control fields. For the moment, suppose that we have stored a signal photon

collinearly, as described in the previous chapter. Confining ourselves to the same

one dimensional model, there are two possible read out geometries: forward, and

backward retrieval.

6.1.1 Forward Retrieval

We can describe forward retrieval using the same equations of motion as we used

for storage — (5.3), or (5.106) in normalized units. For the retrieval process, the

signal field boundary condition is set to zero, and the spin wave boundary condition

is set by the coherence generated in the medium during the storage process. Let us

denote quantities associated with the retrieval process with a superscript r. In this

notation the boundary conditions at read out are

Arin(τ r) = 0, Br

in(z) = Bout(z). (6.1)

The second condition assumes that there is no loss of coherence during the time that

the excitations are stored. This kind of loss is homogeneous in space, and so it would

appear only as a constant factor reducing the amplitude of Brin. The optimization

of the memory is therefore unaffected by the neglect of decoherence.


The first step in analyzing the efficiency of the retrieval process is to examine

the form of the map from the stored excitations to the retrieved signal field,

Arout(τ

r) =∫ 1

0Kr(τ r, z)Br

in(z) dz. (6.2)

In fact, in the adiabatic limit, the retrieval kernel Kr is identical in form to the

storage kernel K, when expressed in terms of the integrated Rabi frequency ωr,

rather than the time τ r. This is guaranteed by the symmetrical form of the equations

of motion in the adiabatic limit, which is discussed in §5.3.3 in Chapter 5. For

completeness, we verify this by explicit calculation. In the notation of §5.2.5 in

Chapter 5, the retrieved signal field is expressed in k-space as

Ar(k, τ r) = −i

√d√

2πk〈↑ |V r−1(k, τ r)V r

in(k)| ↓〉Brin(k). (6.3)

Using V rin = I, along with the adiabatic approximations (5.70) and (5.73), we obtain

Ar(k, τ r) = − 1√2π

√d

ΓrΩr(τ r) exp

[− 1

Γr

∫ τ r

−∞|Ωr(τ ′)|2 dτ ′

]× 1k + i dΓr

exp

[id

Γr2

∫ τ r

−∞ |Ωr(τ ′)|2 dτ ′

k + i dΓr

]Br

in(k). (6.4)

Here Ωr is the Rabi frequency describing the temporal profile of the read out control

pulse, which may differ from that of the storage pulse, and Γr = γ − i∆r allows for

the possibility that the detuning is changed for the readout process. We transform

the time coordinate from τ r to ωr, and take the inverse Fourier transform from k


to z-space. Using the result (D.40), along with the shift and convolution theorems

(D.18), (D.26) in Appendix D, we find that the retrieval kernel is given by

Kr(τ r, z) =Ωr∗(τ r)√W r

×Kr [ωr(τ r), z] , (6.5)

where

Kr(ωr, z) = i

√dW r

Γre−[W rωr+d(1−z)]/Γr

J0

[2i√dW r

Γr

√ωr(1− z)

]. (6.6)

This kernel has precisely the same form as (5.76), except that ω has been switched

for z, and z has been switched for ωr. Now we can write down the adiabatic map

describing the combined processes of storage followed by forward retrieval,

Arout(τ

r) =∫ ∞−∞

Ktotal(τ r, τ)Ain(τ) dτ. (6.7)

The kernel Ktotal describes the entire memory interaction (see §3.4 in Chapter 3),

and is given by

Ktotal(τ r, τ) =Ωr∗(τ r)Ω(τ)√

W rW×Ktotal [ωr(τ r), ω(τ)] , (6.8)

where

Ktotal (ωr, ω) =∫ 1

0Kr(ωr, z)K(z, ω)dz. (6.9)

Here the kernel K is the adiabatic storage kernel given in (5.76) in Chapter 5, and

Kr, given in (6.6), is related to K by the symmetry connecting the storage and


retrieval kernels just derived above.

The input mode that optimizes the total memory efficiency is found from the

SVD of Ktotal in (6.9). The kernel looks superficially similar in form to the normally

ordered kernel KN formed from the storage kernel K. But Ktotal 6= KN because

Kr 6= K∗. If these two kernels were equal to one another, then the optimal input

mode found from the SVD of Ktotal would be equal to the mode found from the

SVD of K; this follows from the properties of KN (see §3.3.1 in Chapter 3). Since

these two kernels are not equal to each other, the optimization of storage followed

by forward retrieval is different to the optimization of storage alone.

Note that the optimal input mode does not depend on the shape Ωr of the read

out control; it only depends on the total energy in the read out pulse, parameterized

by W r. But changing the shape of Ωr changes the temporal profile of the signal field

retrieved from the memory. This is useful: one can optimize the memory efficiency

once, for a fixed energy W r, and then changing the shape of Ωr allows one to produce

an output signal pulse of any shape, within the limits of the adiabatic approximation.

Novikova et al. have already demonstrated optimal storage, followed by shaped

retrieval, experimentally, on resonance [74,150,151], using the theory of Gorshkov et

al. [133], which applies in the light-biased limit (see the end of §5.2.8 in Chapter 5).

In Figure 6.1, we show an example of how the optimal input mode for storage,

followed by retrieval, differs from the optimal storage mode. We used the adiabatic

kernel (5.76) to model the storage interaction, and we evaluated the expression (6.9)

for Ktotal numerically. This is easily done by discretizing the coordinates so that K


becomes a matrix. We use the same control pulse and detuning for both storage

and readout, Ω = Ωr and ∆ = ∆r, and then Ktotal is given by the product of

two copies the matrix K. Taking the SVD of the result yields a radically different

optimal input mode to that found from the SVD of K alone. The difference can be

understood by considering the shape, in space, of the spin wave Bout(z) produced by

the storage interaction. When optimizing storage alone, this is given by the optimal

output mode ψ1(z) found from the SVD of the storage kernel K. The shape of this

mode generally takes the form of a decaying exponential, as shown in part (b) of

Figure 6.1. This shape is consistent with Beer’s law absorption: as the signal pulse

propagates through the ensemble it is increasingly likely to be absorbed, so that at

the exit face, at z = 1, there is very little probability for the signal pulse to excite

an atom. Therefore the spin wave decays in magnitude steadily from the input to

the exit face. But forward retrieval from a spin wave of this shape is problematic:

if the amplitude of the spin wave is concentrated close to the entrance face of the

ensemble, a retrieved excitation, that has been converted into a signal photon, must

propagate a large distance through the ensemble before reaching the exit face. There

is therefore a high probability that the retrieved photon will be re-absorbed, so that

it never emerges from the ensemble, and this greatly reduces the efficiency of the

read out process. For forward retrieval, it is much better that the bulk of the spin

wave is concentrated towards the exit face of the ensemble, so that re-absorption

losses are minimized. In fact, the optimal spin wave mode for retrieval is precisely

the space-reverse of the mode generated by the storage process; this can be derived


from the symmetry relating the storage and retrieval kernels. It is the fact that the

optimal shape of the spin wave for retrieval is improperly matched to the spin wave

mode generated by optimal storage that complicates the combined optimization of

storage followed by retrieval. A compromise between these two shapes must be

found. Such a compromise is shown in part (d) of Figure 6.1: the amplitude of

the spin wave at z = 1 is much larger than for the spin wave that results from only

optimizing storage. And there is a marked suppression of the spin wave amplitude at

z = 0, since even though the storage interaction naturally excites atoms close to the

entrance face, it is extremely deleterious to the retrieval efficiency to concentrate the

stored excitations there. If we could run the storage process backwards, so that the

retrieval process was precisely the time-reverse of the storage process, we should be

able to achieve a retrieval efficiency equal to the storage efficiency, ηretrieve = ηstorage.

It should then be possible to achieve a combined efficiency of storage followed be

retrieval of ηcombined = ηstorage × ηretrieval = η2storage. However, even after optimizing

the combined efficiency, it still falls far short of this optimum, ηcombined η2storage.

In the example shown in Figure 6.1, ηcombined = 28%, whereas η2storage = 80%. The

tension between Beer’s law during storage and re-absorption during retrieval makes

forward retrieval generally inefficient. In the next section we consider backward

retrieval, which performs better.

6.2 Backward Retrieval 197

0

0.5

1Spin waveInput mode

(a) (b)

(d)(c)In

ten

sity

(arb

. u

nits)

Inte

nsity

(arb

. u

nits)

Pha

se

Pha

se

0

2

4

6

−2 0 20

0.5

1

0 0.5 10

2

4

6

Figure 6.1 Forward retrieval. We consider collinear storage, fol-lowed by forward retrieval, using a Gaussian control pulse Ω(τ) =Ωmaxe

−τ2, with W = 9, so that Ωmax = (2/π)1/4

√W = 2.68. The

optical depth is d = 300, and the detuning is ∆ = 15, in normalizedunits. (a) shows the input mode that optimizes just the storage ef-ficiency, and (c) shows the input mode that optimizes the combinedefficiency of storage followed by retrieval. The shape of the controlintensity profile is indicated by the black dotted lines. The formeroptimization gives a storage efficiency of ∼ 90%, whereas the latteroptimization gives a combined efficiency of only 28%. In parts (b) and(d) we show the ‘intensity’ |Bout(z)|2 and spatial phase of the spinwave generated in the ensemble after the storage process is complete,for the storage and combined optimizations, respectively. The com-bined optimization produces a spin wave that reduces re-absorptionlosses by shifting the ‘centre of mass’ of the spin wave away from theentrance face at z = 0.

6.2 Backward Retrieval

The arguments given in the previous section suggest that retrieving the signal field

backwards should be much more efficient than forward retrieval. If the storage in-

teraction produces a spin wave with its amplitude concentrated toward the entrance

face of the ensemble, re-absorption losses are minimized if retrieved signal photons

propagate backwards to re-emerge from the entrance face. This is indeed the case,

but issues of momentum conservation arise if the energy splitting between the ground


and storage states |1〉 and |3〉 is non-zero. To see why, consider the diagram shown

in part (a) of Figure 6.2. Momentum conservation requires that the wavevectors

ks, kc and κ associated with the signal, control and spin wave, respectively, sum

to zero. When this is the case, the storage interaction is said to be phasematched,

since the spatial phases accumulated by the optical fields as they propagate through

the ensemble are ‘matched’ to the spatial phase of the spin wave. This means that

the slowly varying envelopes of these fields are strongly coupled to the spin wave,

as described by the equations of motion for the quantum memory (see (5.106) in

Chapter 5, for instance). As we will see below, when momentum is not conserved,

the interaction is not phasematched, and destructive interference greatly reduces the

strength of the coupling, and with it the memory efficiency. Fortunately, the storage

process is always phasematched, because there is no fixed dynamical relationship be-

tween the spatial phase of the spin wave and its energy. This is because the atoms

comprising the spin wave do not interact with another, so that there is no coupling

at all between the spatial shape of the spin wave and the frequency splitting between

the ground and storage states. In the storage process, a signal photon is absorbed,

and a control photon is emitted. The spin wave therefore acquires a wavevector that

‘takes up the slack’, given by the difference of the signal and control wavevectors,

κ = ks − kc =ωs − ωc

cz, (6.10)


where z is a unit vector pointing along the positive z axis. In the last equality we

used the optical dispersion relation for a plane wave,

k = ω/c, (6.11)

where k is the magnitude of the wavevector and ω is the angular frequency of

the wave (not the integrated Rabi frequency!). This relation fixes the momentum

associated with the spin wave as given by the difference between the signal and

control field frequencies, which is in turn fixed by the energy splitting that separates

the ground and storage states, in order to satisfy two-photon resonance. Even though

there is no intrinsic connection between the spatial phase of the spin wave and its

energy, the kinematics of the scattering process that generates the spin wave does, in

fact, tie these two quantities together. The efficiency of the retrieval process depends

critically on whether it is possible to phasematch the retrieval interaction. When we

attempt to retrieve the spin wave excitations by sending the readout control pulse

in the backward direction, momentum conservation requires that the wavevector of

the retrieved signal field is formed from the sum of the spin wave and the read out

control wavevectors,

krs = κ+ kr

c. (6.12)

Substituting (6.10) into (6.12), we obtain

krs − kr

c = ks − kc. (6.13)


For collinear storage, we have ks = ksz and kc = kcz. For backward retrieval, we

have krs = −kr

sz and krc = −kr

cz. Inserting these expressions into (6.13), and using

the optical dispersion relation (6.11) along with the two-photon resonance condition,

we get to the condition

ωc − ωsc

=ωs − ωc

c. (6.14)

As is clear from part (b) of Figure 6.2, this condition can only be satisfied when

ωs = ωc, and κ = 0, which requires that the ground and storage states are degenerate

in energy. Experimentally, it is important to be able to distinguish the weak signal

field from the bright control pulse, and the ability to spectrally filter these two fields

should not be surrendered lightly. Therefore it is very useful to consider storage in

memories where ωs 6= ωc, and then the issue of momentum conservation becomes

important when considering backward retrieval.

(a) (b)

Figure 6.2 Phasematching considerations for backward retrieval.(a) The momenta of the control field and spin wave must sum tothat of the signal field, from which they are ‘scattered’. The storageprocess is automatically phasematched, since the magnitude of κ isinitially a free parameter, that is determined by ks and kc duringstorage. (b) The spin wave momentum is pointing in the ‘wrongdirection’, when backward retrieval is attempted: it is not possible tosimultaneously satisfy the two-photon resonance condition kr

s − krc =

(ωs − ωc)/c, and the phasematching condition krs = κ+ kr

c.

In order to optimize the combined efficiency of storage, followed by backward


retrieval, we need to find an expression for Ktotal that describes the entire memory

interaction in this case. The SVD of this kernel will then provide us with the optimal

input mode. The equations of motion for the retrieval process have the same form

as those describing storage, but they must describe propagation in the backward

direction, with the z coordinate is reversed. The kernel Ktotal is simply constructed

using the adiabatic solutions for storage and retrieval separately, provided we are

careful about how the boundary conditions are ‘stitched together’. Let us denote the

flipped z coordinate for the retrieval process by zr, so that zr = L − z, in ordinary

units, where z is the coordinate describing propagation during storage. The atomic

coherence at the start of the retrieval process is identical — as always assuming no

decoherence — to that generated by the storage process. We can write this as

σr13,in(zr) = σ13,out(z), (6.15)

where the σ13(x) denotes the Raman coherence associated with an atom located at

the longitudinal position x. The spin wave B, defined in (4.43), makes use of the

slowly varying operators σ13, introduced in (4.22) (see Chapter 4), which incorporate

an exponential factor eiω13τ . Recall that τ is in fact the retarded time, τ = t− z/c,

so there is a spatial phase built into the spin wave; this represents the momentum

imparted to the spin wave by the optical fields that create it during the storage


process. Equating the boundary conditions, as in (6.15), gives the relation

Brin(zr) = Bout(z)× e−iω13τout × eiω13τ r

in ,

= Bout(z)× e−iω13(trin−tout) × e−iω13(z−zr)/c,

= Bout(L− zr)e2iω13zr/c. (6.16)

Here we used the definition τ r = tr − zr/c = tr − L/c + z/c, and in the last line

we dropped some unimportant constant phase factors. The spatial phase factor in

(6.16) represents the phase mismatch shown in Figure 6.2 (b). Note that it vanishes

if ω13 = 0. The phase mismatch causes oscillations of the spin wave in space that

can dramatically reduce the retrieval efficiency. This can be seen by considering the

form of the retrieval map

Arout(τ

r) =∫ 1

0Kr(τ, zr)Br

in(zr) dzr

=∫ 1

0Kr(τ r, zr)eiδkzr

Bout(1− zr) dzr. (6.17)

Here we have switched to normalized units, and defined the dimensionless phase

mismatch

δk =2ω13L

c. (6.18)

Note that δk is a negative quantity if the storage state |3〉 is more energetic than the

ground state |1〉. It is clear that, regardless of the form of the retrieval kernel Kr, or

of the spin wave Bout, the integral can be made to vanish if δk is made large enough.


This explains why a phase mismatch can be very detrimental to the efficiency of

backward retrieval. More precisely, if kmax represents the width in k-space of the

integrand Kr × Bout, the value of the integral becomes strongly suppressed when

δk kmax. It is therefore possible to mitigate the effect of the phase mismatch,

to some extent, by localizing the spin wave strongly in space, so that kmax becomes

large. Below, we show explicitly how to optimize storage, followed by backward

retrieval in the presence of a phase mismatch; the result essentially generates a more

strongly localized spin wave, for exactly this reason.

To perform the optimization, we construct the kernel Ktotal using the solution

(6.4) for the retrieval process, making the replacement z −→ zr. The result is

Arout(τ

r) =∫ ∞−∞

Ktotal(τ r, τ)Ain(τ) dτ, (6.19)

with

Ktotal(τ r, τ) =Ωr∗(τ r)Ω(τ)√

W rW×Ktotal [ωr(τ r), ω(τ)] , (6.20)

and where

Ktotal (ωr, ω) =∫ 1

0Kr(ωr, zr)K(1− zr, ω)eiδkzr

dzr. (6.21)

Suppose that the readout control pulse is identical in shape and frequency to the

storage control pulse, with Ωr = Ω and ∆r = ∆. Suppose also that the storage state

is degenerate with the ground state, so δk = 0. Then consider the kernel Ktotal in


(6.21) with its output time argument flipped around,

Ktotal(1− ωr, ω) =∫ 1

0Kr(1− ωr, zr)K(1− zr, ω) dzr

=∫ 1

0K(1− zr, ωr)K(1− zr, ω) dzr, (6.22)

The second line follows from the symmetry Kr(1−x, y) = K(1−y, x) that connects

the retrieval kernel (6.6) to the storage kernel (5.76). This shows that Ktotal is,

in the case under consideration, very similar in structure to the normally ordered

kernel KN that would be formed from K(1 − zr, ω). The only difference is that

the first instance of K in (6.22) is replaced by its complex conjugate in KN . If K

is real, then this complex conjugation is not important, and Ktotal = KN , which

means that the optimal input modes derived from the SVD of Ktotal are equal to

the optimal input modes derived from the SVD of K alone. This observation shows

that the optimal input mode for storage alone is close to the optimal input mode

for storage followed by backward retrieval, if identical storage and readout control

pulses are used, and if there is no phase mismatch. The two optimizations are the

same whenever K is real, and more generally whenever ψ1(z), the output spin wave

mode generated by optimal storage, is real.

In Figure 6.3 below we show an example of how the optimal input mode for

storage alone can differ from the optimal mode for storage followed by backward

retrieval. Without a phase mismatch (parts (c) and (d)), the difference between the

two optimizations is rather small, though non-negligible, for the example shown.


When a significant phase mismatch is introduced (parts (e) and (f)) the optimiza-

tions differ more markedly. It is a general feature of the optimization of storage

with backward retrieval, that as |δk| is increased, the shape of the optimal input

mode approaches that of the control pulse. The reason is as follows. Recall that

the dispersion experienced by the signal field as it propagates causes its group ve-

locity to differ from that of the control (see the end of §5.4 in Chapter 5). If the

signal initially overlaps with the control, then the coupling between these two fields

is initially high, but as the signal walks off from the control, the coupling decays

away. The bulk of the storage interaction therefore occurs near the entrance face

of the ensemble at z = 0. As is clear from part (f) of Figure 6.3, the spin wave

generated by the optimal input mode is indeed concentrated more closely towards

the entrance face. This makes the spin wave more strongly localized, which reduces

the ‘wash-out’ caused by the phase mismatch, as discussed above, while at the same

time reducing re-absorption losses.

A discussion of similar optimizations can be found in the work of Gorshkov

et al. [133]. In this work time-reversal arguments are employed to show that for

optimized storage with backward retrieval, ηcombined = η2storage if ψ1(z) is real. This

condition is always satisfied in the light-biased limit (see the end of §5.2.8 in Chapter

5), when the anti-normally ordered kernel KA becomes real, independent of whether

or not the adiabatic approximation is satisfied (see (5.27 in Chapter 5)). As shown

above in the adiabatic limit, when ψ1 is not real, implementing ‘true’ time reversal

requires that the phase of the spin wave is conjugated. Without a practical method


0246

0

0.5

1

0

5

10

0

0.5

1

0246

0246

!2 0 20

0.5

1

02468

0 0.5 10246

Spin waveInput mode

Inte

nsity

(arb

. unit

s)In

tens

ity(a

rb. u

nits)

Inte

nsity

(arb

. unit

s)Phase

PhasePhase

(a) (b)

(d)(c)

(f)(e)

Figure 6.3 Backward Retrieval. We consider collinear storage fol-lowed by backward retrieval, using a Gaussian control pulse Ω(τ) =Ωmaxe

−(τ/Tc)2, with Tc = 0.1 and W = 100, so that Ωmax =

(2/π)1/4√W/Tc = 28.25. The optical depth is d = 300, and the

detuning is ∆ = 150, in normalized units. Parts (a) and (b) showthe optimal input mode for storage alone, alongside the spin wavemode ψ1(z) generated in the medium. We use the adiabatic formfor the storage kernel; comparison with the prediction derived fromthe numerically constructed kernel verifies that the adiabatic approx-imation is well satisfied. The optimal efficiency for storage alone isηstorage ∼ 96%. In parts (c) and (d) we show the optimal input modefor storage followed by backward retrieval, and the generated spinwave, with degenerate ground and storage states, δk = 0. The opti-mized total efficiency is ∼ 88%, which is about 5% less than η2

storage.The most notable difference between these two optimizations is theappearance of a ‘wiggle’ in the phase of the input mode for the com-bined optimization. But this occurs while the signal intensity is ratherlow, so it is actually less important than the more subtle re-shapingthat occurs. In parts (e) and (f), we show the optimal input modefor storage and backward retrieval, along with the generated spinwave, with a phase mismatch δk = −5. Note that this optimizationresults in a spin wave with its ‘centre of mass’ concentrated moreclosely towards the entrance face of the ensemble at z = 0 than theother optimizations. This reduces the effective length of the ensemble— the length over which there is significant excitation — which di-minishes the effect of the phase mismatch. The optimized combinedefficiency of storage and retrieval is around 75% in this case.

6.3 Phasematched Retrieval 207

for doing this, ηcombined < η2storage, and the optimal mode for storage alone differs

from the mode that optimizes the combination of storage and backward retrieval.

In this and the previous section we have shown that retrieving the stored excita-

tions from a quantum memory efficiently is a non-trivial problem. Forward retrieval

is plagued by re-absorption losses — a modematching issue. Backward retrieval is

beset by momentum conservation problems — a phasematching issue. In the next

section we present a solution to both of these problems, which requires a departure

from the one dimensional treatment we have worked with thus far.

6.3 Phasematched Retrieval

The best memory efficiency possible is achieved by optimizing collinear storage,

followed by backward retrieval, with δk = 0. But to spectrally filter the signal field

from the control, we should have δk 6= 0. By introducing a small angle between the

signal and control beams, both at storage and at read out, we can maintain proper

phasematching, even when δk 1, which allows for efficient retrieval. This idea

was proposed by Karl Surmacz, and the following treatment is adapted from our

paper [152]; the numerical simulations were performed by me. The principle of the

scheme is easily understood by considering the phasematching diagrams in Figure

6.4. Here we assume for simplicity that all the beams used, both at storage and at

read out, are confined to a common plane, so that we need only consider two space

dimensions. The spin wave momentum is determined by the signal and control field

wavevectors according to (6.10). If we fix the detuning, so that ∆r = ∆, then


the magnitudes of ks and kc are unchanged at read out, and then the condition

(6.12) uniquely defines the direction of krs that is phasematched. By symmetry, the

angle θ between the signal and control beams is the same at read in as at read out.

Provided we adhere to the geometry shown in Figure 6.4 (a), the retrieval process

remains correctly phasematched for all choices of θ. To maximize the efficiency of

the memory, we should reduce θ as far as is possible, so that

1. we approach a collinear geometry, which maximizes the overlap of the signal

and control pulses, and

2. the spin wave generated by the storage process overlaps well with the optimal

spin wave mode for retrieval.

An heuristic choice of θ that satisfies these desiderata is that shown in part (b)

of Figure 6.4, with ks cos θ = kc. The signal and control wavevectors are close to

parallel, but they are arranged so that the spin wave momentum κ is orthogonal

to kc. This way, when the control field direction is reversed for backward retrieval,

no phase mismatch is introduced and the signal field emerges at an angle θ with

respect to the read out control pulse, as shown in part (a) of Figure 6.5. This choice

for θ assumes that the storage state is more energetic that the ground state, so

that δk is negative, and ks > kc. We also consider the possibility that the storage

state lies energetically below the ground state. In many systems the ground state is

prepared artificially by optical pumping (see §10.12 in Chapter 10), and it is then

quite feasible to select the ground state so that ω1 > ω3. In this case, δk > 0 and


ks < kc. We then choose θ so that kc cos θ = ks; the geometry is shown in part (b)

of Figure 6.5.

(a) (b)

Figure 6.4 Non-collinear phasematching. (a) the general geometryrequired to phasematch storage and retrieval, when the signal beammakes an angle θ with the control. (b) a geometry that closely ap-proaches the collinear one, while preserving correct phasematching.In the next section we include dispersion, which modifies the lengthof ks and kr

s.

Storage

(a)

(b)

Retrieval

Figure 6.5 Efficient, phasematched memory for positive and neg-ative phase mismatches. We show the beam directions of the con-trol (green) and signal (blue) fields at storage and retrieval, with (a)δk < 0, and (b) δk > 0. Off-resonance, the angles used depend onboth δk and on the material dispersion (see §6.3.1 below).


6.3.1 Dispersion

So far we have used the dispersion relation (6.11), which applies to light propagation

in vacuo. The phase mismatch δk quantifies the spatial phase imparted to the

spin wave with collinear storage calculated using only this vacuum dispersion. But

even in the case of degenerate ground and storage states, the spin wave is not

always entirely real, as discussed in the previous section (see part (b) of Figure 6.3).

This spatial phase arises dynamically, and has its origin in the material dispersion

experienced by the signal field as it propagates through the ensemble. The control

field is unaffected, since it couples states |2〉 and |3〉, both of which are unpopulated.

But the signal field couples the populated ground state |1〉 to state |2〉, and therefore

it propagates subject to an augmented refractive index. This effect can be described

by incorporating a dispersive term into our definition for the magnitude of the signal

field wavevector,

ks −→ kd = ks − kdisp, (6.23)

where, working in ordinary units, we define

kdisp =|κ|2∆|Γ|2

=dγ∆

(γ2 + ∆2)L. (6.24)

This definition is motivated by inspection of the adiabatic storage kernel (5.77)

derived in Chapter 5, which includes an exponential factor with an exponent whose

imaginary part contains the spatial phase kdispz. The form of kdisp is consistent with

the refractive index associated with an absorptive resonance at ∆ = 0. Note that on


resonance, kdisp vanishes, and the signal field wavevector assumes its vacuum value.

At large detunings, kdisp ∝ 1/∆, so it is generally small in the Raman limit, but not

negligible.

Incorporating the dispersive phase into the signal wavevector allows us to use

the phasematching scheme outlined in the previous section to compensate both for

non-degeneracy of the ground and storage states and for material dispersion in the

ensemble. By eliminating even this latter dynamical phase, we essentially render

the spin wave real, so that the efficiency of backward retrieval (albeit at a small

angle to the z-axis) is equal to the storage efficiency, which is optimal. And with all

spatial phases removed in this way, the optimization of storage alone is the same as

the optimization of storage with backward retrieval, and so it is only necessary to

perform the former, simpler optimization.

6.3.2 Scheme

Including material dispersion, then, the phasematching scheme is summarized by

the following choice for θ,

θ = cos−1 r ; r = maxkdkc,kckd

, (6.25)

where kc = ωc/c, and kd is given by (6.23). This formula holds for arbitrary values

of δk, and for all detunings.

Provided that θ is sufficiently small, we expect that many of the results pertaining

to the optimization of collinear storage remain approximately valid. Our general


strategy is to use the one dimensional theory of Chapter 5 — in particular the

adiabatic storage kernel (5.77) — to find the optimal temporal input mode, and

then implement the above phasematching scheme by introducing the small angle θ

between the signal and control fields.

Note that this scheme is experimentally appealing for three reasons. First, it

allows for storage with non-degenerate ground and storage states, so the signal and

control fields can be spectrally filtered. Second, introducing an angle between the

signal and control beams makes it possible to spatially filter the signal from the

control, with a pinhole or an optical fibre tip, for example. Third, the effect of

unwanted Stokes scattering can be eliminated. This is because any spin wave exci-

tations produced by Stokes scattering — the process shown in part (b) of Figure 4.3

in Chapter 4 — will not have the same momentum as those produced by absorption

of the signal field. At read out, we choose the direction of the control pulse so that

retrieval of the stored signal field is correctly phasematched. This same control pulse

could potentially drive anti-Stokes scattering, converting the unwanted excitations

into noise photons with the same frequency as the retrieved signal. But this process

will generally not be correctly phasematched, and is therefore greatly suppressed.

All these features of the scheme constitute a compelling case for the utility of a

non-collinear phasematched memory protocol.

Here we should remark that this type of phasematching scheme produces a spin

wave with a larger momentum than arises from collinear storage, since |κ| ≥ |δk|. If

the atoms in the ensemble are free to move — for instance if the storage medium is

6.4 Full Propagation Model 213

a warm atomic vapour — then any large spatial frequencies in the spin wave may be

washed out as atoms diffuse over the phase fronts of the spin wave. Therefore our

phasematching scheme may be more susceptible to decoherence, if D > λκ = 2π/|κ|,

where D is the distance over which the atoms diffuse during the storage time (see

§10.5 in Chapter 10). This problem does not affect solid state memories, which are

appealing for this and many other reasons.

To investigate the efficacy of our phasematching scheme, we should consider the

effect of walk-off between the signal and control fields as they propagate, since when

θ 6= 0 their paths inevitably diverge. In the next section we present the results of

numerical simulations that account both for walk-off and diffraction. These vindicate

the scheme (6.25), and they show, as might be expected, that the beams should be

loosely focussed, so as to maximize the overlap of the pulses.

6.4 Full Propagation Model

We model the propagation of the signal and control fields in three dimensions: two

spatial dimensions, and time. The signal field propagates along the z-axis, and the

control field is launched at an angle θ to the z-axis, in the (x, z)-plane, intersecting

the signal beam in the centre of the ensemble. We consider that both fields are

Gaussian beams — defined below — focussed at the centre of the ensemble, as

depicted in Figure 6.6. The equations of motion for the interaction are given in (4.52)

at the end of Chapter 4. For numerical convenience, we work in normalized units that

render all quantities dimensionless, and as close as possible to ∼ 1. We summarize


the normalizations we employ below; they differ slightly from the normalized units

used in the preceeding analytic treatments.

1. The z coordinate is measured in units of L, and the magnitudes ks, kc, are

measured in units of 1/L. We define the z coordinate so that it runs from −12

to 12 , so that the centre of the ensemble lies at z = 0. This definition is useful

for parameterizing the control field.

2. The x coordinate is measured in units ws, where ws is the transverse beam-

waist of the signal field. That is, at z = 0, the transverse signal field amplitude

is a Gaussian e−(x/ws)2in ordinary units, and by definition e−x

2in normalized

units. To capture all relevant dynamics, x runs from −3 to 3.

3. The time coordinate τ , which as usual is the retarded time τ = t − z/c, is

measured in units of Tc, the duration of the control field. This is a departure

from the time normalization used previously, in which times were measured

in units of 1/γ. With this new normalization, the control field has roughly

unit duration, as does the signal field (since the optimal signal pulse is gen-

erally comparable in duration to the control). All relevant dynamics are then

captured by considering times from τ = −3 up to τ = 3, or thereabouts.


The equations of motion, written using these normalized units, are given by

(iα2

2ks∂2x + ∂z

)A = −

√dγP,

∂τP = −ΓP +√dγA− iΩB,

∂τB = −iΩ∗P. (6.26)

As in all of our previous analyses, we have dropped the Langevin noise operators

associated with spontaneous emission and decoherence, since they have no effect

on the efficiency of the memory. The normalization factor α = L/ws is the aspect

ratio of the ensemble, which appears to correctly account for the difference in units

used for the x and z coordinates. Note that the amplitudes A, P and B are not

the averaged quantities defined in (5.1) at the start of Chapter 5. They are closer

to those quantities given in (4.7), (4.42) and (4.43) in Chapter 4, which depend

on the transverse spatial coordinates ρ, as well as τ and z. In fact, since we have

dropped any dependence on the y-coordinate, we have implicitly averaged over this

dimension. But this has no effect on the equations.

6.4.1 Diffraction

The transverse derivative ∂2x in (6.26) describes the diffraction of the signal field

as it propagates. Its importance in the dynamics is set by the ratio of α2 to ks.

Although both of these are generally large quantities, their ratio can be small, or

large, depending on how tightly the signal is focussed: if ws is small, diffraction


Figure 6.6 Focussed beams. The signal (blue) and control (green)beams are cylindrically symmetric Gaussian beams focussed at thecentre of the ensemble, with a small angle between them.

can become significant. In fact, the signal field Rayleigh range, which quantifies the

length of the region over which the focussed signal beam is well-collimated, is given

in our normalized units by zs = ks/(2α2). As expected, the contribution from the

diffractive term becomes significant when the Rayleigh range falls below 1, which is

the length of the ensemble.

6.4.2 Control Field

Walk-off between the signal and control field appears through the spatial depen-

dence of the control field Ω. To construct the correct expression, consider first the

representation of a pulse with a Gaussian transverse profile, and also a Gaussian


temporal profile, propagating along the z-axis,

Ω(x, τ, z) =Ωmax

W (z/zc)exp

[i

R(z/zc)− 1W 2(z/zc)

](x

wc

)2

− i tan−1

(z

zc

)×e−τ2

.

(6.27)

Here zc = kcw2c/(2α

2) is the Rayleigh range of the control field, with wc the beam-

waist of the control, measured in units of ws. The structure of (6.27) is well known

in optics, since the Gaussian transverse profile arises naturally as the lowest order

mode supported by the confocal cavity in a laser. When a laser beam of this form

is directed through a lens, it retains its Gaussian profile, but its width narrows as it

approaches the focal plane, and then widens out aftwerwards. The functions W and

R parameterize the beam size and the radius of curvature of the field phase fronts,

and are given by

W (z) =√

1 + z2,

R(z) = z

(1 +

1z2

). (6.28)

The focus lies at z = 0, at which point R −→ ∞ and W = 1. The term involving

tan−1 is known as the Gouy phase; its effect on the memory efficiency is negligible,

but we include it for completeness. Direct substitution shows that (6.27) indeed

satisfies the paraxial wave equation

(iw2c

4zc∂2x + ∂z

)Ω = 0. (6.29)


When the control propagates at an angle θ to the z-axis, the control amplitude is de-

scribed by a similar expression to (6.27), except that we rotate the (x, z) coordinates

by the angle θ. We should be careful to apply this rotation with t held constant,

not τ , and we must take account of the different units used for z and x. The correct

transformation is found by making the replacements

x −→ x′ = cos(θ)x+ α sin(θ)z,

z −→ z′ = cos(θ)z − sin(θ)α

x

and τ −→ τ ′ = τ +z − z′

c. (6.30)

The factors of α ensure that the units of x and z are interconverted as they should

be, and the modification to τ represents the change in the apparent velocity of the

control pulse in the reference frame of the signal field.

6.4.3 Boundary Conditions

The boundary conditions must be specified in much the same way as in the previ-

ous one dimensional analyses. One additional feature is the dependence on the x

coordinate. The boundary conditions associated with this new degree of freedom

are simply that all the quantities A, P and B vanish as |x| −→ ∞. In practice, this

is achieved by fixing the value of the signal amplitude A to zero at x = ±3 (i.e.

at the edges of the region covered by the numerics). We use Chebyshev spectral

collocation to treat the spatial derivatives when solving the system (6.26), and we


build this boundary condition directly into the differentiation matrices; the method

is explained in Appendix E. Fixing A at these boundaries also fixes P and B, since

the interaction is local: there can be no atomic excitation where there is no input

light. To model the storage process, we launch the signal field along the z-axis with

a Gaussian transverse profile at the entrance face of the ensemble,

Ain(x, τ) = A(z = −1

2 , x, τ)

= exp

[i

R(−1

2/zs) − 1

W 2(−1

2/zs)]x2

φ1(τ),

(6.31)

where φ1(τ) is the optimal input mode for storage alone, found using the one dimen-

sional theory of the previous Chapter — we use the adiabatic kernel (5.77), so that φ1

is given by (5.78). As described above, the phasematching scheme should eliminate

the spatial phase of the spin wave, so that the optimal mode for collinear storage

alone should be close to optimal for phasematched storage and retrieval. Note that

the absolute magnitude of the signal field is irrelevant, because the memory interac-

tion is linear. This boundary condition is also incorporated into the differentiation

matrices we use to solve the dynamics numerically.

For the storage process, both the spin wave B and the polarization P are initially

zero,

Bin(x, z) = B(τ −→ −∞, x, z) = 0, Pin(x, z) = P (τ −→ −∞, x, z) = 0. (6.32)

We use an RK2 method for the time-stepping, and these boundary conditions are

trivially implemented by zeroing the vectors representing B and P at the collocation


points on the first iteration. See Appendix E for a description of the RK2 method,

and how it is used in combination with spectral collocation to arrive at a solution.

6.4.4 Read out

To model the read out process, we solve the equations of motion once more, this time

with no incident signal field, and a pre-existing Raman coherence determined by the

storage process. We are only able to model the build up of the signal field along a

specific direction, the z-axis, and so we must re-orient our coordinate system so that

the z-axis points along the direction in which the retrieval process is phasematched.

This is the direction of krs in Figure (6.4). The angle through which we rotate our

coordinate system is therefore the angle between ks and krs in Figure (6.4), which is

given by

θ′ = −2 sin−1

kc sin(θ)√k2d + k2

c − 2kckd cos(θ)

=

π if kc > kd

2θ − π if kd > kc

. (6.33)

The first result holds for general values of θ, while the second result applies to the

case where θ is chosen according to the scheme (6.25).

At read out, the signal field is initially zero,

Arin(xr, τ r) = 0, (6.34)


as is the polarization,

P rin(xr, zr) = 0. (6.35)

Here xr, zr are the rotated coordinates describing the retrieval process,

xr = cos(θ′)x+ α sin(θ′)z,

zr = cos(θ′)z − sin(θ′)α

x. (6.36)

As in (6.30), the factors of α appear to interconvert the units of x and z.

The initial spin wave at read out is set by the spin wave generated by the storage

process, but we must be careful to include any spatial phase factors arising from

the above coordinate transformation. When the signal and control fields are not

collinear, the definition of the slowly varying coherence σ13 (see (4.22) in Chapter

4) must be modified accordingly,

σ13(r) = σ13(r)eiω13t+i(ks−kc).r. (6.37)

Here the argument r indicates that the coherence refers to an atom at that position

in the ensemble. The boundary condition for B is found by equating the coherences

at the end of storage and the start of read out,

σr13,in(rr) = σ13,out(r). (6.38)

The spin wave amplitudes are built from the slowly varying coherences. Dropping


some unimportant temporal phase factors, we therefore obtain

Brin(rr) = Bout(r)ei[(kr

s−krc).r

r−(ks−kc).r]. (6.39)

Writing this boundary condition out explicitly in terms of the retrieval coordinate

system gives the relation

Brin(xr, zr) = Bout(xrc′ − s′αzr, zrc′ + s′xr/α)× (6.40)

exp

i[(ks − kcc)(1− c′)− kcss′

]zr + i

[(ks − kcc)s′ − kcs(1 + c′)

]xr/α

,

where c and s denote cos(θ) and sin(θ), respectively, and c′, s′ are equal to cos(θ′)

and sin(θ′). Note that in the absence of dispersion, with ks = kd, the phase factor

identically vanishes; it is removed by the phasematching scheme. When dispersion is

significant, the slowly varying envelope Bout itself contains a spatial phase variation.

The phasematching scheme is then designed so that the exponential factor in the

second line of (6.40) cancels this spatial phase, rendering Br as smooth as possible,

for efficient retrieval.

6.4.5 Efficiency

The efficiency of the memory is calculated in two stages. First, we simulate storage

and evaluate the storage efficiency, which is given by

ηstorage =

∫∞−∞

∫ 1/2−1/2 |Bout(x, z)|2 dz dx∫∞

−∞∫∞−∞ |Ain(x, τ)|2 dτ dx

. (6.41)

6.5 Results 223

We then simulate retrieval, and we calculate the retrieval efficiency,

ηretrieval =

∫∞−∞

∫∞−∞ |A

rout(x

r, τ r)|2 dτ r dxr∫∞−∞

∫ 1/2−1/2 |B

rin(xr, zr)|2 dzr dxr

. (6.42)

The total memory efficiency is then given by ηcombined = ηstorage × ηretrieval.

6.5 Results

In Figures 6.7 and 6.8, we present the results of simulations performed to examine the

effectiveness of our phasematching scheme. We simulated both resonant EIT storage,

and also off-resonant Raman storage. Each simulation was run twice, once with a

tightly focussed control field, with wc = 1, and once with a loosely focussed control,

with wc = 2. In the latter case the energy in the control pulse was quadrupled, so

that the intensity of the control was the same in both cases.

Figure 6.7 shows the angle θ at which the combined efficiency of storage and

retrieval is largest, over a range of values of the phase mismatch δk. The numerical

solutions generally bear out the analytic prediction (6.25) of our phasematching

scheme. But the agreement with our prediction is much better when the control is

loosely focussed. This is to be expected, since transverse walk-off is less important

when the control is wider than the signal, and since a wider control diffracts less, so

that its intensity is more homogeneous over the ensemble. For EIT, collinear storage

with θ = 0 is only optimal when δk = 0, since there is no dispersion on resonance.

But for the Raman memory, collinear storage is optimal when δk ∼ −2, when the

6.5 Results 224

−4 −3 −2 −1 0 1 2 3 40

2

4

6

8x 10

−3

θ (

rad

ian

s)

−4 −3 −2 −1 0 1 2 3 40

2

4

6

8x 10

−3

θ (

radia

ns)

(a)

(b)

EIT

Raman

Figure 6.7 Effectiveness of our phasematching scheme. We plotthe angle θ at which the combined efficiency of storage followed byretrieval is optimized. The blue solid line shows the prediction of ourphasematching scheme (6.25). The black filled circles correspond totight control focussing, with wc = 1; the red open diamonds corre-spond to loose control focussing with wc = 2. We plot the results fora typical EIT protocol in (a): we set γ = 1, d = 30 and ∆ = 0, inunits normalized as described in §6.4 above. In (b) we present equiv-alent results for a Raman protocol d = 300, γ = 0.1 and ∆ = 15.The bandwidth of the control is 10 times larger, with respect to thenatural atomic linewidth γ, in this latter protocol. To make up forthe increased detuning in the Raman protocol, the optical depth isalso much larger. In both protocols, the control field amplitude wasgiven by Ωmax = 5.

storage state lies significantly above the ground state in energy. This is because the

material dispersion alters the signal field wavevector, kd 6= ks, so that even with

degenerate ground and storage states, the spin wave acquires a spatial phase.

In Figure 6.8 we plot the optimal memory efficiencies obtained from the numer-

ical simulations alongside the analytic predictions for the best collinear efficiencies,

again over a range of values of δk. The efficiencies obtained using the phasematching

scheme (6.25) are generally very close to the best efficiencies achieved in the sim-

6.5 Results 225

−4 −2 0 2 4

effic

ien

cy

0

0.2

0.4

0.6

0.8

1

−4 −2 0 2 40

0.2

0.4

0.6

0.8

1

−4 −2 0 2 4

effic

ien

cy

0

0.2

0.4

0.6

0.8

1

−4 −2 0 2 40

0.2

0.4

0.6

0.8

1

analytic

collinear backwards

collinear forwards

phasematched

)b()a(

(c) (d)

Narrow EIT Wide EIT

Narrow Raman Wide Raman

Figure 6.8 Comparing phasematched and collinear efficiencies.Plots (a) and (b) contain the results for the EIT protocol, with tightand loose focussing, respectively. Plots (c) and (d) show equivalentresults for the Raman protocol. The parameters used are the sameas those used in produciing Figure 6.7. The solid lines represent theefficiencies obtained if the phasematching scheme (6.25) is used. Theblack filled circles are the best efficiencies achievable. The green dot-ted lines show the optimal efficiencies predicted for collinear storagewith backward retrieval, calculated using the one dimensional adia-batic kernel (6.21). The dashed red lines show the efficiency achievedby collinear storage with forward retrieval, where the input signalpulses were shaped using the adiabatic kernel (6.9), where this im-proved the efficiency.

ulations. These efficiencies are greatest when θ = 0, since at these points walk-off

between the signal and control is eliminated. As δk is increased, so does the angle θ

required for optimal efficiency, and walk-off therefore reduces the memory efficiency.

But the efficiency falls only slowly with increasing δk. With loose focussing it is

possible to exceed the optimal collinear efficiency when δk is large enough, using

either EIT or Raman storage. This demonstrates what we sought to show with our

simulations, that even including the effects of diffraction and walk-off, non-collinear

storage and retrieval, with proper phasematching, is preferable over collinear stor-

6.5 Results 226

age, with either forward or backward retrieval. The lower efficiencies observed for

tight control focussing confirms that diffraction and walk-off can be detrimental, but

provided sufficient laser energy is available, it is possible to reduce their effects by

simply widening the control beam waist.

The red dashed lines in Figure 6.8 are the result of simulating collinear storage

with forward retrieval, using the full numerical propagation model. For the EIT

protocol, we used the one dimensional adiabatic kernel (6.9) to predict the optimal

input profiles, and these input pulse shapes performed well — better than any other

pulse profiles we tried. This at least confirms the utility of the one dimensional

theory in this context. However, for the Raman protocol, this optimization did not

produce the best pulse profiles. In fact, optimal pulse profiles for backward retrieval

worked better, even though the excitation was retrieved in the forward direction.

So these input profiles were used to produce the red dashed lines in parts (c) and

(d). The reason for the failure of the forward retrieval optimization is that the

Raman protocol is more sensitive to the control intensity than is EIT. The control

intensity is highest in the centre of the ensemble, where it is focussed; its intensity

falls off towards the exit face of the ensemble. But as noted in §6.1.1, optimizing for

forward retrieval tends to shift the bulk of the spin wave towards the exit face. When

diffraction is introduced in the full model, this optimization is no longer beneficial,

since it shifts more of the spin wave to where the control intensity is lower. This

explains why the pulses optimized for backward retrieval performed better in the

simulations of forward retrieval for the Raman protocol. If the control beam waist

6.5 Results 227

is increased above 2.4, diffraction effects become small enough that the optimization

for forward retrieval does indeed produce better results than the backward retrieval

optimization, for both the EIT and Raman protocols.

Comparison of Figures 6.7 and 6.8 reveals that the efficiency falls off sharply in

the region kd > kc, to the left of the point at which θ = 0. That is, the EIT efficiency

falls sharply for δk < 0, and the Raman efficiency falls quickly as δk is reduced below

around −2. The efficiency remains high even for large angles when kd < kc. This

is easily explained by considering the geometry in part (b) of Figure 6.4. When

kd < kc, the retrieved signal field propagates exactly backwards with respect to the

direction of the input signal beam, since θ′ = π (see (6.33)). Therefore the region

over which the retrieved signal experiences gain overlaps precisely with the region

in which the spin wave is deposited during the storage process. We could say that

the ‘retrieval overlap’ is high, and the efficiency is correspondingly large. On the

other hand, when kc < kd, θ′ 6= π, and the retrieved signal field propagates at

an angle to the direction defined by the input signal field. The retrieval overlap

suffers as θ is increased, and this explains the sharp drop in memory efficiency in

this regime: although the memory is correctly phasematched, the atomic excitations

are not distributed favourably for efficient retrieval.

The foregoing discussion suggests that, as a rule, positive phase mismatches are

preferable to negative ones, although dispersion complicates things slightly. That is

to say, for efficient phasematched retrieval, it is generally better to arrange for the

ground state |1〉 to have a higher energy than the storage state |3〉. Although this

6.6 Angular Multiplexing 228

may be rather non-standard, when the ground state manifold is prepared by optical

pumping, it is not problematic.

We have succeeded in showing that proper phasematching can indeed boost the

efficiency of retrieval from a quantum memory. In the next section we describe how

the angular selectivity of phasematching offers the possibility of storing multiple

modes in a single ensemble.

6.6 Angular Multiplexing

There is a sense in which an ensemble memory represents a degree of ‘overkill’ if

just a single photon is to be stored. The number of atoms is many times larger

than the number of input photons, so there is an enormous redundancy built into

the memory. A more efficient use of resources would aim to store multiple input

pulses; in this way a single physical memory allows for highly parallel information

processing. Several schemes that make use of the idea of multimode memories have

already been proposed [17,38,153,154], in connection with both quantum computing

and quantum repeaters. A general difficulty with multimode storage in an ensemble

is the possibility that the control field used for storing one signal pulse, may also

retrieve another signal pulse that was previously stored. Phasematching provides a

way to suppress this kind of unwanted coupling.

We consider storing multiple pulses in a single ensemble by changing the angle θ

between the signal and control for each input pulse. After a pulse has been stored,

momentum conservation picks out a single, unique direction, in which the retrieval


process is phasematched, as shown in part (a) of Figure 6.4. As long as the control

field used to store a subsequent pulse propagates in a different direction to this, the

previously stored pulse will not be retrieved. By successively scanning the angle

of either the signal or control field, it is possible to store many pulses in a single

ensemble. And it is also possible to selectively retrieve just one of the stored pulses,

by aligning a retrieval control pulse with the appropriate direction for the chosen

mode.

The directional selectivity afforded by phasematching is demonstrated using our

numerical model in part (a) of Figure 6.9. Here we simulated EIT storage, with θ

chosen using the phasematching scheme (6.25). Upon retrieval, however, we rotated

away from the optimal angle by θd. In the case δk = 0, we have θ = 0 and the spin

wave momentum vanishes, so that all retrieval directions are phasematched. The

efficiency is therefore only limited by walk-off, so it remains high over a large range

of deviation angles θd. But when we set δk = 1, phasematching selects a unique

retrieval direction, and the efficiency falls off quickly as θd is increased. This shows

how it is possible to ‘switch off’ retrieval from a particular spin wave by tuning the

angle of the control.

6.6.1 Optimizing the carrier frequencies

Angular multiplexing requires that we implement storage with a range of angles θ

between the signal and control fields. Now, for a given pair of frequencies ωc, ωs for

the control and signal fields, there is only one angle that satisfies the phasematching


(b)

0 0.02 0.04 0.06 0.08 0.10

0.2

0.4

0.6

0.8

1

θ (radians)

effic

ien

cy

(a)

d

Figure 6.9 Angular multiplexing. In (a) we plot the dependence ofthe memory efficiency on the deviation angle θd, for EIT storage withδk = 0 (dotted line) and δk = 1 (solid line). We used a wide controlbeam, with wc = 9; all other parameters are the same as those usedfor the EIT protocol in Figures 6.7 and 6.8. The efficiency is onlylimited by walk-off when δk = 0, but when δk = 1 phasematchingrestricts the range of angles over which efficient retrieval is possible.In part (b) we illustrate the principle of multimode storage by angularmultiplexing. The number of pulses that can be stored is given byN = ∆θ/δθ, where ∆θ is the largest angle, permitted by walk-off,for which efficient retrieval is possible, and where δθ is the smallestangle separating two modes, such that storage of one mode does notresult in the accidental retrieval of another previously stored mode.Inspecting part (a), we see that ∆θ ∼ 0.04, and δθ ∼ 0.02, so N ∼ 2.

scheme (6.25). As our simulations showed, this angle provides the optimal memory

efficiency. To retain this optimum as θ is scanned, we should change the signal and

control frequencies accordingly. For example, in the case that ωs < ωc (i.e. δk > 0),

the phasematching scheme imposes the relationship ωs = ωc cos(θ) + ckdisp between

θ, ωs and ωc. In the Raman protocol, it is possible to perform this kind of fine

tuning on the frequencies, since the efficiency changes slowly with the detuning. Of

course, EIT is defined by the condition that ∆ = 0, and so for this protocol there is

no freedom to tune the frequencies in this way.


6.6.2 Capacity

How many modes can be stored in such a multiplexed memory? One constraint is

that it must be possible to resolve one retrieved signal field from another. Therefore

the minimum angle δθ by which the direction of phasematched retrieval must change

between two modes must be greater than the angular divergence of the retrieved

signal field. This angular divergence is set by Fraunhofer diffraction, which yields

the condition δθ & λs/ws — the angle between consecutive modes must exceed the

ratio of the wavelength and beam waist of the signal field.

Equally important is the requirement that there is no ‘cross-talk’ between neigh-

bouring modes — that δθ is large enough that retrieval of an undesired mode is

well-suppressed by its momentum mismatch. An estimate of the magnitude of the

momentum mismatch is given by δθks (see Figure 6.10). This momentum mismatch

is directed approximately perpendicular to the z-axis, so we can approximate the

phase accumulated by the mismatch as δθksws (working in ordinary units). For good

suppression of unwanted retrieval, this phase should exceed 2π, roughly, and from

this we again derive the condition δθ & λs/ws. Therefore the multimode capacity of

a multiplexed memory is bounded by the number of diffraction limited modes that

can be addressed. For the example in Figure 6.9 (a) this condition implies δθ ∼ 0.01,

which appears to be broadly correct.

The number of modes that can be stored is given by N = ∆θ/δθ, where ∆θ is the

largest angle at which efficient retrieval is possible, as limited by walk-off between

the signal and control fields. We approximate ∆θ as the angle subtended by the


Figure 6.10 Minimum momentum mismatch. Two signal fieldswith momenta k1 and ks differ in their directions by a small angleδθ. When retrieval for one field is phasematched, retrieval for theother is not, with the momentum difference shown in red, whichpoints essentially across the ensemble. It’s magnitude is roughly δθks(ignoring any dispersion that may slightly alter the lengths of k1, ksaway from ks). The total phase accumulated through propagationacross the ensemble, of width ws, is then roughly δθksws. Whenthis phase is larger than 2π, phasematching is sufficiently selective toeffectively suppress any accidental retrieval of stored excitations.

control waist across the length of the cell, ∆θ ∼ wc/L (in ordinary units). The

multimode capacity is then given by N ∼ wcws/λsL. This is essentially equal to

the geometric mean of the Fresnel numbers Fs, Fc associated with the signal and

control fields,

N ∼√FsFc, where Fs,c =

w2s,c

Lλs,c, (6.43)

where we used the fact that the control wavelength λc is generally rather close to

the signal wavelength. The Fresnel number of a beam quantifies the number of

diffraction limited modes it supports, so this result has the interpretation that the

number of modes that may be stored in a multiplexed memory is given by the average

(in the sense of the geometric mean) number of modes supported by the optical fields

used. For the example used in Figure 6.9 (a), the formula (6.43) predicts N ∼ 3,

and this agrees reasonably well with what might be estimated ‘by eye’ (see caption

of Figure 6.10).


A high capacity multiplexed memory requires wide beams, with Fs,c 1. In

addition, our numerics showed that wc should be at least 2 (better 3) times larger

than the signal beam waist. The feasibility of this kind of memory depends on the

availability of energetic lasers that can maintain a high intensity, even when very

loosely focussed. But laser technology has advanced sufficiently that this is certainly

possible for beams with a width on the order of centimeters, giving multimode

capacities on the order of 100.

In the next chapter we continue to investigate multimode storage, both in the

Raman and EIT protocols covered so far, and in a number of alternative memory

protocols.

Chapter 7

Multimode Storage

At the end of the last Chapter we described a way to store multiple pulses in a single

ensemble memory. We considered scanning the propagation direction of either the

control or the input signal beams: each independently stored pulse represented a

separate spatial mode, characterized by the direction of its wavevector. However, in

many practical situations, it is desirable to fix the alignment of all components in an

optical system. It may therefore be more useful to consider using a different degree

of freedom to define a basis of modes. As discussed at the start of Appendix C, we

might also consider polarization or spectral modes. And in fact since polarization is

only two dimensional, any such consideration of multimode storage must focus on

spectral modes. Equivalently, we might talk about temporal modes, since time and

frequency are Fourier conjugates; they are two interchangeable representations of

the same underlying space. This space can be thought of as the space of all possible

input profiles for the signal field. The method used in the previous Chapters to

7.1 Multimode Capacity from the SVD 235

optimize the storage efficiency of a quantum memory is immediately useful in this

connection. The SVD of the storage kernel provides a ‘natural’ basis for the space

of signal profiles: the input modes φk. In this Chapter we apply the SVD to the

study of multimode storage: we show how to optimize and quantify the multimode

capacity of the Raman, EIT, CRIB and AFC memory protocols. The following

treatment expands upon the account published in Physical Review Letters [155].

7.1 Multimode Capacity from the SVD

Suppose that we would like to store a ‘message’ in a quantum memory. We must

select an alphabet in which to write the message. If we encode the message in an

optical signal, each letter in the alphabet is represented by a mode of the optical

field. The multimode capacity of the quantum memory is the size of the largest

alphabet we can use for our message compatible with efficient storage. In general,

this capacity will depend on the particular alphabet chosen: some modes of the

optical field cannot be stored at all. To take an absurd example, we certainly could

not encode any letters as X-rays! The multimode capacity also depends on the

brightness of the signals used to encode the message. For instance, when two similar

modes are bright, we can distinguish them and encode two letters, whereas when

they are dim we can encode only a single letter (see Figure 7.1).

In a quantum communication protocol, security requires that we encode each

letter using just a single photon (see §1.5 in Chapter 1). This means that there can

be no overlap whatsoever between the modes comprising the alphabet. That is, we


Bright

A B

Dim

Figure 7.1 Bright overlapping modes are distinct. Two “letters”A and B are encoded with similar optical pulses. When they arebright, the letters can be distinguished; not so when their intensityis reduced.

must find an orthonormal basis of modes. The multimode capacity of a quantum

memory in this context is therefore the number of orthonormal modes that can

be stored. It remains true that this capacity depends on the choice of basis. For

example, it may not be the same for time bin modes as it is for frequency modes

(see §C.1 in Appendix C). What, then, is the optimal basis? Which encoding

maximizes the capacity? It is the basis formed by the input modes φk determined

from the SVD of the storage kernel K. That is, for any number N , storage of the

first N input modes φ1, . . . , φN is more efficient than the storage of any other set

of N orthonormal modes. Each input mode φk is stored with efficiency ηk = λ2k. In

the optimal situation of phasematched backward retrieval without dispersion, the

retrieval efficiency is also equal to ηk (see the discussion towards the end of §6.2 in

6). The total memory efficiency for the kth mode is then η2k = λ4

k. The multimode

capacity is given by the width of the ‘distribution’ of efficiencies (see Figure 7.2).

Below we consider two ways of quantifying this width.


Figure 7.2 Visualizing the multimode capacity. The total memoryefficiency — the efficiency of storage followed by retrieval — for thekth mode is η2

k = λ4k. These efficiencies define a decaying distribu-

tion, and the multimode capacity is related to the width, or ‘variance’of this distribution. The Schmidt number quantifies this width, buttakes no account of the absolute efficiencies. If we introduce a thresh-old efficiency ηthreshold, we can define an operational measure N forthe multimode capacity by considering the number of modes withmemory efficiency greater than the threshold.

7.1.1 Schmidt Number

The Schmidt number S is defined as follows [156,157],

S =1∑∞

k=1 η4k

, where η2k =

η2k∑∞

j=1 η2k

. (7.1)

The η2k are defined so that they sum to unity, turning them into effective probabili-

ties. In fact, if we consider that these probabilities are the eigenvalues of a quantum

mechanical density matrix, describing a statistically mixed state, then S is the in-

verse of the purity of this state [158]. From the definition (7.1) it’s clear that S = 1

if there is just one non-zero singular value. In general, if there are N equal singular

values, with all others vanishing, then S = N , so the Schmidt number does indeed

count the number of modes that may be stored with equal efficiency. When the

singular values are not equal, but instead decay in magnitude smoothly as the index


k increases, S still provides a sensible measure of the number of coupled modes. The

normalization of the η’s makes S independent of the overall efficiency of the memory:

it measures how many modes are involved in the memory interaction with compa-

rable strength, independent of what that strength is. This makes S an independent

figure of merit for characterizing the performance of a quantum memory. One might

choose to optimize the efficiency (i.e. the optimal efficiency) of a quantum memory,

or the Schmidt number, or both. In most situations, however, it is more useful to

quantify the ability of a memory to store multiple modes efficiently, so that a metric

that combines these two measures is desirable. We introduce such a metric in the

next section. But before abandoning the Schmidt number entirely, we observe that

it is possible to develop an intuition for the value of S — at least whether or not

it is large — by looking at the form of the Green’s function describing the storage

interaction (see Figure 7.3). The reason is that the contribution of a single mode to

the Green’s function necessarily takes the form of a product of two functions; each

with contours orthogonal to the other’s: one function represents the input mode;

the other the corresponding output mode. Now, the Schmidt number is independent

of the basis of modes we consider, because the singular values are. We are therefore

always free to choose a basis of modes comprised of localized pulses. Each mode

is orthogonal to the others if the pulses do not overlap. The product of two such

modes produces a contribution to the Green’s function in the form of a rectangular

‘blob’, with sides parallel to the coordinate axes, and rounded edges if the pulses

have smooth edges. Counting the number of such blobs required to make up the


Green’s function provides an estimate of the number of modes required to construct

it. Any feature of the Green’s function that traces out a curve is ‘multimode’, as is

any feature that runs at an angle to the coordinate axes. The Schmidt number is

sometimes useful because it is amenable to this kind of dead reckoning, as we will

see in §7.3.2 below.

(a) (b)Single mode Multimode

Figure 7.3 The appearance of a multimode Green’s function. In-spection of the contours of the Green’s function describing a memoryinteraction offers insight into the degree to which it is multimode, asquantified by the Schmidt number S. By reconstructing the approx-imate shape of these contours using non-overlapping rectangles, onecan estimate the number of modes of which it is composed. In (a)we show an archetypal single mode Green’s function, which looks likean isolated ‘hill’; a single elliptical contour is drawn. The contour isroughly approximated by just a single rectangle, indicating that onlyone mode contributes to its structure. In (b) we illustrate how curvedor angled contours admit a decomposition involving more modes.

7.1.2 Threshold multimode capacity

To be more quantitative, we introduce a threshold efficiency ηthreshold that allows

us to delineate those modes that are stored and retrieved with acceptable efficiency

from those that are not. We then define the multimode capacity as the largest

number of modes N for which the mean memory efficiency, averaged over all N


modes, exceeds ηthreshold. The average memory efficiency of the first k input modes

is given by

Λk =

∑kj=1 η

2j

k. (7.2)

Then the multimode capacity can be written as

N =∞∑k=1

βk, where βk =

1 if Λk > ηthreshold

0 otherwise

. (7.3)

A more conservative estimate could replace the average efficiency (7.2) with the

minimum efficiency η2k. The results presented below are not qualitatively altered by

making this alternative definition, but as N becomes large, this definition represents

an increasingly pessimistic estimate of the performance of a memory for storing

random ‘words’ drawn from the alphabet defined by the φk. The capacity defined

in (7.3) is a convenient measure, with a clear interpretation.

In the next section, we investigate the scaling of N for various memory protocols.

We set ηthreshold = 70%; this value makes the form of the scaling apparent for

parameters that are tractable numerically, and indeed experimentally. We return

to the one dimensional model used in Chapter 5, since we are able to write down

explicit expressions for the storage kernels in this case. The simulations presented in

the previous Chapter confirm that the one dimensional approximation works well,

provided that the control field is more loosely focussed than the signal field (see §6.5

in Chapter 7).

7.2 Multimode scaling for EIT and Raman memories 241

7.2 Multimode scaling for EIT and Raman memories

In Chapter 5 we showed that the best possible storage efficiency for a Λ-type quan-

tum memory is found from the SVD of the anti-normally ordered kernel (5.27) [133].

We repeat it here for convenience,

KA

(z, z′

)=d

2e−d(z+z′)/2I0

(d√zz′). (7.4)

Here we are using normalized units for the z coordinates. The kernel (7.4) is valid

for both EIT and Raman protocols, in the light-biased limit where the control pulse

is sufficiently energetic (see §5.2.8 in Chapter 5). The eigenvalues of this optimal

kernel are equal to the largest possible values of the storage efficiencies ηk = λ2k,

so we can calculate the multimode capacity N by diagonalizing KA. This can be

done numerically without difficulty, and we present the results of such a numerical

diagonalization in part (a) of Figure 7.4 below. In part (b) we plot the capacity

predicted by the Raman kernel (5.95) introduced in §5.3 of Chapter 5. In both cases

the capacity rises only slowly with the optical depth d. To explain these results, we

briefly return to the treatment given in §5.2.3 in Chapter 5. There we derived an

approximate expression (5.21) for the storage efficiencies, valid in the limit d 1:

ηk = e−4παk/d, (7.5)


where αk is the kth zero of the function J0(2√

2πx). The zeros of J0 are distributed

approximately linearly along the real line (see §15.4 in Chapter XV in G.N. Watson’s

famous treatise [159]). Therefore the αk increase, roughly, with the square of the index

k. Setting αk ∼ qk2 for some constant q, we find that the index of the mode whose

total efficiency η2k falls below ηthreshold is given approximately by

k ∼√ηthreshold

8πq×√d. (7.6)

The multimode capacity N should scale in the same way, so that we expect

N ∼√d, (7.7)

for large optical depths, regardless of the threshold ηthreshold. Inspection of Figure

7.4, which was generated by setting ηthreshold = 70%, indeed reveals a scaling of

N ≈√d/3.

7.2.1 A spectral perspective

The kernel (7.4) is derived by considering the optimal efficiency for absorption into

the excited state |2〉, and the square root scaling of N with d can be understood by

considering the bandwidth of the absorption line associated with an ensemble of two-

level atoms. If each atom has a Lorentzian lineshape with natural linewidth 2γ, the

absorption profile of the entire ensemble is given by exponentiating the single-atom


profile,

F (ω) = exp[− 2dγ2

γ2 + ω2

]. (7.8)

The full width at half maximum (FWHM) of F is given by

∆ω = 2

√2dγ2

ln 2− γ2 ≈ 2γ

√2dln 2

, (7.9)

where the approximation holds for d 1. An estimate of the multimode capacity is

provided by counting the number of spectral modes with FWHM 2γ that ‘fit’ inside

this absorption profile, which procedure yields

N ∼ ∆ω2γ∼√d. (7.10)

We can therefore identify the origin of the square root scaling of N with d as the

scaling of the bandwidth of the absorption of an ensemble with its optical thickness.

This scaling is rather poor: in order to store two modes in such a memory, we must

quadruple the optical depth required for storing a single mode. If this optical depth

were divided up into four separate ensemble memories, we could store four modes

— one mode in each memory — and so the multimode scaling for EIT and Raman

memories is decidedly sub-optimal. The reason is that when the optical depth is

increased, new atoms are added ‘behind’ the old: they absorb at the same frequencies

as the other atoms, and so they improve the absorption at these frequencies, but

they do not provide much coupling at other frequencies — other modes. In the next


section we will show that adding an inhomogeneous broadening to the ensemble

can improve the multimode scaling, by redistributing optical depth over a range of

frequencies. This turns the square root scaling with d into a linear one, consistent

with what one might expect to achieve from operating separate memories in parallel.

And this is essentially the way such broadened protocols work, albeit within a single

physical ensemble.

0 20 40 60 80 1000 100 200 300 400 5000

2

4

6

8

Optimal kernel Raman kernel(a) (b)

Figure 7.4 Multimode scaling for Raman and EIT memories. (a):the multimode capacity found by numerical diagonalization of theoptimal kernel KA in (7.4). This represents the best possible scalingfor both EIT and Raman memories; the capacity scales only with thesquare root of the optical depth. (b): we also show the multimodecapacity found by diagonalizing the Raman kernel (5.95), which isparameterized by the Raman memory coupling C (see §5.3 in Chapter5). This kernel is valid in the far off-resonant limit — though if thecontrol pulse intensity is increased sufficiently, the optimal kernelKA should be used instead. We plot the Raman capacity against C2,since this quantity is proportional to d. The same square root scalingwith ensemble density is evident. Equivalently, if d is held constant,the Raman capacity scales with the square root of the control pulseenergy. We used ηthreshold = 70% in both plots.

7.3 CRIB 245

7.3 CRIB

The CRIB memory protocol is introduced in §2.3.3 in Chapter 2. Storage is achieved

by direct absorption into the excited state |2〉, which is artificially broadened by

application of an external field. Once storage is complete, the excitation is ‘shelved’

by application of a short, bright control pulse, which transfers the excitation to

the storage state |3〉. To implement retrieval, another control pulse transfers the

excitation back to |2〉, and the inhomogeneous broadening is ‘flipped’, so that the

atomic dipoles re-phase and the signal field is re-emitted.

The same considerations regarding retrieval discussed in the previous Chapter

apply to this protocol, just as they do to Raman and EIT memories: retrieval in

the forward direction is inefficient due to re-absorption losses [160], while backward

retrieval is vulnerable to phasematching problems. However, in this protocol, the

control pulse is applied after the signal has been absorbed, so it is possible to

distinguish the two fields temporally. Spectral and spatial filtering is therefore less

important, and so it is feasible to use an ensemble where the states |1〉 and |3〉 are

degenerate. This allows for efficient collinear storage, followed by phasematched

backward retrieval. Since there is no dispersion on resonance, the stored excitation

has no spatial phase, and the retrieval efficiency is equal to the storage efficiency,

with ηcombined = η2storage. In the following, we restrict our attention to this situation,

which is optimal.

7.3 CRIB 246

7.3.1 lCRIB

We first consider lCRIB, in which the broadening is applied longitudinally [84,161,162].

The resonant frequency of the |1〉 ↔ |2〉 transition varies linearly along the z-axis.

Since the control field is only applied after the signal field has been resonantly

absorbed, the equations of motion for lCRIB are given by the system (5.106), with

Ω = 0, and with the spatial variation of the detuning included.

∂zA(z, τ) = −√dP (z, τ),

∂τP (z, τ) = −Γ(z)P (z, τ) +√dA(z, τ), (7.11)

where Γ(z) = 1 − i∆(z), with ∆(z) = ∆0

(z − 1

2

). Here we have returned to the

normalized units of Chapter 5, with all frequencies measured in units of γ, and with

z running from 0, at the entrance face of the ensemble, up to 1, at the exit face. The

width of the applied spectral broadening is ∆0. As usual, we solve these equations

by applying a unilateral Fourier transform over the z-coordinate. Using the formula

(D.27) from Appendix D, we obtain the transformed system

−ikA− 1√2πAin = −

√dP ,

∂τ P = −(1−∆0∂k + i∆0

2

)P +

√dA, (7.12)

7.3 CRIB 247

whereAin is the temporal profile of the incident signal field. Solving the first equation

for A and substituting the result into the second equation yields

(∂τ −∆0∂k) P = −(1 + i∆0

2 + i dk)P + i

√d√

2πkAin. (7.13)

Now if we temporarily define the composite variable s = k + ∆0τ , we can replace

the derivatives on the left hand side with a single time derivative,

∂τ P = −f(s, τ)P + i

√d√

2π(s−∆0τ)Ain, (7.14)

where the partial derivative ∂τ is taken with s held constant, and where we have

defined the function

f(s, τ) = 1 + i∆02 + i d

s−∆0τ. (7.15)

Integrating (7.14) gives the solution

Pout(s) = Pin(s)e−R T−∞ f(s,τ) dτ + i

√d√

2π

∫ T

−∞Ain(τ)e−

R Tτ f(s,τ ′) dτ ′ dτ, (7.16)

where Pout(s) = P (s, T ) is the atomic excitation at the moment τ = T that the

short control pulse is applied, marking the end of the storage interaction. We do not

explicitly model the atomic dynamics induced by the control; we simply assume that

it is sufficiently intense to transfer — effectively instantaneously — all the atomic

excitations into the storage state |3〉, so that Pout −→ Bout. We set Pin = 0, since no

atoms are excited at the start of the storage process. We then perform the integral

7.3 CRIB 248

in the exponent of the integrand in the second term, and convert back from s to k,

to arrive at the following expression for the lCRIB storage map in k-space,

Bout(k) =∫ ∞−∞

K(k, T − τ)Ain(τ) dτ, (7.17)

with the kernel K defined by

K(k, τ) = i

√d√

2πe−(1+i∆0/2)τ × kid/∆0(k + ∆0τ)−id/∆0−1 (7.18)

for τ ≥ 0, with K = 0 when τ < 0. The multimode scaling of lCRIB is determined

by the singular values of this kernel. Unfortunatey we cannot extract these singular

values directly because there is a singularity at k = −∆0τ . A number of alternatives

are open to us. First, we can return to the equations of motion, and construct the

Green’s function in (z, τ)-space directly by numerical integration (see §5.4 in Chapter

5 and §E.5 in Appendix E). This kernel is not singular, and is therefore amenable to

a numerical SVD. We’ll refer to this as simply the ‘numerical method’. Second, we

can remove the singularity in (7.18) by applying a small regularization to k. That is,

we replace k by k− iε, where ε is some small real number. This shifts the singularity

off the real axis, making the kernel well-behaved. But it also changes the singular

values, since it is not a unitary transformation. We fix this with the following

procedure. We apply a numerical inverse Fourier transform from k-space back to z-

space. This can be performed very efficiently with a fast Fourier transform (FFT), an

implementation of which is standard in Matlab. The Fourier transform is unitary, so

7.3 CRIB 249

it does not affect the singular values. Next, we multiply the result by the exponential

factor e−εz. By the shift theorem (see §D.3.4 in Appendix D), this compensates for

the regularization, so that the result is equal to the Fourier transform of (7.18)

without the regularization applied. Because the Fourier transform is unitary, we

can extract the singular values of (7.18) by taking a numerical SVD of the kernel

generated by this procedure. We will refer to this as the ‘Fourier method’.

We use both of these methods below. But to gain some insight into the form of

the multimode scaling of lCRIB, we now introduce a third approach. The approach is

only valid for very large broadenings, but it clarifies the scaling behaviour exhibited

by the numerical techniques just described.

7.3.2 Simplified Kernel

We perform a series of unitary transformations that simplify the form of the kernel

(7.18), while leaving the singular values unchanged. The first of these transforma-

tions is to trop the exponential factor iei∆0τ/2, since it is just a phase rotation. Next

we drop the factor kid/∆0 . This is, again, a pure phase rotation, which is well defined

as long as ∆0 6= 0 and k 6= 0. The second of these conditions arises because there is a

logarithmic singularity in the phase of kid/∆0 at k = 0. The effect of this singularity

is small when d/∆0 is small, so in the following we assume a large broadening, with

∆0 d. The resulting kernel is

K(k, τ) =

√d√

2πe−τ (k + ∆0τ)−id/∆0−1. (7.19)

7.3 CRIB 250

We now take the inverse Fourier transform, from k-space back to z-space. I have to

confess to not knowing how to perform this transform, but Mathematica provides

an answer! Combining this with the shift theorem gives

K(z, τ) = α (d/∆0)√de−τzid/∆0ei∆0zτ , (7.20)

where α is given by

α(x) = − 1πe−πx/2sinh(πx)Γ(ix), (7.21)

with Γ denoting the Euler Gamma function (not the complex detuning!). Finally,

we note that the factor zid/∆0 is another pure phase rotation, well-defined if ∆0 6= 0

and z 6= 0. We can drop it without affecting the singular values, again in the limit

∆0 d, which gives the simple kernel,

K(z, τ) = α(d/∆0)√de(i∆0z−1)τ . (7.22)

Now we form the anti-normally ordered kernel KA, by integrating the product of

two of these kernels from τ = 0 to τ =∞ (see §3.3.1 in Chapter 3),

KA(z, z′) =d

∆0|α(d/∆0)|2 × 1

2/∆0 − i(z − z′). (7.23)

This kernel has the simple structure we have been seeking. It takes the form of a

Lorentzian peak centred on the line z = z′, with a width set by ∆0, and a height

determined by the ratio d/∆0. We can draw two conclusions from the form of this

7.3 CRIB 251

kernel.

First, in the limit of large broadening ∆0 d, the Schmidt number becomes

independent of the optical depth, being determined only by ∆0. This follows from

the fact that the functional form of (7.23) does not depend on d, which only affects its

overall magnitude. The Schmidt number does not depend on this overall magnitude,

because of the normalization of the η’s in (7.1), and so S is independent of d.

The contours of (7.23) form a strip along the diagonal line z = z′. Application

of the estimation technique described in Figure 7.3 suggests that the multimode

capacity is proportional to the ratio of the length to the width of the strip, so that

S ∝ (2/∆0)−1 ∼ ∆0 (see Figure 7.5). Numerics confirm this to be the case: the

Schmidt number of an lCRIB memory rises linearly with the applied broadening

(see Figure 7.6).

The second conclusion we can draw from the structure of (7.23) is that the

threshold multimode capacity N rises linearly with the optical depth d — a signifi-

cant improvement over the square root scaling derived for EIT and Raman memories

in §7.2. To see why, consider a situation where the optimal memory efficiency η21

exceeds the threshold efficiency ηthreshold by a reasonable margin, so that N ≈ S

(see Figure 7.7). If we double the applied broadening, the Schmidt number doubles.

But the ratio d/∆0 is then halved, so that the overall efficiency falls below ηthreshold,

and N ≈ 0. In fact, the function α(x) ≈ 1 for x 1, so that halving d/∆0 ap-

proximately halves the ηk, and divides the total memory efficiencies η2k by four. To

bring the overall efficiency back to its previous value, above the threshold, we must

7.3 CRIB 252

(a) (b)Many modes Few modes

0 0.5 10

0.5

1

0 0.5 10

0.5

1

Figure 7.5 Scaling of Schmidt number with broadening. In (a)and (b) we plot |KA| for large and small broadenings of ∆0 = 100and ∆0 = 10, respectively, with d = 30 in both cases. Below theseplots we illustrate the mode-counting procedure described in Figure7.3, which provides a way to understand why the Schmidt numberincreases linearly with the applied broadening (see Figure 7.6).

double the optical depth. The multimode capacity N is then doubled, because we

have both doubled S and maintained the correspondence N ≈ S by keeping the

overall efficiency above ηthreshold. This argument is quite general: we can increase

the number of modes contributing to the storage interaction by increasing the ap-

plied broadening, but the coupling is ‘shared’ between all these modes, so we must

increase the optical depth at the same time in order to maintain efficient storage

over all the modes. If our aim is to maximize N , there is an optimal value for the

ratio d/∆0, which depends on ηthreshold. If the ratio is too large, we can afford to

increase the broadening and introduce more modes without bringing the efficiency

below the threshold. Converseley, if the ratio is too small, we should sacrifice some

7.3 CRIB 253

0 100 200 3000

0.5

1

(a) (b)

0 100 200 3000

10

20

30

Figure 7.6 Comparison of the predictions of the kernels (7.23) and(7.18). (a): the optimal total memory efficiency η2

1 predicted by thesimplified kernel (7.23) (green dotted line) is plotted alongside theefficiency predicted by the kernel (7.18), using the ‘Fourier method’(blue solid line), as a function of the applied broadening. The opticaldepth is set at d = 30; the simplified kernel compares well with theFourier method for ∆0 & 3d. (b): the Schmidt number predicted bythe two kernels. The agreement between the two, even at the bound-aries of the regime of validity of the simplified kernel, is excellent.The linear scaling of S with ∆0 expected from the form of (7.23) isclear.

modes and reduce the broadening to boost the efficiency above the threshold. This

analysis shows that the multimode capacity N of an lCRIB memory scales linearly

with d, provided that the broadening ∆0 is increased linearly with d at the same

time.

The preceding discussion applies in the limit ∆0 d. We confirm the persistence

of linear multimode scaling outside this regime using the direct numerical method

described earlier, in which we construct the Green’s function by integrating the

equations of motion. The result is plotted in Figure 7.8 at the end of §7.3.3 below,

alongside the results for EIT and Raman shown earlier, and the results for tCRIB,

which we now turn to.

7.3 CRIB 254

(a) (b) (c)

Figure 7.7 Understanding the linear multimode scaling of lCRIB.In the limit of large broadening, the Schmidt number of the lCRIBstorage interaction depends linearly on the width of the applied spec-tral broadening, but the overall storage efficiency is determined bythe ratio d/∆0. To double the multimode capacity N , given a fixedthreshold efficiency ηthreshold, we need to double the Schmidt number,while keeping the overall efficiency the same. The process is shownin parts (a) to (c). We begin with η2

1 > ηthreshold, so that S ≈ N . In(b) we double the spectral broadening, which doubles S, but reducesthe efficiency (in fact, by roughly a quarter). To return to the sameefficiency as in (a), in part (c) we double the optical depth. Thishas no effect on S, but it returns the ratio d/∆0 to the same valueas in part (a). By doubling the optical depth, we have suceeded indoubling the multimode capacity. These arguments explain why thethreshold capacity N depends linearly on d for lCRIB.

7.3.3 tCRIB

In a tCRIB memory, the direction of the broadening is perpendicular to the z-axis.

That is, the resonant frequency of the atoms varies across the ensemble (see part

(a) of Figure 2.10 in Chapter 2). The theoretical treatment requires that we divide

the ensemble into frequency classes, where all the atoms in one frequency class have

the same detuning from the signal field carrier frequency. Thorough treatments are

given by Gorshkov et al. [163], and also by Sangouard et al. [160], and we adapt their

techniques to our purpose, which is to derive the Green’s function for the tCRIB

memory interaction.

Suppose that the applied broadening produces an inhomogeneous line with spec-

7.3 CRIB 255

tral absorption profile p(∆). This means that a proportion p(∆)d∆ of the the atoms

have their resonant frequencies shifted by ∆ away from their nominal frequency. The

profile is normalized, so that ∫p(∆) d∆ = 1. (7.24)

The total optical depth d is divided amongst all the frequency classes, so that the

optical depth contributed by atoms detuned by ∆ is d(∆)d∆ = dp(∆)d∆. The

equations of motion for the storage process are found by a straightforward general-

ization of the system (7.11) to the case of multiple frequency classes (and with no

variation of ∆ with z in this case, of course),

∂zA = −∫ √

d(∆)P (∆) d∆,

∂τP (∆) = −Γ(∆)P (∆) +√d(∆)A. (7.25)

Here we have emphasized the functional dependence of the complex detuning Γ(∆) =

1− i∆, and we have defined P (∆) as the slowly varying polarization associated with

the frequency class of atoms detuned by ∆. We quickly encounter difficulties if we

attempt to solve this system of equations with our usual trick of Fourier transforming

over z. Instead, we get to the solution by applying a unilateral Fourier transform

over time, from τ to ω. Note that here ω is the frequency conjugate to the retarded

time τ ; it certainly has no relation to the integrated Rabi frequency used in Chapter

7.3 CRIB 256

5! The transformed equations are given by

∂zA = −∫ √

d(∆)P (∆) d∆,

−iωP (∆)− Pin(∆)√2π

= −Γ(∆)P (∆) +√d(∆)A. (7.26)

Solving the second equation, and substituting the result into the first yields the

following equation for the signal field,

[∂z + df(ω)] A(ω, z) = −√

d

2π

∫ √p(∆)Pin(z; ∆)Γ(∆)− iω

d∆, (7.27)

where we have defined the lineshape function

f(ω) =∫

p(∆)1− i(∆ + ω)

d∆, (7.28)

which is essentially the convolution of the Lorentzian spectral response of the atoms

with the inhomogeneous profile p(∆). Recall that in our normalized units the natural

atomic linewidth is defined to be equal to 1. We can immediately solve for the signal

field,

A(ω, z) = e−df(ω)zAin(ω)−√

d

2π

∫ √p(∆)

∫ z

0e−df(ω)(z−z′) Pin(z′; ∆)

Γ(∆)− iωdz′ d∆.

(7.29)

Here Ain is the spectrum of the incident signal field. Now that we are in possession of

a solution for the signal, it is a matter of algebra to construct the Green’s function.

7.3 CRIB 257

But it is not enough to find the storage map alone. We can write down an expression

for the atomic excitation at the end of the interaction, but since it is distributed over

all frequency classes, it is not clear what we should optimize for efficient storage.

There is no single spin wave whose norm represents the storage efficiency. Therefore,

we construct the kernel Ktotal that relates the input signal to the retrieved signal

field (see §3.4 in Chapter 3). The modes found from the SVD of this kernel have a

clear interpretation as those modes that are eventually retrieved from the memory.

To find this kernel, we first solve for the polarization Pout at the end of the storage

process, setting Pin = 0. This requires taking an inverse Fourier transform from ω

back to τ = T , where T marks the end of the storage interaction,

Pout(z,∆) =1√2π

∫ ∞−∞

e−iωT P (ω, z; ∆) dω, (7.30)

where, using the second line of (7.26),

P (ω, z; ∆) =

√d(∆)A(ω, z)Γ(∆)− iω

=

√d(∆)e−df(ω)z

Γ(∆)− iω× Ain(ω), (7.31)

where in the second line we used (7.29). This completes the description of the storage

process. At time τ = T , the control pulse shelves all the excited atoms. To describe

retrieval of the excitations, we again use (7.29). As usual we use a superscript r to

identify quantities associated with the retrieval process. There is no input field for

retrieval, so Arin = 0. The retrieved signal field, at the exit face of the ensemble with

7.3 CRIB 258

zr = 1, is given by

Arout(ω

r) = −√

d

2π

∫ √pr(∆r)

∫ 1

0e−df

r(ωr)(1−zr) Prin(∆r, zr)

Γ(∆r)− iωrdzr d∆r. (7.32)

We ‘stitch’ together the storage and retrieval interactions by making the identifica-

tion

P rin(∆r, zr) = Pout(∆, z). (7.33)

This says that the atomic polarization is the same at the end of the storage process

as it is at the start of the retrieval process. But in between storage and retrieval, the

inhomogeneous profile is flipped, so that red-detuned atoms become blue-detuned

and vice-versa. This is crucial for reversing the atomic dynamics so that they re-emit

the signal field. We model this flipping of the detunings by setting

∆ = −∆r, (7.34)

so that frequency classes with positive and negative detunings swap places. For

backward retrieval with no phase mismatch — the optimal situation — we also

swap the z-coordinate:

z = 1− zr. (7.35)

Note that in this case the frequency ωr is conjugate to the retarded time τ r, which is

the time coordinate in a frame moving at the speed of light backwards with respect to

the initial z-axis used for the storage process. We now combine the relations (7.35)

7.3 CRIB 259

and (7.34) with (7.33), and substitute this into (7.32), using (7.30) and (7.31). The

result is

Arout(ω

r) =∫ ∞−∞

Ktotal(ωr, ω)Ain(ω) dω, (7.36)

where the Green’s function for the memory interaction is given by

Ktotal(ωr, ω) = −e−iωT

∫ √pr(∆r)p(−∆r)

[Γ(∆r)− iωr][Γ(−∆r)− iω]d∆r× d

2π

∫ 1

0e−d(1−zr)[f r(ωr)+f(ω)] dzr.

(7.37)

Now we assume that the inhomogeneous profile is the same for both the storage

and retrieval processes, and symmetric about the unbroadened resonance. then

pr(∆r) = p(∆r) = p(−∆r), and the lineshape is also unchanged, f r = f . Performing

the integrals over ∆r and zr, we find

Ktotal(ωr, ω) =1

2πe−d[f(ωr)+f(ω)] − 1

2 + i(ωr + ω), (7.38)

where we have dropped the exponential factor of e−iωT , since it represents only

an unimportant phase rotation. Here we note that a nearly identical treatment

was first given by Gorshkov et al. [163]. The crucial difference for us is that we

used a unilateral Fourier transform, rather than a Laplace transform, to solve the

equations. This means that the transformed amplitudes Ain and Arout have a natural

interpretation as the spectra of the input and retrieved signal fields. The norm of

the signal spectrum is the same as the norm of the temporal profile, by Parseval’s

theorem (or, alternatively, by energy conservation), and so the SVD of the kernel

7.3 CRIB 260

Ktotal in (7.38) tells us about the storage efficiency of the memory, and indeed, its

multimode capacity.

Now that we have found an explicit form for the Green’s function describing

tCRIB, the multimode capacity can be found by taking its SVD: the singular values

of Ktotal are equal to the ηk. In Figure 7.8 we plot the resulting prediction for the

multimode scaling of tCRIB. We used a rectangular broadening profile with total

width ∆0,

p(∆) =

1/∆0 if |∆| ≤ ∆0/2,

0 otherwise.

(7.39)

And we optimized N with respect to the width ∆0, using a threshold ηthreshold =

70%. It is clear that the scaling of N with d is linear. Furthermore, it is the same

as the scaling of lCRIB, whose multimode capacity is also plotted. This suggests

that the multimode capacity of both CRIB protocols is identical. The scaling of

unbroadened EIT and Raman protocols, as derived from (7.4), is shown for com-

parison. Clearly CRIB dramatically outperforms equivalent unbroadened protocols:

given the same optical depth — the same total number of atoms — more modes of

the optical field can be stored by adding a controlled broadening. We found that for

both tCRIB and lCRIB, the multimode capacity scaled roughly as N ∼ d/25, and

the optimal width for the spectral broadening scaled as ∆opt0 ∼ 9d/5. This confirms

the validity of the arguments given at the end of §7.3.2 for lCRIB.

Just as the poor multimode scaling of unbroadened protocols can be explained

by considering the absorption bandwidth of a homogeneous ensemble (see §7.2.1), so

7.3 CRIB 261

we can understand the improved scaling of CRIB by considering the inhomogeneous

linewidth. Consider the storage of a spectral mode with bandwidth 2γ. An optical

depth of order 10 is required to efficiently absorb the incident light. To store N

such spectral modes ‘side by side’ in frequency, we should have a total optical depth

d ∼ 10N , spread over a spectral width ∆0 ∼ 2γN . That is, the multimode capacity

scales linearly with d, and so does the broadening ∆0, which is precisely what we have

found to be the case for CRIB. And it is clear why there is an optimal broadening:

if ∆0 is too large, there is insufficient optical depth over the bandwidth of a mode

for efficient absorption.

In the next section, we consider a modification to the Raman protocol which

takes advantage of spectral broadening to improve its multimode scaling.

0 100 200 300 400 5000

10

20

Figure 7.8 Multimode scaling for CRIB memories. The multi-mode capacity N is shown as a function of the total optical depth dfor tCRIB (green dashed line), lCRIB (red dotted line) and, for com-parison, unbroadened EIT and Raman protocols (solid blue line).For the tCRIB calculation, a numerical SVD was applied directlyto the kernel (7.38). For lCRIB, we constructed the storage kernelby solving the equations of motion (7.11) numerically — we foundthis to be the most reliable method, and of course no approximationsare required. Nonetheless some numerical error is apparent at largebroadenings, since the numerical problem becomes increasingly stiffas ∆0 increases. For both the CRIB protocols, we optimized N over∆0, and as expected, the optimal broadening width was found toscale linearly with d. We set ηthreshold = 70% for all calculations.

7.4 Broadened Raman 262

7.4 Broadened Raman

Is it possible to improve the multimode capacity of the EIT and Raman protocols by

incorporating a spectral broadening? Here we consider a simple modification to the

Raman protocol, in which a longitudinal broadening is applied to the storage state.

That is, the energy of the state |3〉 varies linearly along the z-axis, covering a range

of frequencies ∆0. The treatment is very similar to that of lCRIB given earlier,

and we will see that it is indeed possible to recover the linear multimode scaling

characteristic of CRIB. As usual, we start with the equations of motion describing

one dimensional propagation. Adapting the system (5.106) to the present scenario,

with an added broadening, we have

∂zA(z, τ) = −√dP (z, τ),

∂τP (z, τ) = −ΓP (z, τ) +√dA(z, τ)− iΩ(τ)B(z, τ),

∂τB(z, τ) = i∆0

(z − 1

2

)B(z, τ)− iΩ∗(τ)P (z, τ). (7.40)

The first term on the right hand side of the Heisenberg equation for B just describes

a position-dependent energy shift of the storage state. We make no attempt to solve

these equations exactly. Rather, we proceed directly to study the adiabatic limit,

in which any driven dynamics are much slower than the timescale 1/∆ set by the

detuning. In this limit, we can eliminate the polarization P , by setting the left hand

side of the second equation to zero (see §5.3.3 in Chapter 5, and also the papers by

Gorshkov et al. [133]). Solving the resulting algebraic equation for P , and substituting


the result into the other two equations yields the system

[∂z +

d

Γ

]A(z, τ) = i

Ω(τ)√d

ΓB(z, τ),[

∂τ +|Ω(τ)|2

Γ− i∆0

(z − 1

2

)]B(z, τ) = −i

Ω∗(τ)√d

ΓA(z, τ). (7.41)

As we did for lCRIB, we now apply a unilateral Fourier transform over z. The

adiabatic equations of motion in k-space are then

A(k, τ) =Γ√2πAin(τ) + iΩ(τ)

√dB(k, τ)

d− ikΓ,[

∂τ −∆0∂k + i∆0

2+|Ω(τ)|2

Γ

]B(k, τ) = −i

Ω∗(τ)√d

ΓA(k, τ). (7.42)

Again we encounter the combined derivatives ∂τ − ∆0∂k, which we deal with by

transforming from the coordinates (k, τ) to (s, τ), where s = k+∆0τ . The derivatives

can then be replaced with ∂τ , where now s is held constant. Substituting the first

equation of (7.42) into the second, and integrating, we arrive at the solution

Bout(s) = Bin(s)e−R∞−∞ f(s,τ) dτ−i

√d

2π

∫ ∞−∞

Ω∗(τ)d− iΓ(s−∆0τ)

e−R∞τ f(s,τ ′) dτ ′Ain(τ) dτ,

(7.43)

where the function f is given by

f(s, τ) = i∆0

2+|Ω(τ)|2

Γ

[1− d

d− iΓ(s−∆0τ)

]. (7.44)


To model the storage process, we set Bin = 0. At the end of the storage interaction,

at time τout −→∞, the coordinate s is given by s = k+∆0τout. That is, k is related

to s by a constant offset. This does not affect the norm of Bout, so it does not affect

the efficiency of the memory. In the following we therefore make the replacement

s −→ k, since k has the clearer physical meaning: it is the spatial frequency of the

spin wave. We also drop the first term i∆0/2 from f , since it represents only a phase

rotation, which also does not affect the memory efficiency. The storage map for a

broadened Raman memory can then be written as

Bout(k) =∫ ∞−∞

K(k, τ)Ain(τ) dτ, (7.45)

where the k-space storage kernel is given by

K(k, τ) = −i

√d

2πΩ∗(τ)g(k, τ)e−

1Γ

R∞τ |Ω(τ ′)|2[1−dg(k,τ ′)] dτ ′ , (7.46)

with g defined by

g(k, τ) =1

d− iΓ(k −∆0τ). (7.47)

Note that the broadening introduces a timescale into the dynamics that is not set

by the control field, so we cannot conveniently write this kernel in terms of the

integrated Rabi frequency used in Chapter 5 (see (5.46)). But if we set ∆0 = 0,

(7.46) does indeed reduce to the standard adiabatic storage kernel (5.74) for a Λ-type

memory.


We can extract the singular values from (7.46) with a numerical SVD. This can be

a little problematic, since for large detunings the kernel becomes nearly singular. But

it is easy to resolve this issue by introducing a regularization, Fourier transforming,

and then compensating — this ‘Fourier method’ is described at the end of §7.3.1

above, where we applied it to the k-space storage kernel for lCRIB, which is also

singular. In Figure 7.9 we show how the square root scaling of the unbroadened

protocol is transformed into linear scaling upon application of a broadening. To

see how this scaling arises from the structure of the storage kernel, we repeat the

arguments used for lCRIB which allow us to simplify the kernel. We consider the

special case of control field with a rectangular profile, since this allows us to perform

the integral in the exponent of K,

Ω(τ) =

Ωmax for 0 ≤ τ ≤ T ,

0 otherwise.

(7.48)

The kernel evaluates to

K(k, τ) = −i

√d

2πΩmaxg(k, τ)e−Ω2

maxT/Γ

[g(k, τ)g(k, T )

]idΩ2

maxΓ2∆0

. (7.49)

In the Raman limit ∆ 1, we can write Γ ≈ −i∆, to obtain

K(k, τ) = −iC∆√2πT

e−iΩ2maxT/∆ × [g(k, T )]i

C2

∆0T × [g(k, τ)]1−i C2

∆0T , (7.50)


where C is the Raman coupling (this is defined in (5.94) in §5.3 of Chapter 5). In

this far-detuned limit, the exponential factor is a pure phase rotation that we are

free to remove, and so is the factor involving g(k, T ), since g is purely real and it is

here raised to a purely imaginary power. As in our treatment of lCRIB, there is a

logarithmic singularity at g(k, T ) = 0, but its effect becomes negligible in the limit

∆0 C2/T of large broadening. Dropping these terms, and some other spurious

phase factors, we find

K(k, τ) =C√2πT

(d

∆− k −∆0τ

)i C2

∆0T−1

. (7.51)

Taking the inverse Fourier transform, with the help of Mathematica, and the shift

theorem, and dropping a further phase factor, we find the z-space kernel

K(z, τ) = α(C2/∆0T

)√C2/Te−i∆0zτ , (7.52)

where the function α is defined in (7.21) in §7.3.2 above. This kernel is nearly

identical in form to (7.22). Instead of being damped in time by the factor e−τ due

to the spontaneous lifetime of the excited state, the kernel is instead truncated at

τ = T , when the control field is switched off. But the functional form is the same:

an exponential parameterized only by the spectral broadening ∆0. The magnitude

of the kernel is also the same as for (7.22), except the optical depth d has been

swapped for the quantity C2/T . By analogy with the facts we know for lCRIB, we

therefore conclude the following. First, in the limit of large broadening, the Schmidt


number of the Raman memory depends only on ∆0. Second, the multimode capacity

N , given some threshold efficiency, scales linearly with C2/T , provided that as this

quantity is increased, the applied broadening ∆0 is also increased proportionately.

Note finally that C2 ∝ d, so the multimode capacity scales linearly with the optical

depth, as it does for CRIB.

These assertions are confirmed by the results of a numerical SVD performed on

the kernel (7.46) using the ‘Fourier method’ described above. We used a Gaussian

control pulse, but the shape of the control makes no difference to the multimode

scaling in the adiabatic limit. This is to be expected, since in this regime the dy-

namics adiabatically follow the control, and so its temporal profile only affects the

shapes of the input modes, not the efficiency of the memory. For the parameters

shown, we found N ∼ d/300, with an optimal broadening width of ∆0 ∼ d/77.

We should remark that the multimode capacity is much smaller than that found

for CRIB, or even than is predicted by the optimal kernel KA in (7.4). This is

because the memory operates far off resonance, so that a much higher optical depth

is required to achieve a strong interaction. Of course the other advantages of Ra-

man storage — for instance, broadband capability, tunability, and insensitivity to

unwanted inhomogeneous broadening — are retained in the present scheme. Since

investigating this protocol, a demonstration has been implemented by Hetet et al.

in Canberra [164], although only a single optical mode was stored and retrieved: this

is quite difficult enough to start with! An analysis of this protocol was also recently

conducted by Moiseev and Tittel [165].

7.5 AFC 268

0 1000 2000 30000

5

10

Figure 7.9 The multimode scaling of a broadened Raman proto-col. The blue solid line shows the multimode capacity calculatedfrom the kernel (7.46) using the ‘Fourier method’, optimized over thewidth ∆0 of the applied broadening. The green dashed line showsthe square-root scaling obtained if ∆0 is set equal to zero. We used aGaussian control pulse, with Ω(τ) =

√10de−(τ/0.1)2 , and a detuning

of ∆ =√

90d. These parameters maintain the adiabatic conditionthat ∆ Ωmax, so that we remain in the Raman limit as the cou-pling is increased. We used a threshold efficiency of 70% for bothcalculations.

The last memory protocol we consider is the AFC memory protocol proposed by

Afzelius et al. in Geneva [91], which is introduced in §2.3.4 of Chapter 2.

7.5 AFC

In the AFC protocol, an ensemble with a naturally broad inhomogeneous absorption

line is prepared by optical pumping, producing an atomic frequency comb. That is,

atoms are removed (or pumped into a ‘shelf’ state) that have resonant frequencies

lying between the ‘teeth’ of a spectral comb, so that the ensemble only absorbs light

at the evenly spaced frequencies of the comb teeth. The great advantage of this

approach, is that the broad spectral bandwidth covered by the ensemble absorp-

tion arises naturally. Adding more teeth to the comb to increase the absorption

7.5 AFC 269

bandwidth only requires that fewer atoms are removed, the density or size of the

ensemble need not be increased. This should be contrasted with CRIB, in which

broadening the absorption bandwidth requires an increase in the total optical depth,

if the same level of absorption is to be maintained. As a result, the multimode ca-

pacity of AFC does not depend on the density of the ensemble, making it by far the

most ‘multimode’ protocol yet proposed.

We model AFC using precisely the same approach as we used for tCRIB. We

assume that we have succeeded in preparing an ensemble so that it has an inho-

mogeneous absorption profile that takes the form of a series of M equally spaced

resonances, covering a total spectral width ∆0, each with optical depth d,

p(∆) =M∑j=1

δ(∆− δj), where δj = ∆0

[j − 1M − 1

− 12

]. (7.53)

Note that we have defined p so that∫p(∆) d∆ = M . The total optical depth associ-

ated with the entire frequency comb is then dtotal = Md. We reserve the designation

d for the optical depth associated with a single tooth of the frequency comb, since

this is set by the density of the ensemble. This definition allows direct comparison

of AFC with the other protocols studied in this Chapter, where d quantifies the

physical resources — density, length — required to build the memory.

The lineshape function for AFC is a sum of Lorentzian lines,

f(ω) =M∑j=1

11− i(δj + ω)

(7.54)

7.5 AFC 270

(again, recall that the ‘1’ in the denominator represents the natural atomic linewidth

in our normalized units). We construct the Green’s function for the AFC memory

in the same way as we did for tCRIB. The only difference is that the inhomogeneous

profile is not flipped around for retrieval: the discrete structure of the comb means

that the phase of the atomic dipoles undergoes periodic revivals, without requiring

any external meddling. We still consider phasematched retrieval in the backward

direction, however, since this is the optimal situation. The equivalent expression to

(7.37) for AFC is therefore

Ktotal(ωr, ω) = −e−iωT

∫p(∆)

[Γ(∆)− iωr][Γ(∆)− iω]d∆× d

2π

∫ 1

0e−d(1−zr)[f(ωr)+f(ω)] dzr,

(7.55)

where we used ∆r = ∆, pr = p and f r = f . Performing the integrals, and dropping

the unnecessary phase factor, we find the Green’s function for AFC to be

Ktotal(ωr, ω) =1

2πf(ωr)− f(ω)

ωr − ω× e−d[f(ωr)+f(ω)] − 1

f(ωr) + f(ω). (7.56)

The first quotient on the right hand side exhibits a removable singularity at ωr = ω,

but this can be dealt with using L’Hopital’s rule:

Ktotal(ω, ω) =1

2πf ′(ω)

e−2df(ω) − 12f(ω)

, where f ′(ω) =M∑j=1

iω[1− i(δj + ω)]2

.

(7.57)

In Figure 7.10 we show the multimode scaling for AFC derived from the SVD of this

kernel. In part (a) we show the multimode capacityN as function of the tooth optical

7.5 AFC 271

depth d, for various numbers of comb teeth. For each number of teeth M , the square

root scaling characteristic of an unbroadened memory is apparent. But it is possible

to increase the multimode capacity arbitrarily, just by adding more teeth to the

comb. In principle, the multimode capacity of this memory protocol is infinite! Of

course, there is a limitation in practice. First, the number of modes stored can never

exceed the number of atoms in the ensemble. A more important restriction however

comes from the spectral width of the initial inhomogeneous line. To achieve efficient

retrieval, the teeth comprising the comb must be ‘well-separated’, so that they do

not overlap in frequency. If they overlap, the re-phasing of the atomic dipoles will

not be complete, and there will be only partial retrieval of the signal field. Because

the lineshape function f is a sum of Lorentzians, which are functions that do not

have compact support, a rather large frequency separation between the comb teeth

is desirable. Therefore, as more teeth are added to the comb, the total width ∆0 of

the comb must be increased, in order to accomodate the increased number of teeth

with the same separation between them. Eventually, the width of the comb will

approach the width of the initial inhomogeneous absorption profile from which the

comb was prepared. More teeth cannot then be added without compromising the

efficiency of the retrieval process. We found that the memory efficiency begins to

suffer seriously if the finesse F falls below around 30, with F defined by

F =∆0

2(M − 1). (7.58)

7.5 AFC 272

The finesse is just the ratio of the tooth separation δj+1− δj and the natural atomic

linewidth 2 (the FWHM in normalized units). For our threshold of ηthreshold = 70%,

a finesse of greater than 100 was required.

In part (b) of Figure 7.10 we show the multimode scaling of AFC as a function

of the total optical depth dtotal, alongside the scaling for CRIB and unbroadened

protocols. For these protocols, dtotal = d; the plots are simply reproduced from

Figure 7.8. This plot shows that, as we might expect, the multimode capacity of

AFC does indeed scale linearly with dtotal. That is, N remains proportional to the

total number of atoms available for the protocol. The advantage of AFC lies in the

use of a ‘natural resource’ of atoms, so that increasing the number of atoms used in

the protocol does not require a higher ensemble density. Note however, that even

as a function of dtotal, the AFC protocol still outperforms CRIB, with a scaling of

N ∼ 2d/25, and an optimal comb width of ∆opt0 ∼ 5d.

This concludes our investigation of multimode storage. The techniques used are

quite general, and are readily applicable to new quantum memory protocols as they

are invented. Experimental implementations of multimode storage will likely by

challenging, but the technical advantages for applications of memory to quantum

communication makes the research a worthwhile endeavour.

Practically, it is hard to imagine how one might encode or detect photons in

the optimal input modes φk(τ). These modes have non-trivial shapes, which do

not necessarily overlap well with the ‘natural’ modes of photon detectors. Gener-

ally these detectors, be they avalanche photodiodes (APDs), photomultiplier tubes

7.5 AFC 273

(PMTs) or superconducting bolometers (SSPDs), are engineered to have a broad,

flat spectral response. This means they couple to a temporal mode that looks like

a short, sharp, spike. The natural modes for single photon detection are therefore

time-bin modes, where information is encoded in the time of arrival of a photon.

These time-bin modes are not usually the same as the φk, so time-bin encoding is

sub-optimal, and the multimode capacity predicted by the SVD cannot be reached.

But the discrepancy between the optimal capacity N and the achieved capacity

depends on the overlap between the subspace of signal profiles spanned by the time-

bin modes with the subspace spanned by the first N optimal modes. As N becomes

large, this overlap grows, and the capacity for time-bin modes rapidly approaches

the optimal capacity. Without attempting a detailed calculation, the plausibility of

this claim can be appreciated by comparing parts (a) and (b) in Figure 7.5 in §7.3.2.

Here, we estimated the Schmidt number by decomposing the Green’s function us-

ing non-ovelapping rectangular pulses as spatial modes. If we consider a Green’s

function defined in the temporal domain, such pulses are precisely time-bin modes.

And it is clear that in part (a), the multimode Green’s function is more faithfully

reconstructed than its few-mode counterpart in part (b). This is quite general. As

a Green’s function becomes multimode, it admits an increasingly fine-grained de-

composition in terms of time-bin modes. This shows that the time-bin basis quickly

becomes ‘just as good’ as the φk. Therefore the multimode capacity for time-bin

modes fast approaches that calculated from the SVD. A similar argument applies

if a frequency-bin encoding is used, or, for that matter, if any other orthonormal

7.5 AFC 274

basis is chosen for the signal alphabet. The multimode capacities calculated in this

chapter can accordingly be characterized as a tight upper bound on the performance

of the protocol concerned.

In the next chapter, we wrap up our analysis of quantum memory by addressing

the optimization of storage when the signal field is given, and cannot be ‘shaped’.

7.5 AFC 275

0 100 200 300 400 5000

50

100

150

2

8

14

20

0 100 200 300 400 500

0

10

20

30

40

(a) (b)

Figure 7.10 The multimode scaling of the AFC memory protocol.(a): We show the multimode capacity of AFC for various numbers ofcomb teeth — indicated by the numbers in the plot — as a functionof the optical depth d associated with a single comb tooth. For eachnumber of comb teeth M , the capacity scales with the square root ofd, just like an unbroadened protocol. But adding more teeth allowsto increase N arbitrarily, so the multimode capacity is not limitedby the optical depth. (b): Here we show the multimode capacityof AFC (green dashed line) as a function of the total optical depthdtotal = Md associated with the entire frequency comb. This showsthat N scales linearly with the total number of atoms involved inthe protocol, just as it does for the CRIB protocols. For comparison,the capacities of tCRIB, lCRIB and EIT/Raman protocols are alsoshown (red dotted line, black dot-dashed line and blue solid line,respectively). For these latter protocols dtotal = d, so these capacitiesare just taken from Figure 7.8. It is interesting that the capacity ofAFC is larger than that of CRIB, even when evaluated as a functionof dtotal. Of course, d is the more relevant physical resource for AFC,since this sets the ensemble density. We optimized the capacitiesplotted for AFC in both (a) and (b) over the spectral width ∆0 of thefrequency comb. The threshold efficiency used was ηthreshold = 70%,as usual.

Chapter 8

Optimizing the Control

The SVD has proved an extremely useful tool for the analysis of quantum storage.

Given a set of parameters, including the ensemble geometry and density, as well

as the detuning and the control pulse profile, it is possible to construct a Green’s

function that contains all the information we might want to know about the memory

interaction. In particular, taking its SVD provides us with the optimal input mode,

so that we can achieve efficient storage by shaping the signal field. But suppose that

we are simply given a signal field, and asked to store it. This is probably the more

likely situation in practical applications, where we have control over our memory,

but not over the source that generates the signal field to be stored. In this case we

are not able to shape the signal so that it matches the optimal mode φ1. How do we

achieve efficient storage in this situation? We must try to deform the optimal mode

φ1 into the shape of our signal! This can be done by shaping the control field in

order to sculpt the Green’s function; in this chapter we will explore how the correct

8.1 Adiabatic shaping 277

shape for the control may be found.

8.1 Adiabatic shaping

The adiabatic limit is precisely the limit in which the natural atomic response is

sufficiently fast that the atoms can ‘follow’ the driving fields. In this limit, the atomic

response function is completely determined by the control field profile. Therefore

the adiabatic limit is the regime we should work in, if our aim is to be able to affect

the Green’s function by shaping the control. Fortunately, another consequence of

adiabatic following is that the Green’s function may be completely parameterized by

the integrated Rabi frequency ω, defined in (5.46) in §5.2.6 of Chapter 5. Later in

Chapter 5 we derived an analytic solution for the adiabatic storage kernel of a Λ-type

ensemble memory (5.77). It is extremely convenient that the optimal input mode

found from the SVD of this kernel applies universally for all control profiles (within

the adiabatic limit, of course). Given a control profile, it is easy to find the temporal

shape of the optimal input mode, simply by converting back from the ω to τ . The

conversion is given by (5.78). The optimization problem in this case is then simple.

We must adjust the control field until the shape of the optimal temporal mode is the

same as the shape of the signal field we intend to store. Changing the shape of the

control pulse has no effect on the singular values, since the storage kernel depends

only the control pulse energy through W . Therefore, when the shaping is complete,

the same storage efficiency is achieved, as would have been if we had shaped the

signal field and left the control fixed.

8.1 Adiabatic shaping 278

In Figure 8.1 we show some examples of this kind of optimization. The control

field is parameterized by a vector Ω of 2N real numbers, giving the real and imagi-

nary parts of Ω(τ) at a set τj of N discrete points in time. We choose the points τj

to lie on a Chebyshev grid, since this makes polynomial interpolation of the control

field to other times numerically stable (see §E.1.2 in Appendix E). Starting with

the initial guess Ω(τj) = Ain(τj), we optimize Ω using a simplex search algorithm —

‘fminsearch’ in Matlab — to minimize the norm of the difference Ain(τ) − φ1(τ).

At each iteration the optimal mode φ1(τ) is determined from Ω by interpolating to

find the envelope Ω(τ), and then using (5.78). Convergence does not take longer

than 1 minute on a 3 GHz machine. After each optimization, we use the optimized

control profile along with the numerical method described in §5.4 of Chapter 5 to

construct the storage kernel without the adiabatic approximation. The resulting op-

timal mode is then shown for comparison with the signal profile to be stored. This

provides a direct way to examine the accuracy of the optimizations. The method

used relies on the validity of the adiabatic kernel (5.77), and it is discernible that the

optimization performs better in the Raman limit than for EIT, since the adiabatic

approximation is less robust on resonance (this is discussed in §5.2.9 of Chapter 5).

As with so much of this thesis, an excellent account of closely related work can

be found in the papers of Gorshkov et al. [133], who studied this problem in the light-

biased limit of large control pulse energy, where the optimal kernel (5.27) is valid (see

§5.2.8 in Chapter 5). They did not use the SVD, or the integrated Rabi frequency

ω, so their approach is a little more convoluted, but nonetheless it is accurate and

8.2 Non-adiabatic shaping 279

ingenious.

8.2 Non-adiabatic shaping

The opposite to the adiabatic limit is the limit of a very short, broadband control

pulse. If the pulse is sufficiently short, there is no time for the coherence induced

on the |2〉 ↔ |3〉 transition to couple to the optical polarization P connecting states

|1〉 and |2〉. In this limit, the shape of the control field makes no difference to the

form of Green’s function describing absorption of the signal. Instead, the control

simply induces Rabi oscillations between |2〉 and |3〉 that transfer population from

the excited state to the storage state. The efficiency of this population transfer

is given by sin2(θ/2), where θ = 2∫

Ω(τ) dτ is the pulse area (note the difference

between this dimensionless quantity and the integrated Rabi frequency ω, which is

relevant in the adiabatic limit). Perfect transfer is achieved by an infinitely short

π-pulse: this kind of instantaneous map is assumed in the CRIB and AFC protocols

treated in the last Chapter, and in the derivation of the optimal kernel (5.27) in

Chapter 5. If we use such a control pulse, the optimal signal input mode is fixed

by the homogeneous and inhomogeneous lineshape of the ensemble. If the signal

field mode does not coincide with this optimal mode, there is nothing we can do to

optimize the storage efficiency.

But there is a large middle-ground between the adiabatic regime and the extreme

case of a π-pulse control. In this middle-ground, the analytic solution for the storage

kernel (5.77) is of limited use, but the shape of the control field still has an effect, at


least to some degree, on the form of the storage Green’s function. Since the analytic

solution breaks down, we must resort to numerical solutions of the equations of

motion to optimize the control. The method we use is the most direct method one

could imagine. Given a vector Ω parameterizing the control, we construct the full

control profile Ω(τ) by interpolation, and then we integrate the equations of motion

numerically, and we extract the storage efficiency η. We then use a simplex search

algorithm to optimize the elements of Ω so as to maximize η — using fminsearch

in Matlab, we minimize −η. This approach works well, but relies on the ability to

repeatedly solve the equations of motion quickly and accurately. Fortunately, this

is possible using the method described in Appendix E, where we use Chebyshev

spectral collocation for the spatial propagation, and RK2 for the time-stepping.

To improve the convergence of the optimization, it helps to start with an initial

guess that is already close to optimal. We can make use of the SVD in this con-

nection. We start with an initial guess Ω, determined by setting Ω(τ) = Ain(τ).

This is motivated by the fact that φ1(τ) = Ω(τ) in the adiabatic limit, when the

coupling is small (i.e. when d or C is small), so it is a reasonable opening gambit.

Next we use the numerical method described in §5.2.9 of Chapter 5 to construct the

resulting Green’s function, and we take its SVD to extract the optimal input mode

φ1(τ). In general, since the coupling is not small, φ1(τ) 6= Ain(τ) — this is why the

optimization is necessary. By fitting a cubic spline interpolant to φ1, and also to

Ain, we are able to build a sequence of n functions φ(k)1 that represent a smooth

deformation of this optimal mode into Ain. That is, φ(1)1 = φ1, and φ(n)

1 = Ain, with


the intervening functions lying ‘between’ these two. This is easily done in Matlab by

building the φ(k)1 using splines with coefficients found by interpolating between the

coefficients for φ1 and Ain. Now, Ω already describes the optimal control for storing

the mode φ(1)1 , by construction. We now use the optimization method described

in the previous paragraph to optimize Ω for storing the mode φ(2)1 . The hope is

that this is sufficiently ‘close’ to φ(1)1 that the optimization will converge quickly.

We then repeat the optimization, this time using the newly optimized vector Ω as

our initial guess, and using the mode φ(3)1 as the target input mode. The pattern

is now clear. We iterate these steps, each time switching the target input mode

from φ(k)1 to φ(k+1)

1 , and using the previous result for Ω as an initial guess. As we

procede, the target modes approach Ain, and Ω approaches the vector describing

the optimal control for storing Ain. The rationale behind this approach is that the

control remains near-optimal at all times. We begin with an optimal control for the

‘wrong’ target, and by degrees we deform this target into Ain, all the time ‘bringing

along’ the control. We found that this method is helpful in the most non-adiabatic

situations, where the initial guess Ω(τ) = Ain(τ) is particularly poor. When the

dynamics are more adiabatic, optimizing for Ain directly in a single step is often

sufficient.

In Figure 8.2 we show some examples of this type of optimization. Clearly the

numerical optimization works well where the analytic optimization described in the

previous section fails. The method is not particularly time consuming on a modern

computer, although it is of course slower than the analytic optimization. It is quite


general though, and it provides an easy way to optimize the storage efficiency for a

given input, and also to check the final proximity of the optimal input mode to the

desired shape.

Gorshkov et al. have also studied the optimization of the control field outside the

adiabatic limit, in the fourth of their series of papers on the subject [166]. They use

an interesting approach involving gradient ascent, in which they derive an explicit

formula for the incremental change in the control profile that will improve the storage

efficiency. Numerical solution of the equations of motion provides them with this

incremental change, and by iterating they are able to generate the optimal control

for storage. This is a robust and efficient optimization that produces very similar

results to the method we describe here. They comment that they are not able to

verify the optimality of their results, and this is true for our optimizations too. But

in both cases, the SVD of the numerically constructed Green’s function does make it

possible to check the resemblance between the desired input mode, and the optimal

mode resulting from the control. When these two modes match up, as they do in

the cases shown in Figure 8.2, it is clear that the optimization has converged. When

these modes do not match, of course, it is not obvious whether the result is a global

or simply a local optimum.

This completes our theoretical treatment of optimal storage and retrieval from

ensemble memories. In the next Chapter we present a derivation of the equations

of motion for the memory interaction in diamond, since this is one of the media we

have used in our experiments, and the theory of Chapter 4 is not directly applicable.


Finally, in Chapter 10 we review the status of our experiments, before concluding

the thesis in Chapter 11.


0

0.5

1Broadband Raman

Inte

nsi

ty (

arb

. un

its)

Inte

nsi

ty (

arb

. un

its)

0

2

4

6

8

Broadband EIT

Ph

ase

Ph

ase

0

2

4

6

−2 0 20

0.5

1Narrowband EIT

0

2

4

6

8

−2 0 2

−0.2 0 0.2 −0.2 0 0.2

Non-adiabatic

0

2

4

6

(a) (b)

(d)(c)

Figure 8.1 Adiabatic control shaping. Given an input signal pro-file, as well as the available control pulse energy W , optical depth dand detuning ∆ we construct the adiabatic kernel (5.77), and extractthe optimal input mode, as a function of the integrated Rabi fre-quency ω. We then optimize the control profile Ω(τ) until the tempo-ral profile of the optimal mode matches the given signal. In parts (a)to (d), the signal is assumed to be a Gaussian, Ain(τ) = e−[(τ−τs)/Ts]2 .Its intensity profile |Ain(τ)|2 is shown as a dashed black line. Thetiming of the signal pulse τs is arbitrary, and in each case we chooseit so as to aid the convergence of the optimization. The temporalintensity profile |Ω(τ)|2 of the optimized control is represented bythe blue solid line, with its temporal phase shown by the red solidline (referred to the axes on the right-hand side). We used N = 21Chebyshev points to parameterize the control in all cases. The greendotted lines show the temporal intensity profiles of the optimal inputmode determined from the optimized control using numerical integra-tion to construct the Green’s function. These should coincide withthe signal mode, if the optimization has succeeded. Part (a) showsthe result for an optimization with W = 2.5, ∆ = 150 and d = 300,and a signal duration of Ts = 0.1, all in normalized units. This de-scribes off-resonant Raman storage. The adiabatic approximation iswell satisfied, and the optimization performs well. In part (b) weshow the result for the same optimization when the detuning is setto zero, which describes broadband EIT. Here the adiabatic approx-imation is not well satisfied, and the optimization performs ratherpoorly. In part (c) we optimize the control for a narrowband signalwith Ts = 1 (note the difference in time scale on the horizontal axis).The optimization performs much better. Finally in part (d) we re-duce the optical depth to d = 10, and we increase the control energyto W = 49. The detuning is set to 0, and the signal duration remainsTs = 1. Despite the narrow signal bandwidth, the adiabatic approx-imation that worked well for part (c) is now ‘broken’, because of thelarge Rabi frequency, and the optimization fails entirely. The storageefficiencies achieved in parts (a) to (d) were 94%, 74%, 94% and 54%,respectively. For comparison, the efficiencies that would have beenachieved if the signal field were exactly equal to φ1(τ) were 97%, 99%,96% and 82%.


0

5

10

0

2

4

6

0

5

10

0

2

4

6

0

0.5

1In

ten

sity

(a

rb. u

nit

s)In

ten

sity

(a

rb. u

nit

s)

0

0.5

1

Ph

ase

Ph

ase

Broadband Raman Broadband EIT

−2 0 2

Narrowband EIT

−2 0 2

−0.2 0 0.2 −0.2 0 0.2

Non-adiabatic

(a) (b)

(d)(c)

Figure 8.2 Non-adiabatic control shaping. Parts (a) to (d) showthe results of the direct numerical optimization described above,where n = 10 steps were used in deforming the initial target Ain(τ) =φ

(1)1 (τ) into the final target Ain(τ) = e−[(τ−τs)/Ts]2 . The optimiza-

tions each ran in around 2 minutes on a 3 GHz machine. The pa-rameters are identical to those used in parts (a) to (d) of Figure 8.1.The numerical optimization copes well with non-adiabatic dynam-ics, and in all cases comparison of the target signal mode with theoptimized input mode shows that the optimizations have met withsome success. In part (b), the broad bandwidth of the signal makesthe adiabatic approximation poor, and it is noticeable that the opti-mized control profile features a small oscillation which is not presentin the control in part (c). The adiabatic approximation fails entirelyin part (d), and here the control involves a large oscillation, with itsenergy distributed into two ‘pulses’. These oscillations are typicalof non-adiabatic shaping, since the ‘ringing’ of the atomic dynam-ics must be compensated by the control to produce a smooth inputmode. The storage efficiencies achieved in parts (a) to (d) were 97%,99.5%, 97% and 81%, respectively. For comparison, the efficienciesthat would have been achieved if the signal field were exactly equalto φ1(τ) were 97%, 99.6%, 97% and 82%.

Chapter 9

Diamond

In this chapter we build a theoretical description of quantum memory in diamond.

The end result is a set of equations with precisely the same form as (5.107) describing

Raman storage in a vapour, so that our analysis of optimal storage and retrieval

applies in diamond just as it does to atomic vapours (see §5.3.3 in Chapter 5). But

some justification of this claim is required, and so we present a derivation below.

9.1 Diamond Scheme

Diamond is a singular material, both physically and aesthetically. It is the hardest

and most transparent mineral, with the highest thermal conductivity of any material,

and a very large refractive index (around 2.4). These properties arise in part from

its extremely simple structure. It is comprised entirely of carbon atoms; each is

connected to four others by strong covalent bonds. The bonds are all equivalent,

and this symmetry produces a tetrahedral arrangement of atoms that is exceedingly

9.1 Diamond Scheme 287

robust. The diamond structure can be visualized by convolving a basis of two carbon

atoms with a face-centred cubic bravais lattice [167], as shown in Figure 9.1.

Figure 9.1 The crystal structure of diamond. The diamond latticeis FCC (face-centred cubic); here we show just a single unit cell,outlined for clarity with thin ‘rods’. Each basis is shown as a pairof atoms connected by a thick ‘tube’, one atom is coloured red; theother grey. The grey atom of each basis is located at the sites of theFCC lattice. Each atom is connected by four covalent bonds (notshown) to its neighbours, forming a tetrahedral pattern.

If the basis is deformed, the bonds produce a large restoring force, and so in

response to an impulse the atoms of the basis can undergo harmonic oscillations

relative to one another. The frequency of these ‘internal’ vibrations is rather high,

because the interatomic bonds are very ‘stiff’. It is these high frequency vibrational

modes we seek to excite when implementing a quantum memory in diamond.

Eventually any relative motion within a basis couples to collective motion of

the basis with respect to its neighbouring bases, and the energy is dissipated as

9.2 Quantization 288

waves of motion of the lattice sites with respect to one another: sound waves. This

process limits the lifetime of a diamond quantum memory, and in fact the lifetime

— on the order of picoseconds — is much too short to make diamond a useful

medium for quantum storage. Nonetheless, it is possible to store very broadband

pulses in diamond, because the oscillation frequency is so high, and a solid-state,

room-temperature, broadband quantum memory is interesting in its own right.

9.2 Quantization

Just as the electromagnetic field can be quantized, revealing photons as the con-

stituents of light, so the harmonic oscillations of a crystal can be quantized. The

quanta of crystal vibrations are known as phonons. Our aim in the present context

is to describe the coherent mapping of a single photon to a single phonon in the

diamond crystal.

The crystal vibrations may be quantized by imposing periodic boundary con-

ditions on the atomic displacements within a cubic crystal of side length L. To

illustrate this procedure we consider a one dimensional wave b(z) = eikz describing

the atomic displacement at position z. Periodic boundary conditions require that

b(0) = b(L), so that we must have kL = 2πm for some integer m. Therefore the

momenta associated with crystal vibrations are quantized, with k = 2πm/L. The

quanta are phonons.

The smallest non-zero wavevector allowed is found by setting m = 1, whence we

obtain the mode separation δk = 2π/L. There is also an upper limit to the allowed

9.2 Quantization 289

wavevectors. The wavelength of any vibration cannot be meaningfully defined if it

falls below 2a, where a is the lattice constant — the distance separating neighbouring

atoms. The wave is only ‘sampled’ at the atomic positions, and the spatial frequency

of the waves cannot exceed the sampling rate. The maximum wavevector, set by

this coarseness of the crystal structure, is ∆k = π/a. As shown in Figure 9.2 below,

any wave with a larger wavevector k is physically indistinguishable from a wave with

momentum k−2∆k, and so we only consider phonons with wavevectors lying within

the range [−∆k,∆k]. This region in k-space is known as the first Brillouin zone,

or just the Brillouin zone. That all physically distinct phonon modes are contained

within the Brillouin zone can be confirmed by counting them: the number of modes

in the Brillouin zone is 2∆k/δk = L/a = N , where N is the number of atoms in

the crystal. So the number of phonons is equal to the number of atoms. This must

be true in one dimension, since each atom has one vibrational degree of freedom.

The generalization to three dimensional vibrations turns the Brillouin zone into a

three dimensional volume in k-space, often with a non-trivial shape. But this is not

important for us. As in most of the rest of this thesis, we will use a one dimensional

model to describe the quantum memory.

The lattice constant in diamond is around 3.6 A, so the Brillouin zone boundary

lies at ∆k ∼ 1010 m−1. By comparison, the wavevector associated with visible light

at around 500 nm is about 107 m−1. Any momentum imparted to the crystal by

interaction with optical fields is therefore very small, on the scale set by the crystal

lattice. The excitations produced by a quantum memory may safely be considered to

9.3 Acoustic and Optical Phonons 290

lie at the zone centre, with δk ∼ 0. This, of course, greatly simplifies our theoretical

description.

dis

pla

cem

en

t

Figure 9.2 Phonon aliasing. A crystal vibration with a wavelengthsmaller than 2a is physically equivalent to one with a longer wave-length. In the example shown, the blue solid line represents the profileof a short wavelength vibration, and the dashed blue line shows theprofile of the equivalent longer wavelength vibration. Note that theatomic positions, indicated by the red circles, are identical for bothwaves. The black circles show the equilibrium positions of the atoms,equally spaced by the lattice constant a.

9.3 Acoustic and Optical Phonons

Phonons come in two varieties, as we have already hinted. Acoustic phonons are

the quanta of sound waves in a crystal. They represent compression and rarefac-

tion within the crystal lattice, and the energy associated with this lattice distortion

clearly vanishes as its wavelength becomes large, since in the limit of infinite wave-

length there is no distortion, and therefore no restoring force. As mentioned above,

optical wavelengths are already much larger than a unit cell, so the energies of op-

tically accessible acoustic phonons are very small. A typical dispersion relation for

acoustic phonons is shown in part (a) of Figure 9.3. The energy of acoustic phonons

rises linearly with their wavevector k for k ∆k. Scattering from low-energy zone-


centre acoustic phonons is conventionally known as Brillouin scattering, but it is not

of interest to us here. We will focus instead on Raman scattering, which in crystals

refers to the scattering of light from so-called optical phonons. These represent the

second variety of phonon; they are the quanta associated with the high-frequency

internal vibrations of the crystal bases. The bases oscillate essentially independently

of their neighbours, provided the wavelength of the oscillation is not so small that

their neighbours are ‘pulling’ on them. In the limit of infinite wavelength, the op-

tical phonon energy does not vanish, but is set by the frequency associated with

the normal modes describing the natural internal oscillations of the basis. A typi-

cal dispersion relation for optical phonons is shown in part (a) of Figure 9.3. The

non-vanishing energy of optical phonons at the zone-centre distinguishes them from

acoustic phonons. The Raman interaction in diamond couples an incident signal

field to these zone-centre optical phonons, and it is these phonons that play the

role of the storage state in a diamond quantum memory. Broadly speaking, these

phonons are like the metastable state |3〉 used for storage in the atomic systems

discussed previously.

9.3.1 Decay

In many crystals, the basis is composed of unlike atoms or ions, and so the optical

phonons are associated with an electric dipole moment. This means they couple

strongly to the electromagnetic field — that is why they are given the designation

‘optical’. However in diamond, which is homopolar, with both atoms in its basis


Optical branch

Acoustic branch

(a) (b)

Figure 9.3 Phonon dispersion. (a): typical dispersion curves foracoustic and optical phonons. The former have negligible energy nearthe zone-centre, which is the region to which we have access withoptical fields. The latter have large energies, and a flat dispersionrelation close to the zone-centre. It is the zone-centre optical phononsthat we use to store an incident photon. (b): The decay of opticalphonons is dominated by the Klemens channel, in which anharmoniccoupling allows a zone-centre optical phonon (black dot) to decayinto a pair of acoustic phonons with large, opposite momenta (reddots). We only show the dispersion in the first Brillouin zone. Thedotted lines on the left and right hand sides indicate the boundariesof this zone, which are identified, meaning that any we could wrapthe plots around a cylinder and stitch these two lines together: theyrepresent physically equivalent momenta.

identical, there is no separation of charge associated with internal oscillations. The

optical phonons in diamond are therefore not, in fact, optically active. That is, they

do not directly radiate or absorb electromagnetic radiation. This is advantageous for

quantum memory, since the optical phonons in diamond are accordingly longer-lived

than in many other materials. As touched upon above, the dominant decoherence

process for these phonons is the decay into acoustic phonons via anharmonic cou-

plings: the covalent bonds do not behave like perfect springs, and their deviation

from Hooke’s law allows the optical and acoustic phonons to exchange energy. To

conserve momentum the acoustic phonons are produced in pairs with approximately

opposite momenta, as illustrated in part (b) of Figure 9.3. This process is known


as the Klemens channel [168,169]. The anharmonicities that give rise to the Klemens

channel are largely geometrical in origin, and the lifetime of optical phonons is only

weakly affected by temperature [170]. The purity and quality of the crystal also con-

tribute, but all diamonds have an optical phonon lifetime1 τp . 10 ps.

9.3.2 Energy

There are three optical phonons in diamond, corresponding to basis oscillations in

three orthogonal directions. The three phonons are degenerate at the zone centre,

because diamond is symmetric with respect to the interchange of these three direc-

tions. With a one dimensional interaction we will excite just a single phonon mode;

the degeneracy of the modes means we need not worry about which mode this is.

The optical phonon energy at zone centre is Ep = 0.17 eV, which corresponds to a

wavenumber of νp = 1332 cm−1, or an angular frequency of ωp = 2.5 × 1014 s−1.

This would correspond to an infra-red wavelength of λp = 7.5 µm. The large phonon

energy in diamond is advantageous for the following reasons:

First, the energy scale kBT associated with room temperature (T ∼ 300 K) is

around 1/40 eV, which is much smaller than the phonon energy. Therefore there

are very few thermally excited optical phonons at room temperature: using the

Boltzmann formula pthermal = e−Ep/kBT we predict a population of around 1.7×10−3

thermal phonons per mode. Therefore demonstrating quantum memory at room1These phonon lifetimes were studied in our research group by Felix Waldermann, and later

by K.C. Lee and Ben Sussman, using a technique they named TCUPS [171]. In these experiments,a pair of delayed pump pulses directed through a diamond crystal produces a corresponding pairof Stokes pulses. The phase coherence, as measured from the visibility of spectral interference,between the two Stokes pulses directly measures the coherence of the optical phonons.

9.4 Raman interaction 294

temperature in diamond is feasible.

Second, the bandwidth of the stored signal field cannot exceed the phonon fre-

quency, since ωp sets the frequency difference between the signal and control fields,

and these should not overlap spectrally. Since the phonon frequency is so large, the

signal bandwidth can be large, meaning that a short pulse can be stored. Taking

τs ∼ 1/ωp as a rough estimate of the shortest signal pulse duration that can be

stored, we find τs ∼ 1000τp. If the ratio of the shortest storable pulse duration

to the maximum storage time τs/τp is taken as a figure of merit for a memory, a

diamond quantum memory is actually rather impressive!

9.4 Raman interaction

9.4.1 Excitons

The Raman interaction in diamond involves an intermediate state, just as it does in

the atomic case considered in earlier chapters. The optical fields are detuned from

resonance with this state, but the interaction nonetheless requires strong coupling

to this state to mediate the storage of the signal field. The relevant intermediate

state in diamond is an exciton. To understand what this is, recall that the electronic

orbitals in an extended crystal arrange themselves into disjoint bands. Electronic

band structure arises from Bragg scattering of electrons from the periodic potential

associated with the regular lattice of atomic nuclei in the crystal. As the De Broglie

wavelength of an electron approaches 2a, the reflected and transmitted components


of the electronic wavefunctions interfere destructively, leading to the appearance of

forbidden energy bands, containing no allowed electronic states, at the edges of the

Brillouin zone. This is illustrated in Figure 9.4.

Free(a) Periodic(b) Bands(c)

Figure 9.4 Band structure. (a): the dispersion relation of a freeelectron is parabolic, since its kinetic energy E = p2/2m is quadraticin k. (b): electrons in a periodic lattice must have a periodic dis-persion relation. To a first approximation, this is found by simplyadding ‘copies’ of the free electron dispersion relation at intervals of2∆k = 2π/a in k-space. (c): scattering of the electrons from the pe-riodic potential produced by the atomic cores results in anti-crossingsin the dispersion relations, leaving energy gaps, shown shaded in gray.Here we have removed any electronic states lying outside the firstBrillouin zone (red shaded area), since they are not physically mean-ingful.

An exciton is produced when an electron in the lower energy band, the valence

band, is promoted into the upper energy band, the conduction band, leaving behind

a hole. The hole in the valence band simply represents the absence of an electron,

but it is convenient to think of it as a particle in its own right, with positive charge,

negative energy and negative mass. An exciton is the combined system of an electron

in the conduction band and a hole in the valence band (see Figure 9.5). In fact, since

these two particles have opposite charge, they attract one another, and it is possible

for them to form a bound system, similar to a hydrogen atom, or the positronium


of particle physics. The binding energy is rather small, however, and here we will

treat the electron and hole as if they were free particles2.

Conduction band

Valence band

Figure 9.5 An exciton. A photon (not shown) promotes an elec-tron (blue dot) from the valence band into the conduction band,leaving behind a positively charged hole (red dot). Note that thecurvature of the valence band for the hole has been reversed, sincethe hole has negative energy. In this picture, the electron and holehave approximately equal and opposite momenta, so that the totalmomentum of the exciton is small, consistent with the small momen-tum of the photon.

9.4.2 Deformation Potential

An incident photon can produce an exciton if it has sufficient energy to breach

the band gap. This process provides the coupling between the optical fields and

the diamond. To excite an optical phonon, there should be some coupling between

excitons and phonons. In polar crystals, there are direct couplings between the

dipole fields of excitons and phonons, in the form of the Frohlich and piezoelectric

interactions [173–176]. In diamond, these long-range electric couplings are absent, but2A precise characterization of the Raman cross-section in diamond does require an account of

bound excitons [172], but our aim is simply to study the feasibility of using this interaction forquantum storage, so we sacrifice rigour for simplicity


there remains a short range coupling known as the deformation potential. The origin

of this interaction can be understood as follows. The diamond structure takes the

form of two interpenetrating FCC lattices, offset from one another. A zone centre

optical phonon — with infinite wavelength — may then be interpreted as the rigid

displacement of one sub-lattice with respect to the other. At any instant, the crystal

structure is accordingly deformed, much as if it were subject to an external strain,

and the electronic band structure, which depends on electron scattering from the

crystal potential, is altered. Therefore electronic energy levels are coupled to crystal

vibrations. More specifically, excitons are coupled to optical phonons. Optical

phonons are not energetic enough to create or destroy an exciton outright, but

an exciton with some momentum k can scatter from the deformation potential to

produce an exciton with a momentum k′, and an optical phonon with momentum κ.

Phasematching of this process over the length of the crystal ensures that momentum

is conserved, with κ = k − k′.

We now have to hand all the ingredients necessary to unpick the Raman inter-

action in diamond, which is shown in Figure 9.6,

1. A signal photon produces a virtual exciton — ‘virtual’ because the energy

of the signal photon is smaller than the band gap. This is analogous to the

virtual, or dressed state to which the signal couples in atomic systems, when

it is detuned from the excited state |2〉.

2. The virtual exciton scatters from the deformation potential to produce an

exciton with a different momentum, and also a zone-centre optical phonon.


3. The remaining virtual exciton recombines — the electron decays back into the

valence band, filling the hole — and emits a control photon.

From this description, it’s clear that the Raman interaction in diamond is third

order, rather than second order as it is in the atomic systems considered in earlier

Chapters. In principle this makes it weaker than in atomic systems, but the ex-

tremely high density of electrons in the solid state more than makes up for the extra

perturbative order, and the Raman cross section in diamond is, in fact, extremely

large.

(a) (b)

Figure 9.6 The Raman interaction in diamond. (a): An energylevel diagram for the Raman quantum memory interaction. The rel-evant states are written in the form |n,m〉, where n is the numberof excitons involved, and m is the number of optical phonons. A sig-nal photon (blue wavy arrow) produces a virtual exciton (indicatedby the dotted line, detuned from the real exciton state). The defor-mation potential interaction then produces another virtual exciton,and an optical phonon (orange arrow). Finally, the virtual excitonrecombines, emitting a control photon (green arrow) and leaving asingle optical phonon behind. (b): A Feynman diagram for the sameprocess. Here the dotted lines indicate the world lines of the virtualelectron and hole comprising the intermediate exciton. The opticalphonon is indicated by the orange spiral.

9.5 Propagation in Diamond 299

9.5 Propagation in Diamond

It is advantageous to describe both the optical fields and the excitations of the

diamond crystal in the Heisenberg picture, in order to treat propagation. This

was done for the atomic case in Chapter 4 by studying the dynamics of the flip

operators σjk describing the atomic evolution. The analysis was greatly simplified

by considering just three discrete atomic states. In diamond however the electronic

states form a quasi-continuum in each energy band. It is therefore not immediately

obvious how the approach of Chapter 4 carries over to the present case.

In addition to this issue, there is the problem of how to describe the local dynam-

ics of the crystal excitations. To examine the spatial distribution of these excitations,

we would like to obtain equations for the phonon or exciton amplitudes at some po-

sition z within the crystal. In the atomic case the interaction was entirely local,

since each atom scattered light at a point, and independently of all other atoms.

The local dynamics was therefore determined by the Hamiltonian of a single atom,

and propagation was treated by summing these local contributions. The situation

in diamond is conceptually different. First, phonons are global excitations of the

crystal. Second, the electrons in a crystal are not localized around the atomic cores;

rather they form a quasi-free Fermi gas distributed over the volume of the crystal.

The above problems are essentially cosmetic, as we’ll see below. Our general

strategy is as follows. To work in the Heisenberg picture, we write the crystal


Hamiltonian H in second-quantized form,

H =∑α,β

Hαβ|α〉〈β|, (9.1)

where in the sum, both α and β run over a complete set of states, and where the

Hαβ = 〈α|H|β〉 are the matrix elements of the Hamiltonian connecting these states.

The dynamics are then determined from the Heisenberg equations of motion for the

flip operators |α〉〈β|. This approach broadly mirrors that used in Chapter 4 (see

§4.3).

To extract the spatial variation of the crystal excitations we seek to express the

crystal Hamiltonian in the form

H =1L

∫ L

0H(z) dz, (9.2)

where H is an effective local Hamiltonian. It turns out that this representation of

the Hamiltonian emerges naturally from the periodic structure of the crystal lattice.

9.5.1 Hamiltonian

The Hamiltonian for a diamond quantum memory is comprised of three parts,

H = HER +HEL +H0. (9.3)


The first two contributions represent the interaction of the electrons in the diamond

with the radiation field and with the lattice respectively. The last part accounts for

the energy of the excitations in the diamond. We neglect the Hamiltonian HL of

the free radiation field, as we did in §4.4 in Chapter 4, since it plays no part in the

equations of motion.

9.5.2 Electron-radiation interaction

The Hamiltonian HER is simply the A.p interaction introduced in §C.4 in Appendix

C. This form of the light-matter interaction is more appropriate than theE.d electric

dipole Hamiltonian, because the electrons in diamond are not localized around the

atomic cores. Instead, they are spread over the entire volume of the crystal, in so

called Bloch waves3.

Signal and control fields We divide the vector potential into two parts; the

weak signal field and the strong classical control,

A = As +Ac. (9.4)

The control field is written as

Ac(t, z) = vcAc(t, z)eiωc(t−nz/c) + c.c., (9.5)3The vector potential is not actually a physical field, and strictly we should apply the PZW

transformation (C.20) to the Bloch states, in order that our treatment is gauge invariant [177,178].However, this transformation simply shifts the electron momentum (c.f. (C.23)), and has no effecton the transition matrix elements. In any case, in the Heisenberg picture we are free to choose agauge such that A(t = 0) = 0, whence the PZW transformation becomes trivial [177].


where we have included the factor n = 2.417 in the exponent, which is the refractive

index of diamond. We treat the signal field quantum mechanically, so it is written

in second-quantized notation as

As(z) = vs

∫g(ω)ω

a(ω)e−iωnz/c dω + h.c., (9.6)

where we have used the one dimensional formula (C.10) along with (C.13) from

§C.4 in Appendix C. The mode amplitude g(ω) =√

~ω/4πε0nAc includes the

refractive index. Just as we did in (4.6) in Chapter 4, we anticipate the compact

spectral support of the signal pulse about its carrier frequency ωs by pulling the

mode amplitude g(ω)/ω out of the integral and defining a slowly varying envelope

operator A,

As(t, z) =vsgsωs

A(t, z)eiωs(t−nz/c) + h.c.. (9.7)

Here gs =√

2πg(ωs). We have also introduced the time dependence of the operators

arising in the Heisenberg picture (see Appendix B).

Electron wavefunctions It is sufficient to consider the interaction of the optical

fields with just a single active electron. This is because the complicated many-body

physics governing the behaviour of all the electrons in the crystal can be swept under

the rug of Fermi-Dirac statistics: the Pauli-exclusion principle prevents more than

one electron from occupying each state, and since all valence band states in the

crystal are initially occupied, any electron-electron scattering in this band is ‘frozen


out’ because no electron can change its state without occupying a previously filled

orbital. The upshot of this is that we may consider p to be the momentum operator

for a single electron.

The wavefunction ψk,n(r) of an electron with wavevector k in the nth energy band

is given by the product of a spatial phase factor with a periodic Bloch function,

ψk,n(r) = eikzuk,n(r). (9.8)

The Bloch functions have the same translational symmetry as the crystal lattice,

uk,n(z + a) = uk,n(z), which is a consequence of Floquet’s theorem. Note that such

states are not exactly eigenstates of the momentum operator; this is why the A.p

interaction can induce electronic transitions.

Matrix elements We are interested in the matrix element 〈α|HER|β〉 connecting

two quantum states. Neglect for the moment the state of the signal mode. This ma-

trix element is then given by the spatial overlap of the electronic orbitals describing

the initial and final states, with the operator A.p inserted between the two orbitals:

〈α|HER|β〉 = − e

m

∫crystal

ψ∗α(r)A(z).pψβ(r) d3r. (9.9)

Here the indices α, β are standing in for the wavevectors and band indices of the

initial and final orbitals. Now, the coordinate representation of the momentum

operator is p = −i∇. Applying this to ψβ(r), and using (9.8), we can write the


matrix element as


m

∫crystal

ei(kβ−kα)zA(z). [u∗α(r)(p+ kβ)uβ(r)] d3r. (9.10)

Since all the momenta are very close to the zone-centre, the spatial variation of

the exponential factors in the integrand is very slow, and this is also true for any

variation of the optical field A(z) (which contains only slowly varying envelopes

or similarly long-wavelength exponential factors). On the other hand, any rapid

spatial variation of the Bloch functions uα,β is periodic, repeated in every unit cell.

We therefore factorize the integral into two parts, as follows,


m

∑j

ei(kβ−kα)zjA(zj).[∫

unit cellu∗α(r)(p+ kβ)uβ(r) d3r

], (9.11)

where zj is the position of the jth unit cell. We define the matrix element pαβ as N

times the overlap integral inside the square brackets, where N = AL/a3 is the total

number of unit cells in the crystal. Taking the continuum limit for the sum over the

zj , we can write


mLpαβ.

∫ L

0A(z)ei(kβ−kα)z dz. (9.12)

We have now succeeded in separating out the local from the bulk dynamics. It only

remains for us to introduce the flip operators |α〉〈β| in a convenient form.


Excitons The momentum operator has negative parity, just as the atomic dipole

operator djk does (see §4.3.1 of Chapter 4). Therefore pαα = 0, so there is no

coupling of any state to itself. As mentioned above, if all electrons are in the valence

band, an electron has ‘nowhere to go’, since all the valence band states are occupied.

The only possibility is to promote an electron into the empty conduction band,

creating an exciton. This process, and its time-reverse — exciton recombination

— are the only important scattering processes involved in HER. Once an exciton

has been created, it is of course possible for either the conduction band electron or

one of the valence electrons to undergo scattering (this latter process is the same as

scattering of the hole) within their respective bands via the A.p interaction. But

the energies involved are much smaller than the photon energies in the signal or

control fields, so these processes do not conserve energy, and may be neglected. The

Hamiltonian can therefore be written as

HER(t) = − e

mL

∫ L

0

A(t, z).∑ν,k

pνk,0s†νke

ikz + h.c.

dz, (9.13)

where the subscript 0 on p denotes the crystal ground state, and where s†νk creates an

exciton with momentum k and energy ων . In the notation of (9.12), k = kβ−kα. The

Hermitian conjugate component destroys an exciton, representing recombination.

The energy of an exciton does not depend on its wavevector k, because the electron

and hole comprising the exciton might have large but opposite momenta, giving

a small total momentum, but a large total energy. Therefore the exciton state


is independently parameterized by the two quantities ν and k. Excitons, being

composed of pairs of fermions (a hole is a quasiparticle obeying fermionic statistics),

are bosons. The annihilation operators sνk therefore satisfy the same commutation

relation as photon mode annihilation operators (see Appendix C),

[sνk, s

†µk′

]= δν,µδk,k′ . (9.14)

A merciful simplification is achieved by neglecting any dependence of the matrix

elements pνk,0 on ν or k, since the dependence of the Bloch functions describing the

electron and hole on wavevector is rather weak. We thus write pνk,0 = ip, where p

is the constant magnitude of the matrix element, and where the factor of i appears

because the momentum operator is purely imaginary. With these simplifications,

the Hamiltonian can now be written in the form of (9.2), with the effective local

Hamiltonian given by

HER(t, z) = − iemp.A(t, z)

∑ν,k

s†νkeikz − h.c.

. (9.15)

As a final step, we can perform the sum over momenta in (9.15) explicitly. We define

the local exciton operator

Sν(z) =1√L

∑k

sνke−ikz, (9.16)


which satisfies the commutation relation

[Sν(z), S†µ(z′)

]= δν,µδ(z − z′). (9.17)

The local Hamiltonian then takes the form

HER(t, z) = − ie√L

mp.A(t, z)

[∑ν

S†ν(z)− h.c.

]. (9.18)

9.5.3 Electron-lattice interaction

The Hamiltonian HEL for the electron lattice interaction is just given by

HEL = V |with phonon − V |no phonon, (9.19)

where V is the potential experienced by an electron, generated by all the atomic

cores and other electrons. Let the displacement between the sub-lattices of diamond

caused by a zone-centre optical phonon by given by u. If this displacement is small,

a Taylor expansion to first order gives

HEL =∂V

∂u.u. (9.20)

When we quantize the lattice vibrations, the leading factor becomes the deformation

potential matrix elements, or just the ‘deformation potentials’ [174,179,180], and the

second factor becomes the operator for the optical phonon amplitude. For vibrations


along the direction vp with wavevector κ the phonon amplitude operator can be

written [180,181]

uκ = gκvpeiκz(b†κ + b−κ). (9.21)

Here b†κ creates an optical phonon with momentum κ. Phonons are bosons, so that

we have (see (C.8) in Appendix C)

[bκ, b†κ′ ] = δκ,κ′ . (9.22)

The phonon mode amplitude gκ, which has the dimensions of length (it is the lattice

displacement due to a single phonon), is given by

gκ =√

~NMωκ

, (9.23)

where M is the mass of a carbon atom and ωκ is the phonon frequency.

For reasons discussed in the previous section, the only states where electronic

scattering can occur are the exciton states, with an electron in the conduction band

and hole in the valence band. The action of the deformation potential is therefore to

destroy an exciton with energy and momentum (ν, k), and to produce a new exciton

with modified parameters (ν ′, k′). Summing over all possibilities gives the expression

HEL =∑

κ,µ,ν,k,k′

∫ L

0

1aLvp.Dνµkk′κs

†µk′sνke

i(k′−k)z × gκeiκz(b†κ + b−κ)

dz. (9.24)

The factor of 1/a appears to give the deformation potentials D the dimensions of


energy. The integral over space, along with the factor of 1/L, arises in precisely

the same way as it did in (9.12) above. A dramatic simplification of (9.24) is pos-

sible, since the deformation potentials D depend only very weakly on the phonon

and exciton momenta so close to the zone centre. By the same token, the phonon

frequency ωκ = ωp is independent of κ close to the zone centre (see Figure 9.3).

Dropping these dependencies, we can perform the summations over k, k′ and κ to

obtain the following expression for the effective local Hamiltonian,

HEL(z) =L3/2g

a

∑µ,ν

DµνS†µ(z)Sν(z)

[B†(z) +B(z)

], (9.25)

where Dµν = Dνµ = vp.Dµν is the real magnitude of the deformation potential

connecting excitons with energies ωµ and ων , and where we have defined the local

phonon operator

B(z) =1√L

∑κ

bκe−iκz, (9.26)

which has the commutator

[B(z), B†(z′)

]= δ(z − z′). (9.27)

9.5.4 Crystal energy

The energy H0 of the excited crystal is simply found by counting the number of

excitons and phonons. Using the number operators for these particles (see (C.9) in

9.6 Heisenberg equations 310

Appendix C), we find

H0 =∑ν,k

ωνs†νksνk +

∑κ

ωκb†κbκ. (9.28)

Or, in terms of the operators S and B, the local energy takes the form

H0

L=∑ν

ωνS†ν(z)Sν(z) + ωpB

†(z)B(z). (9.29)

9.6 Heisenberg equations

Now that we have constructed the Hamiltonians describing the Raman interaction,

we can write down the Heisenberg equations governing time evolution of the opera-

tors Sν and B. The dynamics of A are derived in the next section using Maxwell’s

equations, as was done for the atomic case in Chapter 4.

Commutation of Sν and B with H, using the relations (9.17) and (9.26), yields

the equations

∂tSν = iωνSν +e

m√Lp.A+ i

√Lg

a

∑µ

DµνSµ(B† +B),

∂tB = iωpB + iD√Lg

a

∑µ,ν

DµνS†µSν . (9.30)


9.6.1 Adiabatic perturbative solution

Let us define local operators Sν and B in a rotating frame, so that

Sν = Sνe−iων(t−nz/c),

B = Be−iωp(t−nz/c). (9.31)

For notational convenience, let us also define

B(t, z) = B†(t, z) +B(t, z)

= B†(t, z)e−iωp(t−nz/c) + B(t, z)eiωp(t−nz/c). (9.32)

The equations of motion for the slowly varying operators Sν and B are then given

by

∂tSν =e

m√Lp.Ae−iων(t−nz/c) + i

√Lg

a

∑µ

Dµν Sµeiωµν(t−nz/c)B,

∂tB = i√Lg

ae−iωp(t−nz/c)

∑µ,ν

Dµν S†µSνe

−iωµν(t−nz/c), (9.33)

where ωµν = ωµ − ων . The spatial phase factors are included for convenience when

considering the exponentials within A.

Our aim is to obtain an equation for B, the local phonon amplitude, in terms

of the signal and control fields. We achieve this by eliminating the intermediate

excitons Sν adiabatically. The procedure is related to that used in §5.3.3 in Chapter


5. We start by formally integrating the equation for Sν in (9.33),

Sν(t) =e

m√Lp.

∫ t

0A(t′)e−iων(t′−nz/c) dt′

+i√Lg

a

∑µ

Dµν

∫ t

0Sµ(t′)B(t′)eiωµν(t′−nz/c) dt′. (9.34)

We have set S(t = 0) = 0, since there are no excitons in the crystal initially.

Unfortunately (9.34) does not provide a direct solution for Sν , because it is coupled

to all the other excitons through the summation on the right hand side. We settle

instead for a perturbative solution. Substituting the first term on the right hand

side of (9.34) into the second term yields the approximate result

Sν(t) =e

m√Lp.

∫ t

0A(t′)e−iων(t′−nz/c) dt′ (9.35)

+i√Lg

a

∑µ

Dµν

∫ t

0

[e

m√Lp.

∫ t′

0A(t′′)e−iωµ(t′′−nz/c) dt′′

]B(t′)eiωµν(t′−nz/c) dt′,

This solution is correct to first order in the deformation potential. To perform the

integrals in (9.35), we use the fact that the time variation of the exponential factors

is much faster than the temporal dynamics of the optical fields, and also faster than

the dynamics of the crystal excitations produced by these fields. This adiabatic

approximation requires that the detunings ∆ν = ων − ωs of the excitons from the

signal frequency are all much larger than the bandwidths of the signal or control

fields. In diamond, the bandgap is in the ultraviolet, so this condition is always very

well satisfied if optical frequencies are used. We proceed by pulling all slowly varying


amplitudes out of the integrals and integrating only the exponentials. The resulting

expression for Sν contains 12 terms. Inserting this into the equation (9.33) for B

provides us with the dynamical description we have been looking for, although now

there are 144 terms! Fortunately, of the first order terms in the product S†µSν , only

very few contribute significantly to the memory dynamics: most terms are oscillating

at high frequencies, so that they average to zero. After some legwork, we obtain

∂tB = iKΩ∗A, (9.36)

where we have defined the control field Rabi frequency

Ω(t, z) =evc.pAc(t, z)

m~, (9.37)

and where the coupling constant K, with the dimensions of (length)−1/2×(time)1/2,

is given by

K =g

~a× ep.vsgs

~ωsm× 1√

L

∑µν

[Dµν

(ωµ + ωc)(ων + ωs)+

Dµν

(ωµ − ωc)(ων − ωs)

]. (9.38)

Let us suppose that we are sufficiently close to resonance that ων + ωs ων − ωs.

This need not be the case, because the bandgap in diamond is very large, but it

is a convenient simplification to neglect the ‘counter-rotating’ terms with summed

frequencies in their denominators (c.f. §4.6 in Chapter 4). The coupling now takes

9.7 Signal propagation 314

the approximate form

K =g

~a× ep.vsgs

~ωsm× 1√

L

∑µ,ν

Dµν

(∆µ + ωp)∆ν, (9.39)

where we have used the Raman resonance condition ωs = ωc + ωp. The double

appearance of the detuning in the denominator is characteristic of third-order scat-

tering. Equation (9.36) is very similar in form to the equation for the spin wave

(5.107) derived in Chapter 5. Thus encouraged, we proceed in the next section to

derive the equation describing the propagation of the signal field.

9.7 Signal propagation

The electric field of the signal is the solution to Maxwell’s wave equation, with the

polarization in the diamond acting as a driving term (see (4.34) in Chapter 4). The

polarization is the dipole moment per unit volume. That is, er =∫P dV , where

r is the one-electron position operator. We can relate the matrix elements of r to

those of the momentum operator p, as follows (see (F.6) in §F.2 of Appendix F),

pαβ = −imωαβrαβ, (9.40)


where ωαβ is the frequency splitting between the energy eigenstates |α〉, |β〉. Using

this relation, a second-quantized form for the polarization operator can be found:

P =ei

mAL∑ν,k

(pνk,0ων

s†νkeikz −

p0,νk

ωνsνke

−ikz

)= − ep

mA√L

∑ν

1ων

(S†νe

−iων(t−nz/c) + h.c.). (9.41)

Making the SVE approximation (see (4.39) in Chapter 4), we find the propagation

equation (∂z +

n

c∂t

)A = − µ0ω

2s

2gsksv∗s .Ps, (9.42)

where Ps is the component of the polarization oscillating at the signal frequency ωs.

Substituting the solution (9.35) into (9.41), performing the integrals with the help

of the adiabatic approximation, and retaining only those terms with the appropriate

time dependence, we arrive at the final equation for the signal,

(∂z +

n

c∂t + iχ

)A = iK∗ΩB, (9.43)

where K is very similar to K,

K =g

~a× ep.vsgs

~ωsm× 1√

L

∑µ,ν

ωsDµν

ων(∆µ + ωp)∆ν, (9.44)


and where χ represents a spatial phase picked up by the signal due to the crystal

dispersion,

χ =∣∣∣∣evs.pgs~mωs

∣∣∣∣2 × 1L

∑ν

ωsων∆ν

. (9.45)

The discrepancy between K and K is probably spurious, and may arise from the

secular approximation we use in eliminating terms oscillating at the ‘wrong’ frequen-

cies [177,178]. In any case, the difference between the two expressions is small close

to resonance, and never more than an order of magnitude. In what follows, we set

K −→ K.

The coupling constants are admittedly rather dense combinations of various ma-

terial and optical parameters. In the next section we will try to extract a prediction

for the coupling strength of a diamond quantum memory. But for the moment,

what is important is the form of the equations (9.36) and (9.43). They are es-

sentially identical to the Raman memory equations of §5.3.3 in Chapter 5. We

can make the similarity explicit with a number of coordinate transformations and

re-normalizations. First, we introduce the retarded time τ = t−nz/c (note the pres-

ence of the refractive index n). When the equations are written in terms of τ and

z, the derivative ∂z + nc ∂t becomes simply ∂z, while the time derivative ∂t becomes

∂τ . Next, we remove the dispersive factor χ with a phase rotation, by making the

transformation

A −→ A = Aeiχz,

B −→ B = Beiχz. (9.46)


To remove the dependence on the control field profile, we introduce the normalized

integrated Rabi frequency ω = ω(τ) — defined in (5.46) in Chapter 5 — and the

dimensionless transformed variables

α(z, ω) = i√WA(z, τ)Ω(τ)

, β(z, ω) =√LB(z, τ). (9.47)

Here we have also re-scaled the longitudinal coordinate by L so that z runs from

0 up to 1. As a final simplification, we replace K with K (as mentioned above),

and we assume that K is real. The equations of motion for the diamond quantum

memory then take the form

∂zα = −Cβ, ∂ωβ = Cα, (9.48)

where the dimensionless Raman memory coupling is

C =√LWK. (9.49)

This tells us that a diamond memory will behave in precisely the same way as a

Raman memory based on an atomic vapour (see (5.109) in §5.3.3 of Chapter 5). We

know that efficient storage is possible if the Raman memory coupling C is of the

order of unity (we should have C & 2). And all of the results pertaining to optimal

shaping carry over. So given the coupling C, we can directly find the optimal input

mode φ1 for the signal, using (5.97).

9.8 Coupling 318

9.8 Coupling

Here we show that efficient storage is possible in a small sample of diamond, so that

diamond quantum memory need not be a luxury enjoyed only by the super rich.

To estimate the memory coupling C, we must evaluate the summations in (9.39).

To do this we first write Dµν = Dδµ,ν , which holds approximately, since the wave-

functions of distinct excitons overlap poorly [172]. Next we assume that both con-

duction and valence bands are parabolic around the zone centre (as shown in Figure

9.5), so that we may write the energy of an exciton as [182]

ων −→ ω(k) = ω0 +~k2

2m?, (9.50)

where ~k is the momentum of the electron relative to the hole, and where m? is

an effective reduced mass for the exciton that describes the local band curvature.

Parameterizing the exciton energies in this way, the summations in K may be re-

written approximately as an integral over a sphere in k-space,

∑µ,ν

Dµν

(∆µ + ωp)∆ν−→

∑ν

D

(∆ν + ωp)∆ν(9.51)

≈ DAL

(2π)3

∫ kmax

0

4πk2

(ω0 − ωs + ωp + ~k2/2m?) (ω0 − ωs + ~k2/2m?)dk,

where kmax is an arbitrary, suitably large cut-off. The integral can be performed

9.8 Coupling 319

with a partial fraction expansion and a trigonometric substitution,

∫ κ

0

k2

(a+ bk2)(c+ bk2)dk =

b−3/2

a− c

[∫ √bκ0

a

a+ x2dx−

∫ √bκ0

c

c+ x2dx

](9.52)

=b−3/2

(a− c)

[√a tan−1

(√b/aκ

)−√c tan−1

(√b/cκ

)].

Choosing kmax sufficiently large, we get to the result [182]

∑µν

Dµν

(∆µ + ωp)∆ν≈ ALD

4πωp

(2m?

~

)3/2 (√∆ + ωp −

√∆), (9.53)

where ∆ = ω0 − ωs is the detuning of the signal field from the conduction band

minimum.

We can express W in terms of the energy Ec in the control pulse as follows,

W =∫ ∞−∞|Ω(τ)|2 dτ =

2πα|vc.p|2

~m2ω2cAn

× Ec, (9.54)

where here α = e2/4πε0~c = 1/137 is the fine structure constant.

We estimate the momentum matrix elements as vc.p ≈ vs.p = ~ × 2π/a. This

is justified by noting that the band gap at the zone centre is produced by Bragg

scattering that mixes electrons with k = 0 and k = 2∆k = 2π/a, and it is this latter

component that is responsible for interband transitions.

Other parameters are as follows. The bandgap4 in diamond is ~ω0 = 13 eV, and4This energy corresponds to the direct gap in diamond (at the zone centre, or ‘Γ-point’). The

conduction band minima occur elsewhere in the Brillouin zone (~ω0 ∼ 5 eV at the ‘L-point’), buttransitions to these states are mediated by phonons. They are therefore suppressed somewhat.Although they may not be insignificant, we neglect these processes here.

9.8 Coupling 320

the deformation potential has a value of around D ≈ 7~ω0 ≈ 90 eV. As mentioned

before, the phonons have wavenumber νp = 1332 cm−1. The lattice constant in

diamond is a = 3.6 A. The refractive index is n = 2.417, and the mass of a carbon

atom is M = 12 a.m.u. (indeed, by definition). For simplicity we assume a reduced

exciton mass m? = m. We then consider a crystal of length L = 1 mm, illuminated

by beams with waists 100 µm in diameter. If the control pulse is taken from a

modelocked pulse train with an 80 MHz repetition rate and an average power of

10 mW, it has an energy of 0.13 nJ. Using a signal field with central wavelength

λs = 800 nm, we estimate a memory coupling of

C = 2.4. (9.55)

The optimal storage efficiency is therefore around 99.9% (see Figure 5.5 in §5.3 of

Chapter 5). This demonstrates that efficient storage in a small sample of diamond

is extremely feasible. Of course, the estimate we have made is very crude, but

since neither the laser energy nor the sample size required are close to any practical

limitations, a downward revision of C by a factor as large as 100 could still be

accommodated. In fact (9.55) is something of an underestimate, since we have

neglected the counter-rotating terms in K. When the signal is so far detuned from

the conduction band edge, these terms still contribute significantly.

9.9 Selection Rules 321

9.9 Selection Rules

The large Stokes shift (i.e. the large phonon energy) in diamond makes it easy

to distinguish the signal and control fields spectrally. For instance, if the signal

field wavelength is 800 nm, the control wavelength is around 894 nm. In addition,

however, it turns out that the crystal symmetry requires the signal and control fields

to be orthogonally polarized. The Raman interaction that couples the input and

output fields is constrained to be proportional to an irreducible representation of

the symmetry group associated with the optical phonons. The zone-centre phonons

are always described by a subgroup of the crystal point group — the group of

reflections and rotations that leaves the crystal unchanged. In the case of diamond,

the crystal point group is the cubic group m3m (sometimes written Oh), and the

optical phonons transform as the subgroup Γ+5 (sometimes written T2g). The optical

fields are 3 dimensional vectors, and so the Raman interaction must be proportional

to the 3× 3 irreducible representation of the group Γ+5 , which takes the form [180]

Γ+5 (x) =

1

1

, Γ+5 (y) =

1

1

, Γ+5 (z) =

1

1

,

where zero elements have been left blank for clarity, and we have assumed that the

z-axis is aligned with the [001] direction (that is, parallel to a vertical edge of the

cubic unit cell; see Figure 9.1). The Raman interaction requires that vc = Γ+5 (z)vs,

from which it is clear that vc should be perpendicular to vs. This polarization

9.10 Noise 322

selection rule adds to the experimental attractiveness of a diamond memory.

9.10 Noise

The foregoing analysis of storage in diamond implicitly ignored the possibility of

unwanted couplings. For instance, the intense control pulse can stimulate strong

Stokes scattering, creating optical phonons and Stokes photons in pairs. This is the

same problem as that shown for the atomic case in part (b) of Figure 4.3 in §4.7 of

Chapter 4. Here it is exacerbated because the detuning from resonance is typically

so large that both processes — storage of the signal and Stokes scattering — occur

with roughly equal probabilities. In §6.3.2 of Chapter 6, a solution to this problem

is described that involves introducing a small angle between the signal and control

beams, and this solution certainly carries over to the diamond memory. Another

interesting possibility is that of modifying the optical dispersion so that the unwanted

Stokes light cannot propagate. It would be possible to do this by building a Bragg

grating — alternating layers of diamond and air — with a spatial frequency equal

to that of the unwanted Stokes light. Interference within the Bragg structure would

then suppress any Stokes generation. But we won’t consider this further; a proof-

of-principle demonstration of broadband storage in room temperature diamond is

challenging enough, without engaging in micro-fabrication [183].

In the next chapter, we review the experimental progress made in our group

towards the goal of demonstrating a Raman quantum memory.

Chapter 10

Experiments

Although most of this thesis is theoretical, I had initially intended to build a working

quantum memory. I have not succeeded as yet, but the ‘Memories’ subgroup is

continuing its efforts in this direction. In this chapter we discuss some of the ongoing

experimental work; its goals and future prospects.

10.1 Systems

The experimental programme divides into three projects.

1. Diamond,

2. Quantum dots,

3. Atomic vapour.

My experimental research has been focussed on the last of these; the theoretical

analysis in the preceeding chapter constitutes my contribution to the first. We will

10.1 Systems 324

not describe the quantum dot project here; suffice it to say that quantum dots may

be thought of as artificial atoms, so that a sample containing many dots behaves

like an atomic ensemble, in which Raman storage may be implemented.

Light storage in atomic vapour is becoming standard in quantum optics. In al-

most all cases, resonant EIT is used, and narrowband diode lasers provide the signal

and control fields [74,150,151]. Our research group has some considerable experience

with ultrafast lasers, however, and it was decided that a broadband Raman memory

would be interesting. Atomic vapour is an ideal system for demonstrating such a

memory; indeed this is the system that is considered when deriving the memory

equations in Chapter 4.

A common feature of all the experiments is the requirement of strong Raman

coupling between a laser field and a material sample. The easiest way to verify the

existence of strong coupling is to observe strong Stokes scattering. This is therefore

the first step in all our experiments — the general strategy is shown schematically

in Figure 10.1. Unfortunately, we have not yet been able to achieve this first pre-

requisite in the atomic vapour experiments. This is rather depressing, but we are

persevering, since there are many improvements to be made. In the rest of this chap-

ter, we will introduce the atomic species used in our experiments with atomic vapour.

We then discuss the theory of Stokes scattering, and we estimate the strength of the

coupling we expect to achieve. Finally, we describe the experimental techniques we

have developed in our attempts to see strong Stokes scattering. We finish with a

discussion of the planned realization of a Raman quantum memory.

10.2 Thallium 325

Vapour cellRaman pumpStokes

Filter

Laser

Figure 10.1 Observing Stokes scattering as a first step. StrongRaman coupling between a bright laser and the atomic ensemble isrequired for a Raman quantum memory. The ability to produce,and detect, strong, stimulated Stokes scattering is a sine qua non forimplementing the memory.

10.2 Thallium

The first incarnation of the atomic vapour experiment used thallium (Tl) as the

storage medium. Thallium is an extremely toxic poor metal, with a history of use

in rat poison, and homicide generally. However the atomic structure of thallium

exhibits a well-defined Λ-system, with a large Stokes shift (see Figure 10.2). For

this reason, attempts were made to build a Raman quantum memory with thallium

vapour, provided by means of a heated glass cell containing solid thallium. After

around a year of unsuccessful attempts to observe strong Raman scattering from

thallium vapour, it was realized that the vapour pressure of thallium is too low for

a strong Raman interaction to be engineered (see the discussion in §10.9.1, and part

(a) of Figure F.1 in Appendix F).

10.3 Cesium 326

1/ 27S

1/ 26P

378 nm

1283 nm

F=0

F=1

F=0

F=1

3/ 26P

F=1

F=2

Figure 10.2 Thallium atomic structure. The three levels of the Λ-system are discernible, marked by the thickened line segments. Thehyperfine structure (indicated by the fine branches) is ignored, beingnegligible in the ground P -wave manifolds. The large Stokes splittingbetween the J = 1/2, 3/2 states in the P -state manifold makes it idealfor broadband storage. Unfortunately Thallium has a low vapourpressure, making an efficient thallium quantum memory impractical.

10.3 Cesium

To increase the Raman coupling, an atomic species with a much higher vapour

pressure was required. Our current experiment uses cesium1 (Cs), which is many

orders of magnitude denser than thallium at room temperature (see part (b) of

Figure F.1 in Appendix F). Cesium is a soft, gold-coloured alkali metal that is,

thankfully, non-toxic, although it reacts explosively on contact with water, even the

water vapour in air! Our cesium is sealed in an evacuated glass cell, along with a

small amount of neon (10Ne), which acts as a buffer gas (see §§10.4 and 10.5 below).

There is only one stable isotope of cesium, namely 133Cs. Needless to say we refer

only to this isotope in what follows; our sample is naturally isotopically pure. The1Cesium is the American spelling; the British spelling Caesium retains some of the latinate

flavour if its etymology. The word derives from the latin for ‘sky’, because the 7P ↔ 6S1/2 doubletlines are a brilliant blue colour. However American spell-checkers and journal styles have worn medown, and now the British spelling seems odd.

10.3 Cesium 327

Λ-system is implemented in the so-called cesium D2 line at 852 nm. This is the

second of the strong ‘doublet lines’ (the D1 line is at 894 nm) characteristic of alkali

metals — the same doublet in Sodium illuminates the night-time activities of most

of this planet. The relevant atomic structure is shown in Figure 10.3.

9.2 GHz

852 nm

Figure 10.3 Cesium atomic structure. The F = 3 and F = 4hyperfine levels in the 6S1/2 ground state — the famous ‘clock states’— provide the ground and metastable states for the Λ-type quantummemory. The 6P3/2 manifold collectively provides the excited state.

The upper 6P3/2 state is split by the hyperfine interaction: The nuclear spin

of I = 7/2 combines with the total electronic spin of J = 3/2 to produce four

hyperfine levels with total angular momentum quantum numbers F = 2, 3, 4 and 5.

The interaction is weak however, and the splitting between each state is around 200

MHz. For the purposes of the memory, we therefore treat the excited state manifold

as a single state, which plays the role of |2〉 in the Λ-system.

The hyperfine interaction is much stronger in the 6S1/2 electronic ground state.

The reason is that the ground state is an S-wave, meaning that it has no orbital

angular momentum. It therefore has no azimuthal phase, and so it remains well-

10.4 Cell 328

defined at the origin without vanishing — a wavefunction with such a phase is multi-

valued at the origin unless it is zero, so higher orbitals must disappear at the origin.

In the ground state, then, the electron penetrates into the cesium nucleus. There is

therefore a strong magnetic dipole coupling between the nuclear and electronic spins,

known as the Fermi contact interaction, which acts to separate the two hyperfine

states with F = 3 and F = 4 by an enormous 9.2 GHz. These two states are

sometimes known as the clock states because coherent oscillations between them are

used in cesium atomic clocks. In fact the hyperfine splitting is now defined to be

exactly 9, 192, 631, 770 Hz, with the duration of the second being a derived quantity.

The clock states form the two lower states in the quantum memory Λ-system.

10.4 Cell

We contain the cesium vapour in a glass cell (the TT-CS-75-V-Q-CW from Triad

Technology in Colorado, USA). The cell is 10 cm in length, and is made of glass that

is resistant to heating up to temperatures of 500 C. The cell windows are 25 mm

across, and are made of optically polished quartz, to which an anti-reflection coating

has been applied which reduces reflection losses for light at the D2 wavelength of

852 nm down to around 0.1%. The cell is evacuated, and then a small sample of

solid cesium is introduced. Finally, the cell is backfilled up to a pressure of 20 torr

(∼ 2700 Pa) with neon gas. The reason for introducing this buffer gas is explained in

§10.5. The cell is then hermetically sealed, by pinching shut the glass tube through

which the cell contents are delivered.

10.4 Cell 329

10.4.1 Temperature control

The cell is wrapped in heating tape: a thin weave of high resistance wires surrounded

by thermal insulation. A thermocouple connected to an electronic temperature

controller allows one to set and maintain the cell temperature as desired. The glass

protuberance remaining after the cell is sealed provides a convenient ‘cold finger’

— an unheated region protruding from the cell where the cesium preferentially

condenses. As long as the cell windows remain hotter than this cold finger, cesium

does not collect on the cell windows, and the cell remains transparent to our laser

beams.

10.4.2 Magnetic shielding

The hyperfine levels in cesium are not pure quantum states. A hyperfine state with

total angular momentum quantum number F is (2F+1)-degenerate, being comprised

of Zeeman sublevels with quantum numbers m = −F,−F + 1, . . . , F − 1, F . These

quantum numbers represent the projection of the atomic angular momentum along

some axis, known as the quantization axis (see §F.4 in Appendix F). In the presence

of a magnetic field, the atomic angular momenta precess around the direction of the

field, and it is natural to define the quantization axis as aligned with this direction.

With this definition, the quantum numbers m remain good quantum numbers, but

the degeneracy of the Zeeman sublevels is lifted — this is known as the Zeeman

shift. Optical transitions between the Zeeman sublevels occur subject to selection

rules, which determine whether or not a transition conserves angular momentum,

10.4 Cell 330

based on the polarization of the incident light. With a judicious choice of polarized

lasers, one can prepare the ensemble in just one Zeeman sublevel. Such a ‘spin

polarized’ ensemble is used by Julsgaard et al. when implementing their continuous-

variables memory in cesium [122] (see §2.4 of Chapter 2). This type of ensemble

state is exquisitely sensitive to external magnetic fields, and it is common to build

a magnetic shield of so-called µ-metal (an alloy with a high magnetic permeability

that deflects magnetic field lines) around the vapour cell.

In our system, the spectral bandwidth of the laser pulses comprising the signal

and control fields is much larger than any feasible Zeeman shift one could produce

(see §10.6). Therefore the Zeeman sublevels cannot be resolved in the quantum

memory interaction, and so we neglect the Zeeman substructure of the hyperfine

states. There is no need to spin-polarize the ensemble, and no need for magnetic

shielding.

In §F.4 of Appendix F, we show that orthogonal circular polarizations are not

coupled by the Raman interaction. This is a further reason why it is not useful to

polarize the ensemble: it might have been possible to use the Zeeman selection rules

to our advantage (as is explained in the Appendix), but this result obviates this

possibility.

An unpolarized ensemble is in a mixed quantum state. It can be thought of as

several independent sub-ensembles, each with a different spin polarization. Each sub-

ensemble interacts coherently with the optical fields, however, and the theoretical

description of the memory interaction given in Chapter 4, 5 remains valid for each

10.5 Buffer gas 331

sub-ensemble.

The lack of any magnetic shielding means that there may be stray magnetic

fields that introduce a distribution of frequencies into the evolution of the Raman

coherence through the Zeeman shift. This may cause the spin wave to dephase, so

that the coherence is lost. However, as long as the magnetic fields remain constant

over the memory lifetime, the spin wave will re-phase, because the number of Zeeman

components is finite (this periodic re-phasing of a discrete set of oscillators is the

principle behind retrieval from the AFC quantum memory [91]; see §2.3.4 in Chapter

2). The periodic beating between the Zeeman sublevels restricts the times at which

efficient retrieval is possible to those times where the spin wave is ‘in phase’, but the

efficiency of the memory is not adversely affected.

The above discussion justifies our decision not to build a magnetic shield for our

cesium cell, and indeed not to attempt to polarize the vapour. However, since we

have not yet been able to observe strong Stokes scattering, we cannot be sure that

a magnetic shield would not help. We are investigating the construction of such a

shield.

10.5 Buffer gas

The 20 torr of neon buffer gas is added in order to extend the time that the cesium

atoms spend in the interaction region. The cesium atoms are deflected by collisions

with the neon atoms, so that their motion is diffusive rather than ballistic. The

mean time between collisions is given by 1/γp, as can be verified by computing

10.5 Buffer gas 332

〈τ〉 =∫ τ

0 τps(τ) dτ using the ‘survival’ distribution (F.15) in Appendix F. If we

follow the trajectory of a cesium atom, it will trace out a random walk with an

average step length l = vth/γp, where vth =√

2kBT/M is the average thermal

velocity of the atom. After N such steps, the root-mean-square displacement of the

atom from its starting point is D =√Nl. The beam diameter is roughly

√A, so an

atom escapes the beam after a time tescape, where

D(tescape) ∼√A,

⇒ tescape ∼ AMγp2kBT

. (10.1)

A typical diffusion time for a beam with diameter 100 µm is around 10 µs at room

temperature, which compares with around 0.5 µs in the absence of a buffer gas.

During the course of this Brownian motion, the atom undergoes around 105 col-

lisions. As described in Appendix F, these collisions randomize the phase of the

optical polarization, causing pressure broadening. But the hyperfine coherence —

the spin wave — is unaffected. The ground state hyperfine levels represent different

orientations of the cesium nuclear spin, but neon is spinless, so there is no magnetic

dipole interaction between the atomic nuclei in a collision. The Raman coherence is

therefore maintained, despite the frequent collisions of cesium atoms with the buffer

gas. The natural lifetime of the hyperfine coherence is several thousand years, so

the memory lifetime is set by tescape.

As well as slowing the escape of the cesium atoms, collisions with the buffer gas

10.6 Control pulse 333

change their velocities, so that the cesium atoms diffuse spectrally : as the cesium

atoms’ velocities change, so do their Doppler shifted resonant frequencies. Before

an atom leaves the interaction region, its resonance will ‘wander’ across the entire

Doppler profile (see §10.10 below). This means that the atoms can be optically

pumped (see §10.12 below) using a narrowband laser tuned into resonance with

stationary atoms, since all atoms will eventually drift into resonance with the pump

laser. Without a buffer gas, it is necessary to dither the pump laser frequency in

order to address all the atoms, which is inconvenient (although certainly possible).

10.6 Control pulse

The control pulse, or alternatively the Raman pump light, is sourced from a bespoke

Ti:Sapphire laser oscillator: a Spectra-Physics Tsunami. The oscillator is actively

modelocked using an acousto-optic modulator installed in the laser cavity, and the

laser produces a pulse train with a repetition rate of 80 MHz, which is set by the

cavity round-trip time.

10.6.1 Pulse duration

The bandwidth of the control pulse cannot exceed the 9.2 GHz Stokes splitting be-

tween the two lower hyperfine states, if a quantum memory is to be implemented.

However the gain bandwidth of Ti:Sapphire is very wide (several hundred nanome-

ters!), so some care was taken to limit the bandwidth of the laser. A Gires-Tournois

Interferometer (GTI) is installed inside the laser cavity. This is essentially a Fabry-


Perot etalon, with the rear of the two plates having a reflectivity of 100%. All

incident energy is therefore reflected, but the spectral phase inside the cavity under-

goes periodic jumps, with the free spectral range determined by the plate separation.

Any pulse in the laser cavity with a bandwidth spanning one of these phase jumps

experience catastrophic dispersion, which greatly reduces the efficiency with which

the pulse — being heavily distorted — can extract energy from the Ti:Sapphire

crystal. The bandwidth of the laser is therefore limited to bandwidths smaller than

the free spectral range of the GTI. Use of a GTI with an unusually large plate sep-

aration produces a modelocked pulse train with a spectral bandwidth or around 1.5

GHz (∼ 3.6 pm at 852 nm).

Beamsplitter Retroreflector

Photodiode

Mirror

Mirror

150 200 250 3000

500

1000

x (mm)

x

inte

nsity (

arb

. units)

Figure 10.4 First order autocorrelation. The set up is essentiallywhat is known as a Fourier transform spectrometer. An incidentpulse is split into two components, and one is delayed with respect tothe other. The arrangement with the retroreflector and facing mirrormakes the movable arm insensitive to misalignment as the retroflectoris translated over a large distance.

To characterize the pulse train, measurements of the first and second order cor-

relation functions of the output light were made. A Michelson interferometer was


built, so that interference of the optical field with a delayed copy of itself could

be observed. The first order field autocorrelation, shown in Figure 10.4, produces

a fringe pattern whose envelope is equal to the Fourier transform of the spectral

intensity I(ω) of the laser output (see (F.16) in Appendix F for an example of this

relationship). The spectral intensity is found to be approximately Gaussian, with

a FWHM bandwidth of 1.5 GHz, as mentioned above. This is consistent with a

Fourier-transform-limited pulse duration of 300 ps, but it may be that distortions

to the spectral phase of the pulse train ‘smear out’ the pulses, producing longer du-

rations. To investigate this possibility, we performed a second order interferometric

autocorrelation. The experimental set up is shown in Figure 10.5. Two delayed

copies of an incident pulse are focussed into a non-linear crystal — a small piece of

β-Barium Borate (BBO). Blue light scattered in the forward direction results from

the frequency upconversion of one pulse with its delayed counterpart. The envelope

of the resulting fringe pattern can be related to the pulse duration. Although the

shape of the autocorrelation envelope is not uniquely related to the pulse envelope

(it is not possible to ‘invert’ the autocorrelation to retrieve the pulse profile), it is

possible to infer the presence of spectral phase distortions by inspection of the lower

portion of the interferogram. In the presence of ‘chirp’ (a drift in the carrier fre-

quency of the pulse as a function of time), interference between the leading edge of

one pulse, and the trailing edge of the other, is suppressed, since the carrier frequen-

cies are no longer commensurate. This causes a distinctive narrowing of the lower

portion of the interferogram that is not present in our measurement. On the basis of


this result, we conclude that the pulses produced by our laser source are close to be-

ing transform-limited, with approximately Gaussian spectral and temporal profiles.

The FWHM duration of each pulse is around 300 ps, and the spectral bandwidth is

approximately 1.5 GHz. On the one hand, this is much narrower than the Stokes

splitting, as required. On the other hand, the pulses are much more broadband than

have been used to date in quantum memory experiments (pulse durations of 100’s

of microseconds are the shortest that are employed for EIT [74]).

BBO

Lens

150 200 250 3000

200

400

600

x (mm)

inte

nsity (

arb

. u

nits)

Beamsplitter Retroreflector

Photodiode

Mirror

Mirror

x

Filter

Figure 10.5 Second order interferometric autocorrelation. A non-linear crystal is inserted, and the frequency upconversion of one pulsewith its delayed counterpart is detected. The inset shows the mea-sured envelope of the interferogram. Again, the red lines are Gaus-sian fits. These data, along with the data in Figure 10.4, are the onlyexperimental data in this thesis that were taken by me!

10.6.2 Tuning

The plate separation of the GTI can be temperature tuned using a knob on top of the

laser, and this also provides a convenient way to smoothly tune the laser frequency

over small frequency shifts, of order 10 or 20 GHz (it appears that the spectral phase

10.7 Pulse picker 337

introduced by the GTI introduces differential gain over its free spectral range, which

allows small changes to the FSR to sweep the laser frequency). Larger frequency

shifts can be ‘dialled in’ using a birefringent filter (this is actually an intra-cavity

Lyot filter — see §10.13.2 below).

10.6.3 Shaping

Although a great deal of the theoretical work in this thesis deals with the optimiza-

tion of quantum storage by appropriate shaping of the control or signal pulse profiles,

it is not feasible to shape the pulses used in this experiment. They are too short

to be shaped electronically, and two narrowband to be shaped spectrally! However,

since we are still struggling to demonstrate storage in the first place, we are content

to walk before attempting a steeplechase. The excellent work of Novikova et al. [74]

on resonant shaped storage is a vindication of the theory of Gorshkov et al. [133], and

by extension the theory in this thesis, since they are so closely related (at least at

optical depths less than 50 or so [151]).

10.7 Pulse picker

The time between consecutive pulses in the output of our laser is 12.5 ns. This

is much shorter than the lifetime of the Raman coherence (see §10.5 above), so if

the pulse train from the laser is sent ‘as is’ into the cell, the ensemble does not

recover after each Raman interaction. If one’s aim is to generate strong Stokes

scattering, this may be beneficial. The Stokes process can be stimulated both the

10.8 Stokes scattering 338

presence of Stokes photons, and by the presence of spin wave excitations, so the

Raman coherence produced by a previous pulse may stimulate the Raman interaction

in subsequent pulses. However it is not possible to investigate the efficiency of a

quantum memory if different realizations of the memory interaction are coupled. To

demonstrate the quantum memory, it is necessary to reduce the repetition rate of the

laser so that each implementation of the memory is independent of all others. This

is done with a ‘pulse picker’, which is a fast optical switch based on the electro-optic

effect, known as a Pockels cell. The device is synchronized with the laser output,

and the optical switch is set to transmit every 80, 000th pulse, reducing the laser

repetition rate to 1 kHz. The ensemble now has 1 ms to ‘reset’ between pulses,

which is plenty of time.

In the next section we introduce the theory of Stokes scattering.

10.8 Stokes scattering

The requirements for observing strong Stokes scattering are very similar to the

requirements of a Raman quantum memory. A medium must be prepared in the

ground state of a Λ-system. There should be strong Raman coupling between an

incident laser pulse and the excitations of the medium. And it should be possible

to observe scattering at the Stokes-shifted wavelength by filtering out the strong

Raman pump.

A description of Stokes scattering runs along very similar lines to that of a Raman

quantum memory: the signal and control fields simply trade places, becoming pump


and Stokes fields (see part (a) of Figure 1.6, or part (b) of Figure 4.3). However,

instead of absorption, the Stokes field experiences gain, with energy being transferred

from the pump field into the Stokes field, while at the same time excitations are

generated in the Raman medium.

We are interested in transient Stokes scattering, which refers to the regime in

which the duration of the Raman pump pulse is much shorter than the lifetime of

the excitations in the medium. This is consistent with the requirement that the

same medium should be useful as a quantum memory, which requires that the spin

wave excitations far outlive the optical pulses.

The equations describing transient Stokes scattering in the adiabatic limit can

be written in the following simple form,

∂zα = Cβ†, ∂ωβ = Cα†. (10.2)

Here all the notation has the same meaning as in (5.109) from §5.3.3 in Chapter 5,

except that α is a dimensionless annihilation operator for the Stokes mode. Recall

that ω is the integrated Rabi frequency (defined in (5.46) in §5.2.6 of Chapter 5),

where now the relevant Rabi frequency is that of the Raman pump pulse. The cou-

pling constant C is the same as the Raman memory coupling, so the efficiency of

both Stokes scattering and quantum memory are characterized by the same number.

The equations (10.2) describe a squeezing, or Bogoliubov transformation as opposed

to a beamsplitter interaction: optical and material excitations are produced in cor-


related pairs. This interaction is the basis of the DLCZ quantum repeater protocol,

since the Stokes mode becomes entangled with the ensemble (see §1.6.4 in Chapter

1). The equations can be solved in precisely the same manner as the system (5.109).

The solution for the Stokes field is

αout(ω) = α0(ω) +∫ ω

0

√C2

ω − ω′I1

[2C√ω − ω′

]α0(ω′) dω′ (10.3)

+C∫ 1

0I0

[2C√

(1− z)ω]β0(z) dz.

Here the I’s denote modified Bessel functions, which describe exponential gain. We

are interested in the intensity of spontaneously initiated Stokes scattering, when

there are initially no spin wave excitations and no photons in the Stokes mode.

The average number of Stokes photons produced is given by the normally ordered

product

〈Nout〉 =∫ 1

0〈α†out(ω)αout(ω)〉 dω. (10.4)

On substitution of (10.3) into (10.4), a number of terms vanish. The cross terms

involving the products 〈α†0(ω)β†0(z)〉 and 〈β0(z)α0(ω)〉 are both zero, since in the

first case the inner product between singly-excited and vacuum states is taken, and

in the second case the vacuum state is annihilated. The same is true of the term

involving 〈α†0(ω)α0(ω)〉. The only non-vanishing term involves the anti-normally

ordered combination 〈β0(z)β†0(z)〉. To evaluate this term, we observe that β is a

bosonic annihilation operator which satisfies the same commutation relation as B


(see (4.45) in Chapter 4, or (9.27) in Chapter 9),

[β(z), β†(z′)

]= δ(z − z′). (10.5)

We therefore have that 〈β0(z)β†0(z′)〉 = 〈δ(z−z′)+β†0(z′)β0(z)〉 = δ(z−z′). Inserting

this result into the expression for 〈Nout〉, we obtain2

〈Nout〉 = 2C2[I2

0 (2C)− I21 (2C)

]− CI0(2C)I1(2C). (10.6)

Figure 10.6 shows the average number of scattered Stokes photons as a function of

the coupling C. C = 1 corresponds to 〈Nout〉 ∼ 1, and this may be thought of as

marking the onset of the stimulated scattering regime, when previously scattered

Stokes photons stimulate further scattering, so that the scattering efficiency begins

to grow exponentially with C. For efficient Raman storage, a quantum memory

requires C ∼ 1 also, so the possibility of producing stimulated Raman scattering is

a necessary condition for implementing a Raman quantum memory in any system.

In the next section we explain how to calculate a prediction for the coupling

constant C, when an atomic vapour is used as the memory medium (a calculation

of C for the case of a diamond quantum memory was undertaken at the end of the

previous Chapter).2This result appears as equation (38) in the seminal paper on Stokes scattering by Michael

Raymer and Jan Mostowski [139], who specialized to the case of a square pump pulse.

10.9 Coupling 342

0 2 4 6 810

−5

100

105

1010

Figure 10.6 Stokes scattering efficiency. As the coupling C in-creases, the number of Stokes photons scattered rises, growing expo-nentially for C & 1. The model neglects depletion of the energy inthe pump pulse: of course the number of Stokes photons scatteredcannot exceed the number of photons in the pump pulse, or indeedthe number of atoms in the ensemble.

10.9 Coupling

Throughout this thesis we have made reference to the quantities d, Ω, C etc...

Fortunately these quantities are not difficult to calculate in the context of an atomic

vapour.

10.9.1 Optical depth

Combining the definition (5.12) in §5.2.3 of Chapter 5 with (4.50) in §4.11 of Chapter

4, the optical depth can be written as

d =|d∗12.vs|2ωsnL

2γε0~c. (10.7)

The spontaneous emission rate 2γ can be expressed in terms of the dipole moment,

using Fermi’s golden rule to derive the stimulated emission rate, and then Einstein’s

10.9 Coupling 343

relations to connect this to the spontaneous rate [107]. The result is

2γ =ω3

21|d12|2

3πε0~c3. (10.8)

Here we have neglected any factors arising from degeneracy of the states involved.

The factor of 1/3 represents an average over all spatial direcitons. Substituting

(10.8) into (10.7), and assuming that the signal polarization vs is aligned with the

dipole moment d∗12, gives the result

d =3

4π× nλ2L

∼ nλ2L, (10.9)

where λ = 2πc/ω21 is the wavelength associated with the |1〉 ↔ |2〉 transition, and

where we have made the approximation ωs ≈ ω21 (any detuning is much smaller than

an optical frequency). (10.9) is consistent with the notion that the scattering cross-

section of an atomic transition is roughly λ2. The approximation in the second line of

(10.9) is not generally accurate, but it is extremely useful as an ‘order of magnitude’

estimate for the optical depth. The number density n can be found from the vapour

pressure, given the temperature of the atomic ensemble (see §F.1 in Appendix F).

For an optical transition with λ ∼ 1 µm and a typical ensemble length of L ∼ 1 cm,

we have d ∼ n[m−3]× 10−14. As a rule of thumb, this allows one to easily estimate

the atomic number density required for an efficient memory (with d & 100).

In general, a more accurate value for the optical depth should be calculated by

10.9 Coupling 344

using empirical values for the rate γ (recall that 2γ is the spontaneous emission rate)

and for the dipole moment d12. Sometimes data tables list the values of oscillator

strengths associated with atomic transitions. These are dimensionless numbers that

quantify the dominance of a transition over other possibilities within the atom. The

connection between the oscillator strength and the dipole moment for a transition

is derived in §F.2 of Appendix F. The optical depth can be expressed in terms of

the oscillator strength f12 for the |1〉 ↔ |2〉 transition as follows,

d =(π~αm

)× f12nL

γ, (10.10)

where m is the electron mass and α = 1/137 is the fine structure constant. Figure

10.7 below shows the variation of the optical depth of our cesium cell, with a length

of 10 cm, as a function of temperature.

10.9.2 Rabi frequency

The Rabi frequency is given by

Ω =d23.vcEc

~, (10.11)

where Ec is the electric field amplitude of the control associated with its positive

frequency component. The instantaneous intensity of the control field is given by

Ic =12ε0c× |2Ec|2. (10.12)

10.9 Coupling 345

300 350 400 450 50010

2

104

106

108

Temperature, K

Opt

ical

dep

th

Figure 10.7 Cesium optical depth. This is the optical depth as-sociated with the complete 6P3/2 manifold. The optical depth is thesame whether we consider transitions from the F = 3 ground statelevel, or the F = 4 level. The dipole moment for the cesium D2

line is around 10−29 Cm, according to the data provided by DanielSteck [184] (this figure has been divided by a factor of 3, because weconsider linearly polarized incident fields). The length of our cell is10 cm. The number density is found from the vapour pressure curvein Figure F.1 in Appendix F.

An estimate of the energy in the control pulse is then given by Ec = IcATc. Neglect-

ing any complex phase in Ec, we can express the peak Rabi frequency of the control

in terms of this energy,

Ωmax ≈d23.vc

~×

√Ec

2ε0cATc. (10.13)

This can, of course, be expressed in terms of the oscillator strength for the |2〉 ↔ |3〉

transition using the conversion formula given in Appendix F. If the control pulse

is taken from a pulse train — for example the output of a modelocked laser — we

can write Ec = P/R, where P is the average power in the beam, and where R is the

pulse repetition rate. The pulse duration Tc needs to be measured (see §10.6 above).

10.9 Coupling 346

10.9.3 Raman memory coupling

The Raman memory coupling is defined in ordinary units as C =√dγW/∆. An

approximate expression, assuming a top-hat shape for the control pulse, is

C2 ≈ Tcdγ ×(

Ωmax

∆

)2

. (10.14)

If (10.13) is used in this expression, the factors of Tc cancel, and the assumption of a

top-hat profile can be dropped. Since adiabatic evolution requires that ∆ Ωmax,

the above form of the Raman coupling makes it easy to see why an efficient, adiabatic

Raman memory should have Tcdγ 1 [133]. That is to say, the bandwidth δc ∼ 1/Tc

of the pulses stored (the optimal signal bandwidth is close to the control bandwidth)

should be small compared with dγ.

When estimating the magnitude of the Raman coupling, the following form can

be useful3,

C2 ≈(

~mA∆

)2

× (πα)2 × (f12f23)×NaNc, (10.15)

where Na is the number of atoms in the ensemble that are illuminated by the optical

fields, and where Nc is the number of photons in the control pulse. Putting f12 ≈

f23 ≈ 1 provides an upper limit to the Raman coupling that is achievable with a

given number of atoms and control photons.

As an example, an optical depth of d ∼ 3.8×104 is predicted for our cesium cell,

of length 10 cm, when heated to 90 C. Setting the control field beam waist to 3303This form of the coupling appears in our Rapid Communication on Raman storage [77], but

there is a typo! The form given here is correct.

10.9 Coupling 347

µm gives a Rayleigh range (see §6.4 of Chapter 6) of 5 cm, so that the beam remains

collimated over the full length of the cell. Setting the average power from our laser

oscillator to 600 mW gives a pulse energy of 7.5 nJ. One such pulse, focussed as

just described and detuned by ∆ = 20 GHz from the D2 line is sufficient to produce

a Raman memory coupling of C ∼ 2.4, corresponding to a storage efficiency of

∼ 99.9%. The memory balance is then R ∼ 0.5, so the memory is well within

the Raman regime (i.e. the interaction is balanced; see §5.3.1 in Chapter 5). The

detuning is around 50 times larger than the Doppler linewidth (see §10.10 below),

so the inhomogeneous broadening may be neglected. Adiabaticity is guaranteed

because the detuning is by far the largest frequency involved in the interaction,

∆/γ ∼ 104, ∆Tc ∼ 40, ∆/Ωmax ∼ 6.

10.9.4 Focussing

Substitution of (10.10) and (10.13) into (10.14) reveals that the Raman coupling

depends on the geometry of the ensemble in the following way,

C2 ∝ L

A. (10.16)

This is because the optical depth only depends on the length, and the squared Rabi

frequency depends only on the inverse of the beam area. The ensemble is addressed

by collimated laser beams, which in the ideal case (when the laser is working well)

have a Gaussian transverse profile. The Rayleigh range zR of a focussed Gaussian

10.9 Coupling 348

beam of wavelength λ is related to its cross-sectional area at the focus by diffraction,

zR =Aλ. (10.17)

The beam remains well collimated over a region of length zR either side of the focus,

but after this is quickly diverges, and the intensity drops rapidly. Therefore we may

consider that the length of the ensemble over which the memory interaction is rea-

sonably strong is limited by the Rayleigh range, L ∼ zR. Making this identification

in (10.16), and using the relation (10.17), we find

C2 ∝ 1λ. (10.18)

That is, the geometrical dependence of C drops out. The Raman coupling is inde-

pendent of how the beams are focussed. Loosely focussed beams are better described

by the one-dimensional theory, and indeed numerical simulations show that a loosely

focussed control beam improves the memory efficiency (see 6.5 of Chapter 6). But

if the beams are too loosely focussed, their Rayleigh range will extend beyond the

cell length, and the coupling will be limited by the dimensions of the cell. Therefore

the optimal situation obtains when the Rayleigh range of the beams is matched to

the cell length, zR = L/2.

10.10 Line shape 349

10.10 Line shape

Figure 10.8 shows an absorption spectrum for the D2 line, measured using a laser

with a spectral bandwidth of 1.5 GHz, with the cell heated to 70 C. A number

of factors contribute to the shape of the D2 absorption lines in a warm cesium

vapour. The absorption linewidth of each transition in the D2 line is around 250

MHz at room temperature for a single atom. But the large optical depth widens

the absorption lines of the ensemble (see §7.2.1 in Chapter 7, and §10.11 below).

The dominant line broadening mechanism is Doppler broadening, with pressure

broadening contributing at the level of ∼ 10 MHz. These mechanisms are explained

in §F.3 of Appendix F.

The Doppler width is calculated using (F.12) by setting M = 133 a.m.u. — the

mass of a cesium atom — and ω0 = 2πc/λ0, with λ0 = 852 nm the cesium D2 line

wavelength.

The pressure-broadened linewidth is calculated from (F.18), but the parameters

used are those of the buffer gas. The cesium cell contains neon, which was back-filled

up to a nominal pressure of pbuffer = 20 torr. The buffer gas reduces the mean free

path of the cesium atoms, so that they stay in the interaction region — the volume

illuminated by the light beams — for a longer time than they otherwise would (see

§10.5 below). The number density of neon is much greater than the number density

of cesium, at all reasonable temperatures, so a cesium atom is much more likely

to collide with a neon atom than with another cesium atom. The relevant number

density in estimating the pressure-broadened linewidth is therefore n = pbuffer/kBT0,

10.10 Line shape 350

−10 0 10 200

200

400

600

800

1000

Detuning, GHz

Sig

nal,

mV

Absorption spectrumNo cell

Figure 10.8 Cesium D2 absorption spectrum. This plot appearsthanks to Klaus Reim, a D.Phil student currently dividing his timebetween the cesium and quantum dot projects. A weak laser isscanned across the cesium D2 line at 852 nm. The blue line showsthe signal from a photodiode placed after the cesium cell. The reddotted line indicates the signal detected if the cell is removed. Thetwo dips correspond to the two ground state hyperfine levels: on theleft is the F = 3 state; the F = 4 state is on the right. The 9.2 GHzsplitting between these states is evident. The hyperfine structure ofthe upper 6P3/2 state is not resolved. The laser used has a Gaussianspectrum with a FWHM bandwidth of ∼ 1.5 GHz (see §10.6), andthis contributes to the wide absorption linewidth. Doppler broad-ening, along with the ‘smearing’ effect of the hyperfine splitting inthe upper state, and the large optical depth (d ∼ 104 at 70 C), ac-counts for the remainder of the linewidth, with pressure broadeningcontributing negligibly.

where T0 ∼ 300 K is the temperature at which the cell was filled. The cell is sealed,

so the number density of the buffer gas is fixed. The relevant collision velocity is

the thermal velocity of Neon atoms, calculated using M = 10 a.m.u.. A reasonable

figure for the collision cross-section σ is found by choosing datom = (300 + 50)/2 pm,

which represents an average of the atomic diameters of a cesium and neon atom.

Figure 10.9 shows the variation in the Doppler and pressure-broadened linewidths

with temperature for our cesium cell. The temperature dependence is very weak,

10.11 Effective depth 351

since both linewidths scale linearly with the thermal velocity of the atoms in the

vapour cell, which in turn scales with√T . It’s clear that Doppler broadening dom-

inates over pressure broadening. The linewidth is similar to the hyperfine splitting

in the 6P3/2 excited state manifold, so these hyperfine states are not resolved in our

sample. Fortunately it is not important, either for optical pumping (see §10.12 be-

low), or for the quantum memory interaction, to distinguish the hyperfine structure

of the excited state. Provided that Raman storage is implemented with a detuning

that is large compared to both the Doppler linewidth and the hyperfine splitting,

a theoretical description that ignores these complications remains appropriate. In

fact, Gorshkov et al. have shown that even resonant storage is unaffected by Doppler

broadening provided the ensemble is sufficiently optically thick [163]. However, char-

acterizing the optical depth of the ensemble is not entirely straightforward, in the

presence of line-broadening.

10.11 Effective depth

If a weak probe beam is tuned into resonance with one of the D2 line transitions,

the measured attenuation is much less than would be predicted on the basis of the

optical depth estimated using (10.9). The reason for this is the redistribution of

optical depth over a wide spectral range, because of Doppler broadening. Suppose

one measures the transmitted power Pout, and the input power Pin. The effective

optical depth is defined by

Pout = Pin × e−2deff . (10.19)

10.11 Effective depth 352

300 350 400 450 50010

6

107

108

109

Temperature, K

Lin

ew

idth

, Hz

Pressure

Doppler

Figure 10.9 Absorption linewidth. The pressure (green) andDoppler (blue) linewidths γ/2π for our cesium sample, calculatedusing the formulae (F.12) and (F.18) derived in Appendix F. Thered dotted line shows the half-width at half-maximum of the result-ing Voigt profile, which is the line profile resulting from the convo-lution of the pressure-broadened Lorentzian and Doppler-broadenedGaussian lineshapes. Doppler broadening dominates however, andthe lineshape is essentially Gaussian.

The spectrum of the transmitted beam can be found from (7.29) in Chapter 7,

setting z = 1 and Pin = 0. In the presence of pressure-broadening, the polarization

decay rate is modified,

γ −→ γ′ = γ + γp/2, (10.20)

and the pressure broadened optical depth dp = d×γ/γ′ should be used. The effective

optical depth is then given by

deff =12

ln ∫

Iin(ω) dω∫Iin(ω)e−2dp<[f(ω)] dω

, (10.21)

where Iin(ω) is the spectral intensity profile of the probe beam, and where the

lineshape function f(ω) is defined in (7.28) in Chapter 7, the only difference being

10.12 Optical pumping 353

that all frequencies are normalized by γ′ instead of γ.

Using a beam with an average power of 1 µW, taken from the output of a

modelocked Ti:Sapphire oscillator with a spectral bandwidth of 1.5 GHz (see §10.6),

we measure deff ∼ 3 at a cell temperature of 90 C. If we invert the formula (10.21),

using γ = 16.4 MHz (the spontaneous lifetime of the cesium D2 line is τ = 1/2γ = 30

ns), γ′ ≈ 70 MHz and γd ≈ 260 MHz, we infer a ‘real’ optical depth of d ≈ 3,500.

This is consistent with the prediction d = 3,700, found from (10.7) using the number

density plotted in part (b) of Figure F.1.

10.12 Optical pumping

Even though the hyperfine splitting is extremely large, it represents a very small en-

ergy gap at room temperature, with Esplitting ≈ kBT/680. Therefore the populations

of both the clock states are equal. From the perspective of quantum storage, this

means there is a large thermal background of incoherent material excitations that

would swamp any stored signal. But the situation is worse than this: the memory

interaction is totally destroyed if the populations of the two lower states are equal.

As signal photons are absorbed, they are also produced by Stokes scattering from

the thermal population in the storage state. The gain from thermally seeded Stokes

scattering exactly balances the quantum memory absorption, and the memory is

rendered useless. This effect is demonstrated in Figure 10.10.

For an efficient quantum memory, therefore, one of the lower two states must be

emptied. This is done in the laboratory by by optical pumping.


Inte

nsi

ty(a

rb. u

nit

s)

(a) (b)

Figure 10.10 Equal populations destroy quantum memory. (a):Absorption of a Gaussian signal pulse with the storage state empty.The memory efficiency is 73% (b): With equal populations in bothground and storage states, the absorption vanishes. The memory ef-ficiency here is ∼ 0.01%. These plots are produced by numericallyintegrating the full system of Maxwell-Bloch equations, without mak-ing the linear approximation introduced in §4.5 of Chapter 4. Thereare 8 equations to solve, one for each optical field (i.e. the signaland the control pulses), one for the population of each state in the Λ-system, and three equations describing the coherences between pairsof states. The numerical methods used are described in AppendixE. We consider a Raman quantum memory, with ∆ = 150, d = 300,and W = 122.5, and a Gaussian control pulse with duration Tc = 0.1(working in normalized units with frequencies expressed in terms ofγ, the natural linewidth of the excited state — pressure and Dopplerbroadening are not modelled). The Raman memory coupling is thenC = 1.28, and the balance is R = 0.64 (see §5.3.1 in Chapter 5).The signal field profile is identical to the control profile (no optimiza-tion is applied), but its amplitude is 1000 times smaller than thatof the control (the relative amplitudes of the signal and control areimportant when the equations are not linearized).

Suppose we wish to use the F = 4 state as the ‘ground state’ for the memory

— such a choice allows for efficient phasematched backward retrieval, as discussed

in §6.3 of Chapter 6. We then need to pump all the atoms into the F = 4 state,

leaving the F = 3 state completely empty. We accomplish this by tuning a laser

into resonance with the |F = 3〉 ↔ |2〉 transition, where |2〉 stands for any of the

states in the 6P3/2 manifold. The laser field excites atoms into |2〉, which then decay


by spontaneous emission back into both the F = 3 and F = 4 ground states (the

branching ratio is roughly equal). The laser field then re-excites the atoms, and

the process repeats (see Figure 10.11). Consider the effect of consecutive cycles of

excitation and decay. With each cycle, only around 50% of the initial population

of the F = 3 state ends up back in F = 3. The rest ends up in the F = 4 state.

Since there is no laser field exciting atoms out of this state, population builds up in

F = 4, while the F = 3 state is eventually emptied entirely. This is the principle

behind optical pumping.

Figure 10.11 Optical pumping. A CW diode laser is tuned intoresonance with the |F = 3〉 ↔ |2〉 transition. Spontaneous emissionredistributes the excited population more-or-less equally between thetwo ground hyperfine levels. After several cycles, population buildsup in the un-pumped F = 4 state, while the F = 3 state is emptied.

10.12.1 Pumping efficiency

To characterize the optical pumping efficiency a simple measurement was made (this

was done by Virginia Lorenz and Klaus Reim), based on an experiment performed

by Jean-Louis Picque in 1974 [185]. The set-up is illustrated in Figure 10.12. We use

an external cavity diode laser to provide the pumping light. This is a single-mode

10.13 Filtering 356

diode laser with a diffraction grating placed in front of it in the Littrow orientation:

the first order of diffraction is directed back into the laser. By fine-tuning the

grating angle, it is possible to select those frequencies that experience the greatest

optical gain (other frequencies are misaligned by the grating and are lost). The

diode laser frequency can therefore by stabilized and tuned using the grating. The

specular reflection from the grating provides the laser output, which is a continuous

wave (CW) beam with a linewidth of around 100 MHz. Using the measurement

scheme shown in Figure 10.12, we detect the fluorescence signal from a weak probe

beam. The suppression of this signal with the pump beam set at 30 mW over the

signal with the pump blocked indicates that we achieve a pumping efficiency of

around 95%, with the cell temperature set at T = 50 C. At higher temperatures,

the optical pumping efficiency drops as the energy in the pump is absorbed by the

larger number of atoms. We are currently installing an ECDL with an output power

of 100 mW, which should improve our pumping efficiency for higher temperatures.

10.13 Filtering

The greatest experimental challenge posed by the implementation of a Raman quan-

tum memory is the ability to filter out the very weak — even single photon — signal

field from the strong classical control pulse. The problem is particularly difficult in

cesium, because the Stokes shift of 9.2 GHz corresponds to a very small spectral

shift. At 852 nm — the D2 resonance wavelength — the signal and control frequen-

cies differ by around 20 pm. The required filter contrast is of order 108 or better,

10.13 Filtering 357

Laser diode

Lock-in

Pump

Probe

Cs cell

Heaters

Lens

Photodiode

Chopper

Grating

Beamsplitter

Figure 10.12 Verifying efficient optical pumping. The beam froman external cavity diode laser (ECDL) tuned into resonance withthe |F = 3〉 ↔ |2〉 transition is split into two parts at an asymmetricbeamsplitter (just a microscope slide). 96% of the diode beam is usedto pump the cesium atoms in the heated cell. 4% of the beam is sentthrough an optical chopper, which applies a 1 kHz modulation, beforebeing directed through the cell at a small angle to the pump beam.Fluorescence is collected by a lens and focussed onto a photodiode. Alock-in amplifier is used to isolate the component of the fluorescencedue to the modulated probe beam. This signal quantifies the opticalpumping efficiency. The inset shows the reduction in the fluorescencesignal as the pump power is increased. A pumping efficiency of 95%is achieved. These data appear thanks to Virginia Lorenz and KlausReim.

so that no single filter provides sufficient rejection of the pump: several filters must

be used in tandem. In this section we give details of the various filtering techniques

we have at our disposal.

10.13 Filtering 358

10.13.1 Polarization filtering

The Raman interaction couples orthogonal linear polarizations: a horizontally po-

larized control pulse will store a vertically polarized signal pulse, and vice versa,

provided the detuning is sufficiently large. And any Stokes scattering induced by a

linearly polarized pump pulse is polarized orthogonally to the pump. These facts are

derived in §F.4 in Appendix F. Therefore the pump can be effectively rejected after

the cell using a polarizer aligned orthogonally to the pump polarization. We use

Glan-Laser polarizers from Foctek, which use total internal reflection at the bound-

ary of two calcite crystals with perpendicular optic axes to preferentially transmit

one linear polarization. The extinction ratio of these polarizers is of order 10−6.

However, stress-induced birefringence of the cell windows can distort the pump po-

larization, causing leakage of the pump light through the polarizer. The effective

polarization extinction is therefore of order 10−4.

10.13.2 Lyot filter

A Lyot filter is a polarization-based spectral filter, popular among astronomers be-

cause of its wide working numerical aperture. In general a Lyot filter consists of

a number of concatenated filter stages, with each stage improving the finesse of

the filter (that is, reducing the spectral width of the pass-band, while increasing

the spectral width of the region over which frequencies are blocked). We built the

simplest possible type of Lyot filter, which involves a single stage. The filter then

has a sinusoidal transmission as a function of frequency. Figure 10.13 shows the

10.13 Filtering 359

structure of the filter. It is essentially a Mach-Zender interferometer for polariza-

tion: an incident pulse is split into fast and slow polarization components inside a

birefringent retarder. The polarizations are then combined on a polarizer, where

they interfere to produce spectral fringes. The free spectral range (i.e. the period

of the fringes in frequency) depends inversely on the length and birefringence of the

material used for the retarder. In order to be useful as a spectral filter for Stokes

scattering on the cesium D2 line, a free spectral range of ∆f = 2 × 9.2 GHz is

required, so that transmission of the Stokes light is accompanied by rejection of the

Raman pump light (see Figure 10.14). Calcite has the largest birefringence avail-

able, with |no − ne| = 0.17, but 10 cm of calcite are still required to produce the

required free spectral range! We use three pieces of calcite, each around 3 cm long,

mounted in series, along with a pair of Foctek Glan-laser polarizers (see previous

section). The filter contrast is limited to 99%, however, because of phase distortions

arising from the surface roughness of the calcite faces. The Lyot filter can be tuned

in frequency over one free spectral range by tilting one of the calcite crystals slightly,

which alters the optical path length.

10.13.3 Etalons

Since the Lyot filter does not have sufficient contrast for filtering the Stokes light, we

ordered some custom Fabry-Perot etalons. These are fixed air-gap etalons — pairs

of optically flat glass plates that form a planar cavity with a discrete transmission

spectrum. The free spectral range of the etalons depends on the separation L of

10.13 Filtering 360

optic axis

P1

B1

B2

P2

Retarder

Fringes(a) (b)

Figure 10.13 Lyot filter. (a): The filter consists of a piece ofbirefringent material of length L — a retarder — placed betweenpolarizers P1 and P2. The optic axis of the retarder is aligned at45 to P1, so that an incident pulse of wavelength λ (purple) is splitinto ordinary and extraordinary polarization components (red andblue pulses). One component is delayed with respect to the other,because the refractive indices no, ne associated with each componentare different. The phase retardance is given by φ = 2π|no − ne|L/λ.When the components recombine at P2, they interfere, producingsinusoidal spectral fringes, with a free spectral range ∆f = c/L|no −ne|. (b): Analogy with a Mach-Zender interferometer, in which twodelayed pulses are mixed on a beamsplitter, producing fringes.

the plates as ∆f = c/2L, since 2L is the distance an optical wave must traverse

to make a round-trip of the cavity. Requiring a free spectral range of 18.4 GHz,

as in the case of the Lyot filter, sets the etalon plate separation to be 8.2 mm. As

mentioned in §10.6 below, the laser bandwidth is around 1.5 GHz, so the width of

the pass band should not be smaller than this, in order to transmit the full Stokes

spectrum (which follows the Raman pump spectrum). The maximum finesse is then

F = 18.4/1.5 ∼ 12. The finesse of a Fabry-Perot etalon is fixed by the reflectivity

R of the plates comprising the cavity, according to the formula

F =π√R

1−R. (10.22)

10.13 Filtering 361

9.2 GHz

Stokes

Pump

Figure 10.14 Stokes filtering. A Lyot filter with a free spectralrange of 18.4 GHz can be used to suppress the Raman pump, whiletransmitting the weaker Stokes field. The finesse F = ∆f/δf , whereδf is the FWHM of the pass band, is only 2, so this filter is far fromideal.

The above considerations therefore fix the plate reflectivity to be 78%. This limits

the out-of-band extinction to around 95%, so one of these etalons is not quite as

good a filter as the Lyot filter. However, they are considerably more convenient,

being very easy to align, and much smaller! Six custom etalons with these specifi-

cations, and a large clear aperture of 15 mm, were ordered from CVI Melles Griot.

Initially it was intended to concatenate several etalons together, in series, so as to

combine their extinction. However, when the etalons are placed one after another,

their behaviour becomes complicated by interference effects arising from reflections

between the etalons. This is mitigated somewhat because the etalons we use have

anti-reflection coatings on their outside faces, which are also deliberately wedged.

Nonetheless it can be problematic to use the etalons consecutively.

10.13 Filtering 362

10.13.4 Spectrometer

Most recently, a grating spectrometer was built to filter the Stokes light (this was

done by Virginia Lorenz and Klaus Reim). The resolution required is R = λ/δλ >

852/0.02 ≈ 106. The resolution of a grating spectrometer is limited by the number

of lines ruled on the grating, R ≈ N . The spacing between consecutive lines cannot

fall below half the optical wavelength, or the first order of diffraction does not exist.

For light at λ = 852 nm, this limits the maximum groove frequency to less than

2300 lines mm−1. Using this groove frequency, a grating roughly 40 cm across is

required. Such a large grating is not available, but a spectrometer with a grating

around 6 cm across has been built, using a large off-axis parabolic mirror to focus

the diffracted light onto a photon-counting CCD camera. Although the resolution

of this spectrometer is sub-optimal, the ability to visualize the spectrum of the light

downstream from the cesium cell makes it an extremely useful apparatus.

All the above techniques have been used in an attempt to observe strong Stokes

scattering from the cesium cell. So far we have not been successful, but our combined

filter contrast is still improving.

10.13.5 Spatial filtering

The utility of spatial filtering for a quantum memory is discussed in §6.3.2 of Chapter

6. In a quantum memory, where both signal and control beams are directed by the

experimenter, it is possible to introduce a small angle between the beams so that

the strong control pulse can be blocked after the cell. When looking for stimulated

10.13 Filtering 363

Stokes scattering however, it is necessary to look along the axis of the Raman pump,

since the Raman gain is restricted to this direction. Therefore it is not generally

possible to spatially filter Stokes scattering, although attempting to detect scattered

light at very small angles from the pump direction may be sensible. A pinhole, or

even an optical fibre — fibre in-coupling is very spatially selective — can be used;

we are investigating these possibilities.

It is possible, however, to look for Stokes scattering in the backward direction.

The duration of our pulses is sufficiently long that they extend, in space, over the

length of the cell: cTc ∼ L. This means that a Stokes photon emitted near the exit

face of the cell in the backward direction, while illuminated by the leading edge of

the pump pulse, can propagate backwards, nearly all the way to the entrance face

of the cell, while all the time the cell remains illuminated by the pump, so that the

photon experiences Raman gain all along the cell’s length (see part (a) of Figure

10.15). The experimental set-up shown in part (b) of Figure 10.15 has been used

to look for this type of Stokes scatter. Strong stimulated fluorescence has been

observed, and the signal is extremely clear, because there is no need to filter out

the control pulse: it is propagating in the opposite direction! However the search

is still on for Stokes scattering, the signature being that the Stokes signal should

tune in frequency as the pump frequency is tuned, rather than remaining at the D2

resonance, as it does currently.

Similar arguments would suggest that a Raman quantum memory in which the

control and signal pulses are counter-propagating might be efficient, as long as the

10.14 Signal pulse 364

(a)

(b)

Figure 10.15 Backward Stokes scattering. (a): If the spatial ex-tent of the Raman pump pulse is comparable to the cell length,a backward scattered Stokes photon experiences Raman gain as itpropagates backward. (b): A polarizing beamsplitter can be usedto deliver the control pulse, and separate the orthogonally polarizedstimulated Stokes light that is scattered backwards. This essentiallyeliminates the background from the Raman pump. We are currentlylooking for Stokes light using this method.

pulses are sufficiently long. The propagation theory of Chapters 4 and 5 is not ap-

plicable in this situation, but the numerical model presented in Chapter 6 can be

used to show that the memory efficiency indeed remains high in this case. Such

an arrangement removes the demanding filtering requirements, and is therefore an

appealing possibility. A classical optical memory based on off-resonant Brillouin

scattering using counter-propagating pulses in an optical fibre has in fact been im-

plemented [186], so there is some precedent for this approach.

10.14 Signal pulse

Before implementing a single-photon quantum memory, a proof-of-principle demon-

stration using a weak coherent pulse for the signal field is planned. A single-photon

10.14 Signal pulse 365

source for the signal field is not, therefore, of immediate concern. However it remains

challenging to generate the signal field. Currently we have only one laser oscillator.

It is therefore necessary to generate the signal pulse from the control. The idea is to

sample a small portion of the control field, perhaps using a beamsplitter, and then

to apply a frequency modulation to shift the carrier frequency of the sampled pulse

by 9.2 GHz, so that this frequency shifted pulse can act as the signal.

Raman modulator One way to achieve this frequency modulation is to use the

cesium atoms themselves, by inducing strong Stokes scattering of a control pulse.

The Stokes sideband can then be sent into a second cesium cell along with the re-

maining control, where storage can be implemented. However, the Stokes scattering

process is somewhat aleatoric: large fluctuations in the intensity of the Stokes light

may make it difficult to assess the reliability of the memory.

EOM A second possibility is to use an Electro-Optic Modulator (EOM) to apply

the frequency shear. This is a device containing an electro-optically active crystal,

whose refractive index can be altered by the application of an external voltage.

Subjecting the crystal to a sinusoidally varying potential with frequency Ω will

imprint the waveform onto the phase of an optical wave passing through the crystal,

E(t) −→ E(t)× eiφ sin(Ωt), (10.23)

10.15 Planned experiment 366

where φ quantifies the amplitude of the phase modulation. Fourier transforming

(10.23) reveals the presence of sidebands, separated by the modulation frequency,

E(ω) =∞∑

k=−∞sgn(k)|k|J|k| (φ) E(ω + kΩ), (10.24)

where the sideband amplitudes are given by Bessel functions. With the choice

φ ∼ 0.2, the energy in the first sideband represents around 2% of the total trans-

mitted energy, while all higher order sidebands may be neglected. Use of such a

modulator allows one to reproducibly generate a weak signal pulse with the correct

frequency shift from the control pulse. A modulator with the ability to operate in

the microwave X-band at 9.2 GHz has been ordered from New Focus.

10.15 Planned experiment

The current experimental set-up is in a state of flux, we are continually improving

our filtering contrast and optical pumping efficiency, in the hope of detecting the

strong stimulated Stokes signal that indicates there is sufficient Raman coupling to

implement a Raman memory. Plans for the final implementation of the memory are

tentative, being contingent on the eventual success of these early stages. However in

the spirit of optimism we present in Figure 10.16 below a schematic of the possible

layout of an experimental demonstration of a cesium Raman memory. The optical

pumping beams are not shown, for clarity, although of course efficient optical pump-

ing is critical. The off-axis geometry for phasematched retrieval described in Chapter


6 is used, and we assume that the atoms have been prepared by optical pumping in

the upper F = 4 state for this purpose. The angle between the control and signal

beams is around 2, and we assume that the control is more loosely focussed than

the signal (the signal focus can be tightened by expanding the signal beam before it

enters the confocal system, but we have not shown the beam expander). Since we

seek only to demonstrate the feasibility of the memory, a long memory lifetime is

not important, and so the delay between the storage and retrieval control pulses is

adjusted by a mechanical delay stage. It is only feasible to move such a stage over a

few feet, which corresponds to just a few nanoseconds of variability in the memory

storage time, but this is sufficient for our purposes. The quarter wave plate in the

control beam following the cell rotates the control polarization through 90 (since

the control beam traverses it twice, the combined effect being that of a half-wave

plate). The rotated control then retrieves the signal field into the orthogonal polar-

ization mode to the polarization mode of the incident signal field. This allows the

retrieved signal to be re-directed to a detector using a polarizing beamsplitter. The

efficiency of the memory can be quantified by comparing the energy in the retrieved

signal field to that of the incident signal. The incident signal pulse is generated from

the control using an EOM (see §10.14 above), and an etalon removes the fundamen-

tal (the unmodulated light transmitted through the EOM). A Pockels cell is used

to reduce the repetition rate of the Ti:Sapphire laser, as described in §10.7.

We look forward to overcoming our present difficulties and assembling the above

apparatus, or a variation thereupon, in the near future.


EOM

Ti:Sapphire

Pulse

Picker Block L1 L2

QWP

Delay stage

E1

E2

Cs cell

Retrieved signal

Figure 10.16 A possible design for demonstration of a cesiumquantum memory. The pulse train from a Ti:Sapphire oscillatorpasses through a pulse picker to reduce the pulse repetition rate.Consider a single pulse. A beamsplitter redirects a portion of thepulse into an EOM, and the first sideband, shifted by 9.2 GHz is iso-lated from the output using a Fabry-Perot etalon E1 (see §10.13.3).This is the input signal field. The remainder of the initial pulse is usedas the control field. It is directed through the cesium cell at a smallangle to the signal pulse using a confocal arrangement (lenses L1 andL2): the signal is (hopefully) stored in the cell. The transmitted con-trol field is sent through a variable delay line, and its polarization isrotated through 90, before being sent back through the cell. Thestored signal field is retrieved with the orthogonal polarization to theincident signal, and is sent by a polarizing beamsplitter through anetalon E2 (to remove any residual control) to a detector. Note thatwe have not shown the optical pumping beams, which are critical.

This concludes the thesis. In the next chapter we summarize the results of the

present research.

Chapter 11

Summary

This thesis has been concerned with the problem of storing light. I can recall feeling

some puzzlement, lying in bed on a school night, at how completely my room dark-

ened when the light was switched off. Why did you have to keep pouring more and

more light into a room? Well, light is an ephemeral beast. But the possibility of

a material that remembers its illumination — not with the feeble pallor of glow-in-

the-dark paint, but with the unmistakable vigour of a laser pulse — is remarkable. I

am not the first to study such media, and the current research was undertaken in the

aftermath of the successes of light stopping by EIT. The main contributions of this

thesis are theoretical: a fairly general framework for the analysis and optimization

of light storage has been developed. The framework provides a unified description

of both EIT and Raman storage, and generalizes to tCRIB, lCRIB, AFC and broad-

ened Raman protocols. Further applications of the formalism are expected. The

use of the SVD has been crucial to the success of the theoretical programme. Many

370

of the results in the thesis are simple adaptations of well-known facts from linear-

algebra to the particular case under study. Another important ingredient of the

thesis is numerical simulation. The propagation of optical fields through an atomic

ensemble is always described by a set of coupled linear partial differential equations,

and these are particularly easy to solve on a modern computer.

The ‘take-home’ results are as follows.

1. Any quantum memory is a linear system, with storage and retrieval interac-

tions described by Green’s functions, which are essentially large matrices.

2. The SVD of the Green’s functions provides a complete characterization of the

memory. It allows one to immediately identify the optimal input mode.

3. The singular values of the Green’s function are invariant under unitary trans-

formations. This fact can be very useful in analyzing the memory interaction.

One can work in the Fourier domain, or indeed in an entirely unfamiliar coor-

dinate system.

4. The efficiency of a memory is limited by its optical depth [133].

5. Explicit expressions for the Green’s functions describing storage in a Λ-type

atomic ensemble are provided. The Rosen-Zener kernel holds whenever the

control has a hyperbolic secant temporal profile. The adiabatic kernel holds for

all detunings and control pulse shapes that satisfy the adiabatic approximation.

The Raman kernel holds for adiabatic interactions that are far-detuned and

‘balanced’.

371

6. In the adiabatic limit, a coordinate transformation links the results for different

control profiles. Therefore the SVD only needs to be computed once for a

given control pulse energy. Changes to the control profile simple change the

coordinate transformation.

7. A Raman memory may be characterized as a multimode beamsplitter inter-

action between optical and material modes. A single set of modes describes

both the transmitted fields and the stored excitations. A single number C

characterizes the efficiency of a Raman memory.

8. The Green’s function can always be constructed numerically, so that the op-

timal input modes and memory efficiencies can be found at any detuning,

regardless of whether or not the interaction is adiabatic.

9. Retrieval of the stored excitations can be problematic. Forward retrieval suf-

fers from re-absorption losses. Backward retrieval is not phasematched if the

ground and storage states are non-degenerate. Numerical simulations ver-

ify that an off-axis geometry allows for efficient backward retrieval with non-

degenerate states. Loose focussing of the control field is desirable.

10. The multimode capacity of a quantum memory can be evaluated by consid-

ering the SVD of the Green’s function. The multimode scaling of EIT, Ra-

man, tCRIB, lCRIB, AFC, and a broadened Raman protocol is studied. Un-

broadened ensembles have poor multimode scaling. Adding an inhomogeneous

broadening improves the scaling. The AFC protocol has the best multimode

11.1 Future work 372

scaling of all the protocols studied.

11. If one is not able to shape the signal pulse, optimal storage can still be achieved

by instead shaping the control. But to solve the optimization problem, the

equations of motion for the memory must be solved numerically. This can

be done rather quickly, however. A simple optimization algorithm works well

for finding the optimal control profiles. The SVD allows one to verify the

optimality of the numerical solutions. This optimality suffers as the interaction

becomes less adiabatic.

12. A Raman memory in bulk diamond, based on the excitation of optical phonons,

is feasible. It is shown that the equations of motion describing the Raman

interaction in diamond have precisely the same form as the Raman equations

describing storage in an atomic vapour. The form of the coupling constant C

is derived.

13. Attempts to implement Raman storage in cesium vapour have been made, but

it is proving difficult even to generate and detect Stokes scattering. I am not

a good experimentalist!

11.1 Future work

There is a great deal of experimental work to do. It may be that there is a good

reason why our attempts to build a Raman memory have been unsuccessful: we

should either make the memory work, or find this reason, in the coming year.

11.1 Future work 373

An intriguing theoretical challenge is how to make use of the stored excitations

once they are in place. Is it possible to perform computationally interesting oper-

ations on the spin wave in an atomic ensemble? Can operations be designed that

allow different stored modes in a multimode memory to interact? Work on this front

has begun in the literature [17,187,188], but this is likely to be a rich seam.

If you have survived this far, I am very grateful for your attention! I hope that

some of the results in this thesis are useful to other workers in the field, even if only

as a warning of what not to try.

Appendix A

Linear algebra

Physics is generally concerned with change: the evolution of a system over time, or

the response of a system to an external agent. The easiest, and most uninspiring

situation to analyze, is when there is no change and no response. Linear algebra

is concerned with the much more interesting situation arising when the response

depends linearly on some parameter. On the one hand, this is almost always an

approximation that only holds for small changes in a parameter, and small responses.

So linear algebra rarely provides an exact description. On the other hand, the

linear approximation can be successfully applied to nearly every physical system!

Linear algebra is therefore useful in almost every branch of physics, as well as in

mathematics and science generally. In our case, the linear response of a quantum

memory is certainly an approximation, valid when the signal field does not contain

too many photons.

Here we summarize various concepts that are needed to properly understand the

A.1 Vectors 375

singular value decomposition as it pertains to the optimization of quantum storage.

There is a significant overlap with the formalism of quantum mechanics, so we

will also take this opportunity to review some aspects of that formalism. A clear

and comprehensive introduction can be found in Nielsen and Chuang’s quantum

information bible [158].

A.1 Vectors

A vector is essentially a list of numbers. It also helps to keep in mind the image of

a vector as an arrow (see Figure A.1). This analogy cannot always be made with

rigour, but it provides a convenient visualization. The numbers comprising a vector

are the components of the arrow along the coordinate axes. When writing down

these components, implicit reference is therefore always made to some coordinate

system. It’s clear that we could rotate the coordinate axes — altering the vector’s

components — without changing the arrow in Figure (A.1), and in this sense, a

vector transcends its components. Nonetheless, it will be useful to write down the

vector components for concreteness.

We will use two equivalent sets of notation for a vector labelled ‘v’. Either v, or

|v〉. The first symbol, in bold face, is in general use. The second — a ket — is an

example of ‘Dirac notation’, used only in the context of quantum mechanics. Dirac

notation is at times very convenient, and it will help to be able to use these two

types of notation interchangeably.

The number of components of a vector is called the dimension of the vector. If

A.1 Vectors 376

Figure A.1 A vector. On the left is a representation in ‘componentform’. On the right the same mathematical object is drawn as anarrow. The direction and length of the arrow are determined by itscomponents α, β. Implicitly, a coordinate system (thinner arrows) isused to define the components.

there are n components, the vector is said to be n-dimensional. We will adopt the

convention that v is a column vector;

v = |v〉 =

v1

v2

...

vn

. (A.1)

Vectors can be added together, provided they have the same dimension,

v +w = |v〉+ |w〉 =

v1 + w1

v2 + w2

...

vn + wn

. (A.2)

A.1 Vectors 377

And a vector can be multiplied by a number, say α, like this

αv = α|v〉 =

αv1

αv2

...

αvn

. (A.3)

Using these operations, vectors can be combined together to make new vectors, and

it is convenient to think of all these possible vectors as inhabiting a ‘space’, known

as an n-dimensional vector space. Our own universe, with three spatial dimensions,

can be thought of as a 3-space.

A.1.1 Adjoint vectors

The components of a vector do not have to be real numbers. In quantum mechanics,

and many other applications, they are generally complex numbers. A useful concept

in this case is the Hermitian adjoint v† of a vector v. This is simply another vector,

this time a row vector, with each component equal to the complex conjugate of the

corresponding component of v,

v† = 〈v| =(v∗1 v∗2 . . . v∗n

). (A.4)

The symbol 〈v| is known, rather unfortunately, as a ‘bra’, for reasons that will

become clear. These row vectors (bras) can be added or multiplied by numbers in

the same way as column vectors (kets), and so they form their own vector space,

A.1 Vectors 378

sometimes known as the adjoint space.

Every vector v has a corresponding Hermitian adjoint v†; every ket has its

corresponding bra. And Hermitian conjugation is involutive: The Hermitian adjoint

of v† is v again.

A.1.2 Inner product

Vectors can be multiplied together in ways as various as mathematicians are inven-

tive. The inner product — sometimes scalar product — is defined as the sum of the

component-wise products of a bra and a ket with the same dimension,

v†w = 〈v|w〉 = v∗1w1 + v∗2w2 + . . .+ v∗nwn. (A.5)

This type of product between two vectors is not another vector; it’s just a number.

For some reason it’s rather satisfying to take an inner product, and Paul Dirac’s

notation anticipates something of this satisfaction. When a bra 〈v| encounters a ket

|w〉 they merge to become a ‘braket’ 〈v|w〉, and so fulfill their destiny.

It’s quite common to speak of taking the inner product of two kets, |v〉 and

|w〉. In this case it is understood that one of the kets has to be replaced by its

corresponding bra before using (A.5). Note that 〈v|w〉 = 〈w|v〉∗, so one should be

consistent about which of the two kets is replaced.

A complex vector space with an inner product defined as in (A.5) is known as

a Hilbert space. Quantum mechanics is a theory about vectors in Hilbert space; as

such it is extremely simple. It is the reconciliation of this mathematical structure

A.1 Vectors 379

with what we know about the real world that makes the theory so difficult to pin

down.

A.1.3 Norm

Having defined the inner product, we can now define the norm of a vector. This is

defined as the square root of the inner product of a vector with itself,

v = ||v|| = || |v〉 || =√v†v =

√〈v|v〉. (A.6)

This is a positive, real quantity that grows with the size of the components of v. In

fact, substituting in the definition (A.5) and applying Pythagoras’ theorem shows

that the norm is simply the length of the arrow representing the vector v. Some

further geometrical manipulations reveal that the inner product is related to the

angle θ between the arrows representing two vectors, as follows (see Figure A.7),

v†w = 〈v|w〉 = vw cos θ. (A.7)

An immediate consequence of this is that the inner product 〈v|w〉 vanishes when

θ = π/2, that is, when v is perpendicular to w. Often, the word orthogonal is

used instead of perpendicular, especially in the case that the vectors involved are

complex, or when they have dimension greater than 3, since then the notion of an

angle is less transparent.

A.1 Vectors 380

Figure A.2 The inner product of two vectors.

A.1.4 Bases

We have already mentioned in passing the concept of a coordinate system, with

respect to which the components of a vector are defined. We drew the coordinate

axes in Figure A.1 as black arrows. The axes themselves are therefore described by

a pair of vectors, one pointing along the x-axis; the other along the y-axis. Let’s

call them |x〉 and |y〉. If we fix the length (the norm) of these vectors as 1, then we

can write any vector v directly in terms of |x〉 and |y〉,

|v〉 =

vx

vy

= vx|x〉+ vy|y〉. (A.8)

The set of two vectors |x〉, |y〉 is a basis from which we can construct any other

2-dimensional vector. In fact, it’s clear that any two vectors, as long as they point

in different directions, can serve as a basis. The nice feature of the set |x〉, |y〉 is

that these two vectors are orthogonal to each other, 〈x|y〉 = 0. This is particularly

convenient, because the components of |v〉 can be found directly by taking inner

products, vx = 〈x|v〉, vy = 〈y|v〉. A basis of this kind is almost always preferable to

non-orthogonal bases. Such a basis, with mutually orthogonal basis vectors of unit

A.2 Matrices 381

norm, is called an orthonormal basis. When we speak of a coordinate system, or

coordinate axes, we are implicitly making reference to an orthonormal basis.

A.2 Matrices

A matrix is essentially an array of numbers, laid out on a rectangular grid, as follows:

M =

M11 M12 . . . M1n

M21 M22 . . . M2n

......

. . ....

Mm1 Mm2 . . . Mmn

(A.9)

The numbers Mij comprising a matrix are known as its elements. The dimension

of a matrix is specified by two numbers, the number of rows, and the number of

columns in the matrix. In the example (A.9) M has dimension m× n. Just as with

vectors, matrices are greater than the sum of their parts: the actual values of the

elements of a matrix are not important. To visualize a matrix, one should imagine

a process, in which a vector is transformed into another vector (see Figure A.3).

For this reason, matrices are sometimes referred to as maps, since they map one

vector onto another. The term operator is also used, since a matrix can be viewed

as an operation — rotation, or reflection, say — applied to a vector. The way this

operation is performed mathematically is via matrix multiplication, written like this,

w = Mv, or |w〉 = M |v〉. (A.10)

A.2 Matrices 382

This multiplication is evaluated by combining the elements of M and the components

of v to form w, in the following way. Define a set of column vectors

mj = |mj〉 =

M1j

M2j

...

Mmj

, (A.11)

so that each column of the matrix M is given by one of these vectors,

M =

m1

m2

. . .

mn

. (A.12)

The vector w is then given by a weighted sum of the mj , with coefficients equal to

the components of v,

w = v1m1 + v2m2 + . . .+ vnmn. (A.13)

From (A.13) it is clear that the number of columns of M must be the same as the

dimension of v for this multiplication to be possible. Thus the number of columns

of M sets the dimension of the vectors upon which M can act. Similarly the number

of rows of M sets the dimension of the vectors mj , and therefore the dimension of

the output vector w. So an m×n matrix is an operator that acts on n dimensional

A.2 Matrices 383

vectors to produce m dimensional ones.

Figure A.3 A matrix acting on a vector. Here M maps the initialvector v (black) onto the final vector w (red), via a rotation and a‘dilation’ (length increase). The values of the matrix elements Mij

depend on the components of v and w, which in turn depend on thedirection of the coordinate axes (thin arrows). Rotating the coordi-nate axes would change the Mij , but the transformation representedby M would be the same. In this sense, a matrix is more fundamentalthan its elements.

Matrices with the same dimensions can be added together; M + N is just the

matrix whose elements are given by the sum of the corresponding elements of M

and N . And of course they can be multiplied by numbers. αM is a matrix whose

elements are αMij . Incidentally, these properties mean that the space of all matrices

is actually also a vector space. But this will not be important for us.

Two matrices can be multiplied together to produce a new matrix. In the product

MN = Q, each of the column vectors qj of Q are formed from the corresponding

column vector nj of N , by combining the column vectors mj of M in a weighted

sum like (A.13), with coefficients given by the components of nj . So M acts on each

column of N to produce the columns of Q.

To multiply a row vector (or a bra) by a matrix, we simply treat the row vector

A.2 Matrices 384

as a 1× n matrix, and apply the above rule. We therefore obtain

〈v|M =(〈v|m1〉〈v|m2〉 . . . 〈v|mn〉

). (A.14)

Matrices also have Hermitian adjoints. The Hermitian adjoint M † of M is given

by swapping the rows and columns of M , and taking the complex conjugate of all

its elements,

M † =

M∗11 M∗21 . . . M∗m1

M∗12 M∗22 . . . M∗m2

......

. . ....

M1n∗ M∗2n . . . M∗mn

. (A.15)

Using this definition, it’s easy to check that Q† = N †M † (note the reversed order of

M and N), and that (M |v〉)† = 〈v|M †. These facts are useful when manipulating

expressions involving several matrices.

A.2.1 Outer product

The outer product of two vectors |v〉 and |w〉 is a matrix, written as

|v〉〈w| = vw† =

v1w∗1 v1w

∗2 · · · v1w

∗m

v2w∗1 v2w

∗2 · · · v2w

∗m

......

. . ....

vnw∗1 vnw

∗2 · · · vnw

∗m

. (A.16)

A.2 Matrices 385

Each column of this matrix is just |v〉, multiplied by the corresponding element of

〈w|, so that its structure can be visualized as that of a row vector 〈w| with column

vectors |v〉 ‘hanging’ from it. The Dirac notation is very satisfying in this context,

since the result of applying the operator |v〉〈w| to a third vector |x〉 is written like

this,

|v〉〈w||x〉 = |v〉〈w|x〉 = 〈w|x〉|v〉. (A.17)

That is, the matrix product of |v〉〈w| with |x〉 is just the same as the inner product

of 〈w| and |x〉, multiplied by the vector |v〉. If |x〉 = |w〉, the result is w2|v〉.

As |x〉 deviates away from |w〉, the inner product 〈w|x〉 gets smaller and smaller,

until it vanishes, when |x〉 is orthogonal to |w〉. A natural interpretation for the

operator (A.16) is therefore as a kind of ‘switch’ that maps an input from |w〉 to |v〉.

Operators of this kind are sometimes known as flip operators, or transition operators,

in quantum mechanics. Breaking down larger operators into flip operators can often

provide valuable insights.

A.2.2 Tensor product

A further generalization of the outer product is the tensor product. The tensor

product is used to combine vector spaces together to produce a new, larger space.

Suppose we have an n-dimensional vector space V, and also an m-dimensional space

W. The tensor product V ⊗W of these two spaces would be the space of all vectors

with dimension nm. Vectors v and w from the smaller spaces can be combined

together via the tensor product to produce a vector v⊗w inhabiting the larger space.

A.2 Matrices 386

And similarly matrices M and N acting on the smaller spaces can be combined

together to produce an operator M ⊗N , that acts on vectors in the tensor product

space V ⊗W. The result of applying the combined operator to the combined vector

is the same as the result of applying the operators to the vectors separately, and

then taking the tensor product:

(M ⊗N)(v ⊗w) = (Mv)⊗ (Nw). (A.18)

A common example arising in quantum mechanics is the tensor product of a pair

of 2-dimensional vectors, representing the state of a pair of qubits (a pair of elec-

tron spins perhaps). Suppose one qubit is in the state labelled |0〉, and the other

is in the state |1〉. The combined state |ψ〉 of both is found by taking the ten-

sor product of these two vectors, |ψ〉 = |0〉 ⊗ |1〉. Sometimes the more compact

notation |01〉 is employed, where the meaning should be clear from the context.

But other 4-dimensional vectors, which cannot be represented as tensor products

of 2-dimensional vectors, can exist in the 4-dimensional tensor product space. For

instance, the vector |ψ〉 = (|01〉 + |10〉)/√

2 is a valid state in quantum mechan-

ics (see (1.7) in Section 1.6.4 of Chapter 1). It cannot be written in the form

|state 1〉 ⊗ |state 2〉, but it is a 4-dimensional vector, produced by adding together

two vectors that can be written in this form. Vectors of this kind, that exist in the

tensor product space, but cannot be written as a tensor product of vectors from

the component spaces, are known as non-separable. In quantum mechanics, they

A.2 Matrices 387

represent states that are entangled.

The tensor product of two matrices M (with dimension m × n) and N (with

dimension p×q) is found by ‘attaching’ a copy of M to each element of N , as shown

below,

M ⊗N =

M11

N11 · · · N1q

.... . .

...

Np1 · · · Npq

· · · M1n

N11 · · · N1q

.... . .

...

Np1 · · · Npq

...

. . ....

Mm1

N11 · · · N1q

.... . .

...

Np1 · · · Npq

· · · Mmn

N11 · · · N1q

.... . .

...

Np1 · · · Npq

.

(A.19)

The procedure for vectors is identical; the vectors are just treated as matrices with a

single column (or row, in the case of bras). A bit of head scratching will verify that

this definition, when combined with standard matrix multiplication (A.13), satisfies

the requirement (A.18). An important property of the tensor product is as follows.

If |i〉 is an orthonormal basis for one space, and |j〉 is an orthonormal basis for

a second space, the set of tensor product vectors |i〉 ⊗ |j〉 is an orthonormal basis

for their tensor product space.

A.3 Eigenvalues 388

A.3 Eigenvalues

Consider a matrix R representing a reflection about the x-axis, as shown in Figure

A.4. A vector |1〉 lying along the x-axis is not changed by the action of this matrix.

That is, it is its own reflection. So we have R|1〉 = |1〉. A second vector |2〉 lying

along the y-axis is flipped around by R. Its reflection points in the opposite direction

to itself, so R|2〉 = −|2〉. Other vectors are altered in more complicated ways when

they are reflected, so that the vector resulting from the application of R is not related

to the original vector by a simple numerical factor (1 or −1 in the two cases above).

The vectors |1〉 and |2〉 are examples of vectors for which the action of R is the same

as multiplication by a number. These ‘special’ vectors are known as eigenvectors of

R. In general, any matrix M has a set of eigenvectors |i〉, such that

M |i〉 = λi|i〉. (A.20)

Here the number λi is the eigenvalue corresponding to the eigenvector |i〉. For the

example given above, we had λ1 = 1 and λ2 = −1. The eigenvectors and eigenvalues

contain all the information required to reconstruct the transformation implemented

by M ; they represent the essence of a transformation, and as such they are of

paramount importance in linear analysis, and central to quantum mechanics.

A.3 Eigenvalues 389

Figure A.4 Eigenvectors and eigenvalues. The matrix R representsreflection in the x-axis (horizontal axis). The eigenvectors of R arethose vectors pointing along, or perpendicular to the x-axis, sincethe application of R to one of these vectors produces the same vectoragain, multiplied by some number.

A.3.1 Commutators

In general, matrix multiplication is not commutative. That is, MN 6= NM ; the order

in which matrices are multiplied is important. This makes sense when matrices are

viewed as representing transformations of vectors (see Figure A.5).

Often it is useful to examine the commutator of two matrices, defined by

[M,N ] = MN −NM. (A.21)

If M and N were just numbers, their commutator would always vanish, but for

matrices often it does not. In quantum mechanics, the commutator of different

physical quantities may be non-zero, and this non-vanishing of the commutator

can be viewed as the source of a great many of the strange features of quantum

mechanics.

A.3 Eigenvalues 390

1

2

3

1

23

Figure A.5 Non-commuting operations. Here M represents a re-flection around the y-axis, while N is an anti-clockwise rotationthrough 90 degrees. The red arrows are numbered in order, with1 the initial vector, 2 the result after the application of one of thetransformations, and 3 the result after both transformations havebeen applied. On the left, M is applied first, and then N . On theright, N is applied first, followed by M . The results are differentbecause the matrices M and N do not commute. The notation canbe counter-intuitive: the product MN represents the application ofN first, with M applied afterwards.

If the two matrices N , M do commute, then they have common eigenvectors. To

see this, suppose that |u〉 is an eigenvector of N , with eigenvalue λ. If we take the

product MN |u〉 (that is, we apply N to |u〉 first, and then M), the result is simply

λM |u〉. On the other hand, if [M,N ] = 0, we can swap the order of M and N , to

get NM |u〉. That is,

N(M |u〉) = λ(M |u〉). (A.22)

Therefore the vector M |u〉 is also an eigenvector of N , with the same eigenvalue

λ. M |u〉 must be parallel to |u〉, so that M |u〉 = µ|u〉. That is, |u〉 is also an

eigenvector of M , with some new eigenvalue µ. This fact is intimately connected

with the epistemology of quantum mechanics.

A.4 Types of matrices 391

A.4 Types of matrices

There are some types of matrix that are particularly important, both for the calcu-

lations in this thesis, and for quantum mechanics generally.

A.4.1 The identity matrix

The identity matrix, often denoted by I, is the matrix equivalent of the number 1.

It is the matrix that results in no change when it is multiplied by another matrix —

it represents the operation ‘doing nothing’. That is, IM = MI = M . And of course

the identity does not change a vector either, I|v〉 = |v〉, 〈v|I = 〈v|. The identity

matrix is a square matrix (i.e. dimension m×m), with ones along its main diagonal,

and zeros everywhere else (the zero elements are left blank below to avoid clutter),

I =

1

1

. . .

1

. (A.23)

Sometimes care should be taken to ensure that the correct dimension m of I is used,

so that the multiplication is possible. Usually this is quite clear from the context,

but the symbol Im can be used when the size of I needs to be specified.


A.4.2 Inverse matrix

The inverse M−1 of a matrix M is the matrix that ‘undoes’ the action of M . It is

the matrix equivalent of a reciprocal. The inverse satisfies the relations M−1M =

MM−1 = I. It is clear that taking the inverse of a matrix is also involutive, since the

inverse of an inverse is just the original matrix, (M−1)−1 = M . If M is rectangular,

with m < n, then M describes a map from a larger space into a smaller space,

so that some information is inevitably lost, in the sense that there are different

vectors in the input space that are mapped to the same vector in the output space.

Therefore M cannot have an inverse — it is impossible to ‘undo’ this type of map. It

is generally the case that only square matrices, with m = n, have a matrix inverse.

It is possible to define a pseudo-inverse, that represents the closest approximation

of a true inverse, for any matrix (even rectangular ones), but we will not make use

of the pseudo-inverse [189–192]. Calculating the inverse of a matrix can be rather

involved, and although an algorithm for inverting 3 × 3 dimensional matrices is

taught to students in school, matrix inversion is rarely performed explicitly. Lloyd N.

Trefethen is a prominent numerical analyst who teaches a course on computational

linear algebra at Oxford University. His reaction to a suggestion that students should

consult W. H. Press’s famous book on numerical techniques was

The only way to annoy a numerical analyst more than by inverting

a matrix, is to use Numerical Recipes.


The formula for the inverse of a 2× 2 matrix M is simple however,

M−1 =

a b

c d

−1

=1

ad− bc

d −b

−c a

. (A.24)

The quantity ad−bc is known as the determinant of the matrix — sometimes denoted

by vertical bars, |M | — since it determines whether or not the inverse of M exists:

if |M | = 0, the formula for the inverse ‘blows up’. In this case, the matrix does not

have in inverse, which implies that there exists some vector |v〉 such that M |v〉 = 0.

Clearly it is not possible to invert this expression. This provides a convenient way to

find the eigenvalues of a matrix. If we want to find |u〉 such that M |u〉 = λ|u〉, then

we must have that (M − λI)|u〉 = 0, and therefore we require that |M − λI| = 0.

A.4.3 Hermitian matrices

A Hermitian matrix is equal to its Hermitian adjoint, H = H†. It is the matrix

equivalent of a real number, and in fact its eigenvalues are all real numbers. To see

this, consider the quantity k = 〈i|H|j〉. On the one hand, using the definition of H,

along with (A.20), we have

k = (H†|i〉)†|j〉 = (H|i〉)†|j〉 = (λi|i〉)†|j〉 = λ∗i 〈i|j〉.

On the other hand, we have

k = 〈i|(H|j〉) = λj〈i|j〉.


Taking the difference of these, we get k − k = (λ∗i − λj)〈i|j〉 = 0. If we set i = j,

we must have that λ∗j − λj = 0, since 〈j|j〉 > 0 is the square of the norm of |j〉.

Therefore λj = λ∗j , that is, the eigenvalues of H are real numbers. At the same

time, if we set i 6= j, we must have that 〈i|j〉 = 0, which means that different

eigenvectors of H are all orthogonal to one another. Note that we are free to scale

the eigenvectors |i〉 so that they have length 1. If we do this, the set of eigenvectors

|i〉 of a Hermitian matrix is an orthonormal basis. The eigenvectors define a

‘natural’ coordinate system for the space of vectors upon which H acts. And it’s

very practical to work with this coordinate system, since the effect of H on each

basis vector reduces to multiplication by the corresponding eigenvalue,

H|v〉 = H (v1|1〉+ v2|2〉+ . . .+ vn|n〉) = v1λ1|1〉+ v2λ2|2〉+ . . .+ vnλn|n〉. (A.25)

A.4.4 Diagonal matrices

A diagonal matrix D is a matrix with zeros everywhere except along its main diag-

onal,

D =

D11

D22

. . .

Dmm

. (A.26)

Clearly diagonal matrices must always be square, with n = m. The identity matrix

is a diagonal matrix with Djj = 1. Diagonal matrices are very easy to work with.


For example, the square of a diagonal matrix D2 = DD is another diagonal matrix

with its elements equal to D2jj . The inverse D−1 of a diagonal matrix is just another

diagonal matrix with all its elements equal to 1/Djj . Any two diagonal matrices

commute with one another, [D1, D2] = 0, and the eigenvalues of a diagonal matrix

are just equal to its elements, λj = Djj , with its eigenvectors being the basis vectors

of the coordinate system with respect to which the matrix elements are defined.

This last property is important. Diagonal matrices are wonderfully simple to

manipulate, but surely it is very unlikely that any interesting matrices are diagonal.

The point is that all matrices are diagonal matrices (or more correctly, most square

matrices), as long as you write them down with reference to the correct coordinate

system! This coordinate system is the one defined by the eigenvectors of the matrix,

and when written down using this basis, the elements of the matrix are just its

eigenvalues.

A brief inspection of (A.25) reveals that in fact, when written down with reference

to the coordinate system defined by the eigenvectors |i〉, H is actually diagonal, with

elements H11 = λ1, H22 = λ2, etc...

For this reason, the process of finding the eigenvalues and eigenvectors of a matrix

is sometimes referred to as diagonalization, since this calculation is simply what is

required to convert a matrix into a diagonal one. The eigenvalue decomposition can

be written as

M = WDW−1, (A.27)

where D is a diagonal matrix containing the eigenvalues of M , and where the eigen-


vectors of M comprise the columns of the matrix W . Very efficient algorithms exist

for finding this decomposition; the results in this thesis rely heavily on the speed

and precision of the LAPACK routines implemented in MATLAB.

A.4.5 Unitary matrices

A unitary matrix U is a matrix whose inverse is equal to its Hermitian adjoint,

U−1 = U †. A unitary matrix represents a rotation in space, so that |w〉 = U |v〉 is a

vector pointing in a different direction to |v〉, but with the same norm — the same

length. To see why, consider the norm of |w〉, w2 = 〈w|w〉 = 〈v|U †U |v〉. But since

U † = U−1, we have that U †U = I, so w2 = 〈v|I|v〉 = v2. That is, unitary matrices

preserve the norm of vectors upon which they act.

Figure A.6 A unitary transformation. U represents a rotationfrom an initial (black) into a new (red) coordinate system. Thecolumns of U are unit vectors comprising an orthonormal basis forthe new coordinate system.

A rotation can be thought of as a transformation from one orthonormal coordi-

nate system to another, as shown in Figure A.6. Associated with this new coordinate

system is an orthonormal basis |i〉, and these vectors are actually the columns of

U , ui = |ui〉 = |i〉. To see this, consider the product K = U †U . From the definition


of the Hermitian adjoint, the rows of U † are the bras 〈ui|,

U † =

〈u1|

...

〈um|

. (A.28)

Applying the matrix multiplication described in (A.13), we find that each element

of the product matrix K is given by an inner product, Kij = 〈ui|uj〉. But since U

is unitary, K = I, the identity, so that we must have 〈ui|uj〉 = δij , where δij is the

kronecker delta symbol (δij = 1 if i = j, and 0 otherwise). Therefore the column

vectors |ui〉 form an orthonormal basis. If U is applied to a vector |v〉 pointing

along the x-axis, with components v1 = 1, vj 6=1 = 0, the result is |u1〉. In the same

way, each coordinate axis is mapped by U to a new axis |ui〉, and so U represents a

rotation into a new orthonormal coordinate system defined by its columns.

Incidentally, the above arguments serve to demonstrate that the inner product

〈u|v〉 of two vectors |u〉 and |v〉 is always independent of the coordinate system used

for writing out the components of |u〉 and |v〉. Changing the coordinate system

is done by applying a rotation |u〉 → U |u〉, |v〉 → U |v〉, and the inner product is

then 〈u|U †U |v〉 = 〈u|v〉. Changing coordinates makes no difference. This is to be

expected of course, since (A.7) makes no reference to any coordinates.

It is worth noting that U † is unitary, if U is. Therefore the columns of U † also

form an orthonormal basis, and so the rows of U form an orthonormal basis.

Note also that the product of two unitary matrices U , V is also unitary: UV (UV )† =


UV V †U † = UIU † = I. Two rotations composed together can always be thought of

as a single rotation.

Unitary matrices play a central role in quantum mechanics, and we will encounter

them in the optimization of quantum memories.

Appendix B

Quantum mechanics

In this Appendix we give a brief review of the structure of quantum mechanics. This

is intended as a pedagogical precursor to Appendix C, on quantum optics. We will

make use of the concepts developed in Appendix A.

Quantum Mechanics was developed in the early twentieth century, primarily as a

theory of atomic physics. In the days before Google, interdisciplinary communication

was more difficult, and in fact Werner Heisenberg re-invented matrices in order to

formulate his version of quantum theory [193]. The incarnation we present here uses

the notation introduced by Paul Dirac [194], and we follow broadly the excellent

account given by Nielsen and Chuang [158].

B.1 Postulates 400

B.1 Postulates

B.1.1 State vector

In quantum mechanics, the state of a system is described by a ket |ψ〉. The simplest

vector is a 2-dimensional one, and this describes the simplest type of quantum system

— a qubit. More complicated systems are described by higher dimensional vectors.

B.1.2 Observables

Quantities, like energy, momentum or position — any observable that might be mea-

sured — are represented by matrices that act on the state vector. These matrices are

always Hermitian, and this guarantees that their eigenvalues are real numbers (see

Section A.4.3 in Appendix A). In addition, the eigenvectors of Hermitian matrices

form an orthonormal basis: they define a coordinate system.

B.1.3 Measurements

Quantum mechanics provides the following recipe for making predictions about mea-

surements. The observable being measured is assigned to a Hermitian operator H.

Making this assignment correctly is left up to the skill and imagination of the physi-

cist. This operator is diagonalized, yielding its eigenvalues λi and eigenvectors

|i〉. The eigenvalues are real numbers, and each one represents a possible nu-

merical outcome of the measurement: the number you might see on an oscilloscope

screen, for example. Each eigenvalue λi is associated with an eigenvector |i〉, and

these eigenvectors define a coordinate system. The state vector of the system |ψ〉 is

B.1 Postulates 401

written with reference to these coordinates, known as the measurement basis,

|ψ〉 = ψ1|1〉+ ψ2|2〉+ . . .+ ψm|m〉. (B.1)

The probability pi that the measurement yields the result λi is then given by |ψi|2,

the squared magnitude of the ith component of |ψ〉 in the measurement basis. A

more compact way to write this is

pi = |〈ψ|i〉|2. (B.2)

This is known as the Born rule, after Max Born who proposed it in 1926 [195]. Imme-

diately after the measurement has been completed, the state of the system changes,

essentially instantaneously, according to the measurement result, |ψ〉 → |i〉. This is

known as the collapse postulate.

The average value of the measurement result is often useful. This is sometimes

called the expectation value of the quantity H, since it is the number one would

expect when repeating the measurement many times. The expectation value is

given by 〈H〉 =∑

i piλi, and a bit of thought shows that this is equal to 〈ψ|H|ψ〉.

The fact that 〈H〉 = 〈ψ|H|ψ〉 is another convenience of Dirac notation.

B.1.4 Dynamics

It must always be the case that the probabilities pi sum to unity,∑

i pi = 1. This just

codifies the assertion that we must always get some result from a measurement, even

B.1 Postulates 402

if the result is ‘no signal’. Using the Born rule, this means that∑

i |ψi|2 = 〈ψ|ψ〉 = 1.

That is, the norm of a state vector in quantum mechanics is always exactly equal

to 1. The norm can never be altered by any dynamical process, which immediately

fixes all dynamics in quantum theory to be unitary. In other words, given some

initial state |ψ0〉, and a final state |ψ〉, we must have

|ψ〉 = U |ψ0〉, (B.3)

where U is a unitary operator that advances the system from the initial to the final

state. Differentiating (B.3) with respect to the time t, we obtain the equation of

motion

∂t|ψ〉 = U |ψ0〉

= UU †|ψ〉, (B.4)

where the overdot indicates the time derivative of U . Now, the requirement that

the norm of |ψ〉 does not change can be expressed by the condition ∂t(〈ψ|ψ〉) = 0.

Substituting in (B.3) gives

〈ψ0|U †U + U †U |ψ0〉 = 0, (B.5)

from which we derive the condition that the operator (UU †) is skew-Hermitian,

meaning that it changes sign under Hermitian conjugation. Any skew-Hermitian

B.2 The Heisenberg Picture 403

operator can be represented as the product of the imaginary unit i with a Hermitian

operator H, and making this replacement in (B.4) gives us the Schrodinger equation

∂t|ψ〉 = iH|ψ〉. (B.6)

The operator H is known as the Hamiltonian. Schrodinger’s great insight was to

identify H as the operator associated with the energy of the system. In (B.6) it is the

energy, represented by H, that sets the rate of change of the state vector. Systems

with high energy evolve quickly, with fast oscillations, while low energy systems are

more sluggish.

B.2 The Heisenberg Picture

The above discussion was based on the so-called Schrodinger picture, in which the

quantum state |ψ〉 evolves in time. It is possible to formulate quantum mechanics

differently, and sometimes it is easier to solve a problem by using this different

formulation. The results are identical, regardless of how the calculations are done. In

Heisenberg’s formulation, the quantum state |ψ0〉 of a system at some initial time is

fixed. It does not change with time. Instead, the operators acting on the state vector

evolve in time. As an example, consider a Hermitian operator A associated with

some quantity that we might want to measure. Here we use the symbol A instead

of H; we reserve the symbol H for the Hamiltonian from now on. In the Heisenberg

picture the operator A depends on the time at which we make the measurement.


In order for this formulation to work, we must have that the expectation value

〈A〉 predicted by either formalism is the same. Denoting the fixed operator in the

Schrodinger picture with a subscript S, we must have

〈ψ0|A|ψ0〉 = 〈ψ|AS|ψ〉,

⇒ 〈ψ0|A|ψ0〉 = 〈ψ0|U †ASU |ψ0〉,

∴ A = U †ASU. (B.7)

That is, the time evolution of an operator in the Heisenberg picture is found by sand-

wiching the Schrodinger operator between two copies of U , the same operator that

generates the time evolution of the state in the Schrodinger picture. Differentiating

(B.7) with respect to time, we find

∂tA = U †ASU + U †ASU

= U †UA+AU †U . (B.8)

Note also that [U †, U ] = 0, since

U †U = U †UUU † = U †UUU † = UU †, (B.9)

where we used the fact that [U, U ] = 0 (a little thought shows that an operator must

always commute with its derivative; see Section A.3.1 in Appendix A). Therefore

(B.8) can be re-written in terms of the Hamiltonian, to produce the Heisenberg


equation

∂tA = i[A,H]. (B.10)

This is the fundamental equation of motion in the Heisenberg picture; it plays the

same role as the Schrodinger equation does in the Schrodinger picture — generating

time evolution.

B.2.1 The Heisenberg interaction picture

Often we are interested in analysing the behaviour of a system when it is subjected

to a weak external field. Of specific relevance in this thesis is the case of an atom

illuminated by a laser: the internal electric fields generated by the charges within

the atom are much stronger than the electric fields within the laser beam, so the

laser acts as a weak external perturbation, on top of the much stronger interactions

binding the atom together. In such cases, it is convenient to separate out the strong

and weak contributions to the energy of a system. Suppose that we can divide

the Hamiltonian into two parts, H = H0 + Hint, where H0 dominates, and Hint

represents a comparatively small interaction. The large contribution H0 will make

the operator A change very quickly (as can be seen from the form of (B.10), where a

large energy produces rapid oscillations in time). This rapid oscillation can obscure

any interesting effects arising from the interaction Hamiltonian Hint. To extract

these interesting effects, we define a slowly varying operator A in the following way,

A = U0AU†0 . (B.11)


Here U0 is the time evolution operator associated with the Hamiltonian H0. That

is, U0 satisfies U0U†0 = iH0. Differentiating (B.11) with respect to time, and using

the Heisenberg equation (B.10), we find

∂tA = U0AU†0 + U0(∂tA)U †0 + U0AU

†0 ,

= iH0A+ U0 (i[A,H0 +Hint])U†0 − iAH0,

= −i[A,H0] + i[A,H0] + i(U0AHintU

†0 − U0HintAU

†0

). (B.12)

Conveniently, the first two terms cancel. The last term can be re-written in a

compact form, if we define a modified Hamiltonian H = U0HintU†0 , whence we

obtain the interaction picture equation of motion

∂tA = i[A, H]. (B.13)

Appendix C

Quantum optics

Quantum optics is the study of the quantum features of light. The theory requires a

treatment of ensembles of identical photons, which are easily created and destroyed

in their interaction with atoms. Therefore the techniques of quantum field theory

must be employed, in order to deal with the creation and destruction of identical

particles. In this Appendix we briefly review the quantum mechanical description

of the electric field associated with a propagating light beam, before describing the

form of the interaction between light and matter.

C.1 Modes

Classically, light is a transverse electromagnetic wave. Apart from its amplitude,

it has three degrees of freedom that must be specified to uniquely determine its

properties. These are (i) its polarization, (ii) its frequency and (iii) its propagation

direction.

C.1 Modes 408

The polarization is the direction along which the electric field oscillates; it is

a vector in a plane perpendicular to the propagation direction. It is easy to see

that the space of polarizations is simply a 2-dimensional vector space. In fact,

due to the possibility of phase delays between different polarization directions, it is

actually a complex vector space — a Hilbert space (see Section A.1.2 in Appendix

A). Nonetheless it is a 2-dimensional vector space.

The same is true for the other degrees of freedom. That is, the space of fre-

quencies is a vector space. It is a space with an uncountably infinite number of

dimensions, since the different possible frequencies are infinitely closely spaced, but

it is no different in character to the space of polarizations. And similarly for the

propagation direction: there are an infinite number of infinitely closely spaced prop-

agation directions, and the set of all of these forms a vector space.

Already, in talking of these vector spaces, we have made implicit reference to a

basis for each of them. We talk of two perpendicular directions for polarization. Or

different directions of propagation. These are labels that we use to keep track of

dimensions in a vector space, and they are intuitive and natural. But any basis is as

good as any other. For example, instead of talking about different frequencies, we

could talk about different arrival times. Or we could think of left and right circular

polarizations as the polarization basis. It is useful in quantum optics to be flexible

about the basis we use to describe an optical field. A common concept is therefore

that of the optical mode.

A mode is a member of an orthonormal basis for one of the vector spaces associ-

C.1 Modes 409

ated with a light field. So, horizontal polarization is a polarization mode, since it is

one of a pair of perpendicular polarizations that form a basis for the space of possi-

ble light polarizations. The other, orthogonal mode, is vertical polarization. And a

single frequency ω labels a spectral mode. It is orthogonal to another frequency ω′,

because two plane-waves with these frequencies have a vanishing inner product,

∫eiωτe−iω′τ dτ = 0, (C.1)

when ω 6= ω′. Equivalently, we could label different temporal modes t and t′. These

are orthogonal because two delta-functions with these timings also have a vanishing

inner product, ∫δ(τ − t)δ(τ − t′) dτ = 0, (C.2)

when t 6= t′.

An optical mode is a member of a basis for the full space of all possible optical

fields. This space of all possible fields is just the tensor product of the vector spaces

associated with each degree of freedom. And a basis for the full space is found

by taking the tensor product of the bases used for each degree of freedom (see

Section A.2.2 in Appendix A). That is, an optical mode is the tensor product of a

polarization mode, a spectral mode and a spatial mode.

Once a basis of modes is settled upon, it is possible to introduce the concept of

a photon. A photon is an excitation of an optical mode. Sometimes it is useful to

remember that photons are only defined with respect to a basis of modes. Although

C.2 Quantum states of light 410

photons are often contrasted with waves as an embodiment of the particulate nature

of light, they do not have to be localized, like tiny bullets. The ‘shape’ of a photon

is the shape of the mode of which it is an excitation.

C.2 Quantum states of light

C.2.1 Fock states

Suppose we consider a plane wave optical mode. That is, a mode with a linear po-

larization (horizontal, say), a single frequency ω, and a single propagation direction

k, where k is the wavevector of the mode. An excitation of this mode has a fixed

energy, given by the Planck formula E = ~ω, so a single photon in this mode is an

eigenstate of the Hamiltonian for the field. Similarly, if we excite two photons in

this mode, we have a state with twice the energy, E = 2~ω. This is also an energy

eigenstate, but with a different eigenvalue. It follows that these two states must

be orthogonal. And by extension, each photon number state is orthogonal to every

other photon number state. If we use the notation |n〉 to denote the state with n

photons, we must have

〈n|m〉 = δnm. (C.3)

Changing the basis of optical modes from plane waves to some other basis cannot

change this orthogonality, since the inner product is invariant under unitary trans-

formations. Therefore (C.3) holds generally, for different photon number states of

an arbitrary optical mode.


The orthonormal basis of photon number states |n〉, associated with excitations

of some given optical mode, is known as the Fock basis for that mode. The photon

number states are sometimes known as Fock states, and the space for which they

form a basis is accordingly Fock space. The Fock space represents the final degree of

freedom associated with an electromagnetic quantum state: the amplitude. That is

to say, the more photons in a mode, the more intense the field. Thus the quantum

state of an electromagnetic field is fully specified by the tensor product of 4 vector

spaces: the polarization, spectral and spatial modes (collectively specifying an opti-

cal mode), and finally the Fock space (specifying the photon number: the energy in

the field; its brightness).

C.2.2 Creation and Annihilation operators

A marked difference between optical fields and material systems is the imperma-

nence of photons. Generally the atoms and electrons in a quantum memory are

considered to be indestructible. They are not created or destroyed by their interac-

tions. But photons can be absorbed and re-emitted. So we must describe processes

that change one Fock state into another — processes that change the number of

photons excited into a given mode. This description is accomplished by introducing

a creation operator a†, that adds a single photon to an optical mode. The simi-

larity of the symbol ‘†’ for Hermitian conjugation to a ‘+’ sign serves as a useful

mnemonic for this operator’s function. The effect of applying a† to an empty optical

mode |0〉, containing no photons, is to produce the state |1〉, with a single photon.


Further applications of a† add extra photons, with contributions from all possible

permutations of arranging these photons (see Figure C.1). These contributions must

be included, since photons are bosons, meaning that their state must be unchanged

by swapping any pair of photons. The Fock states created by the action of a† are

not correctly normalized, so that a numerical factor, accounting for the number of

permutations, must be included,

(a†)n|0〉 =√n!|n〉. (C.4)

Another way to write this is

a†|n〉 =√n+ 1|n+ 1〉. (C.5)

Taking the norm of (C.5), we have

〈n|aa†|n〉 = (n+ 1)〈n+ 1|n+ 1〉 = n+ 1,

⇒ aa†|n〉 = (n+ 1)|n〉. (C.6)

That is, the Hermitian conjugate a = (a†)† is an annihilation operator that removes

a photon. And from (C.6) we see that

a|n〉 =√n|n− 1〉. (C.7)


Note that a ‘kills’ the empty vacuum state, a|0〉 = 0, which is fortunate, since there

cannot be fewer than zero photons in a mode! It is often useful, when manipulating

Figure C.1 Symmetrized photons. n applications of the photoncreation operator a† to the vacuum state |0〉 produces a symmetrizedn photon state, with contributions from all n! permutations of the nphotons. Swapping any two photons leaves the state unchanged, asrequired by Bose statistics.

expression involving these operators, to be able to reverse their ordering. This is

done using their commutator which, applying (C.5) and (C.7), is given by

〈n|[a, a†]|m〉 = 0, 〈n|[a, a†]|n〉 = 1,

⇒ [a, a†] = 1. (C.8)

The commutator (C.8) expresses what is known as the canonical commutation rela-

tion; commutators of this form are common to creation and annihilation operators

for all bosonic fields.

Another useful operator is the number operator N = a†a, a Hermitian operator

that satisfies the eigenvalue equation

N |n〉 = n|n〉, (C.9)

so that N counts the number of photons excited into a particular mode.

C.3 The electric field 414

C.3 The electric field

Electric fields are associated with separated charges, while magnetic fields are asso-

ciated with moving charges. Electrons move rather slowly in most ordinary forms

of matter, and accordingly their interaction with light is dominated by its electric

component. In this thesis, we treat light fields as if they were purely electric waves,

an approximation that is very well satisfied provided that light intensities are not

sufficient to produce a relativistic electron plasma.

The electric field associated with a beam of light, as might be generated by a

laser, can be expressed in terms of the annihilation operators a(ω) associated with

plane waves propagating along the beam [107],

E(z) = iv∫g(ω)a(ω)e−iωz/c dω + h.c., (C.10)

where z is the longitudinal position along the beam, v is a unit polarization vec-

tor in the plane perpendicular to the beam and g(ω) =√

~ω/4πε0Ac is the mode

amplitude. Here A is the cross-sectional area of the beam, ε0 is the permittivity

of free space and c is the speed of light. Note that in principle the electric field is

an observable quantity, that we could measure (although at optical frequencies it is

not generally possible to directly measure the electric field, at radio frequencies it

certainly is feasible). And so E is a Hermitian operator, as expected.

The annihilation operators a(ω), labelled by the frequency ω of the mode upon

C.4 Matter-Light Interaction 415

which they act, satisfy the commutation relation

[a(ω), a†(ω′)] = δ(ω − ω′). (C.11)

This expresses the fact that operators for different frequency modes do not ‘see’

eachother, so they commute, while when ω = ω′, the canonical relation (C.8) is

satisfied. The delta function is the appropriate generalization for the case when the

modes are labelled by a continuous parameter, such as ω.

C.4 Matter-Light Interaction

Generally light interacts with matter through electrons. In most quantum memory

protocols these are the optically active outer electrons bound to some atoms. We

will also consider scattering in a diamond crystal, and here the electrons are more

appropriately described as free, or quasi -free particles. The Hamiltonian describing

the interactions are slightly different in these two cases; here we briefly review their

origin, and the relationship between them.

C.4.1 The A.p Interaction

The interaction of an electron with the electromagnetic field is found by incorpo-

rating the appropriate electromagnetic term, associated with so-called U(1) gauge

symmetry, into the Lagrangian density. The effect of this term is to modify the


momentum p of the electron,

p −→ p− eA, (C.12)

where e is the electronic charge, and where A is the magnetic vector potential.

The potential A is not actually an observable field. The electric and magnetic

fields are related to its derivatives, but the absolute value of A is arbitrary to some

extent. Different choices for the functional form ofA— known as different gauges —

produce different Hamiltonians, with differing degrees of calculational convenience;

the physical predictions of the theory are unchanged of course. A standard choice of

gauge in quantum optics is the Coulomb gauge, which requires that A is divergence

free, ∇.A = 0. With this choice, the physical electric and magnetic fields are given,

respectively, by

E = −∂tA, B = ∇×A. (C.13)

Using (C.13) we can express the potentialA in the form (C.10), with the replacement

ig(ω) −→ g(ω)/ω.

The Hamiltonian for an electron, with mass m, in an electromagnetic field is

found by substituting the ‘canonical momentum’, given by (C.12), into the Hamil-

tonian for a ‘bare’ electron,

H =(p− eA)2

2m+A2ε0

∫ [E2 + c2B2

]dz. (C.14)

This is known as the minimal coupling Hamiltonian. The first term is the kinetic


energy of the electron, with the transformation (C.12) included. The second term,

in square brackets, represents the ‘free field’ energy: this is the energy of the electro-

magnetic fields, in the absence of the electron. The integral extends over all space,

or at least, over the entire region occupied by the fields. The contribution from the

magnetic field B is very small, but the contribution from the electric field E is more

significant. Inserting (C.10) for E shows that the free field energy takes the form∫(N + 1

2)~ω dω. The term involving the number operator N simply expresses the

Planck formula E = ~ω, so that the energy in the field increases with the number of

photons excited. The term proportional to 12 is known as the zero-point energy: the

energy of the vacuum. It is rather unfortunate that this energy is infinite (since it is

integrated over all frequencies), but it is possible to work around these technicalities

with some mathematical sleight-of-hand, known as renormalization [196]. In any case

we will not be concerned with the zero point energy.

Multiplying out the first term in (C.14), we obtain a term of the form p2/2m,

which just describes the ‘bare’ kinetic energy of the electron, without the field.

There is a term A2/2m, which describes the field acting back on itself — this type

of non-linear back action is generally negligibly small. And there is a term of the

form −eA.p/m. This describes the coupling of electronic momenta to the vector

potential. In situations where electrons are spread over an extended region, such as

in a crystal, this interaction dominates the atom-light coupling.


C.4.2 The E.d Interaction

When electrons are well-localized, such as when bound into atoms, a more con-

venient form of the interaction Hamiltonian can be derived. This is accomplished

formally by means of a unitary transformation due to Power, Zienau and Woolley

(PZW) [197,198]. In general it is desirable to eliminate explicit reference to the vector

potentialA in the Hamiltonian, since then the equations are manifestly gauge invari-

ant — it is quite clear that there can be no-dependence on the choice of gauge. The

PZW transformation removes A from the Hamiltonian, and introduces interactions

between the physical field E and the moments of the atomic charge distribution. To

see how this is done, we will need two results. The first is the equal-time commutator

of E and A, the amplitudes of the electric field and the vector potential,

[A(z), E(z′)] = −i~e

Aε0δ(z − z′), (C.15)

This is easily derived from (C.10) and (C.13) using (C.11). It is well known that in

quantum mechanics momentum and position generally satisfy the relation [x, p] = i~,

and indeed the form of (C.15) when z = z′ reflects the fact that in the Coulomb

gauge the field E is actually the ‘momentum’ that is conjugate to the ‘coordinate’

A in the electromagnetic Hamiltonian.

The second result we need is that

eCDe−C = D + [C,D], (C.16)


whenever [C,D] is just a number (i.e. not another operator). Here the exponential

of an operator is defined according to the series

eC =∞∑n=1

Cn

n!. (C.17)

The result (C.16) is straightforward to derive. Consider the product CnD. Using

the commutator, we can ‘pull’ the operator D through C, in the following way,

CnD = Cn−1(CD) = Cn−1(DC + [C,D]) = (Cn−1D)C + [C,D]Cn−1. (C.18)

Repeating this procedure recursively, we obtain

CnD = DCn + n[C,D]Cn−1. (C.19)

Re-writing the left hand side of (C.16) using the series (C.17), and applying (C.19),

we arrive at (C.16).

With these preliminaries, we can introduce the PZW transformation. Suppose

that the action of the light is to make an optically active electron oscillate; it re-

mains bound to an atom, but it is ‘wiggled’ by the field. This is certainly what we

expect would happen classically. The atomic polarization, distinct from the opti-

cal polarization, is a useful concept in this situation. It is the ‘dipole moment per

unit volume’, where the dipole moment is the product of the electronic charge and

displacement. Suppose that the electron, with charge −e, is displaced a distance x


along the polarization direction v of the incident light field. The dipole moment is

d = −exv, and the atomic polarization is P (z) = dδ(z)/A, where the delta function

describes a single dipole placed at the position z = 0. To express the interaction en-

ergy associated with the atomic polarization, we introduce a unitary transformation

of the Hamiltonian,

H → UHU †, with U = exp[

iA~

∫P (z′).A(z′) dz′

]. (C.20)

This transformation simply changes the coordinate system with respect to which the

quantum states |ψ〉 of the atom-light system are defined. Essentially it is nothing

more than a cosmetic change, but it has a marked effect on the form of the Hamilto-

nian. Applying the transformation to the free-field part of (C.14), we have UE2U †

= (UEU †).(UEU †), with

UEU † = vUEU † = v

E(z) +

iA~

∫P (z′)[A(z′), E(z)] dz′

= E(z)− 1

ε0P (z). (C.21)

That is, the PZW transformation adds a component proportional to the electron dis-

placement into the electric field. Using (C.21), the free-field Hamiltonian, neglecting

the small contribution from the magnetic field B, becomes

A2ε0

∫E2 dz −→ A

2ε0

∫ [E − 1

ε0P

]2

dz

=A2ε0

∫ [E2 +

1ε20P 2

]dz′ − d.E(z = 0). (C.22)


The term proportional to E2 represents the ‘bare’ free-field energy, with no electron

present. The P 2 term represents an unimportant ‘self-interaction’ of the electron.

But the last term, proportional to E.d, represents the interaction of the physical

electric field, at the position of the electron, with the electronic dipole moment. It

is known as the electric dipole interaction Hamiltonian, and it serves as the basis

for the analysis of all the atomic quantum memory protocols in this thesis.

Finally, we note that the electron momentum acquires a component proportional

to the vector potential under the PZW transformation,

p −→ p+iA~

∫A(z′)[P (z′), p] dz′

= p+ eA(z = 0), (C.23)

where this time the commutator of the electronic momentum and position was used,

[x, px] = i~. In the approximation that the wavelength of the light is much longer

than the spatial extent of the atom — a limit valid for all interactions at optical fre-

quencies — we can set A ≈ A(z = 0), and then the vector potential A is completely

eliminated from the Hamiltonian.

Thus when electrons are tightly bound into atoms, the only significant interaction

with optical fields occurs through the electric dipole interaction.

C.5 Dissipation and Fluctuation 422

C.5 Dissipation and Fluctuation

In this section we address the issue of loss in quantum systems. Specifically, we seek

a theoretical description of the decoherence in a quantum memory: the constituent

atoms may emit photons into random directions, or collide with one another, and

these processes partially destroy the quantum information stored in the memory. In

Chapter 4 we use the Heisenberg picture to describe the propagation of light through

a quantum memory, and so we should account for losses using the Heisenberg picture.

In the following we use a simple model to show how the equations of motion for a

quantum system are modified by the presence of losses. Fortunately it is well known

that the results are not significantly altered by refining the model.

Our model consists of a single bosonic mode, our system, coupled to a bath of

other bosons — a reservoir. We use bosons because their commutation relations

are simple, but this model applies rather well to an ensemble quantum memory.

As shown in §4.11 in Chapter (4), both the atomic polarization and the spin wave

appearing in the equations of motion of an ensemble memory have approximately

bosonic commutation relations. In fact, in diamond, the optical phonons that con-

stitute the spin wave really are bosons (see Chapter 9). The reservoir of modes to

which the system is coupled could be the electromagnetic field, and indeed this is

an excellent description of spontaneous emission losses affecting the atomic polar-

ization.

The Hamiltonian contains the free field energy of the bath, and also terms that

represent processes in which a photon in the system is destroyed, while a photon in


the reservoir is created, or vice versa. Working in the Heisenberg interaction picture,

so that the free-field energy of the system (and zero point energy of the reservoir)

is removed, we have

H =∫ωb†(ω)b(ω) dω + κ

∫ [a†b(ω) + b†(ω)a

]dω, (C.24)

where b(ω) destroys a photon with frequency ω in the reservoir, and where a destroys

a photon in the system. The equal-time commutators of these operators are given

by

[a, a†] = 1, and [b(ω), b†(ω′)] = δ(ω − ω′). (C.25)

Using the Heisenberg equation (B.13), we derive the following equations of motion

for the annihilation operators,

∂ta = −iκ∫b(ω) dω, (C.26)

∂tb(ω) = −iωb(ω)− iκa. (C.27)

Integrating (C.27) and substituting the result into (C.26), we obtain

∂ta(t) = −iκ∫

e−iωtb0(ω)− iκ∫ t

0e−iω(t−t′)a(t′) dt′

dω, (C.28)

where the operators b0(ω) = b(ω, t = 0) represent the initial state of the reservoir.

Performing the frequency integral in the second term produces a delta-function (see


(D.9) in Appendix D), which selects out the time t = t′, and then (C.28) becomes

∂ta(t) = −γa(t) + F (t), (C.29)

where

F (t) = −iκ∫b0(ω)e−iωt dω (C.30)

is known as a Langevin noise operator, and where γ = πκ2 is an exponential de-

cay rate. Equations of this form were first used to describe the classical Brownian

motion of colloidal particles buffeted by the molecules of a warm fluid. There, the

term involving F represents the fluctuating force arising from collisions with the

randomly moving molecules, and a similar interpretation is helpful in the present

case. The operator F mixes a component of ‘white noise’ into the dynamics of a,

which otherwise would simply decay exponentially with a rate γ. That the noise is

white, with a flat power spectrum, can be seen from examining its temporal corre-

lation functions. Suppose that initially the reservoir contains some small thermal

excitations, so that there are n photons, on average, in a mode with frequency ω.

Then we have

〈b†0(ω)b0(ω′)〉 = nδ(ω − ω′). (C.31)


The normally ordered correlation function of the noise operator is then given by

〈F †(t)F (t′)〉 = κ2

∫ ∫nδ(ω − ω′)ei(ωt−ω′t′) dω dω′,

= 2πκ2n× 12π

∫eiω(t−t′) dω (C.32)

= 2γnδ(t− t′), (C.33)

and the anti-normally ordered correlation function is similarly given by

〈F (t)F †(t′)〉 = 2γ(n+ 1)δ(t− t′), (C.34)

where we used the commutator in (C.25). The noise is therefore delta-correlated,

meaning that it is completely random from moment to moment. There is no cor-

relation with earlier times, and the noise changes ‘infinitely quickly’; a signature of

unlimited-bandwidth white noise. The infinite bandwidth is a consequence of the

fact that the reservoir coupling κ was assumed to be frequency independent. This

assumption is known as a Markov approximation, since it means that the dynamics

of a does not explicitly depend on its past values. As is clear from the form of

(C.29), the Markov approximation gives rise to exponential decay of a: the ubiquity

of exponential decay in quantum systems serves to confirm the robustness of this

approximation as a model for a wide variety of dissipative processes.

The noise F is just sufficient to preserve the expectation value of the equal-time

commutator in (C.25), so that a remains a bona fide bosonic operator. This can be


seen by inserting the formal solution to (C.29),

a(t) = a(0)e−γt +∫ t

0e−γ(t−t′)F (t′) dt′, (C.35)

into the commutator (C.25),

〈[a(t), a†(t)]〉 = 〈[a(0), a(0)†]〉e−2γt +∫ t

0

∫ t

0e−γ(2t−t′−t′′)〈[F (t′), F †(t′′)]〉dt′ dt′′

= e−2γt

[1 + 2γ

∫ t

0e2γt′ dt′

]= 1. (C.36)

This close connection between fluctuations, represented by F , and damping, rep-

resented by γ, is a manifestation of the fluctuation-dissipation theorem, discovered

first by Einstein.

Finally, we note that the solution (C.35) allows us to solve for the time evolution

of the number operator for the system, giving the result

〈N〉 = 〈a†(t)a(t)〉 = e−2γt〈a†(0)a(0)〉+(1− e−2γt

)n. (C.37)

This shows that in the infinite future, with t −→ ∞, the system relaxes into ther-

mal equilibrium with the reservoir, with N −→ n. And this thermal equilibrium

condition can often be taken as the initial state of the system, if we want to analyze

processes which drive it out of equilibrium.

Appendix D

Sundry Analytical Techniques

In this Appendix we define the Dirac delta function and the unilateral and bilateral

Fourier transforms used in this thesis. We discuss some of their properties, and

finally we demonstrate some results relating to the Fourier transform of certain

Bessel functions. First though, we introduce a useful technique that expedites these

calculations.

D.1 Contour Integration

Contour integration is a powerful method for evaluating integrals that would oth-

erwise be difficult, or impossible, to perform. To see how it works, we consider the

integral

Ix =∫ b

af(x) dx (D.1)

D.1 Contour Integration 428

of some smooth function f(x). The anti-derivative F (x) of f is the function that

satisfies ∂xF = f , and (D.1) can be simply expressed in terms of F as Ix = F (b)−

F (a). That is, the integral of f between two points is given by the change in

‘height’ of its anti-derivative between these same two points (see Figure D.1 (a)).

Now suppose that we introduce the possibility of complex arguments for f , so that

we consider all the values of f(z), where z = x+ iy is an arbitrary complex number.

We can now think of f as a two dimensional surface, lying above the (x, y)-plane:

as we vary the real and imaginary coordinates x and y, the value of f traces out a

characteristic landscape. We could define an integral

Iz =∫ B

Af(z) dz (D.2)

from some initial point A = xA + iyA to a final point B = xB + iyB. But now that

we are working in a two dimensional plane, the endpoints aren’t enough to specify

the integral completely. We need to know the path that we should take to get from

A to B. On the other hand, the anti-derivative F (z) also describes some kind of

surface in the (x, y)-plane, as shown in part (b) of Figure D.1, and the integral is

again given by Iz = F (B) − F (A). So in fact, the integral does not depend on the

path taken, it only depends on its endpoints. The proof of this fact is known as the

Cauchy integral theorem. Suppose that we integrate along a closed loop, so that the

endpoint B is equal to the starting point A. Then F (B) = F (A) and the integral

vanishes. This is interesting, since we could build the first integral (D.1) into the


complex integral (D.2), and possibly use this result to draw some conclusions. For

instance, suppose we start our integral at A = (a, 0) in the (x, y)-plane, integrate

along the real axis (i.e. the line y = 0) to the point (b, 0), and then loop back in a

semi-circle to finish where we started at B = A. The straight part of the loop, along

the real line, is just equal to Ix. The whole integral must vanish, because we finish

where we started. Therefore if we can find an expression for the curved part of the

integral — call it Ic — then we must have Ix = −Ic, in order that the two parts of

the integral cancel out. In this way, interesting results about real integrals can be

derived by considering their behaviour in the complex plane.

(a) (b)

Figure D.1 Contour integrals. (a) a function f and its anti-derivative F . (b) The anti-derivative is extended into the complexplane, and we consider integrating around a loop that includes theoriginal integral along the real line.

D.1.1 Cauchy’s Integral Formula

So far we have only considered smooth functions. When f contains a discontinuity,

the Cauchy integral theorem does not apply. Suppose that we want to integrate the

function f(z)/(z−z0), which ‘blows up’ when z = z0. We consider integrating along


a circular loop around the singularity at z0. We can parameterize the integral using

polar coordinates z = z0 + reiθ, with dz = ireiθdθ. We are free to make the radius

r as small as we like, since as above the details of the path taken don’t matter; it

only matters that we encircle the singularity. We find

∮f(z)z − z0

dz = limr→0

∫ 2π

0

f(z0 + reiθ)reiθ

× ireiθ dθ = 2πi× f(z0), (D.3)

where the symbol∮

indicates integration around a loop. The singularity is like a

witness to our passage around the loop, so that we do not quite get back to where

we started on completing our roundtrip. The contribution 2πif(z0) to the integral is

known as the residue associated with the singularity at z = z0. In general, wherever

a function blows up, it leaves a residue that contributes to an integral that encloses it.

Differentiating the formula (D.3) with respect to z0 tells us how to handle ‘stronger’

singularities of the form 1/(z − z0)n,

∮f(z)

(z − z0)ndz = 2πi×

∂n−1z0 f(z0)(n− 1)!

. (D.4)

D.1.2 Typical example

We now give an example of how to use these formulas to evaluate an integral. The

method is typical. Consider the integral

Ix =∫ ∞−∞

eix

xdx. (D.5)


This is equivalent to the contour integral

Iz = limR→∞

∮eiz

zdz, (D.6)

where the integration path is a closed semicircle with radius R, such that the straight

portion runs along the real line, as shown in Figure D.2. To see that Iz = Ix, note

that the function eiz is damped with a positive imaginary argument, ei(x+iy) =

eixe−y. Therefore any contribution to Iz from the curved portion of the path, all of

which has y > 0, vanishes in the limit R −→ ∞. Note that if the integrand were

conjugated, so that it contained the exponential e−ix, we would have to choose to

close the contour in the lower half of the plane, in order that the curved part of the

integral should vanish.

Now the integral Iz is not zero, because the integrand has a singularity at z = 0.

It’s a little inconvenient that this lies exactly on the integration path. We deal with

this by moving the singularity inside the integration path by a small amount ε, in

order to apply (D.3); we then take the limit ε −→ 0 when we are done:

Ix = Iz = limε→0

limR→∞

∮eiz

z − iεdz

= limε→0

2πi× e−ε (D.7)

= 2πi. (D.8)

This technique for performing integrals is indispensable in Fourier analysis, as dis-

cussed below. And of course it is quite elegant!

D.2 The Dirac Delta Function 432

Figure D.2 Upper closure. The integration path is comprised of astraight portion running along the real axis, and a semicircle in theupper half of the complex plane. In the limit of infinite radius R, thestraight portion becomes equal to the real integral Ix, and the curvedportion vanishes.

D.2 The Dirac Delta Function

We use the Dirac delta function δ(x) a great deal in this thesis. Where it is not

introduced ‘by hand’, prompted by physical arguments, it is most often encountered

in the form of an integral over plane waves,

δ(x) =1

2π

∫ ∞−∞

eikx dk. (D.9)

This integral is not really well defined, but its properties become apparent from the

following limit,

δ(x) = limT→∞

12π

∫ T

−Teikx dk

= limT→∞

T sinc(xT )π

, (D.10)

where sinc(θ) = sin(θ)/θ. For each value of T , this describes a narrow peak of width

∼ 1/T and height T , centred at x = 0. As T →∞, the peak becomes infinitely tall

D.2 The Dirac Delta Function 433

and narrow. But its integral remains finite. To see this, consider the integral

Ix =∫ ∞−∞

sinc(x) dx =−i2

∫ ∞−∞

eix

xdx−

∫ ∞−∞

e−ix

xdx. (D.11)

The first integral on the right hand side is given by (D.8). Applying the same

technique to the second integral, we must use an integration path that is closed

in the lower half of the complex plane, as explained in §D.1.2. When we shift the

singularity using z → z − iε, it moves into the upper half of the plane, so the

contour used for the second integral does not enclose the singularity, and therefore

the integral vanishes. Therefore we find that

Ix =−i22πi− 0 = π. (D.12)

Integrating the delta function then yields

∫ ∞−∞

δ(x) dx = limT→∞

1π

∫ ∞−∞

T sinc(xT ) dx = 1. (D.13)

The Dirac delta function is an ideal ‘spike’, with unit area, and so it has the extremely

useful property that ∫ ∞−∞

f(x)δ(x− x0) dx = f(x0). (D.14)

D.3 Fourier Transforms 434

D.3 Fourier Transforms

D.3.1 Bilateral Transform

The most familiar Fourier transform is the bilateral transform f(k) of a function

f(z),

f(k) = Fz f(z) (k) =1√2π

∫ ∞−∞

f(z)eikz dz. (D.15)

The name ‘bilateral’ refers to the lower limit of integration, which is −∞ in this

case, so that the whole function f(z), including its values for negative z, is involved

in forming the Fourier transform f .

D.3.2 Unitarity

The factor of 1/√

2π makes the transform unitary, meaning that the norm of the

the transform f is the same as the norm of f ,

∫ ∞−∞|f(k)|2 dk =

12π

∫ ∞−∞

∫ ∞−∞

∫ ∞−∞

f(z)f∗(z′)eik(z−z′) dk dz dz′

=∫ ∞−∞

∫ ∞−∞

δ(z − z′)eik(z−z′)f(z)f∗(z′) dz dz′

=∫ ∞−∞|f(z)|2 dz, (D.16)

where the plane wave expansion (D.9), and the property (D.14) were used.


D.3.3 Inverse

The inverse Fourier transform takes essentially the same form, except that the inte-

gral kernel is conjugated,

f(z) = F−1k

f(k)

(z) =

1√2π

∫ ∞−∞

f(k)e−ikz dk. (D.17)

This is easily demonstrated by substituting (D.15) into (D.17) and using (D.9).

D.3.4 Shift

From the definition (D.15) we can derive a simple formula for the Fourier transform

of a shifted function f(z + a), where a is some constant,

Fz f(z + a) (k) =1√2π

∫ ∞−∞

f(z + a)eikz dz

=1√2π

∫ ∞−∞

f(x)eik(x−a) dx

= e−ikaf(k), (D.18)

where we changed integration variables from z to x = z+ a in the second line. This

shows that a shift in the coordinate z corresponds to a phase rotation in the Fourier

domain. Identical arguments show that for the inverse transform

F−1k

f(k + a)

(z) = eizaf(z). (D.19)


D.3.5 Convolution

The bilateral convolution of two functions f and g is defined by the integral

f ∗ g(z) =1√2π

∫ ∞−∞

f(x)g(z − x) dx =1√2π

∫ ∞−∞

f(z − x)g(x) dx. (D.20)

The Fourier transform of a convolution is given by the product of the Fourier trans-

forms of the convolved functions. That is,

Fz f ∗ g(z) (k) =1

2π

∫ ∞−∞

∫ ∞−∞

f(x)g(z − x)eikz dz dx

=1

2π

∫ ∞−∞

∫ ∞−∞

f(x)eikxg(y)eiky dxdy

= f(k)g(k), (D.21)

where we used a change of variables y = z−x with dy = dz. This shows also that the

inverse Fourier transform of a product of two functions is given by the convolution

of their individual inverse transforms.

D.3.6 Transform of a Derivative

The nth derivative ∂nz f(z) of a function can be expressed using the inverse (D.17) as

∂nz f(z) =1√2π

∫ ∞−∞

f(k)∂nz e−ikz dk

= F−1k

(−ik)nf(k)

(z), (D.22)

D.4 Unilateral Transform 437

and therefore we must have that

Fz ∂nz f(z) (k) = (−ik)nf(k). (D.23)

That is, the Fourier transform converts differentiation into ordinary multiplication;

clearly this is a boon when tackling differential equations.

D.4 Unilateral Transform

The unilateral Fourier transform is identical to its bilateral counterpart, except that

it involves an integral over positive coordinates only,

f(k) = Fz f(z) (k) =1√2π

∫ ∞0

f(z)eikz dz. (D.24)

We have used the same notation to denote both types of transform: we will be

careful to distinguish between them when the difference is important. This type of

transform is appropriate for functions f(z) which vanish, or are undefined, when

z < 0. As an example, the function f might describe the temporal response to

an interaction at ‘time’ z = 0, in which case causality requires that f vanishes for

z < 0. For these types of functions, the unitarity property (D.16) still holds, and

the inverse transform is given precisely by (D.17). That is, the inverse is bilateral.


D.4.1 Shift

The shift theorem (D.18) is not directly applicable to the unilateral transform, be-

cause of the semi-infinite integral domain. But since the inverse transform is bilat-

eral, the shift theorem (D.19) does apply to the inverse transform.

D.4.2 Convolution

The unilateral convolution, or causal convolution of two functions f and g is defined

similarly to the bilateral convolution (D.20), except that the integral is limited by

causal consistency,

f ∗ g(z) =1√2π

∫ z

0f(x)g(z − x) dx =

1√2π

∫ z

0f(z − x)g(x) dx. (D.25)

To calculate the transform of this convolution, we need to take care with the limits

of integration, as shown in Figure D.3. We then obtain

Fz f ∗ g(z) (k) =1

2π

∫ ∞0

∫ z

0f(x)g(z − x)eikx dxdz

=1

2π

∫ ∞0

∫ ∞x

f(x)g(z − x)eikz dzdx

=1

2π

∫ ∞0

∫ ∞0

f(x)eikxg(y)eiky dydx

= f(k)g(k). (D.26)

Just as for the bilateral case above, the unilateral Fourier transform of a causal

convolution is given by the product of the transforms of the convolved functions.


(a) (b)

Figure D.3 Integration limits. (a) Schematic of the integral∫∞0

∫ z0

dxdz. (b) Schematic of the same integral with the order of in-tegration reversed,

∫∞0

∫∞x

dz dx. Each arrow represents an instanceof the inner integral at a fixed value of the outer variable.

D.4.3 Transform of a Derivative

The real utility of the unilateral transform is in the treatment of derivatives. Con-

sider the unilateral Fourier transform of ∂zf . Integrating parts,

1√2π

∫ ∞0

eikz∂zf(z) dz =1√2π

[eikzf(z)

]∞0− 1√

2π

∫ ∞0

f(z)ikeikz dz

= −ikf(k)− 1√2πf(0). (D.27)

This is the same result as (D.23) for the bilateral transform, except for the presence

of the boundary condition f(0). Note that we assumed f(z −→ ∞) = 0: all the

functions we will consider satisfy this property, referred to alternately as boundedness

or integrability. Physically, this simply says that the effects of any interaction fall

away to nothing as we move away from their source.

D.5 Bessel Functions 440

D.4.4 Laplace Transform

The unilateral Fourier transform occupies a middle ground between the bilateral

Fourier transform on the one hand, and the well-known Laplace transform on the

other. The Laplace transform Lz f(z) (s) is generated from the unilateral Fourier

transform by making the replacement ik → s. The Laplace transform is more com-

mon since even unbounded functions generally possess a Laplace transform, due

to the exponential damping provided by the integral kernel. However this damp-

ing destroys the unitarity of the transform. For this reason, the unilateral Fourier

transform better suits our purposes in this thesis, since the efficiency of a quantum

memory is unchanged by working with this type of transform.

D.5 Bessel Functions

In this section we define the ordinary and modified Bessel functions that represent

the propagators for quantum memories operated adiabatically (see Chapter 5). The

nth order ordinary Bessel function of the first kind — denoted by Jn — is defined

by the infinite series

Jn(2z) =∞∑m=0

(−1)mz2m+n

m!(m+ n)!, (D.28)

where n is a non-negative integer. The corresponding modified Bessel funtion In is

defined identically, except that the factor of (−1)m is missing from each term of the

sum. Conversion between ordinary and modified Bessel functions is reminiscent of


the relation between trigonometric and hyperbolic ratios,

Jn(iz) = inIn(z). (D.29)

Indeed the J functions have the appearance of decaying cosines, while the I functions

all grow exponentially with increasing z. Just as sin(x) and cos(x) take the values

0 and 1, respectively, at the point x = 0, so it is also true that J0(0) = I0(0) = 1,

while Jn>0(0) = In>0(0) = 0.

D.5.1 Orthogonality

As can be easily shown using the series (D.28), the Bessel functions Jn(z) satisfy

the differential equation

z∂z (z∂z) Jn = (n2 − z2)Jn. (D.30)

Consider the ith zero ai of Jn, which satisfies Jn(ai) = 0. Defining s = z/ai and

X(s) = Jn(ais), we can re-write (D.30) in the ‘normalized’ form

s∂s (s∂s)X = (n2 − a2i s

2)X, (D.31)

with X(1) = 0. We can repeat this procedure for another zero of Jn, say aj , giving

s∂s (s∂s)Y = (n2 − a2js

2)Y, (D.32)


where now s = z/aj and Y (s) = Jn(ajs), so that Y (1) = 0. We now multiply (D.32)

by X and (D.31) by Y, subtract the two resulting equations and divide through by

s, to obtain

(a2i − a2

j )sXY = ∂s [Xs∂sY − Y s∂sX] .

Integrating this from s = 0 to s = 1, and using the fact that X(1) = Y (1) = 0, the

right hand side vanishes. Dividing through by a2i − a2

j and converting back into the

Jn notation, we find the orthogonality condition

∫ 1

0sJn(ais)Jn(ajs) ds = 0, when i 6= j. (D.33)

Alternatively, writing z = s2, we have

∫ 1

0Jn(ai

√z)Jn(aj

√z) dz = 0, when i 6= j. (D.34)

D.5.2 Memory Propagator

In Chapter 5 we are faced with taking the inverse Fourier transform of the function

f(k) =e−ia/k

kn+1, (D.35)

where a is some constant and n = 0, 1, 2.... This is done with the help of con-

tour integration, as described at the beginning of this Appendix. Using the series


expansion of the exponential, we have

f(z) =1√2π

∫ ∞−∞

1kn+1

[ ∞∑m=0

1m!

(−iak

)m]e−ikz dk. (D.36)

We must therefore perform the integral

Ik =∫ ∞−∞

e−ikz

kn+m+1dk. (D.37)

We can equate this to the complex integral

Iz = limε→0

limR→∞

∮e−ikz

(k + iε)n+m+1dk, (D.38)

where the integration contour is closed in the lower half of the complex plane, as

shown in Figure D.4, and where the regularization ε shifts the singularity into the

interior of the contour. Using Cauchy’s integral formula (D.4), we find

Iz = limε→0

−2πi×

[∂n+mk e−ikz

(n+m)!

]k=−iε

= −2πi× (−iz)n+m

(n+m)!. (D.39)

Substituting this result into the series (D.36), and comparing with (D.28), we obtain

the result

f(z) = (−i)n+1√

2πΘ(z)(za

)n/2Jn(2√az). (D.40)


The function Θ(z) is known as the Heaviside step function, defined so that Θ(z) = 0

when z < 0, and Θ(z) = 1 for z > 0. We include it in the above expression to

represent its causal nature. That f(z) = 0 for negative z can be seen by considering

the integration contour used to evaluate the Fourier transform. When z becomes

negative, the integration contour must be closed in the upper half of the complex

plane, so that it no longer encloses the singularity, which therefore no longer leaves

a residue, and the integral vanishes.

Figure D.4 Lower closure. Here the integrand is damped in thelower half of the complex plane, so we close the integration contourin this region.

We will also need the inverse transform of (D.35) when n = −1,

f(z) = F−1k

e−ia/k

(z). (D.41)

Again, inserting the series expansion for the exponential, we have

f(z) =1√2π

∞∑m=0

(−ia)m

m!

∫ ∞−∞

e−ikz

kmdk. (D.42)

The first term, with m = 0, involves an integral with no singularity. In fact, it is an


integral over plane waves — a delta function. We separate off this first term, and

re-index the remaining terms,

f(z) =1√2π

∫ ∞−∞

e−ikz dk +1√2π

∞∑m=0

(−ia)m+1

(m+ 1)!

∫ ∞−∞

e−ikz

km+1dk. (D.43)

Then we perform the singular integrals using contour integration, as before, to obtain

f(z) =√

2πδ(z)− ia√2π

∞∑m=0

(−ia)m(−iz)m

(m+ 1)!m!Θ(z)

=√

2πδ(z)−Θ(z)

√a

zJ1

(2√az)

. (D.44)

We make use of this result in Chapter 5, where it relates transmitted and incident

signal fields in a Raman quantum memory.

D.5.3 Optimal Eigenvalue Kernel

In Chapter (5) we claim that it is possible to derive a certain kernel KA from a

storage kernel K by direct integration — the result is used in the work of Gorshkov

et al. The Fourier transforms presented above provide one way to confirm it. The

two results we need are as follows. First, we need the unilateral Fourier transform

of an integral over the product of two storage kernels:

A = Fz,z′∫ ∞

0e−ax × J0(2

√bxz)e−uz × J0(2

√cxz′)e−vz

′dx

(k, k′). (D.45)


Using the result (D.40) along with the shift theorem (D.19) we obtain

A =1

2π

∫ ∞0

e−ax × e−ibx/(k+iu)

k + iu× eicx/(k′+iv)

k′ + ivdx

=1

2π× 1akk′ + i(va+ c)k + i(ua+ b)k′ − uva− bv − cu

. (D.46)

The second result we need is the unilateral Fourier transform of an anti-normally

ordered kernel:

B = Fz,z′e−α(z+z′)J0(2

√βzz′)

(k, k′). (D.47)

Transforming over z first, and then z′, we get

B =i√2πFz′e−αz

′ × e−iβz′/(k+iα)

k + iα

(k′)

=i

2π

∫ ∞0

e−[α+iβ/(k+iα)−ik′]z′

k + iαdz′ (D.48)

= − 12π× 1kk′ + iα(k + k′)− β − α2

. (D.49)

Comparing these two results, and grinding through some tedious algebra, it is not

hard to demonstrate the claimed equality.

Appendix E

Numerics

In this Appendix we review the numerical methods used in this thesis. Our aim

is to solve a system of coupled linear partial differential equations (PDEs) in space

and time. For pedagogical purposes, we will consider the simple example presented

in §3.5 in Chapter 3: A resonant signal pulse with amplitude A propagates through

an ensemble of classical atoms, whose response to the optical field is described by

the average displacement B of the atomic electrons from their equilibrium positions.

The equations of motion are given by

∂τB = −iαA, and ∂zA = −iβ

αB. (E.1)

(The second equation follows from the first line of (3.34)). Since we are only con-

cerned with the method of numerical solution, we set α = β = 1, and we normalize

the τ and z coordinates so that they are dimensionless, with z running from 0 to 1,

448

and τ running from 0 to 10. The boundary conditions are Ain = A(z = 0, τ) and

Bin = B(z, τ = 0): the profile of the incident signal pulse, and the initial atomic

excitation, respectively. Generally Ain will take the form of a pulse, reminiscent of a

Gaussian, and Bin = 0, if the atoms are prepared in their ground states. To find the

solution numerically, we discretize the space and time coordinates on a finite grid,

z −→ zj , τ −→ τk, (E.2)

and we represent the continuous functions A(z, τ), B(z, τ) as matrices A, B whose

elements approximate those functions sampled at the grid points,

Ajk = A(zj , τk), Bjk = B(zj , τk). (E.3)

We solve the system (E.1) via the method of lines [199]. The technique is so named

because we integrate the spatial derivative ∂z ‘in one go’, whereas we integrate the

temporal derivative ∂τ incrementally, stepping forwards in time iteratively. The so-

lution, starting as a series of points in space at τ = 0, evolves gradually forwards

in time, and each of the points in space traces out a line, over time, that describes

the solution at that position in space. It is a feature of the implementation of the

technique that the temporal discretization is much finer than the spatial one, and

so the numerics are conducive to this mental picture, which is illustrated schemat-

ically in Figure E.1. The spatial derivative is performed using a spectral method,

which uses the value of the function at all the spatial points in order to compute

E.1 Spectral Collocation 449

Figure E.1 The method of lines. The spatial coordinate is dis-cretized on a coarse grid. The solution at each spatial point tracesout a line as the temporal integration proceeds stepwise on a finegrid.

the derivative. For smooth functions, this is extremely accurate, even when very

few spatial points are used. Before describing the method used for the temporal

derivative, we introduce this spectral method more fully.

E.1 Spectral Collocation

An excellent introduction to this field of numerical analysis is provided in Nick

Trefethen’s book Spectral Methods in Matlab [200] — which is hosted on the internet.

Other treatments include the books by Gottlieb and Orzag [201], and by Boyd [202].

The easiest spectral method to understand is that involving a Fourier transform. As

shown in §D.3.6 in Appendix D, Fourier transformation of a function f from z to

k converts differentiation by z into multiplication by −ik. It is also clear that the

operation of taking a Fourier transform, or indeed its inverse, is a linear one. Suppose

that we discretize the coordinates z and k, so that the function f is approximated by


a vector f , with components fj = f(zj). In this context the points zj at which f(z)

is sampled are known as collocation points. The linearity of the Fourier transform

then allows us to represent it as a matrix acting on f . The composition of the

operations of taking the Fourier transform, multiplying by −ik, and then taking the

inverse Fourier transform, can therefore also be represented as a single matrix, D.

An approximation fz to the derivative of f at the collocation points is then given

by

fz = Df . (E.4)

The matrix D is dense, meaning that there are generally very few elements of D

that are zero, so all of the elements of f are involved in determining fz. As a result,

spectral methods can be very accurate, even if very few collocation points are used.

This accuracy comes at the price of requiring a dense matrix multiplication, which is

computationally more expensive than a more ‘local’ method — such as the method

we employ for the temporal integration (see below). As it happens there is a more

efficient way to implement a discrete Fourier transform (DFT), known as the fast

Fourier transform (FFT), which was developed by Cooley and Tukey in 1965 [203].

The FFT does not use explicit matrix multiplication, and it is generally employed

for Fourier spectral methods. We do not use a Fourier spectral method in this

thesis, however. One reason for this is that a DFT works by fitting a finite number

of complex exponentials — which are periodic — to the vector f . Since the basis

functions are periodic, the vector f must be periodic. Of course any function f(z)

defined on a finite domain [0, 1] can be made periodic by ‘gluing’ copies of the


function onto either end of the domain, as shown in Figure E.2. But if f(0) 6= f(1),

the resulting ‘extended’ function will contain discontinuous jumps, where the copies

were glued together. These discontinuities represent a very rapid change in f , so

that the spectrum f(k) contains very high spatial frequencies. Given that f , and

the approximated Fourier transform f , are comprised of only a finite number of

collocation points, the accuracy of the DFT is therefore severely compromised by

these discontinuities. Fourier spectral methods are not well suited to dealing with

functions that are not ‘really’ periodic, for this reason. Since we are modeling the

spatial distribution of atomic excitations, which need have no intrinsic periodicity,

it behooves us to use a method that is not subject to this limitation. Such a method

is provided by polynomial differentiation matrices.

(a)

(b)

Figure E.2 Periodic extension. (a): the function f satisfies f(0) =f(1), so that its periodic extension is smooth. The Fourier spectralderivative of this function is accurate. (b): now f(0) 6= f(1), and theperiodic extension of f contains sharp discontinuities, which erodethe accuracy of a Fourier spectral derivative.


E.1.1 Polynomial Differentiation Matrices

To generalize spectral differentiation to the case of non-periodic functions, we should

use non-periodic basis functions. Instead of fitting the vector f with complex expo-

nentials, as in the Fourier case, we fit f with an algebraic polynomial p(z). We can

then easily differentiate this to get p′(z), and the vector approximating the deriva-

tive of f is found by evaluating p′ at the collocation points. The basis functions

from which we construct p are specified by requiring that p evaluates to f at the

collocation points. We write p in terms of a set of basis functions pj, and the

elements of f , as

p(z) =∑k

fkpk(z). (E.5)

Fixing p(zj) = fj imposes the condition

pk(zj) = δjk. (E.6)

The interpolant p(z) is a polynomial of degree N − 1, if there are N collocation

points, and therefore the basis functions pk are also polynomials of degree N − 1.

The condition (E.6) means that each of the N collocation points, except for zk, are

roots of pk(z), and therefore

pk(z) =1ak

∏i 6=k

(z − zi), with ak =∏i 6=k

(zk − zi). (E.7)


Here ak is just the normalization required so that pk(zk) = 1. Differentiating the

relation (E.5), and setting z = zj gives

p′(zj) =∑k

p′k(zj)fk. (E.8)

Identifying the elements of fz with p′(zj), and using the definition (A.13) in Ap-

pendix A for matrix multiplication, we see the elements of the differentiation matrix

D are given by

Djk = p′k(zj). (E.9)

Taking the logarithm of (E.7) and differentiating, we find

p′k(z) = pk(z)∑i 6=k

1z − zk

. (E.10)

The elements of D are then given by the general formulae

Djk =

∑

i 6=j1

zj−zi if j = k,

ajak(zj−zk) otherwise.

(E.11)

E.1.2 Chebyshev points

We are now able to construct a differentiation matrix that works on an arbitrary

set of collocation points zj. A grid of equally spaced points is the natural choice,

but this is in fact disastrously unstable: the interpolant p develops pathological

oscillations near the edges of the domain, close to z = 0 and z = 1. This is known


as the Runge phenomenon, and it is dealt with by using a set of collocation points

that cluster together near the domain edges. To develop an intuition for why this

might be, observe that the magnitude of one of the basis functions can be written

in the form

|pk(z)| =eV (z)

|ak|, where V (z) =

∑i 6=k

ln |z − zi|. (E.12)

Notice that V (z) has the same functional form as the electrostatic potential due to

infinite lines of charge intersecting the z-axis at the collocation points zi. There-

fore the size of the basis functions is exponentially sensitive to the potential energy

associated with the ‘charge distribution’ describing the collocation points. The in-

terpolant p is well-behaved when this energy is constant across the domain [0, 1].

If we allowed the collocation points to move along the z-axis according to the mu-

tual repulsion between them represented by V , they would arrange themselves in

a minimum-energy configuration which renders the potential flat, and the inter-

polant would be stable. In this configuration, the points are clustered together at

the domain boundaries, and this explains why choosing a set of collocation points

according to this configuration avoids the Runge phenomenon.

A stable set of collocation points is provided by the Chebyshev points, which are

the projections onto the domain of equally spaced points along a semicircle joining

the domain boundaries (see Figure E.3):

zj = 12

1− cos

[π(j−1)N−1

]. (E.13)

E.2 Time-stepping 455

By substituting the points (E.13) into the formula (E.11), the appropriate Cheby-

collocation points

(a) (b)

Figure E.3 Chebyshev Points. (a) Chebyshev collocation pointsare clustered towards the boundary of the domain. They are thedownward projections of equally spaced points along a semicircle ofradius 1

2 centred at z = 12 . (b) The method of lines with Chebyshev

spectral collocation.

shev differentiation matrix can be calculated. We use Matlab for all of our numer-

ical computations, and the differentiation matrices are conveniently generated by

a simple script called cheb.m, which is available online, and can be found in Nick

Trefethen’s book [200], along with many detailed examples of its application.

Having introduced the method we use for the spatial derivatives, we now discuss

the temporal derivatives, before describing how these two techniques are combined

to solve the system (E.1).

E.2 Time-stepping

Suppose that we would like to solve the differential equation

f ′(τ) = g [f(τ), τ ] , with f(0) = f0, (E.14)

E.2 Time-stepping 456

where the prime now denotes differentiation with respect to τ . We assume that

the boundary condition f0 and the function g are known. Note that g may depend

on f . For instance, in our example, B′ = −iA, but A depends on B. Now, the

simplest numerical approach is to make a finite difference approximation to the

time derivative. In terms of the discretized functions and coordinates, we have

fk+1 − fkδτ

= g (fk, τk)

⇒ fk+1 = fk + δτg (fk, τk) , (E.15)

where δτ = τk+1 − τk is the time step — assumed to be independent of k. The

second line of (E.15) is a recursion relation, relating the future value of f to the

present values of f and g. Starting with the boundary condition at τ = 0, we

can use this relation to step forward in time, gradually building up the solution

for f . This method is known as a first order Euler method, since errors of order

δτ accumulate in the numerical solution. In our numerics, we use a second-order

Runge-Kutta (RK2) method, which is only slightly more complicated, but accurate

up to errors of order δτ2. The Runge-Kutta method reduces the errors by using

values of the known function g at intermediate points, between τk and τk+1. The

recursion formula therefore requires that we discretize our functions on two grids: a

primary grid, with times τk, which we denote with the same notation as used above,

and a secondary, intermediate grid, with times τk = τk + 23δτ . The second order

E.3 Boundary Conditions 457

Runge-Kutta method we employ for time stepping is then written as

fk+1 = fk +δτ

4[g (fk, τk) + 3g

(fk, τk

)], (E.16)

where the value of f on the secondary grid is approximated with the first order Euler

formula (E.15), so that

fk = fk + 23δτg (fk, τk) . (E.17)

E.3 Boundary Conditions

It is clear how to implement the boundary condition Bin on B using the above

time-stepping algorithm. This boundary condition simply tells us the initial values

to use, on the first time step. However it is not so clear, from our discussion of

spectral methods, how to implement the boundary condition Ain on A, which holds

at z = 0 at all times. This is done by incorporating the boundary condition into

the dynamical equation for A. First consider the discretized version of the equation

∂zA = −iB, which is

DA = −iB, (E.18)

where D is a Chebyshev differentiation martrix, which acts on each column of A to

produce the corresponding column of −iB. There is no mention of the boundary

condition so far. The discretized form of the boundary condition on A can be written

E.3 Boundary Conditions 458

as

1× [A11 A12 A13 · · · A1N ] = [Ain(τ1) Ain(τ2) Ain(τ3) · · · Ain(τN )] . (E.19)

That is, 1 multiplied by the first row of A — which corresponds to the values of A

at z = z1 = 0 — is equal to the discretized boundary condition. This equation can

be ‘built in’ to the dynamical equation (E.18) simply by replacing the first row of

D with the first row of the identity matrix I, and by replacing the first row of −iB

with the discretized boundary condition. The equation becomes

DbA = −iBb, (E.20)

where the superscript b — for ‘boundary’ — indicates the modifications

[Db

11 Db12 D

b13 · · · Db

1N

]= [1 0 0 · · · 0] , and[

Bb11 B

b12 B

b13 · · · Bb

1N

]= i× [Ain(τ1) Ain(τ2) Ain(τ3) · · · Ain(τN )] .(E.21)

All the other elements of Db and Bb are the same as those of D and B.

Now that we have fixed the boundary condition for A, we can solve for A in

terms of B. Formally, we can invert the modified differentiation matrix Db to get

A = −iDb−1Bb. (E.22)

E.4 Constructing the Solutions 459

Of course, we do not know all the values of B until we have performed the time-

stepping, and this requires knowledge of A. In order to get to the solutions for A and

B, we must build up B column by column, using the time-stepping, and solving for

each column of A in turn using the above procedure. We are now ready to describe

how this works.

E.4 Constructing the Solutions

Let us denote the columns of A by ak, and similarly the columns of B by bk:

A =

a1

a2

. . .

aM

, B =

b1

b2

. . .

bM

,

(E.23)

where M is the number of temporal discretization points. The algorithm proceeds

as follows. First, we use the boundary condition Bin to set the values of b1,

B11

B21

...

BN1

=

Bin(z1)

Bin(z2)

...

Bin(zN )

. (E.24)

Then, we solve for a1 using the formula

ak = −iDb−1bbk. (E.25)


Here bbk is the kth column of Bb. That is to say, bb

k = bk, except that its first element

is replaced by the signal field boundary condition, (bbk)1 = iAin(τk). Now that we

have both b1 and a1, we implement the first stage of the RK2 iteration, which is to

approximate the first time step on the intermediate grid, b1, using (E.17),

bk = bk − i23δτak. (E.26)

Before we can implement the second part of RK2, we must approximate a1. This is

done by using (E.25) again, this time replacing bbk with bb

k. That is,

ak = −iDb−1bbk. (E.27)

This time, the modified vector contains the signal boundary condition evaluated

at τ = τk. That is,(b

bk

)1

= iAin

(τk + 2

3δτ). Finally, the second part of RK2 is

implemented, which provides us with b2. The formula is

bk+1 = bk − iδτ

4(ak + 3ak) . (E.28)

We have now succeeded in constructing b2 from our knowledge of b1, and along the

way we have found a1. Iterating this procedure, we can construct b3 and a2, and

then b4 and a3, and so on. In this way we proceed forward in time until we reach

the end of our time domain, at τ = τM = 10. This completes the numerical solution.

Here we comment that a dramatic increase in computational speed is achieved


if Gaussian elimination is used instead of matrix inversion in (E.25) and (E.27).

Gaussian elimination is an efficient method for solving the matrix equation Ax =

y to obtain x, without explicitly calculating the inverse A−1. In Matlab, this is

implemented by the ‘backslash’ operator. For example, the Matlab code for (E.25)

might look like

A(:,k)=-i*inv(Db)*Bb(:,k),

where the ‘inv’ function calls a matrix inversion routine. A much more efficient

way to perform the same calculation would be coded as follows,

A(:,k)=-i*( Db\Bb(:,k) ).

This latter method becomes crucial to the feasibility of our chosen spectral

method when modeling complicated dynamics.

In order to produce an informative plot, we can use polynomial interpolation in

order to replace the coarse Chebyshev grid of spatial collocation points with a finer,

equally spaced grid. This utilizes the full accuracy of the spectral method, since the

values at the collocation points are implicitly derived from the global interpolant p(z)

at each time step. In practice, it is often faster, and quite acceptably accurate, to

use a piecewise spline interpolant, which glues together multiple cubic polynomials

in a continuous way, to join the collocation points. Routines for implementing this

are standard in Matlab. In Figure E.4 we plot example solutions for A and B. Even

using only 5 spatial collocation points, the solutions are accurate to ∼ 1%.

E.5 Numerical Construction of a Green’s Function 462

Figure E.4 Example solutions. We plot the the squared moduli|A|2 and |B|2 of the solutions for A and B, found using the methodof lines, as a function of z and τ . We used N = 5 spatial collocationpoints, on a Chebyshev grid, and M = 20 equally spaced time steps.The Matlab code runs in 0.02 seconds on a 3 GHz machine. Thesignal boundary condition is a Gaussian pulse, Ain(τ) = e−(τ−2.5)2 ,and we assume there are no atomic excitations initially: Bin(z) = 0.The black lines indicate the time evolution of the spatial collocationpoints. The smooth surfaces are generated by spline interpolation inbetween these points. The red lines indicate the boundary conditions.The interpolated solutions are accurate to ∼ 1%, even with only 5collocation points - this rapid convergence is a remarkable feature ofspectral collocation.

The behaviour is as might be expected: the signal field is absorbed as it propa-

gates through the ensemble, with the transmitted field having undergone significant

temporal distortion due to the spectral hole burnt by the atomic absorption line.

The atomic excitation grows in time, assuming a roughly exponential shape in space,

at the end of the interaction, consistent with Beer’s law.

E.5 Numerical Construction of a Green’s Function

Now that we are able to solve the system of coupled PDEs describing a quantum

memory, for given boundary conditions, we can find the Green’s function — the

storage kernel — for the interaction. As described in §5.4 in Chapter 5, this is done


by solving with delta function boundary conditions for the signal field, successively

varying the timing of the delta function to construct an approximation to the kernel.

As usual for storage, we set Bin(z) = 0. Of course, it is not possible to implement

a true delta function numerically. Instead, the delta function Ain(τ) = δ(τ − τk) is

represented as

[A11 A12 · · · A1k · · · A1N ] = [0 0 · · · 1 · · · 0] . (E.29)

That is, along the line z = 0, the signal field is identically vanishing at all times,

except at τ = τk, where it takes the value 1. Since the equations are linear, the

absolute magnitude of Ain is not important, it only matters that the incident signal

field is non-zero at just the single time step τ = τk. Due to the causal nature of the

interaction, it is not necessary to numerically integrate from τ = 0 up to τ = τk.

Since the signal field is zero in this region, and since there is no atomic excitation,

the dynamics are trivial in this region. Therefore we only integrate from τ = τk up

to τ = τM .

Let bkM denote the vector representing the atomic excitation at the end of the

interaction at τ = τM , produced by a delta function incident signal at τ = τk. The

numerical approximation to the Green’s function is the matrix whose kth column is


bkM ,

K =

b1M

b2M

. . .

bMM

. (E.30)

Constructing the Green’s function is a more demanding computation than simply

solving an instance of the equations with some particular boundary condition. First,

the system of PDEs must be solved multiple times, and second, the sharp nature of

the delta function boundary condition generally requires a smaller time step than

is required for a smooth boundary condition. Below, in Figure E.5, we plot the

numerically constructed Green’s function for our example system, alongside the

analytic result, expressed as a Bessel function, which is derived in (3.38) in §3.5

of Chapter 3. With a sufficiently fine grid, the numerical and analytic results are

indistinguishable.

numerical analytic

Figure E.5 A numerically constructed Green’s function. We usedN = 30 spatial collocation points, and M = 4000 time steps. Thecomputation takes around 10 minutes on a 3 GHz machine; the agree-ment with the analytic result is excellent.

E.6 Spectral Methods for Two Dimensions 465

E.6 Spectral Methods for Two Dimensions

In Chapter 6 we describe simulations carried out in three dimensions — two space

dimensions, and time. Needless to say, these simulations are much more time con-

suming than those involving only one spatial dimension. The method we use is

essentially identical to that described in §E.4 above, except that now we use spec-

tral collocation to deal with both spatial dimensions: we still use RK2 for the time

stepping. In this section, we explain how to extend the spectral method to two

dimensions. For concreteness, we will consider solving the simple example system

(E.1) using only spectral collocation. The method is rather memory-intensive, and

it quickly becomes unwieldy as the problem becomes complex, but it has the appeal

of being very direct: we encode all of the dynamics into a single matrix equation,

from which we extract the solutions in a single step. To see how this works, consider

the following representation of the dynamical equations (E.1),

∂z i

i ∂τ

A

B

= 0. (E.31)

If we could invert the matrix of derivatives, we might solve for A and B. But

this matrix has no unique inverse: boundary conditions are needed. We proceed by

discretizing the coordinates, and approximating (E.31) numerically using Chebyshev

differentiation matrices to represent the differential operators. We then incorporate

the boundary conditions into the resulting matrix equation, much as we did in §E.3

previously. Finally, we extract the solutions for A and B using Matlab’s backslash


operator.

In order to represent (E.31) numerically, we must convert A and B into column

vectors. First, we discretize both the z and τ coordinates on a Chebyshev grid. We

use N spatial collocation points and M temporal collocation points. The functions

A and B are then represented as N ×M matrices, whose columns describe spatial

variation, and whose rows describe evolution in time. Let us denote the columns of

A and B by ak and bk, as before (see (E.23)). We vectorize the matrices A and

B by concatenating adjacent columns into a single column vector, of total length

NM ,

A = vec(A) =

a1

a2

...

aM

, B = vec(B) =

b1

b2

...

bM

. (E.32)

We now generate the differentiation matrices that will simulate the action of ∂z and

∂τ on A and B. Suppose that Dz is the differentiation matrix designed for the

spatial Chebyshev grid defined on [0, 1] (Dz is therefore exactly the same as the

matrix D discussed in §E.4). This acts on a column ak of A to produce the required

derivative. The matrix Dz, which acts on A as a partial space derivative, is therefore


given by

Dz =

Dz

Dz

. . .

Dz

= I ⊗Dz, (E.33)

where on the right hand side we have used the tensor product notation (see §A.2.2 in

Appendix A). This expresses the fact that Dz acts on the spatial part of the vector

A (within each column of A), whereas the temporal part (across columns of A) is

unaffected. In a similar vein, we assemble the temporal partial derivative ∂τ by first

generating a Chebyshev differentiation matrix Dτ for the temporal Chebyshev grid.

Since the τ coordinate runs from 0 up to 10, the collocation points are given by

τk = 10× 12

1− cos

[π(k−1)M−1

]. (E.34)

Because of the factor of 10 appearing here, we have Dτ = 110D, where D is an

M ×M Chebyshev differentiation matrix generated for the domain [0, 1], like the

one used for the spatial derivative previously. The operator Dτ representing a partial

temporal derivative on B is then given by

Dτ = Dτ ⊗ I. (E.35)

This operator acts only on the rows B, leaving its columns unaffected. We are now


able to write down our numerical approximation to (E.31), which is

Dz i

i Dτ

A

B

= 0. (E.36)

We can express this compactly as

LX = 0, (E.37)

in an obvious notation. In order to solve for X — i.e. for A and B — we must

insert the boundary conditions for the signal field and atomic excitations. As in

§E.3 above, the general approach is to incorporate the equation

1× ( X at the boundary ) = ( appropriate boundary condition ) . (E.38)

For instance, to fix the signal field boundary condition Ain(τ), we identify those

elements of X which describe A at z = 0. This is the first row of the matrix A,

corresponding to the elements

1, N + 1, 2N + 1, . . . , (M − 1)N + 1, (E.39)

of X. The left hand side of (E.38) is then implemented in two steps. First, we set

the rows of L with row indices given by (E.39) to zero. Second, we set the diagonal

elements of L, with both indices given by (E.39), equal to one. The resulting modified


matrix has the property that it maps the signal field at the boundary z = 0 to itself.

The right hand side of (E.38) is realized by introducing a vector C into the right

hand side of (E.37), with all elements zero, except for those with indices given by

(E.39), which are set equal to the discretized boundary condition,

C1 = Ain(τ1), (E.40)

CN+1 = Ain(τ2), (E.41)

... (E.42)

C(M−1)N+1 = Ain(τM ). (E.43)

The same procedure is used to implement the boundary condition Bin for the atomic

excitations. This time the boundary is the first column of B, which is indices 1 to

N of B, and therefore indices NM + 1 to NM +N of X. Finally, we are left with

a modified equation

LbX = C, (E.44)

where the superscript b distinguishes the modified matrix, and where C contains all

the relevant boundary conditions. The formal solution is then

X = Lb−1C. (E.45)

As mentioned previously, this is calculated much more efficiently if Gaussian elim-

ination, rather than explicit matrix inversion, is used. In Matlab, we invoke the


backslash operator. This completes the numerics. We are now in possession of

approximations to both A and B at all spatial and temporal collocation points.

Polynomial interpolation, or piecewise spline interpolation, provides us with an ac-

curate representation of the solution for arbitrary z and τ . In Figure E.6 we plot

the solutions for A and B found using this method.

Figure E.6 Spectral methods in two dimensions. We use N = 5spatial collocation points, andM = 15 temporal collocation points, tosolve the example system (E.1), with the same boundary conditionsas shown previously in Figure E.4. The code runs in 0.01 seconds ona 3 GHz machine, and the interpolated solutions shown are identicalto those generated using spectral collocation and RK2 time stepping.

Appendix F

Atomic Vapours

In this final Appendix we cover some results pertaining to atomic vapours, and in

particular to our ongoing experiments with cesium vapour. We review the behaviour

of vapour pressure with temperature, the concept of oscillator strength and its rela-

tion to the dipole moment associated with a transition, and the various mechanisms

of line broadening. We finish with an analysis of the polarization of Stokes scattering

in cesium.

F.1 Vapour pressure

No solid is infinitely sticky. Although the constituents may be bound tightly, there

remains a finite probability that an atom will escape. Every solid is therefore ac-

companied by a diffuse cloud of free atoms, the pressure of which is known as its

vapour pressure. It is this diffuse vapour that is used for the storage medium in our

experiments (and many others). The density of the vapour determines its optical

F.1 Vapour pressure 472

depth, and with it the achievable storage efficiency (see Chapter 5). To estimate

this density, we treat the atoms as a classical statistical ensemble. The probability

that an atom occupies a state with energy E is given by the Boltzmann distribution,

p(E ) ∼ e−E /kBT . (F.1)

The energy E required for an atom to escape depends on the potentials within the

solid, and is largely independent of temperature. The ratio between the probabilities

for escape at two temperatures T0, T , is therefore

p

p0= e− EkB

(1/T−1/T0). (F.2)

This is, in fact, a good approximation for the vapour pressure above a solid, if p is

now taken to be the pressure. The relation can be derived much more rigourously

from the Clausius-Clapeyron equation [204]. Generally the reference state temper-

ature T0 is taken to be the boiling point of the material, since then the vapour

pressure p0 must be equal to the ambient pressure at this temperature. The energy

E is commonly given as the molar enthalpy of sublimation ∆H. Boltzmann’s con-

stant kB is then switched for the molar gas constant R = 8.31 JK−1mole−1. Part

(a) of Figure F.1 shows the vapour pressure of thallium — the storage medium used

in our preliminary experiments — using the values ∆H = 182 kJmole−1, T0 = 1746

K and p0 = 5.8 × 105 Pa. Application of the ideal gas law p = nkBT allows us to

convert the vapour pressure into the atomic number density, which is also shown in

F.1 Vapour pressure 473

the plot.

Calculation of the vapour pressure of cesium, which is shown in part (b), is

complicated by the fact that cesium melts at 28 C. The qualitative behaviour is

unchanged, but we use an empirical formula taken from the excellent document by

Daniel Steck [184], which is available online.

300 350 400 450 50010

−30

10−20

10−10

10

1610

510

010

510

1010

1510

1810

2010

2210

0

300 350 400 450 500 300 350 400 450 50010

−4

10−2

100

102

300 350 400 450 500

Pressure (Pa)

Number density (m

-3)

(a) (b)Thallium Cesium

Temperature (K) Temperature (K)

Figure F.1 The vapour pressure of (a) thallium, and (b) cesium.The black curves show the vapour pressure in Pascals (left axis) asa function of temperature, measured in Kelvin. The red curves showthe corresponding number density (right axis), calculated using theideal gas law.

As described in §10.9.1 of Chapter 10, a rough estimate of the optical depth

of an ensemble is given by d = nλ2L ∼ n × 10−14, assuming optical wavelengths

and a vapour cell a few centimeters long. From this it is clear that thallium has

insufficient density for efficient storage at reasonable temperatures. Raising the

temperature further starts to introduce a thermal background into the signal field

from blackbody radiation! On the other hand, cesium has an optical depth of order

100, even at room temperature.

F.2 Oscillator strengths 474

F.2 Oscillator strengths

The oscillator strength fjk is a dimensionless measure of the dominance of an atomic

transition |j〉 ↔ |k〉, compared to all other possible transitions. The oscillator

strengths for all transitions from the state |j〉 sum to unity,

∑k

fjk = 1, ∀j. (F.3)

The relation between the fjk and the dipole matrix elements djk = 〈j|er|k〉 may be

derived solely from this, and one other condition, which is that

fjk ∝ |djk|2. (F.4)

This simply relates the f ’s to the transition probabilities. The following derivation is

due to Charles Thiel at Montana State University. To determine the proportionality

constant, we insert (F.4) into (F.3).

∑k

fjk = 1 =∑k

αjk |〈j|er|k〉|2

=∑k

e2αjk ×12

[〈j|r|k〉〈k|r|j〉+ 〈j|r|k〉〈k|r|j〉] . (F.5)

The development in the second line seems a little obtuse, but we now make use of

the following relation between the matrix elements of the position and momentum

F.2 Oscillator strengths 475

operators,

pjk = 〈j|m∂tr|k〉

= im〈j|[r,H]|k〉

= −imωjkrjk, (F.6)

where m is the electron mass, and where ωjk is the frequency splitting between

the states |j〉, |k〉, which are taken to be energy eigenstates of the Hamiltonian H.

Substitution of this relation into (F.5) yields

1 =∑k

ie2αjk2mωjk

[〈j|r|k〉〈k|p|j〉 − 〈j|p|k〉〈k|r|j〉]

=−ie2α

2m〈j| [r, p] |j〉

=e2~α2m

, (F.7)

where in the penultimate line we used the decomposition of the identity I =∑

j |j〉〈j|,

and where we somewhat heuristically set αjk = αωjk. In the final line we made use

of the canonical commutation relation [r, p] = i~. The resulting expression for the

oscillator strength is

fjk =2mωjk

~e2|djk|2. (F.8)

The inverse relation is useful for calculating the dipole moment from the oscillator

strengths listed in data tables.

F.3 Line broadening 476

F.3 Line broadening

There are three processes that increase the absorption linewidth in our cesium

vapour: Doppler broadening, pressure broadening and power broadening. An ex-

cellent treatment of these effects can be found in The Quantum Theory of Light by

Loudon [107]. Here we briefly review the physics of line broadening.

F.3.1 Doppler broadening

Doppler broadening refers to the variation in the resonance frequencies of atoms

moving with different velocities. Consider an atom emitting light of wavelength λ0

while moving at a velocity v (see Figure F.2). Over the course of an optical period

T0, the atom moves so that the wavelength appears compressed, λ = λ0−vT0. Using

the relations λ0 = 2πc/ω0, λ = 2πc/ω and T0 = 2π/ω0, we derive the Doppler shift

δω = −ω0v

c, (F.9)

where δω = ω − ω0 is the shift in angular frequency caused by the motion of the

atom. The spectral intensity is then given by the distribution of atomic velocities

in the vapour. An atom with velocity v and mass M has a kinetic energy

E = 12Mv2. (F.10)


Using the Boltzmann distribution (F.1), and substituting for v using (F.9) gives the

spectrum

I(δω) ∝ e−(δω/γd)2

, (F.11)

where the Doppler linewidth is given by

γd =

√2kBT

M× ω0

c. (F.12)

Figure F.2 The Doppler shift. An atom moving with velocityv ‘catches up’ with the light it emits, so that the wave appearssquashed.

F.3.2 Pressure broadening

Collisions between atoms in a vapour cause disruptions to the wavetrain of light

emitted by them. In particular, the phase of the light is randomized by each collision.

To understand the effect on the spectrum of the light we should consider the statistics

of collisions.

An atom with cross sectional area σ travelling at velocity v sweeps out a volume


Figure F.3 Collisions in a vapour. An atom with cross section σtravelling with velocity v sweeps out a volume vσdτ in a short timedτ . Any other atom within this volume gets hit!

σvdτ in a short time dτ (see Figure F.3). The probability of a collision is just the

probability of finding another atom within this volume, which is given by n×σvdτ ,

where n is the number density of atoms in the vapour. Now consider ps(τ), the

probability that an atom ‘survives’ without colliding for a time τ . We do not know

this probability yet, but we can say the following,

ps(τ + dτ) = ps(τ)× (1− nσvdτ) . (F.13)

That is, the probability of surviving a further time dτ after τ is given by the proba-

bility of surviving for a time τ , and then not colliding during dτ . Taylor expanding

(F.13) gives

ps(τ) + ∂τps(τ)dτ = ps(τ)− nσvps(τ)dτ,

⇒ ∂τps(τ) = −nσvps(τ). (F.14)


The survival probability is therefore given by an exponential distribution,

ps(τ) = γpe−γpτ , (F.15)

where γp = nσv, and where the preceeding factor of γp ensures that the distribution

is correctly normalized.

We are now in a position to derive the form of a collision-broadened spectrum.

The spectral intensity profile of the emitted light is given by

I(ω) =∣∣∣E(ω)

∣∣∣2= Fτ

1√2π

∫E∗(t)E(t+ τ) dt

(ω), (F.16)

where E(ω) and E(t) = E0eiω0t+φ(t) are the spectral and temporal electric field

amplitudes of the light, and where in the second line we have used the convolution

theorem (see §D.3.5 in Appendix D). The convolution inside the curly braces is the

first order correlation function of E(t). In the absence of a collision, there is no

change to E, and this correlation function is constant. After a collision, the phase

φ is randomized, and when this is averaged over all atoms, the correlation function

vanishes. The average correlation function is thus proportional to the probability

that there is no collision in the interval [0, τ ]. Factorizing out the carrier frequency


ω0 using the shift theorem (see §D.3.4 in Appendix D), the spectrum is given by

I(δω) =|E0|2√

2π×Fτ

1−

∫ τ

0ps(τ ′)dτ ′

(δω)

=|E0|2√

2π×Fτ

e−γp|τ |

(δω)

=|E0|2

2π× γp

γ2p + δω2

, (F.17)

where in the second line the modulus sign indicates that the correlation function is

symmetric in time. (F.17) describes a Lorentzian lineshape with a width γp. Al-

though this spectrum was derived by considering emission, the absorption spectrum

is identical, since absorption is simply the time reverse of emission. An estimate of

the collision-broadened linewidth is given by

γp ≈ n(πd2atom)×

√2kBT

M. (F.18)

The number density n can be obtained from the vapour pressure as described in the

start of this Appendix. The term in brackets approximates the collision cross-section

σ as the area of a circle with radius given by the atomic width datom. The last factor

is found by setting the kinetic energy (F.10) equal to the thermal energy kBT and

solving for the average atomic velocity.

F.3.3 Power broadening

In the presence of an intense laser field, an atomic absorption line broadens. One

way to understand this effect is to consider that the presence of light at the atomic

F.4 Raman polarization 481

resonance frequency stimulates emission from the excited state, and this reduces

the lifetime of the state, which introduces a concomitant broadening. Alternatively,

consider that the atomic dynamics are governed by Hamiltonian evolution. In the

loosest possible sense, the resulting behaviour can be characterized as a mixing of all

the frequencies in the Hamiltonian. Thus if there exists an optical frequency ω and

a Rabi frequency Ω, there will be driven oscillations at frequencies of ω ± Ω. Such

oscillations may be resonant with the atomic transition at ω0, even when neither

frequency is separately. Therefore absorption is possible at large detunings, provided

the laser is sufficiently intense. The atomic transition is rendered broader by the laser

field. In fact, the power-broadened linewidth is indeed given by γpb = Ω, as a more

detailed derivation from the damped optical Bloch equations shows [107]. This effect

is essentially the same as the dynamic Stark effect that generates the instantaneous

frequency shift in (5.74) in Chapter 5, and the Autler-Townes splitting discussed in

§2.3.1 in Chapter 2.

F.4 Raman polarization

In Chapter 10 we describe the preliminary steps taken toward building a Raman

quantum memory in cesium vapour. Here we explain how we arrive at the conclusion

that the Stokes light scattered from a linearly polarized pump will be polarized

orthogonally to the pump, when far detuned.

We consider Raman scattering from the 6S1/2 ↔ 6P3/2 D2 line in cesium, at 852

nm. There are two hyperfine levels in the ground state, with F = 3 and F = 4.


Atoms prepared in one of these levels are transferred by the Raman interaction to

the other level, with the transfer mediated by one of the excited states in the 6P3/2

manifold. In the process, a photon from the Raman pump is absorbed, and a Stokes

photon is emitted1. Each photon carries only one quantum of angular momentum,

so the intermediate state cannot differ in its angular momentum quantum number

by more than 1 from the initial and final states. Of the four hyperfine levels —

F = 2, 3, 4, 5 — in the excited state manifold, only the central two with F = 3, 4 are

compatible with this requirement, so we consider only these two intermediate levels.

Within each hyperfine level, there are 2F + 1 Zeeman sublevels, with magnetic

quantum numbers m = −F,−F+1, . . . , F−1, F . We specify one of these states with

the notation |F,m〉. The magnetic quantum numbers correspond to the component

of the atomic angular momentum along some axis, known as the quantization axis.

We must choose the direction for this axis before proceeding further. If a magnetic

field is applied to the atoms, the only sensible choice is to align the quantization

axis with the field. But in the absence of a magnetic field, the choice is arbitrary,

and we may choose the quantization direction in whichever way is convenient for

our calculations. As mentioned above, a single photon carries one unit of angular

momentum, and so it can change the magnetic quantum number by at most ±1, or

indeed not at all, if the spin of the photon has no component along the quantization1Strictly the term ‘Stokes’ refers to Raman scattered photons with less energy than the pump

photons. When they have more energy than the pump, they are commonly termed anti-Stokesphotons. In the present case, Stokes scattering only occurs if the atoms are prepared in the lower ofthe two ground states, with F = 3. The distinction between Stokes and anti-Stokes is not importantfor us, however. For simplicity, we always refer to the light emitted in Raman scattering as Stokeslight, regardless of whether or not its frequency is lower than that of the Raman pump.


axis. Figure F.4 shows how the direction and polarization of a photon with respect to

the quantization axis is connected with the changes in m induced by its absorption.

quantization axis

Figure F.4 Polarization selection rules. Right and left circularpolarizations propagating along the quantization axis change m by+1 and −1; they are known as σ+ and σ− polarizations, respectively.It is intuitive that as the electric field follows a corkscrew trajectory, itapplies a torque to the atoms about the quantization axis, increasingor decreasing m appropriately. π polarized light does not change m.It is polarized linearly along the quantization axis and propagatesat right angles to it, so it cannot induce a turning moment. Theorthogonal linear polarization can, however, change m by either +1or −1.

We choose the quantization axis so that the linearly polarized Raman pump

is π-polarized. With this choice, absorption of a pump photon cannot change the

magnetic quantum number m. Given the initial state |Fi,mi〉 of an atom, this

narrows down the number of possible intermediate states to two: either |Fint =

3,mint = mi〉 or |Fint = 4,mint = mi〉 (the only exception is if mi = 0 initially. Then

F must change, so there is only one possible intermediate state, with Fint 6= Fi). The

atom now decays from the intermediate excited state into the ‘final’ ground hyperfine

level with Ff 6= Fi. There is no restriction on the polarization of the emitted Stokes

photon, so there are three possibilities for this latter part of the interaction. The


possible final states are |Ff ,mf = mi〉 and |Ff ,mf = mi±1〉. As shown in Figure F.5,

each final state |Ff ,mf〉 can be reached from the given initial state |Fi,mi〉 by two

alternative ‘paths’. The first via the excited state with Fm = 3, and the second with

Fm = 4. When two different paths connect the same starting and ending points,

there is the possibility of interference. The present case is archetypical: π-polarized

Stokes emission is forbidden because of destructive interference between the two

interaction pathways. To see this, we explicitly evaluate the quantum mechanical

amplitude Aππ for π-polarized Stokes emission given a π-polarized pump, given the

initial state |Fi,mi〉,

Aππ(Fi,mi) =∑

Fm=3,4

d(J = 1/2, Fi,mi → J = 3/2, Fm,mm = mi)× (F.19)

d(J = 3/2, Fm,mm = mi → J = 1/2, Ff 6= Fi,mf = mi).

Here d denotes a dipole matrix element connecting the states with angular momen-

tum quantum numbers indicated by its arguments. Recall that J = 3/2 describes

the intermediate excited states. The selection rules set by fixing both photon polar-

izations to π keep the magnetic quantum number equal to its initial value throughout

the interaction. We have factorized out any dependence of the dipole matrix ele-

ments on the radial (or principle) quantum number. That this is always possible

is a consequence of the Wigner-Eckhart theorem. There exists a sophisticated and

confusing mathematical apparatus for dealing with the resulting ‘reduced’ matrix


elements. In standard notation, we can write [184]

d(J, F,m→ J ′, F ′,m′) = (−1)F+J+I+1√

(2F + 1)(2J + 1)

J J ′ 1

F ′ F I

C1FF ′

(m′−m)mm′ ,

(F.20)

where the curly braces denote a Wigner 6j symbol, and where Cj1j2Jm1m2Mis a Clebsch-

Gordan coefficient. Here I = 7/2 is the cesium nuclear spin. The 1’s appearing

in various places represent the angular momentum of the photon involved in the

transition. Angular momentum conservation requires that the lower indices of the

Clebsch-Gordan coefficient obey the sum rule m1 +m2 = M . The symbols in (F.20)

can be evaluated by standard routines in Mathematica. Matlab routines have also

been written, and are available freely on the web. Performing the sum in (F.19)

using (F.20), we find that Aππ is identically zero for all choices of the initial state

(i.e. either Fi = 3 or 4, and any value of mi). No doubt there is a deep group-

theoretical reason for this, but I do not know what it is! Nonetheless the calculation

shows that Stokes scattered photons emerge with the orthogonal linear polarization

to that of the pump. A very fortunate outcome for the experimenter who wishes to

filter the weak Raman signal out from the bright pump.

In evaluating the scattering amplitude, we have given equal weight to both path-

ways. This implicitly assumes that the optical fields are sufficiently far detuned from

the excited state manifold that the small energy splitting between the intermediate

hyperfine levels — around 200 MHz — makes a negligible difference to the Raman

coupling. If instead we tune into resonance with one of the levels, we can neglect


-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3

-3 -2 -1 0 1 2 3

-4 4

-3 -2 -1 0 1 2 3-4 4

Figure F.5 Alternative scattering pathways. Each initial state iscoupled by the Raman interaction to a final state via two possibleinteraction pathways; the first involves an intermediate excited statewith Fm = 3 (black-blue), the second with Fm = 4 (black-red). Thefigure illustrates this with the example of an atom initially preparedin the state Fi = 4,mi = 1. Absorption of a π-polarized Raman pumpphoton couples this state to the intermediate states |Fm = 3,mm = 1〉and |Fm = 4,mm = 1〉. Finally emission of a Stokes photon reducesthe magnetic quantum number by one: both paths end with the finalstate |Ff = 3,mf = 0〉.

scattering involving the other level, and there is no longer any interference. Isolating

just a single term from the sum in (F.19), we calculate a Stokes π-polarization of

40% on resonance with the Fm = 3 state, averaged over all possible initial states.

This averaging assumes an unpolarized ensemble, with atoms populating all Zeeman

substates in the initial hyperfine level equally. If we tune into resonance with the

Fm = 4 state, on average only 26% of the Stokes light is π-polarized. These calcu-

lations show that even on resonance, there remains a significant proportion of the

Stokes scattering that is polarized orthogonally to the pump, so that polarization

filtering is always feasible.

Note that these conclusions hold equally well for the polarizations of the the

signal and control fields in a cesium quantum memory, which has at its heart the very


same Raman interaction. Therefore a vertically polarized signal field can be stored

by a horizontally polarized control pulse, when both are tuned far from resonance.

If we choose a circularly polarized Raman pump — this time aligning the quanti-

zation axis with the pump propagation direction for convenience — we find that, far

from resonance, all of the Stokes light is emitted into the same circular polarization

as the pump. That is, right-circular scatters into right-circular, and left-circular into

left-circular. This is rather unfortunate, because it precludes the use of what might

otherwise have been a rather ingenious ‘trick’2: Suppose that the atomic ensemble

is prepared in the the state |Fi = 4,mi = 4〉. Then a σ+-polarized photon can

only couple to a state with mm = 5, which lies within the Fm = 5 hyperfine level

in the excited state. This level cannot participate in the Raman interaction (see

the discussion at the start of this section), and so a σ+-polarized photon cannot

act as a Raman pump. If the ensemble is used as a quantum memory, then a σ+-

polarized control pulse cannot cause spontaneous Stokes scattering. This eliminates

the unwanted ‘noise’ process mentioned in §4.6 of Chapter 4 (see part (b) of Figure

4.3). Unfortunately, the signal photon must be σ−-polarized, and then the memory

coupling vanishes because quantum interference between the two possible scatter-

ing pathways destroys any coupling between orthogonal circular polarizations. This

explains why we make no attempt to spin-polarize our cesium ensemble: it is more

trouble than it is worth!

2This was suggested by Jean Dalibard when he came to Oxford to give a colloquium.

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Quantum Memory in Atomic Ensembles

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