THEORY OF LIGHT!MATTER INTERACTIONS IN CASCADE ...

THEORY OF LIGHT-MATTER INTERACTIONS INCASCADE AND DIAMOND TYPE ATOMIC

ENSEMBLES

A ThesisPresented to

The Academic Faculty

by

Hsiang-Hua Jen

In Partial Ful�llmentof the Requirements for the Degree

Doctor of Philosophy in theSchool of Physics

Georgia Institute of TechnologyDecember 2010

THEORY OF LIGHT-MATTER INTERACTIONS INCASCADE AND DIAMOND TYPE ATOMIC

ENSEMBLES

Approved by:

Prof. T. A. Brian Kennedy, AdvisorSchool of PhysicsGeorgia Institute of Technology

Prof. Carlos Sa de MeloSchool of PhysicsGeorgia Institute of Technology

Prof. Alex KuzmichSchool of PhysicsGeorgia Institute of Technology

Prof. Ken BrownSchool of Chemistry and BiochemistryGeorgia Institute of Technology

Prof. Michael S. ChapmanSchool of PhysicsGeorgia Institute of Technology

Date Approved: November 5, 2010

To grandfather,

who supports and believes in me unconditionally throughout the study,

and in memory of grandmother and father.

v

ACKNOWLEDGEMENTS

I am grateful to my thesis advisor, Professor Brian Kennedy, for his instruction and

support of my research. With his guidance and encouragement, I learned and gained

insights along the way of study. I am thankful to Professor Alex Kuzmich for his

direction in experimental perspective and to Dr. S. D. Jenkins for his helpful dis-

cussions on theoretical background. I am also thankful to the thesis committee,

Professor Michael Chapman, Professor Carlos Sa de Melo, and Professor Ken Brown.

Throughout this work, I had many useful discussions with quantum optics group

members, and I am thankful to them: Dr. D. N. Matsukevich, Dr. T. Chanelière,

Dr. S. Y. Lan, O. A. Collins, C. Campbell, Dr. R. Zhao, and A. Radnaev. I am

also appreciative to Professor Li You, Dr. P. Zhang, and Dr. D. L. Zhou for the

training in my early graduate studies. A special thank you is due for the support of

my friends, Shenshen Lin, Professor I-Tang Yu, Dr. S. C. Lin, Dr. Yu Tsao, Dr. Pei

Lin, Dr. T. Lee, and Dr. A. Liang.

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TABLE OF CONTENTS

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 DLCZ Protocol for the Quantum Repeater . . . . . . . . . . . . . . 2

1.1.1 Correlated cascade emission in quantum telecommunication . 2

1.2 Quantum Memory with Light Frequency Conversion . . . . . . . . . 3

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

II REVIEW OF THEORETICAL AND NUMERICAL METHODS 7

2.1 Quantum and C-number Langevin Equations . . . . . . . . . . . . . 7

2.1.1 Quantum Heisenberg-Langevin equations . . . . . . . . . . . 9

2.1.2 C-number Langevin equation . . . . . . . . . . . . . . . . . . 11

2.2 Fokker-Planck Equations and Stochastic Di¤erential Equations . . . 13

2.2.1 Characteristic functions in P-representation . . . . . . . . . . 13

2.2.2 A Complimentary Derivation of C-number Langevin Equations 15

2.3 Kubo Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

III SUPERRADIANTEMISSION FROMACASCADEATOMIC EN-SEMBLE: ANALYTICAL METHOD . . . . . . . . . . . . . . . . . 21

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 The example of two-state atoms interacting with a pump �eld . . . . 22

3.3 Theory of Cascade Emission . . . . . . . . . . . . . . . . . . . . . . 26

3.3.1 Probability amplitudes for signal and signal-idler emissions . 28

3.4 A Correlated Two-photon State . . . . . . . . . . . . . . . . . . . . 32

3.5 Schmidt Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 34

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IV SUPERRADIANTEMISSION FROMACASCADEATOMIC EN-SEMBLE: NUMERICAL APPROACH . . . . . . . . . . . . . . . . 43

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Theory of Cascade emission . . . . . . . . . . . . . . . . . . . . . . . 44

4.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3.1 Shooting and secant method . . . . . . . . . . . . . . . . . . 48

4.3.2 Outline of the numerical solution . . . . . . . . . . . . . . . . 50

4.3.3 Results for signal, idler intensities, and the second-order cor-relation function . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

V SPECTRALANALYSIS FORCASCADE-EMISSION-BASEDQUAN-TUM COMMUNICATION . . . . . . . . . . . . . . . . . . . . . . . 61

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2 DLCZ Scheme with Cascade Emission . . . . . . . . . . . . . . . . . 61

5.3 Entanglement Swapping . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4 Polarization Maximally Entangled State (PME State) and QuantumTeleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

VI EFFICIENCY OF LIGHT-FREQUENCY CONVERSION IN ANATOMIC ENSEMBLE . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.3 Optimal Conversion E¢ ciency . . . . . . . . . . . . . . . . . . . . . 84

6.4 Pulse Conversion: Solution of the Maxwell-Bloch Equations . . . . . 89

6.5 Discussion of Quantum Fluctuations . . . . . . . . . . . . . . . . . . 92

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

VII CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

APPENDIX A � DERIVATION OF A SCHRÖDINGER WAVEEQUATION FOR SPONTANEOUS EMISSIONS FROM A CAS-CADE TYPE ATOMIC ENSEMBLE . . . . . . . . . . . . . . . . . 101

x

APPENDIX B � DERIVATION OF A C-NUMBER LANGEVINEQUATION FOR THE CASCADE EMISSION . . . . . . . . . . 113

APPENDIXC � MULTIMODEDESCRIPTIONOFCORRELATEDTWO-PHOTON STATE . . . . . . . . . . . . . . . . . . . . . . . . . 145

APPENDIXD � HAMILTONIANANDEQUATIONOFMOTIONFORFREQUENCYCONVERSION INADIAMONDTYPEATOMICENSEMBLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

xi

LIST OF TABLES

4.1 Numerical simulation parameters for di¤erent atomic densities �. Cor-responding optical depth (opd), time and space grids (Mt �Mz) withgrid sizes (�t;�z) in terms of cooperation time (Tc) and length (Lc),and the �tted characteristic time Tf for Gs;i (see text). . . . . . . . . 58

xiii

LIST OF FIGURES

2.1 The two-level atomic ensemble interacts with a classical and quantum�eld. (a) An elongated atomic ensemble of length L is excited by apump �eld of Rabi frequency a and emits a propagating quantized�eld denoted by annihilation operator E+: (b) Two-level structure foran atomic ensemble with the ground (j0i) and excited (j1i) state. Thedetuning of the pump �eld is �1. . . . . . . . . . . . . . . . . . . . . 8

2.2 Kubo oscillator simulation. The time evolution of Rehz(t)i (dashed-red) averaged from an ensemble of 1024 simulations. z(0) = 1: Wecompare with the exact solution, z(0)e�t=2 (solid-black), and �nd goodagreement. A demonstration of one stochastic realization (dashed-circle blue) shows large �uctuation around the averaged and exact re-sults. Note that the imaginary part of the solution is almost vanishingas it should be, and is not shown here. . . . . . . . . . . . . . . . . . 19

3.1 Single- and double-excitation populations as a function of distanceds: (a) The populations of the symmetric state for a single excitationP s1 (dashed-red) and the sum of non-symmetric single-excitation statesP ns1 (dashed-dotted black). (b) The populations of the symmetric statefor double excitations P s2 (dashed-blue) and the sum of non-symmetricdouble-excitation states P ns2 (dashed-dotted black). P s, ns1 and P s, ns2

are normalized respectively by the solutions of non-interacting atomsP(0)1 (solid-red) and P (0)2 (solid-blue). P

(0)1 = 1:58 � 10�3 and P (0)2 =

9:4 � 10�7; are the single- and double-excitation probabilities for in-dependent atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Time evolution of populations for symmetric states P s1 and Ps2 . The

population of the symmetric state for a single excitation is P s1 (dashed-red), and that for the symmetric state for double excitations is P s2(dashed�dotted blue). The pump condition is the same as in Figure3.1 for ds = 3�: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Four-level atomic ensemble interacting with two driving lasers (solid)with Rabi frequencies a and b: Signal and idler �elds are labelledby as and ai; respectively and �1 and �2 are one and two-photon laserdetunings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 The superradiance decay factor N�+1 (� = ��) for a cylindrical ensem-ble of length h and radius a in unit of transition wavelength �. Theatomic density is 8� 1010 cm�3 and � = 795 nm corresponding to theD1 line of 85Rb. See the text for the explanation of the arrows. . . . 30

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3.5 (a) Absolute value of the spectrum for two-photon state probability am-plitudeDs;i and (b) the second-order correlation functionG

(2)s;i (�ts;�ti):

(c) A normalized G(2)s;i (�ts = 0;�ti) with �3� = 0:2. The exponentialdecay corresponds to the superradiant decay factor N ��+ 1 = 5: . . . 35

3.6 Schmidt mode analysis with pulse width � = 0:25 and superradiancedecay factor N �� + 1 = 5: (a) Schmidt number and (b) signal modefunctions: Re[ 1] (solid-red) and Re[ 2] (solid-blue). Imaginary partsare not shown, then are zero. (c) Real (solid) and imaginery (dotted)parts of �rst (red) and second (blue) idler mode functions, �1 and �2.(d) The absolute spectrum jf(�!s;�!i)j. . . . . . . . . . . . . . . . 38

3.7 Absolute spectrum of two-photon state and the eigenvalues of Schmidtdecomposition. N ��+1 = 5 for both (a) � = 0:25 (b) � = 0:5. N ��+1 =10 for (c) � = 0:25. The von Neumann entropy (S) is indicated in theplots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.1 Schematic illustration of the principle of the shooting method for two-point boundary value problems. . . . . . . . . . . . . . . . . . . . . . 49

4.2 Secant method. The root is bracketed by two initial guesses of x1and x2 and an updated guess xi is located at the intersection of twostraight lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Time-varying pump �elds and time evolution of atomic populations.(Left) The �rst pump �eld a (dotted-red) is a square pulse of duration50 ns and b is continuous wave (dotted-blue). (Right) The timeevolution of the real part of populations for three atomic levels �11 =h~�13i (dotted-red), �22 = h~�12i (dotted-blue), �33 = h~�11i (dotted-green) at z = 0; L, and almost vanishing imaginary parts for all threeof them. indicate convergence of the ensemble averages. Note thatthese atomic populations are uniform as a function of z: . . . . . . . . 54

4.4 Temporal intensity pro�les of counter-propagating signal and idler �elds.(a) At z = 0; real (dotted-red) and imaginary (diamond-red) parts ofsignal intensity. (b) At z = L; real (dotted-blue) and imaginary(diamond-blue) parts of idler intensity. Both intensities are normal-ized by the peak value of signal intensity that is 7:56�10�12 E2c . Notethat the idler �uctuations and its non-vanishing imaginary part indi-cate a relatively slower convergence compared with the signal intensity.The ensemble size was 8�105; and the atomic density � = 1010cm�3. . 55

xvi

4.5 Second-order correlation function Gs;i(ts; ti): The 2-D contour plot ofthe real part of Gs;i with a causal cut-o¤at ts = ti is shown in (a). Theplot (b) gives a cross-section at ts = tm � 75 ns, which is normalizedto the maximum of the real part (dotted-blue) of Gs;i: The imagi-nary part (diamond-red) of Gs;i is nearly vanishing, and the number ofrealizations is 8�105 for � = 1010cm�3: . . . . . . . . . . . . . . . . . 57

4.6 Characteristic timescales, Tf and T1 vs atomic density � and the super-radiant enhancement factor N� (� = ��). Tf (dotted-blue) is the �ttedcharacteristic timescale for Gs;i(ts = tm; ti = tm+�) where tm is chosenat its maximum, as in Figure 4.5. The errorbars indicate the �ttinguncertainties. As a comparison, T1= �103 =(N� + 1) (dashed-black) isplotted where �103 = 26 ns is the natural decay time of D1 line of

87Rbatom, and � is the geometrical constant for a cylindrical atomic ensem-ble, as discussed in Chapter 3. The number of realizations is 4�105for � = 5� 108, 5� 109 cm�3 and 8�105 for � = 1010, 2� 1010 cm�3: 59

5.1 Entanglement generation in the DLCZ scheme using the cascade andRaman transitions in two di¤erent atomic ensembles. Large whitearrows represent laser pump excitations corresponding to the dashedlines in either cascade or Raman level structures. Here ays representsthe emitted telecom photon. B.S. means beam splitter that is usedto interfere the incoming photons measured by the photon detector D.The label A refers to the pair of ensembles for later reference. . . . . 63

5.2 Entanglement swapping of DLCZ scheme using the cascade transition.The site A is described in detail in Figure 5.1 and equivalently for thesite B. The telecom signal photons are sent from both sites and in-terfere by B.S. midway between with detectors represented by cy1 andcy2. Synchronous single clicks of the detectors from both sites (my

1;2,ny1;2) and the midway detector (c

y1;2) generate the entangled state be-

tween lower atomic ensembles at sites A and B. The locally generatedentanglement is swapped to distantly separated sites in this cascade-emission-based DLCZ protocol. . . . . . . . . . . . . . . . . . . . . . 64

5.3 Fidelity F , heralding PH , and success PS probabilities of entangle-ment swapping versus relative e¢ ciency �r with perfect detection e¢ -ciency �t = 1: Column (a) NRPD and (b) PNRD. Solid-red, dashed-blue, and dotted-green curves correspond to the pulse width parame-ters � = (0:1; 0:5; 0:5) and superradiant factor N �� + 1 = (5; 5; 10)(see Chapter 3 and Appendix A): The von Neumann entropy is S =(0:684; 2:041; 2:886); respectively. . . . . . . . . . . . . . . . . . . . . 69

5.4 Fidelity F , heralding PH , and success PS probabilities of entanglementswapping versus telecom detector quantum e¢ ciency � for the case of(a) NRPD and (b) PNRD. Solid-red, dashed-blue, and dotted-greencurves correspond to the same parameters used in Figure 5.3. . . . . . 71

xvii

5.5 PME projection (a) and quantum teleportation (b) in the DLCZ scheme.Four atomic ensembles (A,B,C,D) are used to generate two DLCZ en-tangled states at (A,B) and (C,D). PME state is projected probabilis-tically conditioned on four possible detection events of (Dy

A or DyC)

and (DyB or D

yD) in (a). In the quantum teleportation protocol (b),

another two ensembles (I1;I2) are used to prepare a quantum state thatis teleported to atomic ensembles B and D conditioned on four possibledetection events of (DI1 or DA) and (DI2 or DC). . . . . . . . . . . . 72

5.6 Success probability of quantum teleportation as a function of the prob-ability amplitude of teleported quantum state with �r = 0:5 and a per-fect detector e¢ ciency �t = 1: Solid-red, dashed-blue, and dotted-greencurves correspond to the same parameters used in Figure 5.3. . . . . . 75

6.1 The diamond con�guration of atomic system for conversion scheme.Two pump lasers (double line) with Rabi frequencies a;b and propa-gated probe �elds (single line)E+s ; E

+i interact with the atomic medium.

Various detunings are de�ned in the Appendix D, and the atomic lev-els used in the experiment [25] are (j0i; j1i; j2i; j3i) = (j5S1=2;F =1i; j5P3=2;F = 2i; j6S1=2;F = 1i; j5P1=2;F = 2i): . . . . . . . . . . . . . 79

6.2 Dressed-state picture from the perspective of the probe idler transi-tion between atomic levels j0i and j3i: Two strong �elds a;b shiftthe levels with energy �Ea;b and wavy lines represent the idler �eldresonances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3 Self-coupling coe¢ cients �s; �i and cross-coupling coe¢ cient �s. Di-mensionless quantities (a) �sL, (b) �iL and (c) �sL with real (solidblue) and imaginary (dashed red) parts are plotted as a dependence ofidler detuning�!i [same label in (b)] showing four absorption peaks toconstruct three parametric coupling windows. A black dashed-dot lineof the constant �=2 is added in (c) to demonstrate the crossover with=(�sL) indicating the ideal conversion e¢ ciency condition in the leftwindow. The parameters we use are (a, b, �1, �b, �!i) = (33,20, 39, 2, �21) 03 for optical depth ��L = 150 with L = 6mm.Various natural decay rates are 03 = 1=27:7ns, 01 = 1=26:24ns, 12 = 03=2:76; and 32 = 03=5:38 [97]. . . . . . . . . . . . . . . . . . 85

6.4 Down conversion e¢ ciency �d vs optical depth (opd) from 1 to 600:Each dotted point is the maximum for �ve variational parameters a,b, �1, �b, and �!i. . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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6.5 Conversion e¢ ciency �d, �u and transmission Td vs �!i for opd=150.�d and �u are indistinguishable and shown in solid red line, and Td isin dashed blue line. High transmission e¢ ciency corresponds to lowconversion e¢ ciency indicating the approximate conservation conditionwithin each parametric coupling window. The maximum conversione¢ ciency is found in the left window at around �!i = �20 03 andother relevant parameters are the same as in Figure 6.3. . . . . . . . . 88

6.6 Time-varying pump �elds of Rabi frequenciesa;b(t) and down-convertedsignal intensity (jE+s (t; z = L)j2) from an input idler pulse (jE+i (t; z =0)j2). Here we let t = � that is the delayed time in co-moving frame.Pump-b (dotted green) is a continuous wave and pump-a (dashedblack) is a square pulse long enough to enclose input idler pulse with(a) 100 ns and (b) 15 ns (dashed-dot blue). Output signal intensity(solid red) at the end of atomic ensemble z = L is oscillatory due to thepump �elds. The square pulse in rising region (tr� ts

2< t < tr+

ts2) has

the form of 12[1 + sin(�(t�tr)

ts)] that in (a) (tr; ts)=(10,10)ns for pump-a

and (tr; ts)=(20,20)ns for input idler; (b) (tr; ts)=(10,5)ns for pump-aand (tr; ts)=(15,10)ns for input idler where tr is the rising time indi-cating the center of rising period ts: Note that the falling region ofsquare pulse is symmetric to the rising one. . . . . . . . . . . . . . . . 91

6.7 Three-dimensional line plots of converted signal and input idler inten-sities in t (ns) and L (mm). Here we let t = � that is the delayed timein co-moving frame, and the parameters are the same as in Figure 6.6(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

C.1 Model of quantum e¢ ciency of detector. . . . . . . . . . . . . . . . . 146

D.1 Self-coupling coe¢ cient �i: A dimensionless quantity �iL is plottedwith real (solid blue) and imaginery (dashed red) parts as a depen-dence of idler detuning �!i showing a normal dispersion inside theEIT window. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

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SUMMARY

In this thesis, we investigate the quantum mechanical interaction of light with

matter in the form of a gas of ultracold atoms: the atomic ensemble. We present

a theoretical analysis of two problems, which involve the interaction of quantized

electromagnetic �elds (called signal and idler) with the atomic ensemble (i) cascade

two-photon emission in an atomic ladder con�guration, and (ii) photon frequency

conversion in an atomic diamond con�guration. The motivation of these studies

comes from potential applications in long-distance quantum communication where it

is desirable to generate quantum correlations between telecommunication wavelength

light �elds and ground level atomic coherences. In the two systems of interest, the

light �eld produced in the upper arm of an atomic Rb level scheme is chosen to lie in

the telecom window. The other �eld, resonant on a ground level transition, is in the

near-infrared region of the spectrum. Telecom light is useful as it minimizes losses in

the optical �ber transmission links of any two long-distance quantum communication

device.

We develop a theory of correlated signal-idler pair correlation. The analysis is

complicated by the possible generation of multiple excitations in the atomic ensemble.

An analytical treatment is given in the limit of a single excitation assuming adiabatic

laser excitations. The analysis predicts superradiant timescales in the idler emission

in agreement with experimental observation. To relax the restriction of a single

excitation, we develop a di¤erent theory of cascade emission, which is solved by

numerical simulation of classical stochastic di¤erential equation using the theory of

open quantum systems. The simulations are in good qualitative agreement with

the analytical theory of superradiant timescales. We further analyze the feasibility

xxi

of this two-photon source to realize the DLCZ protocol of the quantum repeater

communication system.

We provide a quantum theory of near-infrared to telecom wavelength conversion in

the diamond con�guration. The system provides a crucial part of a quantum-repeater

memory element, which enables a "stored" near-infrared photon to be converted to a

telecom wavelength for transmission without the destruction of light-atom quantum

correlation. We calculate the theoretical conversion e¢ ciency, analyzing the role of

optical depth of the ensemble, pulse length, and quantum �uctuations on the process.

xxii

CHAPTER I

INTRODUCTION

A quantum communication network based on the distribution and sharing of entan-

gled states is potentially secure to eavesdropping and is therefore of great practical

interest [1�3]. A protocol for the realization of such a long distance system, known

as the quantum repeater, was proposed by Briegel et al. [4, 5]. A quantum re-

peater based on the use of atomic ensembles as memory elements, distributed over

the network, was subsequently suggested by Duan, Lukin, Cirac and Zoller [6]. The

storage of information in the atomic ensembles involves the Raman scattering of an

incident light beam from ground state atoms with the emission of a signal photon.

The photon is correlated with the creation of a phased, ground-state, coherent excita-

tion of the atomic ensemble. The information may be retrieved by a reverse Raman

scattering process, sending the excitation back to the initial atomic ground state and

generating an idler photon directionally correlated with the signal photon [7�15]. In

the alkali gases, the signal and the idler �eld wavelengths are in the near-infrared

spectral region. This presents a wavelength mismatch with telecommunication wave-

length optical �ber, which has a transmission window at longer wavelengths (1.1-1.6

um). It is this mismatch that motivates the search for alternative processes that can

generate telecom wavelength photons correlated with atomic spin waves [16].

This motivates the research presented in this thesis where we study multi-level

atomic schemes in which the transition between the excited states is resonant with a

telecom wavelength light �eld [16]. The basic problem is to harness the absorption

and the emission of telecom photons while preserving quantum correlations between

the atoms, which store information and the photons that carry along the optical �ber

1

channel of the network. In this thesis, we theoretically study atomic cascade and

diamond con�gurations in this context.

1.1 DLCZ Protocol for the Quantum Repeater

A long-distance quantum repeater must overcome the exponential losses in the optical

�ber. To overcome this problem, the use of quantum memory was proposed [6]. For

a practical system, it is essential to maximize quantum memory time, to preserve

coherence during protocol operations, and connect the memory elements by light

signals in the low-loss window of the optical �ber medium. The telecom wavelength

range (1.1-1.6 �m) has a loss rate as low as 0.2 dB/km.

It is not common to have a telecom ground state transition in atomic gases except

for rare earth elements [17, 18] or in an erbium-doped crystal [19]. However, a

telecom wavelength (signal) can be generated from transitions between excited levels

in the alkali metals [16].

1.1.1 Correlated cascade emission in quantum telecommunication

The ladder con�guration of atomic levels provides a source for telecom photons (sig-

nal) from the upper atomic transition. For rubidium and cesium atoms, the signal

�eld has the range around 1.3-1.5 �m that can be coupled to an optical �ber and

transmitted to a remote location. Cascade emission may result in pairs of photons,

the signal entangled with the subsequently emitted infrared photon (idler) from the

lower atomic transition. Entangled signal and idler photons were generated from a

phase-matched four-wave mixing con�guration in a cold, optically thick 85Rb ensem-

ble [16]. This correlated two-photon source is potentially useful as the signal �eld

has telecom wavelength.

The temporal emission characteristics of the idler �eld, generated on the lower

arm of the cascade transition, were observed in measurements of the joint signal-

idler correlation function. The idler decay time was shorter than the natural atomic

2

decay time and dependent on optical thickness in a way reminiscent of superradiance

[20�24].

We will develop an analytical theory of the cascade emission in an atomic ensemble

in Chapter 3. The in�uence of electromagnetic dipole-dipole interactions between

atoms is important to account for the idler �eld�s temporal pro�le. By developing the

theory on the assumption of weak adiabatic laser excitation, we are able to calculate

the spectral characteristics of the signal and idler �elds, and make a connection with

the traditional theory of superradiance.

In Chapter 4, , we develop a more elaborate theory of the cascade emission under

similar physical conditions to Chapter 3, but without the assumption of single atomic

excitations. The theory is based on numerical solutions of stochastic di¤erential

equations derived using open-systems methods of quantum optics. We limit our

analysis to the con�rmation of the superradiant emission of the idler �eld predicted

in the simple theory and observed experimentally.

In Chapter 5, we use this theory to discuss a potential application of the cas-

cade emission process in the DLCZ protocol, and discuss the role of time-frequency

entanglement.

1.2 Quantum Memory with Light Frequency Conversion

It is not su¢ cient to generate telecom wavelength light for quantum communication.

The light �eld must be quantum correlated with atomic excitations stored in memory

[16].

Recently there has been a breakthrough in this direction using a pair of cold, non-

degenerate rubidium gas samples [25]. A correlated pair of atomic spin wave and

infrared �elds are generated by conventional Raman scattering in one ensemble. The

light �eld is directed onto a second ensemble where it is frequency converted to the

telecom range by four-wave mixing using a diamond con�guration of atomic levels.

3

The experiments were designed to measure quantum correlations between the stored

atomic excitation and the telecom �eld.

The conversion scheme exploits an e¢ cient low-noise parametric conversion process

that is facilitated by operating in the regime of high transparency [25]. This provides

a basic quantum memory element for a scalable, long distance quantum network. In

Chapter 6, we investigate conditions required to maximize the conversion e¢ ciency

as a function of optical thickness of the atomic ensemble. The in�uence of the probe

pulse duration on the conversion e¢ ciency is studied by numerical solution of the

Maxwell-Bloch equations.

1.3 Outline

The remainder of this thesis is organized as follows.

In Chapter 2, we review some theoretical methods to provide background for the

theories developed in Chapter 4 and 6. In particular we discuss the derivation of

quantum Heisenberg-Langevin equations for the interaction of a group of atoms with

a quantized propagating electromagnetic �eld. We illustrate the connection of these

operator equations with related classical (c-number) stochastic Langevin equations.

The latter have the useful property that they may be numerically simulated, under

certain conditions, and we provide the Kubo oscillator as a numerical test case.

In Chapter 3, we present a theory of cascade two-photon emission in an atomic

ensemble. The radiative atomic dipole-dipole coupling is shown to in�uence the

emission of the idler photon, resulting in the appearance of superradiant time scales.

The theory is developed on the basis of Schrödinger probability amplitudes assuming

single atomic excitations. This approach allows a straightforward treatment of the

spectral entanglement properties of the signal-idler photons.

In Chapter 4, we relax the assumption of single atomic excitations and develop a

theory based on c-number stochastic partial di¤erential equations, derived using the

4

methods reviewed in Chapter 2. Numerical solutions of the equations are used to

compare with the superradiant timescales derived in the analytical theory.

In Chapter 5, the analysis of Chapter 3 is used to discuss the behavior of the

cascade emission on the DLCZ protocol for the quantum repeater. Entanglement

swapping and quantum teleportation are investigated, and the in�uence of time-

frequency entanglement is discussed.

In Chapter 6, the use of the diamond con�guration in frequency up and down

conversion is analyzed using quantum-Heisenberg Langevin and Maxwell-Bloch equa-

tion methods. We present results for the optimal conversion e¢ ciency as a function

of optical thickness of the atomic ensemble. The role of pulse length and quantum

�uctuations are discussed.

In Chapter 7, we present some conclusions.

In Appendixes A-D, we present a great deal of supporting information on the theo-

retical derivations that are quite lengthy on account of both the multimode treatment

of the light �elds and the complicated atomic level schemes and atomic dipole-dipole

interactions.

5

CHAPTER II

REVIEW OF THEORETICAL AND NUMERICAL

METHODS

In this Chapter, we review the derivations of quantum-Heisenberg equations and c-

number Langevin equations for light-atom interactions. The reason for focusing on

these methods is, in the �rst place that they are less familiar than Schrödinger picture

methods (see Chapter 3 and 5) and that our applications of these methods (Chapter

4 and 6) involve rather long derivations that may obscure the basic ideas.

We provide two methods of deriving the c-number Langevin equations and their

noise correlations. The equations may be found from the quantum Heisenberg-

Langevin equations using a "quantum-classical" correspondence [26]. Alternatively,

c-number Langevin equations are deduced by a Schrödinger-picture approach that

employs characteristic equation and coherent state phase space methods. In the

�nal step the Langevin equations are deduced from a Fokker-Planck equation for a

generalized statistical distribution. Such methods were initially applied in quantum

laser theory in the 1960�s by Haken [27]. The independent derivations will be used

to check the lengthy derivations involved in the case of cascade emission.

2.1 Quantum and C-number Langevin Equations

Langevin equations were initially derived to describe Brownian motion [28]. A �uc-

tuating force is used to represent the random impacts of the environment on the

Brownian particle. A given realization of the Langevin equation involves a trajec-

tory perturbed by the random force. Ensemble averaging such trajectories provides

a natural and direct way to investigate the dynamics of the stochastic variables.

7

Figure 2.1: The two-level atomic ensemble interacts with a classical and quantum�eld. (a) An elongated atomic ensemble of length L is excited by a pump �eld ofRabi frequency a and emits a propagating quantized �eld denoted by annihilationoperator E+: (b) Two-level structure for an atomic ensemble with the ground (j0i)and excited (j1i) state. The detuning of the pump �eld is �1.

8

In this section, we review quantum and c-number Langevin equation approaches

for a two-level atomic ensemble interacting with a quantized electromagnetic �eld.

As shown in Figure 2.1, the atoms are excited by a pump �eld of Rabi frequency a,

and a propagating quantized �eld E+ is considered to be emitted along the direction

z of the ensemble with length L.

2.1.1 Quantum Heisenberg-Langevin equations

We consider the Hamiltonian of N two-level (ground and excited states j0i, j1i)

atoms interacting with one pump �eld and a multimode quantized �elds with mode

annihilation operators al that satisfy the commutation relation [al; ayl0 ] = �ll0 for the

lth section along the propagation direction. The propagation length L is discretized

into 2M + 1 elements [29]. In the electric dipole approximation and rotating wave

approximation, the interaction is given by �~d� ~E, Appendix B.1. The Hamiltonian H

includes the free evolution (H0) of atoms with transition frequency !1, the quantized

�eld of central frequency !; and the dipole interaction (HI),

H = H0 +HI , (2.1)

H0 =MX

l=�M

~!1�l11(t) + ~!MX

l=�M

ayl (t)al(t) + ~Xl;l0

!ll0 ayl (t)al0(t) , (2.2)

HI = �~MX

l=�M

hL�

ly01(t)e

ikLzl�i!Lt + gp2M + 1�ly01(t)al(t)e

ikzl + h:c:i(2.3)

where �l01(t) �PNz

� j0i�h1j��r�=zl

. The Rabi frequency L = d10E(kL)=(2~) is one-

half the conventional de�nition. The dipole matrix element d10 � h1jdj0i, coupling

strength g � d10E(k)=~ where E(k) =p~!=2�0V is the electric �eld per photon,

and zp =pL

2M+1; p = �M; :::;M . The matrix !ll0 �

PMn=�M kne

ikn(zl�zl0 )=(2M + 1)

accounts for �eld propagation by coupling the local mode operators.

The dynamical equations including dissipation due to spontaneous emission can

be treated by introducing the reservoir �eld that interacts with the system [30]. After

9

introducing the coupling to the reservoir, we may write down by inspection the dissi-

pation terms. We de�ne 01 to be the spontaneous emission rate from j1i ! j0i: In

the co-moving frame coordinates z and � = t�z=c, the quantum Heisenberg-Langevin

equations are

@

@�~�01 = (i�1 �

012)~�01 + ia(~�00 � ~�11) + ig(~�00 � ~�11) ~E+ + ~F01; (2.4)

@

@�~�11 = � 01~�11 + iL~�

y01 � i�L~�01 + ig~�y01 ~E

+ � ig�~�01 ~E� + ~F11; (2.5)

@

@z~E+ =

iNg�

c~�01 + ~FE+ ; (2.6)

where various Langevin noises ~F associated with atomic operators ~�01, ~�11 and �eld

operator ~E+ are necessary to preserve equal time commutation relations. The detun-

ing of the pump �eld is �1 = !L � !1 and the slowly-varying operators are de�ned

as ~�01 � �l01e�ikazl+i!at=Nz, ~�11 � ~�l11=Nz, and ~E

+(z; t) �p2M + 1ale

i!at where we

let ! = !L. The time evolution of atomic coherence (~�01) depends on the popu-

lation di¤erence (~�00 � ~�11), and in turn atomic population is in�uenced by atomic

coherence and the classical and quantized �elds. The atomic coherence couples to

the quantized �eld along the propagation direction, z.

The noise operator correlations are related to the dissipation through the �uctuation-

dissipation theorem [30, 31]. If we have a quantum Langevin equation for variable

x

_x(t) = Ax(t) + Fx(t) (2.7)

where Ax is so-called the drift term for x, and the corresponding Langevin noise

operator is Fx, the quantum noise correlation functions can be derived from the

generalized Einstein relation,

hFx(t)Fy(t)i = �hx(t)Ay(t)i � hAx(t)y(t)i+d

dthx(t)y(t)i: (2.8)

where the bracket denotes the quantum mechanical ensemble average.

10

With the above recipe, we have the non-vanishing normally ordered quantum noise

correlation function from Eq. (2.6),

D11;11 = 01~�11: (2.9)

whereD~F y11(t; z) ~F11(t0; z0)

E= L

N�(t � t0)�(z � z0)

DD11;11

E, and D is also referred to

as a di¤usion matrix element by analogy with classical di¤usion processes.

Even for this relatively simple light-matter interaction, there is no analytical solu-

tion possible. The c-number Langevin equation approach, below, provides a possible

way to attack the problem numerically by stochastic simulation and to calculate

normally-ordered quantities by ensemble averaging, although we will not pursue such

simulations here.

2.1.2 C-number Langevin equation

A c-number Langevin equation approach may be suitable for stochastic simulation

[28, 32], and utilizes the methods developed by Lax, Louisell, and Haken to describe

the dynamics of the interaction [26, 27]. Their recipe involves a normal ordering

procedure and a so-called "quantum-classical correspondence" to derive the c-number

Langevin equations [26, 31, 33]. The normal ordering chosen is ~�y01, ~�11, ~�01, ~E�, ~E+

where the creation operators always appear to the left of the annihilation operators.

The population operator is put between the atomic coherence operators since it is

self conjugate.

The c-number Langevin equations are then derived from Eq. (2.6) by making the

quantum-classical correspondence that we denote as

~�y01 ! �5; ~�11 ! �4; ~�01 ! �3; ~E� ! E�; ~E+ ! E+: (2.10)

Similarly for the Langevin noises,

11

~F y01 ! F5; ~F11 ! F4; ~F01 ! F3; ~FE� ! F2; ~FE+ ! F1; (2.11)

where the notation is chosen to facilitate the comparison with an alternative approach

that we will discuss in the next Section.

The classical noise correlation functions are also derived from an Einstein relation.

Consider the c-number Langevin equation for the variables x and y;

_x(t) = Ax(t) + Fx(t); (2.12)

_y(t) = Ay(t) + Fy(t): (2.13)

From the requirement of equivalent time evolution of normally-ordered operators and

their c-number counterparts, we have for example

d

dthxyi = d

dthxyi: (2.14)

Classical noise correlations can be derived from the quantum ones using

hFxFyi = hFxFyi+ hxAyi+ hAxyi � hxAyi � hAxyi: (2.15)

where the quantum and classical noise correlations are formally quite di¤erent. For

non-normally-ordered operators xz, we may use the commutator to substitute that

hxzi = hzxi+ h[x; z]i: (2.16)

The drift term of the c-number Langevin equations are closely related to the corre-

sponding term in the quantum Heisenberg-Langevin equations. After the quantum-

classical correspondence is made, we derive the coupled equations with c-number

variables (E+; E�; �3; �4; �5;) and Langevin noises (F1;2;3;4;5) that satisfy

@

@��3 = (i�1 �

012)�3 + ia(�0 � �4) + ig(�0 � �4)E

+ + F3; (2.17)

@

@��4 = � 01�4 + ia�5 � i�a�3 + ig�5E

+ � ig��3E� + F4; (2.18)

@

@��5 = (�i�1 �

012)�5 � i�a(�0 � �4)� ig�(�0 � �4)E

� + F5; (2.19)

@

@zE+ =

iNg�

c�3 + F1;

@

@zE� = �iNg

c�5 + F2; (2.20)

12

The associated non-vanishing di¤usion matrix elements, however look quite dif-

ferent to their quantum counterparts

D3;3 = �i2a�3 � i2g�3E+;

D4;4 = ia�5 � i�a�3 + i�5E+ � i�3E

� + 01�4: (2.21)

The di¤usion matrix elements are de�ned as hFi(t; z)Fj(t0; z0)i = LN�(t � t0)�(z �

z0) hDiji in the continuous limit. For the more complicated light-matter interactions

we will encounter in Chapter 4 involving four atomic levels interacting with two prop-

agating quantized light �elds, the di¤usion matrix calculation is much more intricate.

It is therefore important to have an independent check of the c-number equations

and the associated di¤usion matrix. In the following Section we review the Fokker-

Planck equation approach based on a Schrodinger picture treatment of the quantized

light-atom interaction.

2.2 Fokker-Planck Equations and Stochastic Di¤erential Equa-tions

Here we review the alternative method, due to Haken [27], to derive the c-number

Langevin equations or equivalently stochastic di¤erential equations via Fokker-Planck

equations [27, 32, 34, 35].

The Fokker-Planck equation is used to describe the �uctuations in Brownian mo-

tion [28], and its solution for probability distribution f(x; t) of Brownian particles in

space x and time t is determined by the drift and di¤usion properties of the particles.

2.2.1 Characteristic functions in P-representation

The Characteristic function � is convenient for the derivation of Fokker-Planck equa-

tion, and it is the distribution function of the Fokker-Planck equation in Fourier space.

We follow the same procedure of P-representation laser theory [27].

The relevant operators of our system are atomic coherences (~�ly01, ~�l01), population

13

(~�l11) and �eld operators (ayl , al). The normally ordered exponential operator is

chosen to be

E(�) =Yl

El(�);

El(�) = ei�l5~�

ly01ei�

l4~�

l11ei�

l3~�

l01ei�

l2ayl ei�

l1al ; (2.22)

where E(�) the complete exponential operator and is decomposed into products of

El(�) for each section l of the propagation direction. We note that the ordering of

operators is the same as we chose for the quantum-classical correspondence in the

previous Section. The complex parameters �li are classical counterparts of operators

in Fourier space, as will become clear when we derive the Fokker-Planck equation.

Then characteristic function � can be calculated from a density matrix �;

� = Tr fE(�)�g , (2.23)

@�

@t= Tr

�E(�)

@�

@t

�=Xm

�@�

@t

�m

; m = A;L; I; sp (2.24)

and time evolution of � is

@�

@t=

1

i~[H; �] +

�@�

@t

�sp

; H = H0 +HI ;�@�

@t

�sp

=MX

l=�M

NzX�

012

h2��;l01 ��

�;ly01 � ��;ly01 �

�;l01 �� ;ly01 �

�;l01

i;

where H0 = HA + HL. HA is the Hamiltonian for atomic free evolution, HL is the

Hamiltonian for the pump �eld, and the dipole interaction Hamiltonian is HI : The

dissipation from spontaneous emission is denoted as sp.

The contribution from�@�@t

�spis calculated up to the second order in �i. The

validity of truncation to second order is due to the expansion in the small parameter

1=Nz. The dissipative contribution, identi�ed by superscript (2), takes the form,

14

01Tr�E(�)

��01��

y01 �

1

2�11��

1

2��11

��(2)=

01

��i�32

@

@(i�3)� i�5

2

@

@(i�5)� i�4

@

@(i�4)+(i�4)

2

2

@

@(i�4)

��: (2.25)

where we drop the summation over spatial slices l; which we will retrieve later. Col-

lecting together all contributions to the characteristic function, we may proceed to

write down a Fokker-Planck equation that leads to the c-number Langevin equation.

2.2.2 A Complimentary Derivation of C-number Langevin Equations

The time derivative of the distribution function f is found from the Fourier trans-

form of the characteristic function @f@t= 1

(2�)n

R:::Re�i~��

~� @�@td�1:::d�n: Separating the

di¤erent contributions we may write

@f

@t= Lf =

Xl;l0

[LA�ll0 + LL + LI�ll0 + Lsp�ll0 ] f: (2.26)

The details of the L operators can be found in Appendix B. Here we show LI as

an example,

LI =

iaeikazl�i!at

�� @2

@�l3@�l3

(�l3)�@

@�l3(�2�l4 +Nz) + e

� @

@�l4 (�l5)

�� iae

ikazl�i!at(�l5)

+igp2M + 1eikzl

�� @2

@�l3@�l3

(�l3)�@

@�l3(�2�l4 +Nz) + e

� @

@�l4 (�l5)

��l1

+ig�p2M + 1e�ikzl(�l3)

��l2 �

@

@�l1

�+ (C 0)�; (2.27)

where C 0 is the correspondence that ��3 $ �5; ��4 $ �4; �

�1 $ �2; and � denotes

complex conjugation. The results is a Fokker-Planck equation of the form

@f

@t= � @

@�A�f �

@

@�A�f +

1

2

�@2

@�@�+

@2

@�@�

�D��f (2.28)

15

where A�;� and D�� are drift and di¤usion terms. The corresponding c-number

Langevin equations may be derived rigorously when D is positive de�nite, and take

the form

@�

@t= A� + ��,

@�

@t= A� + �� (2.29)

with a classical noise correlation h��(t)��(t0)i = �(t � t0)D��. Higher order deriv-

atives (third order and higher, from the Taylor expansions of e� @

@�l4 ) are ignored as

they involve the small parameter 1=Nz: The corresponding c-number Langevin, or

stochastic di¤erential, equations are

_�l3 = (�i!1 � 012)�l3 + iae

ikazl�i!at(�l0 � �l4)

+igp2M + 1eikzl(�l0 � �l4)�

l1 + �

l3; (2.30)

_�l4 = � 01�l4 + iaeikazl�i!at�l5 � i�ae

�ikazl+i!at�l3

+igp2M + 1eikzl�l5�

l1 � ig�

p2M + 1e�ikzl�l3�

l2 + �

l4; (2.31)

_�l1 = �i!�l1 � iXl0

!ll0�l0

1 + ig�p2M + 1e�ikzl�l3 + �

l1: (2.32)

We can retrieve the continuous limit with the slowly varying variables, �3(z; t) �

�l3e�ikazl+i!at=Nz, �4(z; t) � �l4=Nz, E+(z; t) �

p2M + 1�l1e

i!at, and note that

�iP

l0 !ll0�l01 = �c @

@zl�l1 and �l0 = Nz � �l4. De�ne also the slowly-varying

Langevin noises,

F3(z; t) =1

Nz�l3e

�ikazl+i!at;F4(z; t) =1

Nz�l4;

F1(z; t) =p2M + 1ei!t�l1: (2.33)

Finally, in the co-moving frame coordinates z and � = t � z=c; the c-number

16

Langevin equation becomes

@

@��3 = (i�1 �

012)�3 + ia(�0 � �4) + ig(�0 � �4)E

+ + F3; (2.34)

@

@��4 = � 01�4 + ia�5 � i�a�3 + ig�5E

+ � ig��3E� + F4; (2.35)

@

@��5 = (�i�1 �

012)�5 � i�a(�0 � �4)� ig�(�0 � �4)E

� + F5; (2.36)

@

@zE+ =

iNg�

c�3 + F1;

@

@zE� = �iNg

c�5 + F2; (2.37)

where �1 = !a � !1. The non-vanishing di¤usion coe¢ cients extracted from the

Fokker-Planck equation are

D3;3 = �i2ae�3� i2e�3E+i ; D4;4 = iae�5� i�ae�3+ ie�5E+i � ie�3E�i + 01e�4: (2.38)Comparing with the results in the previous Section and Eq. (2.21), we �nd com-

plete agreement. As the c-number Langevin equations are derived from a Fokker-

Planck equation, they should be interpreted as Ito-type stochastic di¤erential equa-

tions (SDE), and this is important in the numerical solution method [32]. In nu-

merical simulation it is common to �rst transform from the Ito equation to its corre-

sponding Stratonovich form.

2.3 Kubo Oscillator

We present an example of the Kubo oscillator to illustrate numerical simulation of

a multiplicative noise stochastic di¤erential equation. A Kubo oscillator provides

a good test case in the numerical solution of stochastic di¤erential equations. The

Langevin equation of the dimensionless Kubo oscillator with amplitude z(t) is given

by the Stratonovich equation,

d

dtz(t) = i�(t)z(t) (2.39)

where �(t) is a delta-correlated real Gaussian distributed noise with zero mean,

h�(t)i = 0 , and h�(t)�(t0)i = �(t�t0): The bracket denotes an ensemble average. The

17

exact analytical solution for the �rst moment is hz(t)i = hz(t = 0)ie�t=2. To numer-

ically simulate the Stratonovich equation (2.39), we use the following discretization

in time [36, 37]

z(tm) = zn�1 + i�(tn�1)z(tm)�t

2(2.40)

where z(tm) is evaluated at the midpoint, tm = (tn + tn�1)=2 and �t = tn � tn�1 is

the time step. In this speci�c case where the noise is linear in z(t), we may solve Eq.

(2.40) to give z(tm) = zn�1=(1� i�(tn�1)�t=2). Setting z(tn+1) = 2z(tm)� z(tn), we

use z(tn+1) for the next time step of the integration.

The Langevin noise is numerically simulated as �(t) = randn(t)=p�t, where

randn(t) is a random number generated from a Gaussian distribution with zero mean

and unit variance. In Figure 2.2, we compare the analytical and numerical results for

the Kubo oscillator. The initial condition is set as z(0) = 1, and we use 1024 realiza-

tions for the converged numerical result hz(t)i with �t = 0:01. The numerical result

is in good agreement with the exact solution, z(0)e�t=2. The temporal evolution of

one typical realization of the stochastic process �uctuates signi�cantly, as shown.

We will use the approach demonstrated here to simulate the more complicated

c-number Langevin noises in our investigation of cascade emission from an atomic

ensemble in Chapter 4 (see also Appendix B).

18

Figure 2.2: Kubo oscillator simulation. The time evolution of Rehz(t)i (dashed-red)averaged from an ensemble of 1024 simulations. z(0) = 1: We compare with theexact solution, z(0)e�t=2 (solid-black), and �nd good agreement. A demonstrationof one stochastic realization (dashed-circle blue) shows large �uctuation around theaveraged and exact results. Note that the imaginary part of the solution is almostvanishing as it should be, and is not shown here.

19

CHAPTER III

SUPERRADIANT EMISSION FROM A CASCADE

ATOMIC ENSEMBLE: ANALYTICAL METHOD

In this Chapter, we use Schrödinger�s equation to investigate cascade emission from a

four-level atomic ensemble. Quantum communication has opened up the possibility

to transmit quantum information over long distance. Due to the transmission loss

in long distance �ber-based quantum communication, telecommunication (telecom)

wavelength light is important to maximize the transmission e¢ ciency. The alkali

atomic cascade transition shown in Figure 3.3 is able to generate telecom wavelength

light, the signal, from the upper transition and a near-infrared �eld, the idler, from the

lower one. The telecom light can travel through the �ber with minimal loss, while the

near-infrared �eld is suitable for storage and retrieval in an atomic quantum memory

element. Their use in a quantum information system requires quantum correlations

between stored excitations and the telecom �eld.

We develop a quantum theory to characterize the properties of the correlated sig-

nal and idler photons and study how the laser excitation pulse modi�es their spectral

pro�le. The wave packets of this entangled source are found, and Schmidt decom-

position provides the basis for engineering a pure photon source that is crucial in

quantum information processing.

3.1 Introduction

The spontaneous emission from an optically dense atomic ensemble is a many-body

problem due to the radiative coupling between atoms. This coupling is responsible

for the phenomenon of superradiance �rstly discussed by Dicke [23] in 1954.

21

Since then, this collective emission has been extensively studied in two atom sys-

tems indicating a dipole-dipole interaction [20, 21], in the totally inverted N atom

systems [38, 39], and in the extended atomic ensemble [22]. The emission intensity

has been investigated using the master equation approach [40�42] and with Maxwell-

Bloch equations [43, 44]. A useful summary and review of superradiance can be found

in the reference [45, 46]. Recent approaches to superradiance include the quantum

trajectory method [47, 48] and the quantum correction method [49].

In the limit of single atomic excitation, superradiant emission characteristics have

been discussed in the reference [50] and [51]. For a singly excited system, the basis

set reduces to N rather than 2N states. Radiative phenomena have been investigated

using dynamical methods [52�54] and by the numerical solution of an eigenvalue

problem [55�58]. A collective frequency shift [59, 60] can be signi�cant at a high

atomic density [61] and has been observed recently in an experiment where atoms are

resonant with a planar cavity [62].

3.2 The example of two-state atoms interacting with a pump�eld

The atomic dynamics of N two-state atoms interacting with a pump �eld generally

requires a basis of 2N orthogonal states. In this Section we investigate multiple ex-

citations by a laser by solving numerically the master equation for few atom systems

(N = 2; 3; 4), using the quantum optics toolbox [63]. The complete orthogonal states

may be chosen as 1 symmetric state and (CNn �1) non-symmetric states for any excita-

tion number n where CNn is the combination coe¢ cient. It is natural to construct the

complete orthogonal states using this decomposition because the interaction Hamil-

tonian of the pump �eld, HI = [�~a2PN

� j1ih0jei~k�~r� + c:c:]�~�1

PN� j1ih1j, has the

same form for each atom.

For the example of two two-state atoms, there are 4 orthogonal basis states: the

ground state j00i; the symmetric state of a single excitation (ei~k�~r1j10i+ei~k�~r2j01i)=p2,

22

the associated anti-symmetric state (ei~k�~r1j10i � ei~k�~r2j01i)=

p2, and the state of two

excitations ei~k�(~r1+~r2)j11i: Note that the spatial phase factor for di¤erent atomic

position ~r is included due to the pump �eld of the wavevector ~k that is directed along

the z axis. If more atoms are involved, the complete states of multiple excitations

can be derived by extending the results of reference [52], and here we list the states

of four atoms (N = 4),

n = 0; j�1i = j0; 0; 0; 0i;

n = 1

8>>>><>>>>:j�2i = 1p

N

PN�=1 e

i~k�~r�j1i�j0i�6=�;

j�l+2i =PN�1

j=1

�1+1=

pN

N�1 � �jl

�ei~k�~rj j1ijj0i�6=j � ei

~k�~rNpNj1iN j0i�6=N ;

l = 1; 2; :::N � 1;

n = 2

8>>>>>>>><>>>>>>>>:

j�N+2i = 1pN(N�1)=2

PN�>�

PN�=1 e

i~k�(~r�+~r�)j1i�j1i� j0i�6=�;� ;

j�m+N+2i =PN

l>j

PN�2j=1

�1+1=

pN2

N2�1 � �m;(j;l)

�ei~k�(~rj+~rl)j1ijj1ilj0i�6=j;l

� ei~k�(~rN�1+~rN )

pN2

j1iN�1j1iN j0i�6=N�1;N ; m = 1; 2; :::N2 � 1,

where N2 � N(N � 1)=2;�

�

�

n = N; j�2N i = j1iNNYj=1

ei~k�~rj ; (3.1)

where (j; l) in the subscript of the Kronecker delta function of two excitation states

is de�ned so that (1; 2) = 1; (1; 3) = 2; (1; 4) = 3; (2; 3) = 4; and (2; 4) = 5: Note that

the n = 0; j�1i; and n = N; j�2N i; states are symmetric. For n 6= 0; N excitations,

the states are constructed from one symmetric state and CNn non-symmetric states.

To investigate the probability of multiple atomic excitations in conditions of weak

o¤-resonant excitation, we choose a con�guration of four atoms that sit on the vertices

of a square with side ds: The atomic density matrix includes a laser excitation term

23

Figure 3.1: Single- and double-excitation populations as a function of distance ds: (a)The populations of the symmetric state for a single excitation P s1 (dashed-red) and thesum of non-symmetric single-excitation states P ns1 (dashed-dotted black). (b) Thepopulations of the symmetric state for double excitations P s2 (dashed-blue) and thesum of non-symmetric double-excitation states P ns2 (dashed-dotted black). P s, ns1 andP s, ns2 are normalized respectively by the solutions of non-interacting atoms P (0)1 (solid-red) and P (0)2 (solid-blue). P

(0)1 = 1:58� 10�3 and P (0)2 = 9:4� 10�7; are the single-

and double-excitation probabilities for independent atoms.

24

in addition to the one and two-atom dissipation terms; these arise from spontaneous

emission and radiative coupling due to dipole-dipole interaction [21]; see Eq. (A.15)

in Appendix A. We numerically solve for the time evolution of the density matrix.

The result of steady state single- and double-excitation populations are shown in

Figure 3.1 as a function of ds: We have assumed a continuous laser �eld with peak

Rabi frequency a = 0:2 and detuning �1 = 5 ; where is the single-atom sponta-

neous decay rate for the excited state. The populations of the symmetric states are

P s1 �Tr(�j�2ih�2j) for a single excitation and P s2 �Tr(�j�6ih�6j) for double excitations

where � is the density operator of the atomic system. The total populations of the

non-symmetric excitation states are P ns1 �Tr��P5

x=2 j�xih�xj��; for a single exci-

tation, and P ns2 �Tr��P11

x=7 j�xih�xj��; for double excitations, respectively. The

probabilities of three and four excitations are negligible under the weak excitation

conditions we consider.

As ds approaches and exceeds � (the transition wavelength); the populations tend

to the independent atom limit when dipole-dipole coupling is omitted. In this limit,

the probability of exciting any non-symmetric states goes to zero. The single and

double excitation probabilities, P s1 and Ps2 , are normalized to their independent atom

values, P (0)1 = PeP3gC

41 and P

(0)2 = P 2e P

2gC

42 where Pe = 2a=(4�

21 + 2) [64]; Pg =

1�Pe; and Cni is the combination coe¢ cient. For ds � �, the populations of the non-

symmetric states are comparable to the symmetric ones, indicating the importance of

dipole-dipole interactions. We see no evidence of a dipole blockade e¤ect in this limit

for four atoms, but we have observed it in the case of two atoms. Dipole blockade

refers to the predominance of single excitations as dipole shifts detune double and

higher excitation states.

In Figure 3.2, we show the time evolution of P s1(t) and Ps2(t) for ds = 3� (this

corresponds to an atomic density 8�1010 cm�3). The period of the Rabi oscillation is

determined by 2�=�1, and the asymptotic steady state value for P s2 is about 1:6�10�3:

25

This coincides with the approximate result jpNa=(2�1)j2 that is found when we

truncate the basis to the ground state and the orthogonal states of a single atomic

excitation.

We also numerically solve a line of atoms (N = 2; 3; or 4) with an equal separation

from ds = 1 to 5�; and the results of steady state populations indicate the condition for

truncation of the basis set at a single atomic excitation is valid when j�1j �pNa=2:

If the condition of a single atomic excitation �1 �pNa=2 is relaxed, we will also

have dynamical couplings between symmetric and non-symmetric states (at least for

ds . 3�). It is the dipole-dipole interaction that couples the non-symmetric and

symmetric states in the presence of the pump laser.

3.3 Theory of Cascade Emission

We consider N cold atoms that are initially prepared in the ground state interacting

with four independent electromagnetic �elds. As shown in Figure 3.3, two driving

lasers (of Rabi frequencies a and b) excite a ladder con�guration j0i ! j1i ! j2i:

Two quantum �elds, signal as and idler ai; are generated spontaneously. The atoms

adiabatically follow the two excitation pulses and decay through the cascade emission

of signal and idler photons. Based on the discussion in the previous Section, we permit

only single atomic excitations under the condition of large detuning, �1 �pNa=2.

The Hamiltonian and the coupled equations of the atomic dynamics are detailed in

Appendix A.

To correctly describe the frequency shifts arising from dipole-dipole interactions,

we do not make the rotating wave approximation on the electric dipole interaction

Hamiltonian. The frequency shift has contributions from the single atom Lamb shift

and a collective frequency shift. The Lamb shift is assumed to be renormalized into

the single atom transition frequency distinguishing it from the collective shift due to

the atom-atom interaction.

26

Figure 3.2: Time evolution of populations for symmetric states P s1 and P s2 . Thepopulation of the symmetric state for a single excitation is P s1 (dashed-red), and thatfor the symmetric state for double excitations is P s2 (dashed�dotted blue). The pumpcondition is the same as in Figure 3.1 for ds = 3�:

27

Figure 3.3: Four-level atomic ensemble interacting with two driving lasers (solid)with Rabi frequencies a and b: Signal and idler �elds are labelled by as and ai;respectively and �1 and �2 are one and two-photon laser detunings.

3.3.1 Probability amplitudes for signal and signal-idler emissions

Writing the state-vector j (t)i in a basis restricted to single atomic excitations, and

single pairs of signal and idler photons, we can introduce the probability amplitudes,

Cs;ki(t) =NX�=1

e�i~ki�~r�h3�; 1ks;�s j (t)i (3.2)

and

Ds;i(t) = h0; 1ks;�s ; 1ki;�ij (t)i (3.3)

de�ned in Appendix A. Note that Cs;ki(t) is an amplitude for a phased excitation of

the ensemble of atoms subsequent to signal photon emission.

After adiabatically eliminating the laser excited levels in the equations of motion,

we are able to simplify and derive the amplitude Cs;ki and the signal-idler (two-

photon) state amplitude Ds;i as shown in Appendix A,

28

Cs;ki(t) = g�s(��ks;�s � ds)

X�

ei�~k�~r�

Z t

0

dt0ei(!s�!23��2)t0e(�

�N32+i�!i)(t�t0)b(t0) (3.4)

Ds;i(t) = g�i g�s(�

�ki;�i

� di)(��ks;�s � ds)X�

ei�~k�~r�

Z t

0

Z t0

0

dt00dt0e(��N32+i�!i)(t

0�t00)

ei(!i�!3)t0ei(!s�!23��2)t

00b(t00): (3.5)

The factorP

� ei�~k�~r� re�ects phase-matching of the interaction under conditions

of four-wave mixing when the wavevector mismatch �~k = ~ka+~kb�~ks�~ki ! 0: The

radiative coupling between atoms results in the appearance of the superradiant decay

constant

�N3 = (N ��+ 1)�3 (3.6)

where �3 is the natural decay rate of the j3i ! j0i transition, and �� is a geometrical

constant depending on the shape of the atomic ensemble. An expression for the

collective frequency shift �!i is given in the Appendix A. As shown in Figure 3.4,

we numerically calculate the geometrical factor ��; Eq.(A.12), to demonstrate how the

decay factor N ��+1 depends on the height and radius of a cylindrical ensemble. The

arrows in the �gure point out the contour lines (yellow and green) of N ��+1 � 4 and

6 which are comparable to the operating conditions of the experiment [16].

In the above expressions for the probability amplitudes, b(t) = a(t)b(t)4�1�2

is propor-

tional to the product of the Rabi frequencies. We use normalized Gaussian pulses

as an example where a(t) = 1p��~ae

�t2=�2, b(t) = 1p��~be

�t2=�2, so that the two

pulses are overlapped with the same pulse width. ~a;b is the pulse area, and let

�!s � !s�!23��2��!i; �!i � !i�!3+�!i. We have the probability amplitude

for signal photon emission and atoms in a phased state,

29

Figure 3.4: The superradiance decay factor N�+1 (� = ��) for a cylindrical ensembleof length h and radius a in unit of transition wavelength �. The atomic density is8 � 1010 cm�3 and � = 795 nm corresponding to the D1 line of 85Rb. See the textfor the explanation of the arrows.

30

Cs;ki(t;�!s)

=~a ~bg

�s(�

�ks;�s

� ds)4�1�2

X�

ei�~k�~r� 1

�� 2e(�

�N32+i�!i)t

Z t

�1dt0e

�N32t0ei�!st

0e�2t

02=�2

=~a ~bg

�s(�

�ks;�s

� ds)4�1�2

X�

ei�~k�~r� 1

�� 2�

2

r�

2e(�

�N32+i�!i)te(

�N32+i�!s)2�2=8 �

�1 + erf(

4t� (�N3

2+ i�!s)�

2

2p2�

)�; (3.7)

and the two-photon probability amplitude is

Dsi(t;�!s;�!i)

=~a ~bg

�i g�s(�

�ki;�i

� di)(��ks;�s � ds)4�1�2

X�

ei�~k�~r� 1

�� 2

r�

2

�e��N32t

2(�N32� i�!i)

n�ei�!it+(

�N32+i�!s)2�2=8

�1 + erf(

4t� (�N3

2+ i�!s)�

2

2p2�

)�

+e�(�!s+�!i)2�2e

�N32t�1 + erf(

4t� i(�!s +�!i)�2

2p2�

)�o; (3.8)

where erf is the error function

erf(x) =2p�

Z x

0

e�t2

dt: (3.9)

Asymptotically Dsi approaches the value,

Dsi(�!s;�!i) =~a ~bg

�i g�s(�

�ki;�i

� di)(��s � ds)4�1�2

P� e

i�~k�~r�p2��

e�(�!s+�!i)2�2=8

�N32� i�!i

; (3.10)

indicating a spectral width �N3 =2 for idler photon in a Lorentzian distribution mod-

ulating a Gaussian pro�le with a spectral width 2p2=� for signal and idler. Energy

conservation of signal and idler photons with driving �elds at their central frequen-

cies corresponds to !s + !i = !a + !b, which makes �!s + �!i = 0; the collective

frequency shifts cancel.

31

3.4 A Correlated Two-photon State

Using the asymptotic form of the two-photon state given in Eq. (3.10), the second-

order correlation function G(2)s;i is calculated as [30]

G(2)s;i = h (t!1)jE�s (~r1; t1)E�i (~r2; t2)E+i (~r2; t2)E+s (~r1; t1)j (t!1)i = j�s;ij2

(3.11)

�s;i = h0jE+i (~r2; t2)E+s (~r1; t1)j (t!1)i (3.12)

E+s (~r1; t1) =Xks;�

r~!s2�0V

aks;�~�ks;�sei~ks�~r1�i!st1 (3.13)

E+i (~r2; t2) =Xki;�

r~!i2�0V

aki;�~�ki;�iei~ki�~r2�i!it2 (3.14)

where j (t ! 1)i denotes the state vector in the long time limit that involves the

ground state and two-photon state vectors. Free electromagnetic �elds, signal and

idler photons, at space (~r1; ~r2) and time (t1; t2) are E+s and E+i where (+) denotes

their positive frequency part. For second order correlation function, onlyDsi; derived

in the previous Section contributes to it, then we have,

�s;i =Xks;�s

Xki;�i

!s2�0V

!i2�0V

(~ds �~�ks;�s)~�ks;�s�

(~di �~�ki;�i)~�ki;�i~a ~b4�1�2

P� e

i�~k�~r�p2��

e�(�!s+�!i)2�2=8

�N32� i�!i

ei~ks�~r1�i!st1ei

~ki�~r2�i!it2

=~a ~b4�1�2

P� e

i�~k�~r�p2��

j~dsjj~dij4�20c

6(2�)6

Zds[ds � ks(ks � ds)]Z

di[di � ki(ki � di)]Zd!i!

3i e�i!i(t2�

~r2�kic) (!23 ��!i)3

�N32� i�!i

e�i(!23+�2)(t1�~r1�ksc)

ei(�!i��!i)(t1�~r1�ksc)

Zd�!se

�i�!s(t1�~r1�ksc)e��!

2s�2=8 (3.15)

where we have used the change of variables in the �rst step, replaced !s = !23+�2+

�!s + �!i, and changed the variable �!s ! �!s ��!i. Solid angle integration is

32

denoted as ds;i for signal (idler) photon. The divergent part of !3s (which varies

relatively slowly) has been moved out from the integral of d�!s, and we replace !s

with the signal transition frequency !23: We then have

�s;i

=~a ~b4�1�2

P� e

i�~k�~r�p2��

j~dsjj~dij!33!3234�20c

6(2�)6

Zdsdi[ds � ks(ks � ds)][di � ki(ki � di)]

2p2�

�

Zd�!i

e�i�!i(t2�~r2�kic�t1+~r1�ks

c)

(�N32� i�!i � i�!i)

e�i(!23+�2)(t1�~r1�ksc)e�i!3(t2�

~r2�kic)e�2(t1�

~r1�ksc)2=�2

(3.16)

where we replace !i = !3 + �!i � �!i and change the variable �!i ! �!i + �!i.

The divergent part of !3i is again moved out from the integral of d�!i and replace

!i with the signal transition frequency !3: Finally we have

�s;i

=~a ~b4�1�2


6� 2(2�)6

X�

ei�~k�~r�


e�2(t1�~r1�ksc)2=�2e�i(!23+�2)(t1�

~r1�ksc)e�i!3(t2�

~r2�kic)e(�

�N32+i�!i)(t2�

~r2�kic�t1+~r1�ks

c)

�(t2 �~r2 � kic

� t1 +~r1 � ksc

) (3.17)

where the complex integral with the pole at �!i = �i�N3

2� �!i in the lower half

plane leads to a step function � that shows the causal connection between signal and

idler emission. The emission time for the signal �eld (t1 � ~r1�ksc) is within the pulse

envelope of width � , and the idler photon decays with a superradiant constant �N3 =2.

Note that the collective frequency shift �!i appears in the signal (!23+�2+�!i) and

idler (!3 � �!i) frequency consistent with energy conservation. Let �ts � t1 � ~r1�ksc

and �ti � t2 � ~r2�kic, we then have

33

j�s;i(�ts;�ti)j

=~a ~b4�1�2


6� 2(2�)6

X�

ei�~k�~r�


e�2(�ts)2=�2e�

�N32(�ti��ts)�(�ti ��ts): (3.18)

If we let �t � �ti � �ts and choose �ts = 0 as the origin in time (idler gating

time), then we have the second-order correlation function

G(2)s;i (�t) = j�s;i(�t)j2 / e��

N3 �t where �t � 0: (3.19)

It resembles the result for the second-order correlation function in the case of single

atom, whereas here we have an enhanced decay rate due to the atomic dipole-dipole

interaction.

In Figure 3.5, we plot out the absolute value of spectrum Dsi(�!s;�!i) and the

second-order correlation function G(2)s;i (�ts;�ti). In (c), we show for �3�ti = 0:2:

The width of 1=�ti = 5�3 corresponds to �N3 = (N ��+ 1)�3 = 5�3.

3.5 Schmidt Decomposition

Correlated photon pairs may be generated by parametric down conversion (PDC) [65�

67]. The degree of entanglement can be quanti�ed by Schmidt mode decomposition

[68, 69], allowing the in�uence of group-velocity matching [70] to be assessed. A

pure single photon source is a basis element for quantum computation by linear

optics (LOQC) [71], and it can be conditionally generated by measurement [72]. A

similar approach can be applied to the study of the transverse degrees of freedom

in type-II PDC [73] and PDC in a distributed microcavity [74]. In photonic-crystal

�ber (PCF), a factorizable photon pair can be generated by spectral engineering [75].

The spectral e¤ect has been discussed in relation to a quantum teleportation protocol

[76] as a �rst step toward quantum communication.

34

Figure 3.5: (a) Absolute value of the spectrum for two-photon state probabilityamplitude Ds;i and (b) the second-order correlation function G

(2)s;i (�ts;�ti): (c) A

normalized G(2)s;i (�ts = 0;�ti) with �3� = 0:2. The exponential decay correspondsto the superradiant decay factor N ��+ 1 = 5:

35

We would like to perform an analysis of entanglement properties of our cascade

emission source. In addition to polarization entanglement, a characterization of

frequency space entanglement is required to clarify its suitability in, for example, the

DLCZ protocol [6].

In the long time limit, the state function is given by, Eq. (3.10),

j i = j0; vaci+Xs;i

Ds;ij0; 1~ks;�s ; 1~ki;�ii (3.20)

where s = (ks; �s), i = (ki; �i); and j0;vaci is the joint atomic ground and photon

vacuum state.

The spatial correlation of two-photon state in FWM condition can be eliminated

by pinholes or by coupling to single mode �ber so we consider only the continuous

frequency space. For some speci�c polarizations �s and �i, we have the state vector

ji,

ji =Zf(!s; !i)a

y�s(!s)a

y�i(!i)j0id!sd!i; (3.21)

where

f(!s; !i) =e�(�!s+�!i)

2�2=8

�N32� i�!i

: (3.22)

The quanti�cation of entanglement can be determined in the Schmidt basis where

the state vector is expressed as

ji =Xn

p�nb

yncynj0i; (3.23)

byn �Z n(!s)a

y�s(!s)d!s; (3.24)

cyn �Z�n(!i)a

y�i(!i)d!i; (3.25)

where byn; cyn are e¤ective creation operators. Eigenvalues �n, and eigenfunctions n

and �n; are the solutions of the eigenvalue equations,

36

ZK1(!; !

0) n(!0)d!0 = �n n(!); (3.26)Z

K2(!; !0)�n(!

0)d!0 = �n�n(!); (3.27)

where K1(!; !0) �

Rf(!; !1)f

�(!0; !1)d!1 and K2(!; !0) �

Rf(!2; !)f

�(!2; !0)d!2

are the kernels for the one-photon spectral correlations [68, 69]. Orthogonality of

eigenfunctions isR i(!) j(!)d! = �ij,

R�i(!)�j(!)d! = �ij; and the normalization

of quantum state requiresP

n �n = 1.

In the Schmidt basis, the von Neumann entropy may be written

S = �1Xn=1

�nln�n: (3.28)

If there is only one non-zero Schmidt number �1 = 1, the entropy is zero, which

means no entanglement and a factorizable state. For more than one non-zero Schmidt

number, the entropy is larger than zero and bipartite entanglement is present.

The kernel in Eq. (3.22) has all the frequency entanglement information, entan-

glement means f(!s; !i) cannot be factorized in the form g(!s)h(!i); a multiplication

of two separate spectral functions. By inspection the Gaussian pro�le of signal and

idler emission is a source of correlation. The joint spectrum �!s +�!i is con�ned

within the width of order of 1=� . The Lorentzian factor associated with the idler

emission has a width governed by the superradiant decay rate.

In Figure 3.6, we show the Schmidt decomposition of the spectrum. We use a

moderate superradiant decay constant N ��+ 1 = 5; comparable to the reference [16],

and a nanosecond pulse duration � = 0:25 (� 26=4 ns), and �3=2� = 6 MHz. Due to

slow convergence associated with the Lorentzian pro�le, we use a frequency range up

to �1200 (in unit of �3) with 2000�2000 grid. The numerical error in the eigenvalue

calculation is estimated to be about 1% error. In this case, the largest Schmidt

number is 0:8 and corresponding signal mode function has a FWHM Gaussian pro�le

37

Figure 3.6: Schmidt mode analysis with pulse width � = 0:25 and superradiancedecay factor N ��+1 = 5: (a) Schmidt number and (b) signal mode functions: Re[ 1](solid-red) and Re[ 2] (solid-blue). Imaginary parts are not shown, then are zero. (c)Real (solid) and imaginery (dotted) parts of �rst (red) and second (blue) idler modefunctions, �1 and �2. (d) The absolute spectrum jf(�!s;�!i)j.

38

4p2 ln(2)=� � 19�3. The idler mode function �1 re�ects the Lorentzian pro�le in

the spectrum at the signal peak frequency (�!s = 0),

f(�!s = 0;�!i) =e��!

2i �2=8

(N�+ 1)�3=2� i�!i(3.29)

where a relatively broad Gaussian distribution is overlapped with a narrow spread of

superradiant decay rate [FWHM > (N ��+ 1)�3=2].

Figure 3.7 shows that the cascade emission source is more entangled if the super-

radiant decay constant, or the pulse duration increases. We note that the Gaussian

pro�le aligns the spectrum along the axis �!s = ��!i and the spectral width for

signal photon at the center of the idler frequency distribution (�!i = 0) is determined

by pulse duration � . For a shorter pulse ��1 > (N ��+1)�3=2, the joint Gaussian pro-

�le has a larger width, and the spectrum is cut o¤by the Lorentzian idler distribution.

A larger width leads to a less entangled source and distributes the spectral weight

mainly along the crossed axes �!s = 0 and �!i = 0. A narrow Lorentzian pro�le

cuts o¤ the entanglement source term e�(�!s+�!i)2�2=8 tilting the spectrum along the

line �!s+�!i = 0: In the opposite limit, ��1 < (N ��+1)�3=2, the spectrum is highly

entangled corresponding to tight alignment along the axis �!s = ��!i (Figure 3.7

(c)).

Note that the short pulse duration (� � 0:25 (6:5 ns)) should not violate the

assumption of adiabaticity � & 1=�1 or 1=�2.

The Schmidt analysis and calculation of von Neumann entropy shows that signal-

idler �elds are more entangled if the ensemble is more optically dense, corresponding

to stronger superradiance. For the DLCZ protocol, we wish to avoid frequency

entanglement. The superradiance may be reduced with smaller atomic densities but

good qubit storage and retrieval e¢ ciency require a moderate optical thickness [16].

A better approach involves using short pulse excitation ��1 > (N �� + 1)�3. We will

investigate the spectral properties in more details for the DLCZ scheme in Chapter

39

Figure 3.7: Absolute spectrum of two-photon state and the eigenvalues of Schmidtdecomposition. N �� + 1 = 5 for both (a) � = 0:25 (b) � = 0:5. N �� + 1 = 10 for (c)� = 0:25. The von Neumann entropy (S) is indicated in the plots.

40

5.

41

CHAPTER IV

SUPERRADIANT EMISSION FROM A CASCADE

ATOMIC ENSEMBLE: NUMERICAL APPROACH

In this Chapter, we investigate the cascade emission (signal and idler) from an atomic

ensemble using a numerical approach. In Chapter 3, we studied the correlated emis-

sion using Schrödinger�s equation assuming single atomic excitations. To relax the

assumption of single atomic excitations, we derive a set of c-number stochastic di¤er-

ential equations derived using the quantum statistical methods reviewed in Chapter

2. We solve numerically for the dynamics of the atoms and counter-propagating

signal and idler �elds. The signal and idler �eld intensities are calculated, and the

signal-idler correlation function is studied for di¤erent optical depths of the atomic

ensemble, and compared with the analytical results of Chapter 3.

4.1 Introduction

To account for multiple atomic excitations in the signal-idler emission from a cascade

atomic ensemble, the Schrödinger�s equation approach becomes cumbersome. An

alternative theory based on c-number Langevin equations as discussed in Chapter

2, is suitable for solution by stochastic simulations. An essential element in the

stochastic simulations is a proper characterization of the Langevin noises. These

represent the quantum �uctuations responsible for the initiation of the spontaneous

emission from the inverted [44, 77�79], or pumped atomic system [80, 81] as in our

case.

The positive-P phase space method [29, 32, 82�86] is employed to derive the

Fokker-Planck equations that lead directly to the c-number Langevin equations. The

43

classical noise correlation functions, equivalently di¤usion coe¢ cients, are alterna-

tively con�rmed by use of the Einstein relations reviewed in Chapter 2. The c-number

Langevin equations correspond to Ito-type stochastic di¤erential equations that may

be simulated numerically. The noise correlations can be represented either by using

a square [87] or a non-square "square root" di¤usion matrix [84]. The approach en-

ables us to calculate normally-ordered quantities, signal-idler �eld intensities, and the

second-order correlation function. The numerical approach involves a semi-implicit

di¤erence algorithm and shooting method [88] to integrate the stochastic "Maxwell-

Bloch" equations.

Recently a new positive-P phase space method involving a stochastic gauge func-

tion [89] has been developed. This approach has an improved treatment of sam-

pling errors and boundary errors in the treatment of quantum anharmonic oscillators

[90, 91]. It has also been applied to a many-body system of bosons [92] and fermions

[93]. In this Chapter, we follow the traditional positive-P representation method

[94].

4.2 Theory of Cascade emission

The complete derivation of the c-number Langevin equations for cascade emission

from the four-level atomic ensemble is described in detail in Appendix B. After

setting up the Hamiltonian, we follow the standard procedure to construct the char-

acteristic functions [27] in Appendix B.2 using the positive-P representation [32]. In

Appendix B.3.1, the Fokker-Planck equation is found by directly Fourier transforming

the characteristic functions, and making a 1=Nz expansion.

Finally the Ito stochastic di¤erential equations are written down from inspection

of the �rst-order derivative (drift term) and second-order derivative (di¤usion term) in

the Fokker-Planck equation. The equations are then written in dimensionless form by

introducing the Arecchi-Courtens cooperation units [115] in Appendix B.3.2. From

44

Eq. (B.73) and the �eld equations that follow, these c-number Langevin equations in

a co-moving frame are,

@

@��01 = (i�1 �

012)�01 + ia(�00 � �11) + i�b�02 � i�y13E

+i + F01 (I),

@

@��12 = i(�2 ��1 + i

01 + 22

)�12 � i�a�02 + ib(�11 � �22) + i�13E+s e

�i�kz

+F12;@

@��02 = (i�2 �

22)�02 � ia�12 + ib�01 + i�03E

+s e

�i�kz � i�32E+i + F02;

@

@��11 = � 01�11 + 12�22 + ia�

y01 � i�a�01 � ib�

y12 + i�b�12 + F11;

@

@��22 = � 2�22 + ib�

y12 � i�b�12 + i�y32E

+s e

�i�kz � i�32E�s e

i�kz + F22;@

@��33 = � 03�33 + 32�22 � i�y32E

+s e

�i�kz + i�32E�s e

i�kz + i�y03E+i � i�03E

�i

+F33;@

@��13 = �(i�1 +

01 + 032

)�13 � i�a�03 � ib�y32 + i�12E

�s e

i�kz + i�y01E+i

+F13;@

@��03 = � 03

2�03 � ia�13 + i�02E

�s e

i�kz + i(�00 � �33)E+i + F03;

@

@��32 = i�2 �

03 + 22

�32 + ib�y13 � i(�22 � �33)E

+s e

�i�kz � i�02E�i + F32;

@

@zE+s = �i�32ei�kz

jgsj2jgij2

�Fs;@

@zE+i = i�03 + Fi;

(4.1)

where (I) stands for Ito type SDE. �ij is the stochastic variable that corresponds to

the atomic populations of state jii when i = j and to atomic coherence when i 6= j,

and Fij are c-number Langevin noises. The remaining equations of motion, which

close the set, can be found by replacing the above classical variables, ��jk ! �yjk;

(�yjk)� ! �jk; (E

+s;i)

� ! E�s;i; (E�s;i)

� ! E+s;i , and F�jk ! F y

jk. Note that the atomic

populations satisfy ��jj = �jj: The superscripts, dagger (y) for atomic variables

and (�) for �eld variables, denote the independent variables, which is a feature of the

positive-P representation: there are double dimension spaces for each variable. These

45

variables are complex conjugate to each other when ensemble averages are taken, for

example h�jki =D�yjk

E�and

E+s;i

�=E�s;i

��: The doubled spaces allow the variables

to explore trajectories outside the classical phase space.

Before going further to discuss the numerical solution of the SDE, we point out

that the di¤usion matrix elements have been computed using Fokker-Planck equa-

tions and by the Einstein relations described in Appendix B.3.3. This provides the

important check on the lengthy derivations of the di¤usion matrix elements we need

for the simulations.

The next step is to �nd expressions for the Langevin noises, and the details are

given in Appendix B.3.4 in terms of a non-square matrix B [35, 84]. The matrix B is

used to construct the symmetric di¤usion matrix D(�) = B(�)BT (�) for a Ito SDE,

dxit = Ai(t;�!xt )dt+

Xj

Bij(t;�!xt )dW j

t (t) (I) (4.2)

where �idt = dW it (t) (Wiener process) and

�i(t)�j(t

0)�= �ij�(t � t0): Note that

B ! BS; where S is an orthogonal matrix (SST = I), leaves D unchanged, so B is

not unique. We could also construct a square matrix representation B [28, 32, 87].

This involves a procedure of matrix decomposition into a product of lower and upper

triangular matrix factors. A Cholesky decomposition can be used to determine the B

matrix elements successively row by row. The downside of this procedure is that the

B matrix elements must be di¤erentiated in converting the Ito SDE to its equivalent

Stratonovich form for numerical solution.

The Stratonovich SDE is necessary for the stability and the convergence of semi-

implicit methods. Because of the analytic di¢ culties in transforming to the Stratonovich

form, we use instead the non-square form of B [84] that is shown explicitly in Ap-

pendix B.3.4.

In this case a typical B matrix element is a sum of terms, each one of which is a

product of the square root of a di¤usion matrix element with a unit strength real (if

46

the di¤usion matrix element is diagonal) or complex (if the di¤usion matrix element

is o¤-diagonal) Gaussian unit white noise. It is straightforward to check that a B

matrix constructed in this way reproduces the required di¤usion matrix D = BBT .

As pointed out in the reference [86], the transverse dipole-dipole interaction can

be neglected and nonparaxial spontaneous decay rate can be accounted for by a single

atom decay rate one if the atomic density is not too high. We are interested here

in conditions where the ensemble length L is signi�cant and propagation e¤ects are

non-negligible, and the average distance between atoms d = 3pV=N is larger than

the transition wavelength �: The length scales satisfy � . d� L; and we consider a

pencil-like cylindrical atomic ensemble. The paraxial or one-dimensional assumption

for �eld propagation is then valid, and the transverse dipole-dipole interaction is not

important for the atomic density we focus here.

4.3 Numerical Simulation

In this Section, we discuss the numerical integration of the atomic and �eld equations

derived given in the last Section.

There are several possible ways to integrate the di¤erential equation numerically.

Three main categories of algorithm used are forward (explicit), backward (implicit),

and mid-point (semi-implicit) methods [88]. The midpoint method is in a sense

between the explicit and implicit methods, and we will use an algorithm of this type

in the following. Let tm = tn+�t2for nth segment and iterate (m denotes mid point)

x(tm) = x(tn) + f [tm; x(tm)]�t

2(4.3)

until convergence is reached. Then step forward with x(tn+1) = 2x(tm)� x(tn):

The forward di¤erence method, which Euler or Runge-Kutta methods utilizes, is

not guaranteed to converge in stochastic integrations [37]. There it is shown that the

semi-implicit method [95] is more robust in Stratonovich type SDE simulations [36].

47

More extensive studies of the stability and convergence of SDE can be found in the

reference [96]. The Stratonovich type SDE equivalent to the Ito type equation (4.2),

is

dxit = [Ai(t;�!xt )�

1

2

Xj

Xk

Bjk(t;�!xt )

@

@xjBik(t;

�!xt )]dt

+Xj

Bij(t;�!xt )dW j

t (Stratonovich), (4.4)

which has the same di¤usion terms Bij; but with modi�ed drift terms. This "cor-

rection" term arises from the di¤erent de�nitions of stochastic integral in the Ito and

Stratonovich calculus.

At the end of Appendix B 3.3, we derive the Stratonovich SDE with the (under-

lined) "correction" terms noted above. We then have 19 classical variables including

atomic populations, coherences, and two counter-propagating cascade �elds. With 64

di¤usion matrix elements and an associated 117 random numbers required to represent

the instantaneous Langevin noises, we are ready to solve the equations numerically

using the robust midpoint di¤erence method.

4.3.1 Shooting and secant method

The problem we encounter here involves counter-propagating �eld equations in the

space dimension and initial value type atomic equations in the time dimension. The

initial value problem is addressed by the di¤erence method discussed in the previous

Section.

The counter-propagating �eld equations have a boundary condition speci�ed at

each end of the medium. This is a two-point boundary value problem, and a numerical

approach to its solution, the shooting method [88], is illustrated in Figure 4.1.

Consider the set of di¤erential equations dXi(z)=dz = gi(z; ~X). A subset A of

{Xi} satisfy boundary conditions at z = 0, and the complementary subset B satisfy

boundary conditions at z = L:

48

Figure 4.1: Schematic illustration of the principle of the shooting method for two-point boundary value problems.

49

The shooting method augments the set A with a set of "guesses" A0; so that A[A0

enable the di¤erential equations to be integrated as an initial value problem (from

z = 0 to z = L). The idea is that A0 is the correct choice when the integrated

values at z = L reproduce the true boundary conditions, set B; within a permissible

tolerance. The set A0 is updated to enable convergence of the output at z = L to

the set B:

The secant method that is used to update each element of A0 takes two guesses

x1 and x2 for each variable of A0 and returns an updated value xi;

xi = x2 � f2f2 � f1x2 � x1

: (4.5)

where f1 and f2 are the di¤erences between the required values of that variable in

set B and the numerically computed values assuming x1 and x2 values at z = 0:

This method is iterated until convergence to all values in B is obtained. The secant

method is illustrated in Figure 4.2.

4.3.2 Outline of the numerical solution

We use Matlab to perform the numerical integrations. For simplicity, we label the

atomic and �eld variables as ai and ei. The counter-propagating �eld (�z direction)

variables are e1 and e2 (signal �elds) and e3 and e4 (idler �elds) propagate in the +z

direction. We set the local time � ! t in the following description of the algorithm.

We initialize 15 ai(z; t), 4 ei(z; t) in time t 2 (0; T ) and space z 2 (0; L); and select

19 Gaussian random numbers ni(z; t): Set time and space grids with spacings �t;�z

respectively. For each realization among R statistical ensemble averages, we update

the variables governed by the symbolic equations of motion,

@

@zei = Pi(~e;~a; nei); (4.6)

@

@tai = Ai(~e;~a; nai); (4.7)

50

Figure 4.2: Secant method. The root is bracketed by two initial guesses of x1 andx2 and an updated guess xi is located at the intersection of two straight lines.

51

where Pi and Ai are in general the functions of variables that are denoted as vectors

~e and ~a. Each variable has its own stochastic source term as nei or nai :

The algorithm proceeds by using the midpoint di¤erence method for the evolutions

in space and time and the shooting method for ei;

ei(zm; t) = ei(z; t) +�z

2Pi[~e(zm; t);~a(z; t); nei(z; t)];

ai(z; tm) = ai(z; t) +�t

2Ai[~e(z; t);~a(z; tm); nai(z; t)];

where zm = z + �z=2 and tm = t + �t=2: The two guesses required in the secant

method used in the shooting method are chosen as x1 = fe1(0; t); e2(0; t)g and x2 =

fe3(L; t); e4(L; t)g:

Any normally-ordered quantity hQi can be derived by ensemble averages that

hQi =PR

i=1Qi=R where Qi is the result for each realization. Note that the update

for �eld variables in space precedes the update for atomic variables, which takes into

account that �eld variables evolve faster than atomic variables. The order should

not matter when �ner grids are used.

4.3.3 Results for signal, idler intensities, and the second-order correlationfunction

In this subsection, we present the second-order correlation function of signal-idler

�elds, and their intensity pro�les. We de�ne the intensities of signal and idler �elds

by

Is(t) =E�s (t)E

+s (t)

�; Ii(t) =

E�i (t)E

+i (t)

�; (4.8)

respectively, and the second-order signal-idler correlation function

Gs;i(t; �) =E�s (t)E

�i (t+ �)E+i (t+ �)E+s (t)

�(4.9)

52

where � is the delay time of the idler �eld with respect a reference time t of the signal

�eld. Since the correlation function is not stationary [64], we choose t as the time

when Gs;i is at its maximum.

We consider a cigar shaped 85Rb ensemble of radius 0:25 mm and L = 3 mm.

The operating conditions of the pump lasers are (a; b; �1; �2) = (0:4; 1; 1; 0) 03

where a is the peak value of a 50 ns square pulse, and b is the Rabi frequency of

a continuous wave laser. The four atomic levels are chosen as (j0i; j1i; j2i; j3i) =

(j5S1=2;F=3i; j5P3=2;F=4i; j5P3=2;F=4i; j4D5=2;F=5i). The natural decay rate for

atomic transition j1i ! j0i or j3i ! j0i is 01 = 03 = 1=26 ns and they have a

wavelength 780 nm. For atomic transition j2i ! j1i or j2i ! j3i is 12 = 32 =

0:156 03 [97] with a telecom wavelength 1.53�m. The scale factor of the coupling

constants for signal and idler transitions is gs=gi = 0:775:

We have investigated four di¤erent atomic densities from a dilute ensemble with

an optical density (opd) of 0.11 to a opd = 4.35. In Figure 4.3, 4.4, and 4.5, we

take the atomic density � = 1010 cm�3 (opd = 2.18) for example, and the grid sizes

for dimensionless time �t = 4 and space �z = 0:0007 are chosen. The convergence

of the grid spacings is �xed in practice by convergence to the signal intensity pro�le

with an estimated relative error less than 0.5%.

The temporal pro�les of the exciting lasers are shown in the left panel of Figure

4.3. The atomic density is chosen as � = 1010 cm�3; and the cooperation time Tc is

0.35 ns. The right panel shows time evolution of atomic populations for levels j1i,

j2i; and j3i at z = 0; L; that are spatially uniform. The populations are found by

ensemble averaging the complex stochastic population variables. The imaginary parts

of the ensemble averages tend to zero as the ensemble size is increased, and this is a

useful indicator of convergence, see Appendix B.2 for a discussion. In this example,

the ensemble size was 8�105: The small rise after the pump pulse a is turned o¤ is

due to the modulation caused by the pump pulse b; which has a generalized Rabi

53

Figure 4.3: Time-varying pump �elds and time evolution of atomic populations.(Left) The �rst pump �eld a (dotted-red) is a square pulse of duration 50 ns andb is continuous wave (dotted-blue). (Right) The time evolution of the real partof populations for three atomic levels �11 = h~�13i (dotted-red), �22 = h~�12i (dotted-blue), �33 = h~�11i (dotted-green) at z = 0; L, and almost vanishing imaginary partsfor all three of them. indicate convergence of the ensemble averages. Note that theseatomic populations are uniform as a function of z:

54

Figure 4.4: Temporal intensity pro�les of counter-propagating signal and idler �elds.(a) At z = 0; real (dotted-red) and imaginary (diamond-red) parts of signal intensity.(b) At z = L; real (dotted-blue) and imaginary (diamond-blue) parts of idler intensity.Both intensities are normalized by the peak value of signal intensity that is 7:56�10�12E2c . Note that the idler �uctuations and its non-vanishing imaginary part indicate arelatively slower convergence compared with the signal intensity. The ensemble sizewas 8�105; and the atomic density � = 1010cm�3.

frequencyp�22 + 4

2b . This in�uences also the intensity pro�les and the correlation

functions.

In Figure 4.4, we show that counter-propagating signal (�z) and idler (+z) �elds

at the respective ends of the atomic ensemble. The plots show the real and imaginary

parts of the observables, and both are normalized to the peak value of signal intensity.

Note that the characteristic �eld strength in terms of natural decay rate of the idler

transition ( 03) and dipole moment (di) is (di=~)Ec � 36:3 03. The �uctuation in the

real idler �eld intensity at z = L and non-vanishing imaginary part indicates a slower

convergence compared to the signal �eld that has an almost vanishing imaginary part.

55

The slow convergence is a practical limitation of the method.

In Figure 4.5 (a), we show a contour plot of the second-order correlation function

Gs;i(ts; ti) where ti � ts: In Figure 4.5 (b), a section is shown through ts � 75 ns

where Gs;i is at its maximum. The approximately exponential decay of Gs;i is clearly

superradiant consistent with the theory of Chapter 3 and the reference [16]. The

non-vanishing imaginary part of Gs;i calculated by ensemble averaging is also shown

in (b) and indicates a reasonable convergence after 8�105 realizations. In Table 4.1,

we display numerical parameters of our simulations for four di¤erent atomic densities.

The number of dimensions in space and time is Mt �Mz with grid sizes (�t;�z) in

terms of cooperation time (Tc), length (Lc). The superradiant time scale (Tf) is

found by �tting Gs;i to an exponential function (e�t=Tf ), with 95% con�dence range.

In Figure 4.6, the characteristic time scale is plotted as a function of atomic den-

sity and the factor N ��, and shows faster decay for optically denser atomic ensembles.

We also plot the timescale T1 = �103 =(N� + 1) (ns) that is derived from the theory

of Chapter 3, in which �� is the geometrical constant for a cylindrical ensemble, Eq.

(A.12). The natural decay time �103 = 26 ns corresponds to the D2 line of 87Rb.

The error bar indicates the deviation due to the �tting range from the peak of Gs;i

to approximately 25% and 5% of the peak value. The theory and simulations are in

good qualitative agreement, approaching independent atom behavior at lower densi-

ties. For larger opd atomic ensembles, larger statistical ensembles are necessary for

numerical simulations to converge. The integration of 8�105 realizations used in the

case of � = 1010 cm�3 consumes about 14 days with Matlab�s parallel computing tool-

box (function "parfor") with a Dell precision workstation T7400 (64-bit Quad-Core

Intel Xeon processors).

56

Figure 4.5: Second-order correlation function Gs;i(ts; ti): The 2-D contour plot of thereal part of Gs;i with a causal cut-o¤ at ts = ti is shown in (a). The plot (b) gives across-section at ts = tm � 75 ns, which is normalized to the maximum of the real part(dotted-blue) of Gs;i: The imaginary part (diamond-red) of Gs;i is nearly vanishing,and the number of realizations is 8�105 for � = 1010cm�3:

57

Figure 4.6: Characteristic timescales, Tf and T1 vs atomic density � and the super-radiant enhancement factor N� (� = ��). Tf (dotted-blue) is the �tted characteristictimescale for Gs;i(ts = tm; ti = tm + �) where tm is chosen at its maximum, as inFigure 4.5. The errorbars indicate the �tting uncertainties. As a comparison,T1= �103 =(N� + 1) (dashed-black) is plotted where

�103 = 26 ns is the natural decay

time of D1 line of 87Rb atom, and � is the geometrical constant for a cylindricalatomic ensemble, as discussed in Chapter 3. The number of realizations is 4�105 for� = 5� 108, 5� 109 cm�3 and 8�105 for � = 1010, 2� 1010 cm�3:

58

Table 4.1: Numerical simulation parameters for di¤erent atomic densities �. Corre-sponding optical depth (opd), time and space grids (Mt�Mz) with grid sizes (�t;�z)in terms of cooperation time (Tc) and length (Lc), and the �tted characteristic timeTf for Gs;i (see text).

�(cm�3) opd Mt �Mz�t(Tc);�z(Lc)

Tc(ns);Lc(m)

�tted Tf(ns)[95% con�dence range]

5�108 0:11 101� 44 0.9, 1.5�10�4 1.55, 0.46 24:6 [24:2; 25:0]5�109 1:09 101� 42 2.8, 4.5�10�4 0.49, 0.15 14:8 [14:4; 15:3]1�1010 2:18 101� 42 4, 7�10�4 0.35, 0.10 9:4 [9:2; 9:7]2�1010 4:35 101� 42 5.5, 1�10�3 0.24, 0.07 5:0 [4:6; 5:5]

4.4 Conclusion

We have derived c-number Langevin equations in the positive-P representation for the

cascade signal-idler emission process in an atomic ensemble. The complete c-number

Langevin noise correlations are derived and con�rmed by an alternative theoretical

method. The equations are solved numerically by a stable and convergent semi-

implicit di¤erence method, while the counter-propagating spatial evolution is solved

by implementing the shooting method.

We investigate four di¤erent atomic densities readily obtainable in a magneto-

optical trap experiment. Signal and idler �eld intensities and their correlation func-

tion are calculated by ensemble averages. Vanishing of the unphysical imaginary

parts within some tolerance is used as a guide to convergence. We �nd an enhanced

characteristic time scale for idler emission in the second-order correlation functions

from a dense atomic ensemble, consistent with the superradiance timescales predicted

by the analytical method in Chapter 3, and observed experimentally [16].

59

CHAPTER V

SPECTRAL ANALYSIS FOR

CASCADE-EMISSION-BASED QUANTUM

COMMUNICATION

Cascade emission in alkali atoms is a source of telecommunication photons. In this

Chapter, we investigate the DLCZ [6] scheme using the cascade emission from an

atomic ensemble.

5.1 Introduction

Long distance quantum communication based on atomic ensembles was proposed by

Duan, Lukin, Cirac, and Zoller [6]. This scheme involves Raman scattering of light by

the atoms. The cascade transitions investigated in Chapter 3 and 4 provide a source

of telecommunication wavelength photons. It is interesting to assess the cascade

scheme in the DLCZ protocol given that it could potentially reduce transmission

losses in a quantum telecommunication system. The DLCZ scheme is based on

entanglement generation and swapping and quantum state transfer.

In this Chapter, we �rst discuss entanglement generation and then investigate how

frequency entanglement of the cascade photon pair in�uences entanglement swapping.

5.2 DLCZ Scheme with Cascade Emission

In the DLCZ protocol, a weak pump laser Raman scatters a single photon generating

a quantum correlated spin excitation in the ensemble. By interfering the Raman

photons generated from two separate atomic ensembles on a beam splitter (B.S.), the

DLCZ entangled state (j01i+ j10i)=p2 [98] is prepared conditioned on one and only

61

one click of the detectors after the B.S. Hence j0i and j1i represent the state of zero or

one collective spin excitations stored in the hyper�ne ground state coherences. This

state originates from an indistinguishable photon paths. The error from multiple

excitations can be made negligible if the pump laser is weak enough.

As shown in Figure 5.1, we consider instead that one of the ensembles employ

cascade emission. The idea is for cascade emission to generate a telecom photon (ays)

for transmission in the optical �ber, and an infrared photon that interferes locally

with the Raman photon generated in the �-type atomic ensemble. In this way

interference of the infrared photons generate the entangled state,

ji = 1p2(j01ia;s + j10ia;s); (5.1)

similar to the conventional DLCZ entanglement generation scheme. Now, however,

instead of a stored spin excitation, we generate a telecom photon.

The entanglement swapping with the cascade emission may be implemented as

shown in Figure 5.2, and will be discussed in detail in the next Section. The initial

state is a tensor product of two state vectors generated locally at the sites A and B.

ji = (p1� �1Aj0i+

p�1Aj1iAi j1iAs ) (

p1� �2Aj0i+

p�2Aj1iAr j1iAa )

(p1� �1Bj0i+

p�1Bj1iBi j1iBs ) (

p1� �2Bj0i+

p�2Bj1iBr j1iBa ); (5.2)

where (s, i) represent the signal and idler photons from the cascade emission, and (r,

a) are Raman scattered photon and the collective spin excitation. Here �1 and �2 are

e¢ ciencies to generate cascade and Raman emission. Since �1 and �2 � 1; multiple

atomic excitations or multi-photon generation can be excluded.

5.3 Entanglement Swapping

Consider the product state generated from A and B, Figure 5.2,

62

Figure 5.1: Entanglement generation in the DLCZ scheme using the cascade andRaman transitions in two di¤erent atomic ensembles. Large white arrows representlaser pump excitations corresponding to the dashed lines in either cascade or Ra-man level structures. Here ays represents the emitted telecom photon. B.S. meansbeam splitter that is used to interfere the incoming photons measured by the photondetector D. The label A refers to the pair of ensembles for later reference.

63

Figure 5.2: Entanglement swapping of DLCZ scheme using the cascade transition.The site A is described in detail in Figure 5.1 and equivalently for the site B. Thetelecom signal photons are sent from both sites and interfere by B.S. midway betweenwith detectors represented by cy1 and c

y2. Synchronous single clicks of the detectors

from both sites (my1;2, n

y1;2) and the midway detector (c

y1;2) generate the entangled state

between lower atomic ensembles at sites A and B. The locally generated entanglementis swapped to distantly separated sites in this cascade-emission-based DLCZ protocol.

64

ji = (j10ias + j01iasp

2)A (

j10ias + j01iasp2

)B

=1

2(j1010iasas + j1001iasas + j0110iasas + j0101iasas); (5.3)

where the subscript (a) represents a stored local atomic excitation, and (s) means a

telecom photon propagating toward the B.S. in the middle. We can tell from this

e¤ective state that the �rst component (j1010iasas) contributes no telecom photons

at all (two local excitations) and can be ruled out by measuring a "click" at one of

the middle detectors. The second and the third components have components of the

entangled state of quantum swapping, and the fourth one is the source of error if the

photodetector cannot resolve one from two photons. The error could be corrected

by using a photon number resolving detector (PNRD) if other drawbacks like dark

counts, photon losses during propagation, and detector ine¢ ciency are not considered.

Now we will formulate the entanglement swapping including the spectral e¤ects

discussed in Chapter 3. We ignore pump-phase o¤sets, assuming 50=50 B.S. and a

symmetric set-up (�1A = �1B = �1; �2A = �2B = �2) for simplicity. Expand the

previous joint state, Eq. (5.2) and keep the terms up to the second order of �1;2 that

can contribute to detection events (my1;2; n

y1;2),

jieff

= �1(1� �2)j1iAi j1iAs j1iBi j1iBs + �2(1� �1)j1iAr j1iAcsj1iBr j1iBcs+p�1�2(1� �1)(1� �2)j1iAi j1iAs j1iBr j1iBcs +

p�1�2(1� �1)(1� �2)j1iAr j1iAcsj1iBi j1iBs ;

(5.4)

where the cascade emission state j1isj1ii �Rf(!s; !i)a

y�s(!s)a

y�i(!i)j0id!sd!i has

the spectral distribution f(!s; !i) as derived in Chapter 3.

As shown in Figure 5.2, entanglement swapping protocol is ful�lled by measuring

three clicks from the three pairs of the detectors respectively (my1;2; n

y1;2; c

y1;2). The

65

quantum e¢ ciency of the detector is considered in the protocol, and we describe a

model for quantum e¢ ciency in Appendix C.1. We then use this model to describe

photodetection events registered by non-resolving photon detectors (NRPD). Starting

with the input density operator �in = jieffhj; we derive the projected density

operator, Eq. (C.16), conditioned on the three clicks of my1; n

y1; and c

y1 in Appendix

C.2. We use the Schmidt decomposition of the projected density operator and assume

a single mode for the Raman scattered photon. We �nd the un-normalized density

operator �(2)out given in Eq. (C.16),

�(2)out =

�21(1� �2)2

16(2� �t)�t�

2eff

�1 +

Xj

�2j

�j0ih0j+ �1�2(1� �1)(1� �2)

8�t�

2eff��

SyBj0ih0jSB + SyAj0ih0jSA�+Xj

�j

Z�j(!i)�

�j(!

0i)�

�(!i)��(!0i)d!id!

0i�

SyBj0ih0jSA + SyAj0ih0jSB��

; (5.5)

where �t and �eff are quantum e¢ ciencies of the detectors at the telecom and infrared

wavelengths respectively. The �rst term in Eq. (5.5) is the atomic vacuum state at

sites A and B and contributes an error to the output density operator. The second

term contains the components of the DLCZ entangled state.

We can de�ne the �delity F , the success probability PS of entanglement swapping

of the entangled state jiDLCZ = (SyA+SyB)j0i=

p2; and the heralding probability PH

for the third click as

F � Tr(�(2)outjiDLCZhj)Tr(�(2)out)

; (5.6)

PH = P1 + P2; P1 = P2 =Tr(�(2)out)N ; (5.7)

PS = P1 � F1 + P2 � F2; F1 = F2 = F; (5.8)

66

where P1;2 is the heralding probability of the single click from the midway detector

(cy1;2) as shown in Figure 5.2, and a trace (Tr) is taken over atomic degrees of freedom.

The normalization factor N is calculated in Eq. (C.9) and is given by

N =�21(1� �2)

2

4�2eff +

�1�2(1� �1)(1� �2)

2�2eff +

�22(1� �1)2

4�2eff : (5.9)

We have used the following properties for the calculation of �(2)out and N ,

Zd!sd!ijf(!s; !i)j2 = 1; (5.10)

where orthonormal relations in the mode functions are used, andZd!sd!

0sd!id!

0if(!

0s; !

0i)f

�(!0s; !i)f(!s; !i)f�(!s; !

0i) =

Xj

�2j : (5.11)

Note that the single mode spectral function for the Raman photon satis�esRd!j�(!)j2 =

1:

The �delity, heralding, and success probability become

F =1 +

Pj �j

R�j(!i)�

�j(!

0i)�

�(!i)��(!0i)d!id!

0i

�r(2� �t)(1 +P

j �2j)=2 + 2

; (5.12)

PH =�r�t(2� �t)(1 +

Pj �

2j)=2 + 2�t

(p�r + 1=

p�r)

2; (5.13)

PS = �t1 +

Pj �j

R�j(!i)�

�j(!

0i)�

�(!i)��(!0i)d!id!

0i

(p�r + 1=

p�r)

2; (5.14)

where 1��21��1

� 1 and �r = �1=�2.

The �delity depends on a sum of square of Schmidt numbers in the denominator

and the mode mismatch between the idler and Raman photons in the numerator. Let

us assume that the Raman photon mode is engineered to be matched with the idler

photon mode of the largest Schmidt number (�1(!i) in our case), which is required

to have a larger �delity (so is the success probability) compared to other modes. We

may also compare the NRPD with the performance of PNRD in the midway detectors,

then we have the �delity, heralding, and success probability,

67

F =

8><>:1+�1

�r(2��t)(1+Pj �

2j )=2+2

; NRPD

1+�1�r(1��t)(1+

Pj �

2j )+2

; PRND(5.15)

PH =

8><>:�r�t(2��t)(1+

Pj �

2j )=2+2�t

(p�r+1=

p�r)

2 ; NRPD�r�t(1��t)(1+

Pj �

2j )+2�t

(p�r+1=

p�r)

2 ; PRND(5.16)

PS =

8><>:�t(1+�1)

(p�r+1=

p�r)

2 ; NRPD

�t(1+�1)(p�r+1=

p�r)

2 ; PRND: (5.17)

When the relative e¢ ciency is made arbitrarily small, the �delity approaches

(1+�1)=2 for both types of detectors. It reaches one if a pure cascade emission source

is generated (von Neumann entropy E = 0 and �1 = 1). When �r = 1 with a pure

source using NRPD with a perfect quantum e¢ ciency, F = 2=3; PH = 3=4; PS = 1=2;

which coincide with the results of the reference [99] (with perfect quantum e¢ ciency).

We discuss the frequency entanglement for various pulse widths and superradiant

decay rates in Chapter 3.4. We �nd that for shorter driving pulses and smaller

superradiant decay rates, the cascade emission source is less spectrally entangled.

That means when �r is �xed, a shorter driving pulse heralds a higher �delity DLCZ

entangled state.

In Figure 5.3, we numerically calculate the entropy and plot out the �delity from

Eq. (5.15), the heralding probability from Eq. (5.16), and the success probability

from Eq. ( 5.17) as a function of the relative e¢ ciency �r: With a perfect detection

e¢ ciency (� = 1), we �nd that at a smaller �r; the less entangled source gives us a

higher �delity DLCZ entangled state but with a smaller success probability. Small

generation probability for cascade emission (�r < 1) reduces the error of NRPD from

two telecom photons interference, but it reduces the successful entanglement swapping

at the same time.

The optimal success probability occurs by using the same excitation e¢ ciency

for both cascade and Raman con�gurations. For PNRD, the �delity is higher than

68

Figure 5.3: Fidelity F , heralding PH , and success PS probabilities of entanglementswapping versus relative e¢ ciency �r with perfect detection e¢ ciency �t = 1: Col-umn (a) NRPD and (b) PNRD. Solid-red, dashed-blue, and dotted-green curvescorrespond to the pulse width parameters � = (0:1; 0:5; 0:5) and superradiant factorN �� + 1 = (5; 5; 10) (see Chapter 3 and Appendix A): The von Neumann entropy isS = (0:684; 2:041; 2:886); respectively.

69

NRPD, and the heralding probability is the same independent of the degree of fre-

quency space entanglement. The success probabilities for both types of detectors are

equal. The advantage of PNRD shows up in the �delity of quantum swapping.

In Figure 5.4, we show that the measures improve monotonically with the quantum

e¢ ciency (� = �t) of the detector at telecom wavelength, with �r = 0:5. The success

probabilities for both types of detectors are the same and again the advantage of

PNRD shows up in the �delity.

5.4 Polarization Maximally Entangled State (PME State)and Quantum Teleportation

In Figure 5.5, we illustrate schematically a scheme for probabilistic PME state prepa-

ration and quantum teleportation. Four ensembles (ABCD) are used to generate

two entangled pairs of DLCZ entangled states, and another two ensembles (I1; I2)

are used to prepare a quantum state to be teleported.

With the conditional output density matrix from Eq. (C.16), we proceed to con-

struct the PME state jiPME =1p2(SyAS

yD+S

yBS

yC)j0i where (C;D) represents another

parallel entanglement connection setup, Figure 5.5 (a). This PME state is useful in

entanglement-based communication schemes [6], and we will here calculate its success

probability. The normalized density matrix for the AB system is from Eq. (5.5) (let

�t = �),

�(2);ABout;n =

a

a+ 4j0ih0j+ 2

a+ 4

�SyBj0ih0jSB + SyAj0ih0jSA

+ �1SyBj0ih0jSA + �1S

yAj0ih0jSB

�; (5.18)

where the largest Schmidt number (�1) of mode overlap is chosen and a � �r(2 �

�)�1 +

Pj �

2j

�.

A parallel pair of entangled ensembles (C,D) is introduced, and the joint density

operator is �(2);ABout;n �(2);CDout;n : The latter expression is developed mathematically in

70

Figure 5.4: Fidelity F , heralding PH , and success PS probabilities of entanglementswapping versus telecom detector quantum e¢ ciency � for the case of (a) NRPD and(b) PNRD. Solid-red, dashed-blue, and dotted-green curves correspond to the sameparameters used in Figure 5.3.

71

Figure 5.5: PME projection (a) and quantum teleportation (b) in the DLCZ scheme.Four atomic ensembles (A,B,C,D) are used to generate two DLCZ entangled statesat (A,B) and (C,D). PME state is projected probabilistically conditioned on fourpossible detection events of (Dy

A or DyC) and (D

yB or D

yD) in (a). In the quantum

teleportation protocol (b), another two ensembles (I1;I2) are used to prepare a quan-tum state that is teleported to atomic ensembles B and D conditioned on four possibledetection events of (DI1 or DA) and (DI2 or DC).

72

Appendix C.3.

With projection of the PME state, we have the post measurement success proba-

bility [a click from each side; the side of (A or C) and (B or D)],

PS;PME = hj�(2);ABout;n �(2);CDout;n jiPME;

=4(1 + �21)

[�r(2� �t)(1 +P

j �2j) + 4]

2: (5.19)

For �r � 1, PS;PME reaches the maximum of 1=2 when a pure source (�1 = 1) is

used.

For an arbitrary quantum state transfer to long distance, quantum teleportation

scheme may be used. Another two ensembles (I1; I2) are introduced [6], and the

quantum state can be described by ji = (d0SyI1+ d1S

yI2)j0i with jd0j2 + jd1j2 = 1.

The joint density matrix for quantum teleportation is

�QT = (d0SyI1+ d1S

yI2)j0ih0j(d�0SI1 + d�1SI2) �

(2);ABout;n �

(2);CDout;n : (5.20)

Atomic ensembles (A,B) in parallel with (C,D) provide a scheme for PME state

preparation. Retrieve the quantum state [ensemble (I1; I2)] into photons and interfere

them at B.S., respectively, with photons from A and C. We have the teleported

quantum state at B and D conditioned on the single click of (DI1 or DA) and (DI2 or

DC).

Consider single detection events at DI1 and DI2 as an example. With the NRPD

measurement operators MI1;I2 � (IyD1�j0iD1h0j)j0iDAh0j(I

yD2�j0iD2h0j)j0iDC h0j

(we use D1; D2 for DI1 ; DI2), the density matrix after the measurement becomes

73

�1 � Tr(�QT;effMI1;I2) =

a+ 2

2(a+ 4)2j0iABCDh0j+

4

(a+ 4)2

� jd0j24SyBj0ih0jSB +

jd1j24SyDj0ih0jSD+

�21d0d�1

4SyBj0ih0jSD +

�21d�0d14

SyDj0ih0jSB�; (5.21)

where �QT;eff is calculated in Eq. (C.18), and the trace is taken over the electromag-

netic �eld degrees of freedom.

For a successful transfer of the quantum state j�i = (d0SyB+d1SyD)j0i, the �delity

F1 = h�j�1j�i=Tr(�1); and the heralding probability is P1 = Tr(�1), with the trace

over all atomic degrees of freedom. Except for the detection event we consider here,

there are three other detection events including (DA; DC), (DI1 ; DC) and (DA; DI2).

The teleported state from the detection events (DI1 ; DC) and (DI2 ; DA) requires a

� rotation correction on the relative phase (d0 ! d0; d1 ! �d1).

The �delity and heralding probabilities conditioned on the other three pairs of

clicks are the same as F1 and P1 respectively, so the success probability is

PS;QT =4Xi

PiFi = 4P1F1;

=F 2

(1 + �1)2[1 + (2�21 � 2)jd0j2jd1j2]; (5.22)

where F is the �delity of entanglement swapping for NRPD, Eq. (5.15). For PNRD,

the success probability for quantum teleportation is unchanged.

The success probability for quantum teleportation depends on the probability

amplitude of the quantum state and the �delity F of the entanglement swapping. In

Figure 5.6, for �r = 0:5 and �t = 1, we can see in the region jd0j � 0:3 � 0:9, higher

success probability requires a less entangled cascade emission source. Outside this

region, it prefers a more entangled source. When a pure source is used (�1 = 1) and

let �r � 1; �t = 1, we can achieve the maximum of the success probability PS;QT =14

74

Figure 5.6: Success probability of quantum teleportation as a function of the prob-ability amplitude of teleported quantum state with �r = 0:5 and a perfect detectore¢ ciency �t = 1: Solid-red, dashed-blue, and dotted-green curves correspond to thesame parameters used in Figure 5.3.

when F = 1, which is also achieved in the traditional DLCZ scheme with perfect

quantum e¢ ciencies [99].

5.5 Conclusion

We have described probabilistic protocols for the DLCZ scheme implementing the

cascade emission source. We characterize the spectral properties of the cascade emis-

sion by Schmidt mode analysis and investigate the �delity and success probability

of the protocols using photon resolving and non-resolving photon detectors. The

75

success probability is independent of the detector type, but photon number resolving

detection improves the �delity.

The performance of the protocol also depends on the ratio of e¢ ciencies in gen-

erating the cascade and Raman photons. The success probability is optimized for

equal e¢ ciencies while the �delity is higher when the ratio is smaller than one for

non-resolving photon detectors.

The frequency space entanglement of telecom photons produced in cascade emis-

sion deteriorates the performance of DLCZ protocols. The harmful e¤ect can be

diminished by using shorter pump pulses to generate the cascade emission. A state

dependent success probability of quantum teleportation was calculated, and in some

cases a more highly frequency entangled cascade emission source teleports more suc-

cessfully. An improved performance could be achieved if the error source (vacuum

part) were removed. This could be done by entanglement puri�cation [3] at the stage

of entanglement swapping and then using the puri�ed source to teleport the quantum

state.

76

CHAPTER VI

EFFICIENCY OF LIGHT-FREQUENCY CONVERSION

IN AN ATOMIC ENSEMBLE

In this Chapter 1, the e¢ ciency of frequency up and down conversion of light in an

atomic ensemble, with a diamond level con�guration, is analyzed theoretically. The

conditions of pump �eld intensities and detunings required to maximize the conversion

as a function of optical thickness of the ensemble are determined. The in�uence of

the probe pulse duration on the conversion e¢ ciency is investigated by the numeric

solution of the Maxwell-Bloch equations. The set of equations are similar to those

in Chapter 4, but a c-number version of the interaction is considered here. The

properties of absorption and dispersion of �elds are extracted from the steady state

solutions to demonstrate the parametric coupling between the �elds. We will show

that, in calculating conversion e¢ ciency, a quantum version of the equation including

Langevin noises is equivalent to the c-number one. Frequency conversion provides

the bridge for transmitted qubit (telecommunication wavelength) and local quantum

memory (near-infrared light), in which a large scale quantum communication can be

ful�lled.

In Section II, we discuss the four-wave mixing process and present solutions for

the up- and down- converted �elds. The dressed state picture is used as a guide to

understand the characteristic features of the absorption and signal-idler �eld coupling.

In Section III, we present the results of an optimization in conversion e¢ ciency as a

function of the optical depth of the atomic ensemble. In Section IV, we investigate

the e¤ects of a �nite pulse duration by numerically integrating the Maxwell-Bloch

1This Chapter is based on reference [100].

77

equations. Section V demonstrates the results of Langevin noise correlations and

we conclude in Section VI. The derivations of the Maxwell-Bloch and parametric

equations are relegated to the Appendix D.

6.1 Introduction

The frequency conversion of light �elds has been an important theme in optical physics

for around half a century. In quantum information physics the conversion of single

photons to and from the telecom wavelength band is a topic of more recent vintage,

and is motivated by the desire to minimize optical �ber transmission losses when

distributing entangled states over distant quantum memory elements in a quantum

repeater [4].

An associated technical problem is that telecom light is not readily stored in

ground level atomic memory coherences. Retrieval processes in atomic ensembles, for

example using electromagnetically induced transparency [101], or more speci�cally

the dark-polariton mechanism [102, 103], generate shorter wavelength radiation cor-

related to the stored atomic excitation by Raman scattering. Such radiation, optically

resonant to the ground level of typical atoms and ions, has been retrieved in numerous

experiments [7�11, 13�15, 104, 105]. An important advance would involve generation

of atomic memory coherences quantum-correlated with telecom wavelength radiation,

thereby minimizing transmission losses over long distances. Recently there has been

a breakthrough in this direction using a pair of cold, non-degenerate rubidium gas

samples [25]. The stored excitation is correlated with an infra-red �eld (idler) in

one gas sample, and the idler is then frequency converted to a telecom wavelength

signal �eld in the other ensemble. The frequency conversion mechanism involves the

diamond con�guration of atomic levels shown in Figure 6.1.

In a probabilistic protocol it is important to maximize all e¢ ciencies, e.g., �ber

transmission, single-photon detection, and quantum memory lifetime [106]. In the

78

Figure 6.1: The diamond con�guration of atomic system for conversion scheme.Two pump lasers (double line) with Rabi frequencies a;b and propagated probe�elds (single line) E+s ; E

+i interact with the atomic medium. Various detunings are

de�ned in the Appendix D, and the atomic levels used in the experiment [25] are(j0i; j1i; j2i; j3i) = (j5S1=2;F = 1i; j5P3=2;F = 2i; j6S1=2;F = 1i; j5P1=2;F = 2i):

present work we investigate the e¢ ciency of frequency up- and down- conversion in

the diamond atomic con�guration [16, 107], as a function of the ensemble�s optical

thickness, and the intensity and detuning of the pump �elds involved in the near-

resonant, four-wave mixing process.

6.2 Theory

We consider a cold and cigar-shaped 87Rb atomic ensemble with co-propagating light

�elds similar to the experimental setup in the reference [25].

The conversion scheme shown in Figure 6.1 involves two pump lasers with fre-

quencies !a and !b, respectively; their Rabi frequencies are given by a and b. Two

weak probe �elds, signal and idler, with frequency !s and !i, respectively, propa-

gate through the optically thick atomic medium. Unlike the cascade driving scheme,

where two-photon excitation generates a photon pair spontaneously [16], pump laser

79

b experiences a transparent medium if both the signal and idler �elds are in the vac-

uum state. With an incident signal �eld, four-wave mixing with the pumps generates

an up-converted idler �eld, while an incident idler �eld generates a down-converted

signal.

The Maxwell-Bloch equations for the interacting system of light and four light

�elds is derived in the Appendix D. By linearizing the equations with respect to

the signal and idler �eld amplitudes, and adiabatically eliminating the atoms, one

arrives at coupled parametric equations for the signal and idler �elds. We discuss

their solution in this section, and leave numerical solutions of the Maxwell Bloch

equations to Section IV.

The calculation of conversion e¢ ciencies can also be carried out with the quan-

tized Heisenberg-Langevin version of the coupled parametric equations, which we will

show in Section V. The resulting conversion e¢ ciencies are identical to the semiclas-

sical treatment; the additional quantum noise contributions vanish as the j2i ! j3i

transition driven by pump laser b has vanishing populations and atomic coherence. A

similar simpli�cation occurs in the calculation of the storage e¢ ciency of spin waves

in a system of atoms in the � con�guration [108, 109].

The co-moving propagation equation for c-number signal and idler �elds (respec-

tively, E+s and E+i ) under energy conservation (�! = !a + !s � !b � !i = 0) and

four-wave mixing conditions (�k = ka � ks + kb � ki = 0) are

@

@zE+s = �sE

+s + �sE

+i

@

@zE+i = �iE

+s + �iE

+i : (6.1)

The coupled equations are similar to those found for the double � system [110,

111]. The self-coupling coe¢ cients �s; �i and parametric coe¢ cients �s, �i are de�ned

in Appendix D. The set of equations can be simpli�ed as

80

@

@zx(z) = Ax (6.2)

where

x =

0B@E+sE+i

1CA ; A =

0B@ �s �s

�i �i

1CA : (6.3)

The equations are solved by considering a similarity transformation S that � =

S�1AS is diagonalized and y = S�1x such that

@

@zy = �y (6.4)

y(z) = e�(z�z0)y(z0) (6.5)

where y(z0) is the boundary condition. With the known boundary condition x1(0)

and x2(0) where we choose the input face of propagation as z0 = 0, we have

x(z) = Se�zS�1x(0): (6.6)

And the diagonalized and transformation matrix are

� =

0B@ (�i + �s)=2 + w 0

0 (�i + �s)=2� w

1CA ; (6.7)

S =

0B@ q + w �s

�i �q � w

1CA ; (6.8)

S�1 =1

2w(w + q)

0B@ q + w �s

�i �q � w

1CA (6.9)

where w �pq2 + �s�i, and q � (��i + �s)=2.

The solution of �elds from down conversion is

81

264E+s (L)E+i (L)

375 = Se�LS�1

264 0

E+i (0)

375 = E+i (0)e(�i+�s)L=2

2w

264 �s(ewL � e�wL)

1(w+q)

[�s�iewL + (q + w)2e�wL]

375 :(6.10)

Similarly, the solution of �elds from up conversion is

264E+s (L)E+i (L)

375 = Se�LS�1

264E+s (0)0

375 = E+s (0)e(�i+�s)L=2

2w

264 1(w+q)

[(q + w)2ewL + �s�ie�wL]

�i(ewL � e�wL)

375 :(6.11)

We de�ne the down conversion e¢ ciency �d and transmission of input idler �eld

Td as

�d =

��E+s (L)E+i (0)

��2 = �� s2we(�i+�s)L=2(ewL � e�wL)��2 (6.12)

Td =

��E+i (L)E+i (0)

��2 = �� e(�i+�s)L=22w(w + q)[�s�ie

wL + (q + w)2e�wL]

��2 : (6.13)

Similarly the up conversion e¢ ciency �u and transmission of input signal �eld Tu

is

�u =

��E+i (L)E+s (0)

��2 = �� i2we(�i+�s)L=2(ewL � e�wL)��2 (6.14)

Tu =

��E+s (L)E+s (0)

��2 = �� e(�i+�s)L=22w(w + q)[(q + w)2ewL + �s�ie

�wL]

��2 : (6.15)

The above is the central result of this section. The up and down conversion

e¢ ciencies di¤er only in the parametric coupling coe¢ cients �i and �s: In the

strong parametric coupling regime where j�ij; j�sj � j�ij; j�sj, the coe¢ cients can

be simpli�ed to �u 'q

�i�ssinh(

p�s�iL); �d '

q�s�isinh(

p�s�iL) and Tu = Td '

cosh(p�s�iL): Under the further assumptions �i = �s = 0 and �i; �s are pure imag-

inary, we �nd �u = �d = sin2[Im(�sL)] and Tu = Td = cos2[Im(�sL)]. This result

82

Figure 6.2: Dressed-state picture from the perspective of the probe idler transitionbetween atomic levels j0i and j3i: Two strong �elds a;b shift the levels with energy�Ea;b and wavy lines represent the idler �eld resonances.

was recently derived by Gogyan using a dressed state approach [112], in the case of

resonant pump �elds �1 = �b = 0 [113]. In this ideal limit there is a conservation

condition �u+Tu = �d+Td = 1: The parametric coupling coe¢ cients are not identical,

but in the regime of strong coupling they approach each other. As noted by Gogyan,

when the pump-a intensity is large (a >> j�1j; 03) the j0i $ j1i is saturated, the

atomic coherence is negligible and �s � �i / ~�00;s(�abT02

+ �abT13). Alternatively, in the

limit b >> a; 32 and j�1j >> 01 the atomic coherence of j0i $ j1i dominates

and once again �s � �i /ibjbj2~�y01;s

T13T02. Note that this scheme is also similar to the

frequency conversion in nonlinear materials [114].

The ac-Stark splitting induced by the pump lasers shifts the resonant absorp-

tion condition for the idler and signal �elds. The idler and signal experience reso-

nant absorption at the transition frequency of the dressed atom. The corresponding

transitions for the idler are shown in Figure 6.2. The bare states are shifted by

83

�Ea =��1 �

p�21 + 4

2a

�� =2 and �Eb = ��b �p�2b + 4

2b

�� =2; respectively. Notethat our Rabi frequencies are smaller by a factor 2 than the standard de�nitions to

avoid a plethora of prefactors in the equations of the Appendix.

For resonant pump �elds, �Ea;b = �a;b . The idler transition resonances are

at �!i = �(a + b); � ja � bj ; ja � bj ; (a + b) and these delineate three

windows separated by these four absorption peaks. For a > b the centers of these

windows are at �a; 0; and a, respectively. Choosing the idler detuning �!i = �a

as in Ref. [113], the idler interacts with the atomic medium at the center of the left

or right window.

As an example of the strong coupling windows created by intense pump lasers,

we show in Figure 6.3 the self and cross coupling coe¢ cients for the signal and idler

�elds as a function of the idler frequency. Note that the corresponding frequency of

signal �eld is determined by �!s = �!i � �1 + �b. The dimensionless quantities

�iL, �sL and �sL are shown under the conditions of maximum conversion e¢ ciency

to be discussed in the next section. We choose the optical depth (opd) ��L = 150

where � is the number density, � � 3�2=(4�) the resonant absorption cross-section,

and L the atomic ensemble length in the propagation direction. Three parametric

coupling windows are separated by two strong absorption peaks on the left and two

relatively weak ones on the right. The imaginary part of the self-coupling coe¢ cients

are seen to vanish in each window at a certain point, while the real parts are small

away from resonances. At the same time the cross-coupling coe¢ cients have a large

imaginary part. The positive gradient of =(�sL) and =(�iL) inside the windows is

indicative of normal dispersion.

6.3 Optimal Conversion E¢ ciency

It is important to ascertain the parameters that allow maximum e¢ ciency of conver-

sion due its potential in practical quantum information processing. In principle we

84

Figure 6.3: Self-coupling coe¢ cients �s; �i and cross-coupling coe¢ cient �s. Dimen-sionless quantities (a) �sL, (b) �iL and (c) �sL with real (solid blue) and imaginary(dashed red) parts are plotted as a dependence of idler detuning �!i [same labelin (b)] showing four absorption peaks to construct three parametric coupling win-dows. A black dashed-dot line of the constant �=2 is added in (c) to demonstratethe crossover with =(�sL) indicating the ideal conversion e¢ ciency condition in theleft window. The parameters we use are (a, b, �1, �b, �!i) = (33, 20, 39, 2,�21) 03 for optical depth ��L = 150 with L = 6mm. Various natural decay rates are 03 = 1=27:7ns, 01 = 1=26:24ns, 12 = 03=2:76; and 32 = 03=5:38 [97].

85

Figure 6.4: Down conversion e¢ ciency �d vs optical depth (opd) from 1 to 600: Eachdotted point is the maximum for �ve variational parameters a, b, �1, �b, and �!i.

need to search the three parametric coupling windows to �nd the optimum conditions

for an atomic ensemble of a given optical thickness.

In the previous section we have discussed how three parametric coupling windows

appear for some particular values of pump laser parameters. In the search for the

maximal conversion e¢ ciency, �ve parameters a, b, �1, �b, and �!i are varied to

maximize the conversion e¢ ciency for a �xed optical depth of atomic ensemble, using

functional optimization.

The optical depth ��L appears through the dependence on atomic number N in

the Arecchi-Courtens cooperation time Tc [115]

T�2c � N jgij2 = 03c

2L��L:

In Figure 6.4, we show the maximum of down conversion e¢ ciency using Eq. (6.12)

86

for di¤erent optical depths from 1 to 600. The maximum is found by varying �ve pa-

rameters mentioned above and the conversion e¢ ciency reaches 100% asymptotically

when the optical depth becomes larger. In the strong parametric coupling regime

as we discussed in the previous section, �d ' sin2[Im(�sL)] and it has a maximum

when Im(�sL) = �2; see Figure 6.3. Since Im(�sL) is proportional to optical depth

and inversely proportional to the Rabi frequencies of the driving lasers, an order of

magnitude estimate of the optical depth necessary for near unit conversion e¢ ciency

is opd' �2a;b= 03 >> 1.

The behavior of the cross-coupling coe¢ cient Im(�sL) as a function of idler de-

tuning indicates where large conversion is to be found, as a comparison with Figure

6.5 shows. The maximum e¢ ciency of about 0:92 is located in the left parametric

coupling window at the intersection of Im(�sL) and �2. Inside the windows the trade-

o¤ between conversion and transmission is clear. In the region where absorption is

large, on the sides of the window (especially for the left window), the e¢ ciency and

the transmission are both low although the valley in conversion e¢ ciency corresponds

to a peak in transmission as expected in parametric coupling. The transmission ap-

proaches unity when the incident idler �eld is far o¤-resonance.

We note that the symmetry (�1;�b;�!i) ! �(�1;�b;�!i) gives degenerate

optimal conversion conditions.

Moreover, for the region where absorption is large on the sides of the window

(especially for the left window), the e¢ ciency and the transmission are both low

but the valley of e¢ ciency corresponds to a peak for transmission indicating the

feature of parametric coupling. The plateau for e¢ ciency in absorption region can

be estimated as �d �� s2w

��2 where �i�s � �s�i. Based on the study of �nding the

maximum e¢ ciency for frequency conversion, we will investigate the situation when

the input is a pulse and numerical integration of full equation of motion is required

in the next section.

87

Figure 6.5: Conversion e¢ ciency �d, �u and transmission Td vs �!i for opd=150. �dand �u are indistinguishable and shown in solid red line, and Td is in dashed blue line.High transmission e¢ ciency corresponds to low conversion e¢ ciency indicating theapproximate conservation condition within each parametric coupling window. Themaximum conversion e¢ ciency is found in the left window at around �!i = �20 03and other relevant parameters are the same as in Figure 6.3.

88

6.4 Pulse Conversion: Solution of the Maxwell-Bloch Equa-tions

The e¤ect of �nite-duration input probe pulses, which are often employed in practice,

can be assessed by numerically solving the Maxwell-Bloch equations for the coupled

atoms-�elds system. The characteristic scales of time and length are given by the

Arecchi-Courtens time Tc and Lc = cTc, respectively, which are inversely proportional

to the square root of the opd. The cooperative electric �eld is the product of the

atomic number and the idler electric �eld per photon, i.e., Ec =p�~!i=(2�0):

Scaling the space, time, electric �eld amplitude, various detunings, and natural

decay rates accordingly, indicated by tildes, the Maxwell-Bloch equations of Eqs.

(D.4,D.6,D.7) under energy conservation (�! = !a+!s�!b�!i = 0) and four-wave

mixing conditions (�k = ka � ks + kb � ki = 0) become

@

@~�~�01 = (i ~�1 �

~ 012)~�01 + i~a(~�00 � ~�11) + i~�02 ~E

�s � i~�y13 ~E

+i ;

@

@~�~�12 = (i�~!s �

~ 01 + ~ 22

)~�12 � i~�a~�02 + i(~�11 � ~�22) ~E+s + i~b~�13;

@

@~�~�02 = (i ~�2 �

~ 22)~�02 � i~�12 ~a + i~�01 ~E

+s + i~�03 ~b � i~�32 ~E

+i ;

@

@~�~�11 = �~ 01~�11 + ~ 12~�22 + i~a~�

y01 � i~�a~�01 � i~�y12 ~E

+s + i~�12 ~E

�s ;

@

@~�~�22 = �~ 2~�22 + i~�y12 ~E

+s � i~�12 ~E

�s + i~b~�

y32 � i~�b~�32;

@

@~�~�33 = �~ 03~�33 + ~ 32~�22 � i~b~�

y32 + i~�b~�32 + i~�y03 ~E

+i � i~�03 ~E

�i ;

@

@~�~�13 = (i�~!i � i ~�1 �

~ 01 + ~ 032

)~�13 � i~�a~�03 � i~�y32 ~E+s + i~�b~�12 + i~�y01 ~E

+i ;

@

@~�~�03 = (i�~!i �

~ 032)~�03 � i~a~�13 + i~�b~�02 + i(~�00 � ~�33) ~E+i ;

@

@~�~�y32 = (�i ~�b �

~ 03 + ~ 22

)~�y32 � i~�13 ~E�s + i~�b(~�22 � ~�33) + i~�y02 ~E

+i ; (6.16)

and@

@~z~E+s = i~�12

jgsj2jgij2

,@

@~z~E+i = i~�03 (6.17)

where ~z = z=Lc, ~� = �=Tc, ~a;b = a;bTc, ~E+s;i = E+s;i=Ec; and jgsj2=jgij2 is a factor of

unit transformation from signal to idler �eld strength. Natural life time [97] for signal

89

and idler transitions is used to calculate the ratio of coupling strength gs=gi = 1:035:

The above equations were integrated with a semi-implicit �nite di¤erence method

[95]. The midpoint integration method is stable and has high accuracy without

sacri�cing memory for �ner grids [88]. The algorithm has been tested by comparing

with the parametric equations� solutions in appropriate limits, and these solutions

are recovered when �ne enough grids are employed.

To illustrate the in�uence of �nite pump pulse duration, we compute the down

conversion e¢ ciency

�d =

RjE+s (z = L; �)j2d�RjE+i (z = 0; �)j2d�

: (6.18)

In Figure 6.6, we show the computed values of �d for two di¤erent input idler

pulse durations. We �x the opd=150 and use the near optimum parameters (a,

b, �1, �b, �!i) = (33, 20, 39, 2, �21) 03 determined from the coupled parametric

equations. The temporal shape of the pump laser intensities is also shown. Pump

laser b is taken to be continuous wave, while pump a is a square pulse with duration

large enough to completely overlap the input idler pulse. To compare with the

steady state solutions, we choose the Rabi frequency of idler as 0:1 03; which is small

compared to those of the pumps. We �nd that the conversion e¢ ciency is reduced for

shorter idler pulse inputs. A 100 ns idler pulse is long enough that it has a almost the

same maximum conversion e¢ ciency of 0:92 as in Figure 6.4 for opd=150. While for

the shorter idler pulse of 15 ns, the signal develops signi�cant temporal modulation,

and this reduces the conversion e¢ ciency, although it is still quite appreciable. The

modulation frequency is at the generalized Rabi frequency of pump-ap�21 + 4

2a.

We note the characteristic time and space scales of the calculations are Tc = 0:086 ns

and Lc = 26 mm for a moderate atomic density � = 1:7� 1011cm�3 and L = 6 mm.

The grid size for dimensionless time �~t = 0:5 and space �~z = 0:001 were chosen

for both 100 and 15 ns idler pulse durations, and the convergence is reached with an

estimated relative error less than 1%.

90

Figure 6.6: Time-varying pump �elds of Rabi frequencies a;b(t) and down-convertedsignal intensity (jE+s (t; z = L)j2) from an input idler pulse (jE+i (t; z = 0)j2). Herewe let t = � that is the delayed time in co-moving frame. Pump-b (dotted green)is a continuous wave and pump-a (dashed black) is a square pulse long enough toenclose input idler pulse with (a) 100 ns and (b) 15 ns (dashed-dot blue). Outputsignal intensity (solid red) at the end of atomic ensemble z = L is oscillatory due tothe pump �elds. The square pulse in rising region (tr� ts

2< t < tr+

ts2) has the form

of 12[1 + sin(�(t�tr)

ts)] that in (a) (tr; ts)=(10,10)ns for pump-a and (tr; ts)=(20,20)ns

for input idler; (b) (tr; ts)=(10,5)ns for pump-a and (tr; ts)=(15,10)ns for input idlerwhere tr is the rising time indicating the center of rising period ts: Note that thefalling region of square pulse is symmetric to the rising one.

91

Figure 6.7: Three-dimensional line plots of converted signal and input idler intensitiesin t (ns) and L (mm). Here we let t = � that is the delayed time in co-moving frame,and the parameters are the same as in Figure 6.6 (a).

Moreover, we show in Figure 6.7 of three dimensional plots of signal jE+s (z; t)j2 and

idler intensities jE+i (z; t)j2. A 100 ns input idler pulse is demonstrated in time-space

propagation which is converted to signal pulse at the output surface of the ensemble.

6.5 Discussion of Quantum Fluctuations

In this Section, we derive quantized Heisenberg-Langevin equations by adding corre-

sponding Langevin noises to the coupled equations. Similar to the results of Appendix

92

D, we have

@

@zE+s = �sE

+s + �sE

+i + fs; (6.19)

@

@zE+i = �iE

+s + �iE

+i + fi (6.20)

where signal and idler �elds (respectively, E+s and E+i ) are now quantized, and

Langevin noises (fs and fi) in linearized equations are

fs =iNg�scD

[(T03 +jaj2T13

+jbj2T02

) ~F12 + (�abT02

+�abT13

) ~F03

+ibT13(T03 +

jbj2 � jaj2T02

) ~F13 +i�aT02(�T03 +

jbj2 � jaj2T13

) ~F02] + ~Fs;

(6.21)

fi =iNg�icD

[(a

�b

T02+a

�b

T13) ~F12 + (T12 +

jaj2T02

+jbj2T13

) ~F03

+iaT13(�T12 +

jbj2 � jaj2T02

) ~F13 +i�aT02(T12 +

jbj2 � jaj2T13

) ~F02] + ~Fi

(6.22)

where various atomic and �eld Langevin noises ~Fy are associated with coupled equa-

tions of atomic operators �y and �eld operators E+s;i when y = s; i.

Note that various normal correlation functions of quantum Langevin noises haveD~F yi (t; z)

~Fj(t0; z0)E= L

N�(t� t0)�(z� z0)Dij in continuous limit. If ensemble average

is taken over the above �eld equations, with the property of Langevin noises thatD~F yi (t; z)

E=D~Fi(t0; z0)

E= 0; the �eld equations are reduced to c-number ones where

�uctuations due to Langevin noises do not matter. For calculation of normally-

ordered operators, semi-classical approximation is valid even in quantum regime. We

demonstrate in the following and include Langevin noises in derivation of solutions

of �eld operators.

The set of equations can be written as

@

@zx(z) = Ax+ f (6.23)

93

where x(z) =

264E+s (z)E+i (z)

375 and f =264fs(z)fi(z)

375 :Consider a similarity transformation S that S�1AS = � and y = S�1x, then we

have

@

@zy = �y + S�1f; (6.24)

y(z) = e�(z�z0)y(z0) +

Z z

z0

dz0e�(z�z0)S�1f(z0): (6.25)

With the boundary condition x1(0) and x2(0), we have

x(z) = Se�zS�1x(0) +

Z z

0

dz0Se�(z�z0)S�1f(z0): (6.26)

So we have correspondingly

A =

0B@ �s �s

�i �i

1CA ; � =

0B@ (�i + �s)=2 + w 0

0 (�i + �s)=2� w

1CAS =

0B@ q + w �s

�i �q � w

1CA ; S�1 =1

2w(w + q)

0B@ q + w �s

�i �q � w

1CA (6.27)

where w �pq2 + �s�i, and q � (��i + �s)=2.

The solutions of �elds from down conversion for example are

94

264E+s (L)E+i (L)

375 =

264q + w �s

�i �q � w

375264e[(�i+�s)=2+w]L 0

0 e[(�i+�s)=2�w]L

375 1

2w(w + q)�

264q + w �s

�i �q � w

375264 0

E+i (0)

375+

Z L

0

dz0S

264e[(�i+�s)=2+w](L�z0) 0

0 e[(�i+�s)=2�w](L�z0)

375S�1f(z0);=

264 �s2w(e[(�i+�s)=2+w]L � e[(�i+�s)=2�w]L)

12w(w+q)

[�s�ie[(�i+�s)=2+w]L + (q + w)2e[(�i+�s)=2�w]L]

375E+i (0)+

Z L

0

dz0S

264e[(�i+�s)=2+w](L�z0) 0

0 e[(�i+�s)=2�w](L�z0)

375S�1f(z0);=

264p�dei arg(p�d)pTde

i arg(pTd)

375E+i (0) + Z L

0

dz0

264�s(z0)fs(z0) + �i(z0)fi(z

0)

�s(z0)fs(z

0) + � i(z0)fi(z

0)

375 (6.28)where �d and Td are conversion e¢ ciency and transmission for down conversion as

derived in Section II, arg represents the argument for complex numbers, and we see

that extra terms involve Langevin noises. �s;i and �s;i are de�ned in the below,

�s =1

2w(w + q)[(w + q)2e[(�i+�s)=2+w](L�z

0) + �s�ie[(�i+�s)=2�w](L�z0)]; (6.29)

�i =�s2w[e[(�i+�s)=2+w](L�z

0) � e[(�i+�s)=2�w](L�z0)]; (6.30)

�s =�i2w[e[(�i+�s)=2+w](L�z

0) � e[(�i+�s)=2�w](L�z0)]; (6.31)

� i =1

2w(w + q)[�s�ie

[(�i+�s)=2+w](L�z0) + (w + q)2e[(�i+�s)=2�w](L�z0)]: (6.32)

The down conversion e¢ ciency �down is de�ned as

95

�down =hE�s (L)E+s (L)iE�i (0)E

+i (0)

�= �d +

1E�i (0)E

+i (0)

� ��Z L

0

dz0[��s(z0)f ys (z

0) + �i(z0)f yi (z

0)]

Z L

0

dz00[�s(z00)fs(z

00) + �i(z00)fi(z

00)]

�(6.33)

in which expectation value of normally-ordered operators involves contributions from

semi-classical treatment and normally-ordered noise correlation functions. The rele-

vant normally-ordered quantum di¤usion coe¢ cients Dij from Einstein�s relation are

(note that Dij = Dyji)

(i) D12;12 = 01 h~�22i � 01~�22;s = 0;

D12;03 = D12;02 = 0;

D12;13 = 01 h~�23i � 01~�23;s = 0;

D12;s = D12;i = 0; (6.34)

(ii) D03;03 = 32 h~�22i � 32~�22;s = 0;

D03;02 = D03;13 = 0;

D03;s = D03;i = 0; (6.35)

(iii) D02;02 = D02;13 = 0;

D02;s = D02;i = 0; (6.36)

(iV) D13;13 = 01 h~�33i+ 32 h~�22i � 01~�33;s + 32~�22;s = 0;

D13;s = D13;i = 0; (6.37)

(V) Ds;s = Ds;i = 0; (6.38)

(Vi) Di;i = 0; (6.39)

where we have approximated various nonvanishing quantum di¤usion coe¢ cients by

96

zeroth order properties of atomic operators (the steady state solutions). The above

normally-ordered correlation functions give zero contributions in the linearized equa-

tions of motion, so c-number Langevin equation is su¢ cient to derive the conversion

e¢ ciency. The normally-ordered noise correlations are zero because the population

(~�22;s; ~�33;s) and coherence (~�23;s) properties are zero for the atomic level driven by

pump-b. The linearized �eld equations in diamond structure have similar noise prop-

erties to � system in which most atoms are on the ground state, and Langevin noise

can be neglected if normally-ordered quantities, say storage e¢ ciency, are considered

[108, 109].

6.6 Conclusion

We have studied light frequency conversion in an atomic ensemble with a diamond

con�guration of atomic levels such as 87Rb. The motivation stems from the need to

e¢ ciently convert light resonant with ground state transitions (storable in the sense of

quantum memories) to and from the telecom wavelength band for low-loss quantum

network communication. The optically thick atomic sample is driven by two strong

co-propagating pump �elds, and a probe idler or signal �eld depending on whether

we consider down- or up-conversion. Parametric equations for the probe �elds are

derived and used to compute conversion e¢ ciencies. They can be understood by

dressed-state picture where we can visualize four absorption lines due to two strong

pump lasers and thus three parametric coupling windows are created. There are two

major contributions to the conversion e¢ ciency, which are related to atomic popula-

tions and coherences in the lower arm of the diamond level driven by laser pump-a.

When this transition is saturated by a large pump Rabi frequency or when the co-

herence dominates due to a large pump-b Rabi frequency in the upper transition, the

cross-coupling coe¢ cients and hence the conversion e¢ ciencies are equal.

97

By performing a global parameter search we �nd conditions of pump Rabi frequen-

cies, detunings, and signal/idler input frequency to maximize the conversion e¢ ciency

as a function of optical depth of the ensemble. Only in the limit of very large optical

depth does the maximum e¢ ciency approach the ideal strong coupling result [113].

Under conditions routinely obtained in cold, non-degenerate rubidium gas, with opd

' 100 � 200, optimal conversion e¢ ciencies of the order 80% to 90% are predicted.

Numerical solution of the Maxwell-Bloch equations con�rms the solution of the para-

metric equations in the limit of long pulse duration, and indicates that for shorter

pulses, pump pulse induced modulation may reduce the conversion e¢ ciency.

98

CHAPTER VII

CONCLUSION

We provide a theoretical study of light-matter interactions in cascade and diamond

type atomic ensembles. A correlated two-photon (telecom signal-infrared idler) state

vector is derived in the long time limit within the adiabatic approximation. The

second-order correlation function is calculated, and shows a superradiant time scale

in the infrared idler emission. The entanglement in frequency space for such a two-

photon state is analyzed by Schmidt decomposition. We are able to derive the mode

functions and investigate the in�uence of pump pulse duration and superradiant decay

rate that depends on optical density and ensemble geometry.

To investigate multiple atomic excitations on the correlated emission from the

atomic cascade transitions, we use the coherent state positive-P representation and

derive an equivalent Ito type stochastic di¤erential equation (SDE). The equations

are solved numerically by a stable and convergent semi-implicit di¤erence method,

while the counter-propagating spatial evolution is solved by implementing the shoot-

ing method. We �nd an enhanced characteristic time scale for idler emission in

the second-order correlation functions, consistent with the superradiance timescales

predicted by the analytical method in Chapter 3, and observed experimentally.

In Chapter 5, the correlated two-photon state derived in Chapter 3 is used to

investigate the spectral e¤ects on DLCZ protocols involving entanglement generation,

swapping, and quantum teleportation. We analyze the performance of the protocol

using, photon-number resolving and non-resolving photon detectors. We �nd that

a more genuine and high �delity protocol requires a source with reduced frequency

space entanglement.

99

In Chapter 6, we present the analytical results on the e¢ ciency of light-frequency

conversion in a diamond atomic con�guration. We �nd the optimum e¢ ciency as a

function of optical density. We �nd the maximum conversion e¢ ciency by studying

parametric coupling windows that are created by strong pump �elds, and provide

numerical solutions for the pulse conversion.

100

APPENDIX A

DERIVATION OF A SCHRÖDINGER WAVE EQUATION

FOR SPONTANEOUS EMISSIONS FROM A CASCADE

TYPE ATOMIC ENSEMBLE

In this appendix, we derive the Hamiltonian for the cascade emission (signal-idler)

from a four-level atomic ensemble. We use Schrödinger�s equation to study the

correlated two-photon state from a two-photon laser excitation. Apart from the ro-

tating wave approximation, non-rotating wave probability amplitudes are introduced

to take into account the proper frequency shift. The adiabatic approximation on

laser-excited states is used to simplify the atomic dynamics and solve for the signal-

idler probability amplitude.

A.1 Hamiltonian and Equation of Motion

Consider an ensemble of N four-level atoms interacting with two classical �elds and

spontaneously emitted signal and idler photons as shown in Figure 3.3. These iden-

tical atoms distribute randomly with a uniform density. Use dipole approximation

of light-matter interactions, �~d � ~E where ~E is classical or quantum electric �eld, and

include non-rotating wave approximation (RWA) terms in the interaction of quantum

�elds, the Hamiltonian in interaction picture is

101

VI(t) =

�~�1

NX�=1

j1i�h1j � ~�2

NX�=1

j2i�h2j �~2

NX�=1

haj1i�h0jei

~ka�~r� + bj2i�h1jei~kb�~r�

+h:c:i� i~

Xks;�s

gks

h~�ks;�s aks;�se

�i!kst+i~ks�~r� �~��ks;�s ayks;�s

ei!kst�i~ks�~r�

i� ds

NX�=1h

j2i�h3jei(!23+�2)t + j3i�h2je�i(!23+�2)ti� i~

Xki;�i

gki

h~�ki;�i aki;�ie

�i!kit+i~ki�~r� �

~��ki;�i ayki;�i

ei!kit�i~ki�~r�

i� di

NX�=1

�j3i�h0jei!3t + j0i�h3je�i!3t

�; (A.1)

where the time dependence of laser frequency is absorbed into interaction terms of

signal and idler �elds. Single photon detuning �1 = !a � !1, two-photon detuning

�2 = !a + !b � !2; and !23 = !2 � !3. Rabi frequencies are a � (1jjdjj0)E(ka)=~,

b � (2jjdjj1)E(kb)=~; and coupling coe¢ cients are gks � (3jjdjj2)E(ks)=~, gki �

(0jjdjj3)E(ki)=~. The double matrix element of the dipole moment is independent of

the hyper�ne structure, and E(k) =q

~kc2�0V

. Polarizations of signal and idler �elds

are ~�ks;�s , ~�ki;�i, and the unit direction of dipole operators are ds, di.

In the limit of large detuned and weak driving �elds, �1 �pNa2

; that is discussed

in Chapter 3.2, we consider only single excitations and ignore the spontaneous decay

during the excitation process. The state function can be written as

j (t)i =

E(t)j0; vaci+NX�=1

A�(t)j1�; vaci+NX�=1

B�(t)j2�; vaci+NX�=1

Xks;�s

C�s (t)j3�; 1~ks;�si

+Xks;�ski;�i

Ds;i(t)j0; 1~ks;�s ; 1~ki;�ii+NX�=1

Xki;�i

C�i (t)j3�; 1~ki;�ii+NX�=1

C�(t)j3�i| {z }+

NX�=1

Xks;�s

B�s (t)j2�; 1~ks;�si+

X�<�

NX�=1

Xks;�ski;�i

C��s;i (t)j3�; 3� ; 1~ks;�s ; 1~ki;�ii

| {z }(A.2)

102

where jvaci is the photon vacuum state, s � (ks; �s), i � (ki; �i), jm�i � jm�ij0iN�1� 6=� ,

m = 1; 2; 3 and j3�; 3�i � j3�ij3�ij0iN�2�6=�;� . The probability amplitudes coupled

from rotating wave terms in the Hamiltonian are E(t); A�(t); B�(t); C�s (t); Ds;i(t);

which indicate the complete cycle of single excitation process from the ground state,

intermediate, upper excited state, intermediate excited state with emission of a signal

photon, and the ground state with the signal-idler emission. Note that the states

underlined are coupled through non-RWA terms that describe a transition from upper

excited state to intermediate one by absorbing a photon for B�s (t) and C

�(t) and a

transition from the ground state to the intermediate one by emitting a photon for

C�i (t) and E(t) or C�s (t) and C��s;i (t). Apply the Schrödinger equation i~ @@t j (t)i =

VI(t)j (t)i, and we have the coupled equations of motion,

103

i _E = ��a

2

X�

e�i~ka�~r�A� � i

Xi;�

gi(~�i � d�i )ei~ki�~r�e�i(!ki+!3)tC�i| {z };

i _C�i = ig�i (~��i � di)e�i

~ki�~r�ei(!i+!3)tE| {z };i _A� = ��1A� �

a2ei~ka�~r�E �

�b

2e�i

~kb�~r�B�;

i _B� = ��2B� �b2ei~kb�~r�A� � i

Xs

gs(~�s � d�s)ei~ks�~r�e�i(!s�!23��2)tC�s ;

i _C�s = ig�s(��s� ds)e�i

~ks�~r�ei(!ks�!23��2)tB� � iXi

gi(~�i � d�i )ei~ki�~r�e�i(!i�!3)tDs;i

� iXi

gi(~�i � d�i )ei(!i+!3)thX�<�

C��s;i ei~ki�~r� +

X�>�

ei~ki�~r�C��s;i

i| {z };

i _C��s;i = ig�i (��i� di)ei(!i+!3)t

he�i

~ki�~r�C�s + C�s e�i~ki�~r�

i| {z }

��<�

;

i _Ds;i = ig�i (��i� di)

X�

e�i~ki�~r�ei(!ki�!3)tC�s ;

i _C� = � iXs

gs(~�s � d�s)ei~ks�~r�e�i(!s+!23+�2)tB�

s| {z };i _B�

s = ig�s(��s� ds)e�i

~ks�~r�ei(!ks+!23+�2)tC�| {z } : (A.3)

The Lamb shift for the atomic transition j3i ! j0i with the optical frequency !3

and spontaneous decay rate � isR10d! �

2�[P.V.(! � !3)

�1�P.V.(! + !3)�1] that can

be identi�ed partly within the substitution of these non-RWA terms. We substitute

C�i into E , C��s;i into C

�s , and B

�s into C�, and they are

_E =i�a2

X�

e�i~ka�~r�A� �N

Xi

jgij2j(~�i � d�i )j2Z t

0

dt0ei(!i+!3)(t0�t)E(t0)

=i�a2

X�

e�i~ka�~r�A� �NE

Idi[1� (ki � di)2]

V

(2�)3

Z 1

0

dkik2i

~!i2�0V

jdij2~2

�[��(!i + !3)� iP.V.(!i + !3)�1]

=i�a2

X�

e�i~ka�~r�A� + iNE

Z 1

0

d!i�i2�P.V.(!i + !3)

�1; (A.4)

104

_C�s = g�s(��s� ds)e�i

~ks�~r�ei(!ks�!23��2)tB� �Xi


�Xi

jgij2j(~�i � d�i )j2Z t

0

dt0ei(!i+!3)(t0�t)E(t0)

nX�<�

hC�s (t

0) + ei~ki�(~r��~r�)C�s (t

0)i

+X�>�

hei~ki�(~r��~r�)C�s (t

0) + C�s (t0)io

= g�s(��s� ds)e�i

~ks�~r�ei(!ks�!23��2)tB� �Xi


+i(N � 1)C�sZ 1

0

d!i�i2�P.V.(!i + !3)

�1 �Xi

jgij2j(~�i � d�i )j2 �Z t

0

dt0ei(!i+!3)(t0�t)

X� 6=�

ei~ki�(~r��~r�)C�s (t

0); (A.5)

where we have used the symmetric property of +��(�) = +��(�) �P

i jgij2j(~�i �

d�i )j2R t0dt0ei(!i+!3)(t

0�t)ei~ki�(~r��~r�) [21]. The spontaneous decay rate for the idler tran-

sition is �i � jdij2!3i3�~�0c3 [30, 116], and the same thing for signal transition �s �

jdsj2!3s3�~�0c3

that

_C� = iC�

Z 1

0

d!s�s2�P.V.(!s + !23 +�2)

�1: (A.6)

It is now clear the contribution from non-RWA terms to the Lamb shift of the

idler transition resides in _E and _C�s , which are proportional to N and N � 1. The

di¤erence of the level shifts then gives rise to �R10d! �

2�P.V.(!+!3)�1, and the other

part can be derived from substitutions of RWA terms. The signal transition has the

same e¤ect as shown in _C�. The frequency shift due to dipole-dipole interaction also

appeared in _C�s that has the contribution of interactions from other atoms. There

also will be contributions from RWA terms, and we will show the complete expression

for collective decay rate and frequency shift.

De�ne Cs;qi =P

�C�s e�i~qi�~r�, substitute Ds;i into C�s ; and we have

105

_Cs;qi = g�s(��s� ds)

X�

e�i(~ks+~qi)�~r�ei(!ks�!23��2)tB� �

Xi

jgij2j(~�i � d�i )j2 �

X�

ei(~ki�~qi)�~r�

Z t

0

dt0ei(!i�!3)(t0�t)Cs;ki(t

0)�Xi

jgij2j(~�i � d�i )j2 �Z t

0

dt0ei(!i+!3)(t0�t)hX

�

ei(~ki�~qi)�~r�Cs;ki � Cs;qi

i+i(N � 1)Cs;qi

Z 1

0

d!i�i2�P.V.(!i + !3)

�1

= g�s(��s� ds)

X�

e�i(~ks+~qi)�~r�ei(!ks�!23��2)tB� �

3

8�

Idi[1� (ki � di)2]

�32

�X�


��j~kij=k3

Cs;k3ki + i3

8�

Idi[1� (ki � di)2]

Z 1

0

d!i�i2�h

P.V.(!i � !3)�1 + P.V.(!i + !3)

�1ihX

�

ei(~ki�~qi)�~r�Cs;ki � Cs;qi

i+iCs;qi

Z 1

0

d!i�i2�P.V.(!i � !3)

�1

+i(N � 1)Cs;qiZ 1

0

d!i�i2�P.V.(!i + !3)

�1: (A.7)

Renormalize the Lamb shift (last two lines in the above) and useP

qiCs;qie

i~qi�~r� =

NC�s then we have

106


X�

e�i(~ks+~qi)�~r�ei(!ks�!23��2)tB� �

3

8�

Idi[1� (ki � di)2]

�32�

X�


X�

e�i~ki�~r� 1

N

Xq0i

ei~q0i�~r�Cs;q0i

��j~kij=k3

+ i3

8�

Idi[1� (ki � di)2]�Z 1

0

d!i�i2�

hP.V.(!i � !3)

�1 + P.V.(!i + !3)�1ihX

�


X� 6=�

e�i~ki�~r�C�s

i= g�s(�

�s� ds)

X�

e�i(~ks+~qi)�~r�ei(!ks�!23��2)tB� �

3

8�

Idi[1� (ki � di)2]

�32

� 1N

X�


Xq0i

X�

ei(~q0i�~ki)�~r�Cs;q0i

��j~kij=k3

+ i3

8�

Idi[1� (ki � di)2]

�Z 1

0

d!i�i2�

hP.V.(!i � !3)

�1 + P.V.(!i + !3)�1i 1N

X�


Xq0ihX

�

ei(~q0i�~ki)�~r� � ei(~q

0i�~ki)�~r�

iCs;q0i : (A.8)

Due to the summation of exponential factors from the above, the coupling from

the other modes q0i is signi�cant only when q0i = ki = qi, so �nally we have


X�

e�i(~ks+~qi)�~r�ei(!ks�!23��2)tB� �

�32(N ��+ 1)Cs;qi

+i�!iCs;qi (A.9)

where the collective decay rate is [22]

�32(N ��+ 1) � �3

2

3

8�

Idi[1� (ki � di)2]

1

N

X�;�

ei(~ki�~qi)�(~r��~r�); (A.10)

and the collective frequency shift expressed in terms of the continuous integral over

a frequency space is

107

�!i �Z 1

0

d!i�i2�

hP.V.(!i � !3)

�1 + P.V.(!i + !3)�1iN ��(ki);

=

Z 1

0

d!i�i2�

hP.V.(!i � !3)

�1 + P.V.(!i + !3)�1i 1N

X�;� 6=�

ei(~ki�~qi)�(~r��~r�):

(A.11)

The geometrical constant �� for a cylindrical ensemble (of height h and radius a)

is

��(k3) =6(N � 1)NA2H2

Z 1

�1

dx(1 + x2)

(1� x)2(1� x2)sin2[

1

2H(1� x)]J21 [A(1� x2)1=2] (A.12)

whereH = k3h andA = k3a are dimensionless length scales, and circular polarizations

are considered [22]. J1 is the Bessel function of the �rst kind.

The alternative way to express the collective decay rate and shift is [21]

�N32

=�32(N ��+ 1) � �3

2

1

N

X�;�

F��(k3r��)e�i~qi�(~r��~r�); (A.13)

�!i = ��32

2

N

X�;� 6=�

G��(k3r��)e�i~qi�(~r��~r�)

=�3Nk33

P.V.Z 1

�1

dk

2�

k3

k � k3

X�;� 6=�

F��(kr��)e�i~qi�(~r��~r�): (A.14)

where

F��(�) =3

2f[1� (p � r��)2]

sin �

�+ [1� 3(p � r��)2](

cos �

�2� sin �

�3)g;

G��(�) =3

4f�[1� (p � r��)2]

cos �

�+ [1� 3(p � r��)2](

sin �

�2+cos �

�3)g; (A.15)

and note that � = k3r��.

A.2 Adiabatic Approximation

Under the conditions of large detuned laser excitations, we may use the adiabatic

approximation to eliminate the laser-excited states and solve for the signal-idler prob-

ability amplitude. Before proceeding to the adiabatic approximation, we solve Cs;qi

�rst and substitute it to solve B�.

108

Cs;qi(t) = g�s(��s� ds)

X�

e�i(~ks+~qi)�~r�

Z t

0

dt0ei(!s�!23��2)t0e(�

�N32+i�!i)(t�t0)B�(t

0): (A.16)

Let Bka+kb �P

� e�i(~ka+~kb)�~r�B� and Aka �

P� e

�i~ka�~r�A�, we have

_Bka+kb = i�2Bka+kb + ib2Aka �

Xks;�s

jgsj2j��ks;�s � dsj2

Z t

0

dt0ei(!ks�!23��2)(t0�t)

e(��N32+i�!i)(t�t0)Bka+kb(t

0)

= i�2Bka+kb + ib2Aka �

�22Bka+kb + iBka+kb

Z 1

0

d!s�s2��

P.V.(!s � !23 ��2)�1 (A.17)

where the Weisskopf-Wigner approach is used to derive the decay rate for the signal

transition, and in conjunction with the result of _C�; the Lamb shift is also derived as

the di¤erence of level shifts thatR10d!s

�s2�[P.V.(!s � !23 � �2)

�1�P.V.(!s + !23 +

�2)�1]. We then renormalize it and apply the adiabatic approximation.

When the detunings are large enough that

j�1j; j�2j �jaj2;jbj2;�22:

We can solve the coupled equations of motion by adiabatically eliminating the

intermediate and upper excited states in the excitation process. The adiabatic ap-

proximation requires that the driving pulses are smoothly turned on, and we will

show under what condition of the pulses that the approximation is valid.

First we use integration by parts to solve the probability amplitudes in the adi-

abatic approximation (zeroth order) and their �rst-order correction. Note that we

allow time-varying Rabi frequencies.

109

Aka(t) = ei�1th i2

Z t

�1e�i�1t

0a(t

0)E(t0)dt0 + i

2

Z t

�1e�i�1t

0�b(t

0)Bka+kb(t0)dt0

i� �Na(t)E(t)

2�1

� �b(t)Bka+kb(t)

2�1

+i

2�21

d

dt

�a(t)E(t)

�+

i

2�21

d

dt

��b(t)Bka+kb(t)

�; (A.18)

Bka+kb(t) = ei(�2+i�2=2)th i2

Z t

�1e�i(�2+i�2=2)t

0b(t

0)Aka(t0)dt0

i� � b(t)Aka(t)

2(�2 + i�2=2)+i ddt

�b(t)Aka(t)

�2(�2 + i�2=2)2

(A.19)

where higher order terms involving a second derivative of the �elds are neglected due

to their feature of slow variation. The initial conditions are used in the below,

Bka+kb(�1) = Aka(�1) = 0;d

dt0

�a(t

0)E(t0)��1

= 0;

d

dt0

��b(t

0)Bka+kb(t0)��1

=d

dt0

�b(t

0)Aka(t0)��1

= 0:

With conditions in the following (i) to (iii),

(i)

��ddt

�a(t)E(t)

��1a(t)E(t)

�� 1; (A.20)

(ii)

��ddt

��b(t)Bka+kb(t)

��1�b(t)Bka+kb(t)

�� 1; (A.21)

(iii)

��ddt

�b(t)Aka(t)

�(�2 + i�2=2)b(t)Aka(t)

�� 1; (A.22)

we can derive Aka(t), Bka+kb(t); and E(t) in the adiabatic approximation,

Aka(t) =�Na(t)E(t)

2�1

1� jb(t)j24�1(�2+i�2=2)

� �Na(t)2�1

E(t); (A.23)

E(t) = e� iN4�1

R t�1 ja(t0)j2dt0 � 1� iN

4�1

Z t

�1ja(t0)j2dt0; (A.24)

Bka+kb(t) =

Na(t)b(t)4�1�2

E(t)1� jb(t)j2

4�1(�2+i�2=2)

� Na(t)b(t)

4�1�2

� Nb(t); (A.25)

110

where the probability amplitude of the �rst excited state follows the �rst laser �eld,

and the upper excited state follows the products of two laser �elds. The AC Stark

shift is present in the ground state that can be ignored if �1 � NR t�1 ja(t

0)j2dt0=4.

This condition is also required for the assumption of single excitations states we

consider.

Finally, we have the probability amplitudes associated with the signal Cs;ki(t) and

signal-idler photons Ds;i(t),

Cs;ki(t)

= g�s(��s� ds)

Z t

0

dt0ei(!s�!23��2)t0e(�

�N32+i�!i)(t�t0)

X�

ei�~k�~r�e�i(

~ka+~kb)�~r�B�(t0)

= g�s(��s� ds)

1

N

X�

ei�~k�~r�

Z t

0

dt0ei(!s�!23��2)t0e(�

�N32+i�!i)(t�t0)Bka+kb(t

0); (A.26)

Ds;i(t) = g�i g�s(�

�ki;�i

� di)(��ks;�s � ds)X�

ei�~k�~r�

Z t

0

Z t0

0

dt00dt0e(��N32+i�!i)(t

0�t00)

ei(!i�!3)t0ei(!s�!23��2)t

00b(t00): (A.27)

Note that e�i(~ka+~kb)�~r�B�(t0) does not depend on the atomic index � under the

adiabatic approximation, and �~k = ~ka + ~kb � ~ks � ~ki is the phase mismatch.

The above expressions are the main results of this Appendix and we proceed to

investigate their properties when Gaussian pump pulses are used in Chapter 3.

111

APPENDIX B

DERIVATION OF A C-NUMBER LANGEVIN EQUATION

FOR THE CASCADE EMISSION

In this appendix, we show the details in the derivations of c-number Langevin equa-

tions that are the foundation for numerical approaches of the cascade emission in

Chapter 4. First we describe how to quantize the free electromagnetic �eld [29],

and we formulate the Fokker-Planck equation for our system using the positive P-

representation. We derive the Fokker-Planck equations by characteristic functions

[27], and the corresponding c-number Langevin equations are derived. The noise

correlations are found from the di¤usion coe¢ cients in Fokker-Planck equations.

B.1 Quantized Electromagnetic Field

To describe the propagating quantum �elds in one dimension, we take the approach

of the reference [29]. Before proceeding, we specify the positive frequency of a free

propagating �eld operator in the discrete space,

E+(~x) = iXk;�

r~!k2�0V

ak;�~�k;�ei~k�~x (B.1)

where ~�k;� and ~��(~k) specify polarizations of the �eld, and the interchange of discrete

and continuous space has relation,

Xk

! V

(2�)3

Zd3k ; ak;� !

sV

(2�)3

Zd3ka�(~k);

where creation and annihilation operators satisfy commutation relations,

[ak;�; ayk0;�0 ] = ��;�0�k;k0 : (B.2)

113

For the purpose of describing one-dimensional propagating �eld (paraxial approxi-

mation), we discretize the space along the propagation (z) and denote ~r as the vectors

on the cross section. We then have

E+(z; ~r) = i

MXn=�M

r~!s;n2�0

ei(ks+kn)z1pV

X�

Xkn?

ei~kn?�~r~�kn;�ak;� (B.3)

kn =2�n

L; !s;n = !s + knc ; !s = ksc ; n = �M; :::;M

where L is the length of propagation that is equally split into 2M + 1 elements,

and the center of the interval is z = zm = mL2M+1

with m = �M; :::;M . ks is the

central longitudinal mode of the �eld. Note that the polarization ~�k;� with paraxial

approximation has k � kn.

The next step is to characterize the transverse mode of propagating �eld, and we

introduce a set of orthonormal transverse mode functions (fi;kn?) thatP

kn?f �i;kn?fj;kn? =

�ij. A longitudinal annihilation operator is de�ned as

cn;i;� �Xkn?

f �i;kn? ak;�;

which also satis�es commutation relations [cn;i;�; cyn0;j;�0 ] = �nn0�ij��0. We can sub-

stitute ak;� =P

i cn;i;�fi;kn? that

E+(z; ~r) = i

MXn=�M

r~!s;n2�0

ei(ks+kn)z1pV

X�

Xkn?

ei~kn?�~r~�kn;�fi;kn? cn;i;�: (B.4)

Let the spatial transverse mode function

ui(~r) �ipV

Xkn?

ei~kn?�~rfi;kn? ; where

Zd2rdzu�iui = 1

and we have

E+(z; ~r) =

MXn=�M

r~!s;n2�0

ei(ks+kn)zXi;�

~�kn;�cn;i;�ui(~r): (B.5)

An approximation of a single transverse mode can be applied if only single mode

is collected for the experiment, and a �at transverse mode can be assumed (ui = 1pV)

114

if the collected mode has a narrower spatial bandwidth than the mode function.

Finally, we have

E+(z; ~r) =MX

n=�M

r~ws;n2�0V

ei(ks+kn)zX�

~�kn;�cn;�: (B.6)

For a demonstration of deriving an interaction Hamiltonian and Maxwell-Bloch

equations, we use a two-state system (j0i and j1i), and the polarization is not con-

cerned here. The free �eld and interaction Hamiltonian (interacting with atomic

ensemble with N atoms) is

H0 =Xn

~!s;ncyncn + ~!1j1ih1j; (B.7)

V = �NX�

~d� � ~E(r�) ="�~g

X�

MXn;m=�M

��;mycnei(ks+kn)zm + h:c:

#RWA

;(B.8)

g � d

~

r~!s2�0V

; ~d� � �� + ��;y; ~E � E+ + E� (B.9)

whereP

� sums overN

2M+1atoms in the cross sections and the index m on raising and

lowering atomic operators � characterizes the position of the atoms. The rotating

wave approximation (RWA) is made in the interaction Hamiltonian and slowly varying

coupling constant g is taken out of the discrete mode sum and is assigned a central

frequency, which is the narrow band assumption for the �eld.

Now we introduce a new operator

bl =1p

2M + 1

MXn=�M

cneiknzl ; l = �M; :::;M; (B.10)

which satis�es commutation relations [bl; byl0 ] = �ll0 ; and the Hamiltonian can be re-

expressed as

H0 = ~!sXl

byl bl + ~Xll0

!ll0 byl bl0 + ~!1j1ih1j; (B.11)

V = �~gX�;l

p2M + 1��;lyble

ikszl + h:c:: (B.12)

115

The Heisenberg equation of slowly varying �eld operators (~bl � blei!st) is

_~bl = �iXl0

!ll0~bl0 + ig�X�

p2M + 1��;le�ikszl+i!st ; !ll0 �

Xn

knc

2M + 1eikn(zl�zl0 );

(B.13)

and we may use the limit of M !1 that

zm =mL

2M + 1! z ;

p2M + 1 ~bl ! ~E+s (z; t) ; �i

Xl0

!ll0~bl0p2M + 1! �c d

dz~E+s (z; t)

(B.14)

where the derivative can be shown from

�iXl0

!ll0~bl0 = �iXl0

Xn

knc

2M + 1eikn(zl�zl0 )~bl0 = �

c

2M + 1

Xl0

Xn

d

dzl[eikn(zl�zl0 )]~bl0

= �cXl0

d

dzl�ll0~bl0 = �c

d

dzl~bl: (B.15)

In the end, we have

(@

@t+ c

@

@z) ~E+s (z; t) = ig�limM!1(2M + 1)

X�

��;le�ikszl+i!st��zl!z

; (B.16)

and use the limit,

limM!12M + 1

L�ll0 ! �(z � z0);

then we have (de�ne slowly varying atomic operators ~��;l = ��;le�ikszl+i!st)

limM!12M + 1

L

NX�=1

~��;l�z�;zlL��zl!z

!NX�=1

~��(z� � z)L =N

Nz

NzX�

~��: (B.17)

The �eld propagation equation in Maxwell-Bloch equations becomes

(@

@t+ c

@

@z) ~E+s (z; t) = ig�

NX�=1

~��(z� � z)L = ig�N

Nz

NzX�=1

~��: (B.18)

B.2 Positive P-representation

The phase space methods [32] that mainly include P-, Q-, and Wigner (W) rep-

resentations are techniques of using classical analogues to study quantum systems,

116

especially harmonic oscillators. The eigenstate of harmonic oscillator is a coher-

ent state that provides the basis expansion to construct various representations. P

and Q-representation are associated respectively with evaluations of normal and anti-

normal order correlations of creation and destruction operators. W-representation

is invented for the purpose of describing symmetrically ordered creation and destruc-

tion operators. Since P-representation describes normally ordered quantities that

are relevant in experiments, we are interested in investigating one class of generalized

P-representations, the positive P-representation that has semi-de�nite property in the

di¤usion process, which is important in describing quantum noise systems.

Postive-P representation [35, 94] is an extension to Glauber-Sudarshan P-representation

that uses coherent state (j�i) as a basis expansion of density operator �. In terms

of diagonal coherent states with a quasi-probability distribution, P (�; ��), a density

operator in P-representation is

� =

ZD

j�ih�jP (�; ��)d2�; (B.19)

where D represents the integration domain. The normalization condition of �; which

is Tr{�}= 1; indicates the normalization for P as well,RDP (�; ��)d2� = 1.

Positive P-representation uses a non-diagonal coherent state expansion and the

density operator can be expressed as

� =

ZD

�(�; �)P (�; �)d�(�; �); (B.20)

where

d�(�; �) = d2�d2� and �(�; �) =j�ih��jh��j�i ; (B.21)

and h��j�i in non-diagonal projection operators, �(�; �); makes sure of the normal-

ization condition in distribution function, P (�; �):

Any normally ordered observable can be deduced from the distribution function

P (�; �) that

117

h(ay)mani =ZD

�m�nP (�; �)d�(�; �): (B.22)

A characteristic function �p(��; ��) (Fourier-transformed distribution function in

Glauber-Sudarshan P-representation but now is extended into a larger dimension)

can help formulate distribution function, which is

�p(��; ��) =

ZD

ei��+i��P (�; �)d�(�; �): (B.23)

It is calculated from a normally ordered exponential operator E(�);

�p(��; ��) = Tr{�E(�)}, E(�) = ei��ayei��a: (B.24)

Then a Fokker-Planck equation can be derived from the time derivative of char-

acteristic function,

@�p@t

=@

@tTr{�E(�)}=Tr{

@�

@tE(�)} (B.25)

by Liouville equations,@�

@t=1

i~[H; �]: (B.26)

In laser theory [27], a P-representation method is extended to describe atomic and

atom-�eld interaction systems. When a large number of atoms is considered, which

is indeed the case of the actual laser, a macroscopic variable can be de�ned. Then

a generalized Fokker-Planck equation can be derived from characteristic functions

by neglecting higher order terms that are proportional to the inverse of number of

atoms. It is the similar to our case when we solve light-matter interactions in an

atomic ensemble that the large number cuts o¤the higher order terms in characteristic

functions, which we will demonstrate in the next subsection.

118

B.2.1 Hamiltonian

The Hamiltonian is in Schrödinger picture, and we separate it into two parts where

H0 is the free Hamiltonian of the atomic ensemble and one dimensional counter-

propagating signal and idler �elds, and HI is the interaction Hamiltonian of atoms

interacting with two classical �elds and two quantum �elds (signal and idler). Dipole

approximation of �~d � ~E and rotating wave approximation (RWA) have been made to

these interactions. Similar to the previous Appendix, we have

H = H0 +HI ,

H0 =3Xi=1

MXl=�M

~!i~�lii + ~!sMX

l=�M

ays;las;l + ~Xl;l0

!l0lays;las;l0

+~!iMX

l=�M

ayi;lai;l + ~Xl;l0

!ll0 ayi;lai;l0 , (B.27)

HI = �~MX

l=�M

ha(t)~�

ly01e

ikazl�i!at + b(t)~�ly12e

�ikbzl�i!bt + h:c:i

(B.28)

�~MX

l=�M

hgsp2M + 1~�ly32as;le

�ikszl + gip2M + 1~�ly03ai;le

ikizl + h:c:i(B.29)

where ~�lmn �PNz

� ��;lmn =PNz

� jmi�hnj��r�=zl

; a(t) � fa(t)d10E(ka)=(2~); and fa is

slow varying temporal pro�le without spatial dependence (ensemble scale much less

than pulse length). gs � d23E(ks)=~; E(k) =p~!=2�0V and zm = mL

2M+1; m =

�M; :::;M; and L is the length of propagation. Note that the Rabi frequency is half

of the standard de�nition.

The normally ordered exponential operator is chosen as

119

E(�) =Yl

El(�);

El(�) = ei�l19~�

ly01ei�

l18~�

ly12ei�

l17~�

ly02ei�

l16~�

ly13ei�

l15~�

ly03ei�

l14~�

ly32ei�

l13~�

l11ei�

l12~�

l22ei�

l11~�

l33ei�

l10~�

l32

ei�l9~�

l03ei�

l8~�

l13ei�

l7~�

l02ei�

l6~�

l12ei�

l5~�

l01ei�

l4ays;lei�

l3as;lei�

l2ayi;lei�

l1ai;l : (B.30)

Aside from the atom-�eld interaction @�@t= 1

i~ [H; �]; when dissipation from vacuum

is considered (single atomic decay), we can express them in terms of a Lindblad form

where we have for the four-level atomic system,

�@�@t

�sp

=MX

l=�M

NzX�

n 012[2��;l01 ��

�;ly01 � ��;ly01 �

�;l01 �� ;ly01 �

�;l01 ]

+ 122[2��;l12 ��

�;ly12 � ��;ly12 �

�;l12 �� ;ly12 �

�;l12 ]

+ 322[2��;l

32��;ly

32� ��;ly

32��;l32�� ;ly

32��;l32]

+ 032[2��;l03 ��

�;ly03 � ��;ly03 �

�;l03 �� ;ly03 �

�;l03 ]o: (B.31)

The characteristic functions can be calculated,

� = Tr{E(�)�}, (B.32)

@�

@t= Tr{E(�)

@�

@t} =

�@�@t

�A+�@�@t

�L

+�@�@t

�A�L +

�@�@t

�sp; (B.33)�@�

@t

�A= Tr{E(�)

1

i~[HA; �]},

�@�@t

�L= Tr{E(�)

1

i~[HL; �]},�@�

@t

�A�L = Tr{E(�)

1

i~[HA�L; �]},

�@�@t

�sp= Tr{E(�)

�@�@t

�sp} (B.34)

where H0 = HA + HL, HA is the atomic free evolution Hamiltonian, HL is the

Hamiltonian for laser �elds, and HA�L = HI : Now we continue to derive the time

derivative in each part of characteristic functions.

B.2.2 Characteristic function - atomic part

The atomic part in characteristic function is deduced from�@�@t

�A

120

�@�@t

�A= Tr{E(�)

�@�@t

�A},�@�

@t

�A=1

i~[HA; �]

=Xl

h� i!1(~�

l11�� ~�l11)� i!2(~�

l22�� ~�l22)� i!3(~�

l33�� ~�l33)

i(B.35)

so various components in�@�@t

�Aare

Tr{E(�)Xl

~�l11�} =Xl

Tr{E(�)~�l11�}

=Xl

[i�5@

@(i�5)� i�6

@

@(i�6)� i�8

@

@(i�8)+

@

@(i�13)]l�;

Tr{E(�)Xl

�~�l11} =Xl

[i�19@

@(i�19)� i�18

@

@(i�18)� i�16

@

@(i�16)+

@

@(i�13)]l�;

Tr{E(�)Xl

~�l22�} =Xl

[i�6@

@(i�6)+ i�7

@

@(i�7)+ i�10

@

@(i�10)+

@

@(i�12)]l�;

Tr{E(�)Xl

�~�l22} =Xl

[i�18@

@(i�18)+ i�17

@

@(i�17)+ i�14

@

@(i�14)+

@

@(i�12)]l�;

Tr{E(�)Xl

~�l33�} =Xl

[i�8@

@(i�8)+ i�9

@

@(i�9)� i�10

@

@(i�10)+

@

@(i�11)]l�;

Tr{E(�)Xl

�~�l33} =Xl

[i�16@

@(i�16)+ i�15

@

@(i�15)� i�14

@

@(i�14)+

@

@(i�11)]l�

(B.36)

where the subscript l on the bracket reminds us the derivatives inside the bracket

operate on lth component of the characteristic functions.

B.2.3 Characteristic function - �eld part

The �eld part in characteristic function is deduced from�@�@t

�L

121

�@�@t

�L= Tr{E(�)

�@�@t

�L},�@�

@t

�L=

1

i~[HL; �] =

Xl

h� i!s(a

ys;las;l�� ays;las;l)� i!i(a

yi;lai;l�� ayi;lai;l)

i+X

l;l0

h� i!l0l(a

ys;las;l0�� ays;las;l0)� i!ll0(a

yi;lai;l0�� ayi;lai;l0)

i; (B.37)

and various components in�@�@t

�Lare

Tr{E(�)Xl

ays;las;l�} =Xl

[@2

@(i�4)@(i�3)+ i�3

@

@(i�3)]l�;

Tr{E(�)Xl

�ays;las;l} =Xl

[@2

@(i�4)@(i�3)+ i�4

@

@(i�4)]l�;

Tr{E(�)Xl

ayi;lai;l�} =Xl

[@2

@(i�2)@(i�1)+ i�1

@

@(i�1)]l�;

Tr{E(�)Xl

�ayi;lai;l} =Xl

[@2

@(i�2)@(i�1)+ i�2

@

@(i�2)]l�; (B.38)

and

Tr{E(�)Xl;l0

!l0lays;las;l0�} =

Xl;l0

!l0l[@2

@(i�l4)@(i�l03 )+ i�l

0

3

@

@(i�l03 )]�;

Tr{E(�)Xl;l0

!l0l�ays;las;l0} =

Xl;l0

!l0l[@2

@(i�l4)@(i�l03 )+ i�l4

@

@(i�l4)]�;

Tr{E(�)Xl;l0

!ll0 ayi;lai;l0�} =

Xl;l0

!ll0 [@2

@(i�l2)@(i�l01 )+ i�l

0

1

@

@(i�l01 )]�;

Tr{E(�)Xl;l0

!ll0�ayi;lai;l0} =

Xl;l0

!ll0 [@2

@(i�l2)@(i�l01 )+ i�l2

@

@(i�l2)]�: (B.39)

B.2.4 Characteristic function - atom-�eld part

The atom-�eld interaction part in characteristic function is deduced from�@�@t

�A�L;

and we denote part (a) for the classical �eld interaction.

122

�@�@t

�(a)A�L = Tr{E(�)

�@�@t

�(a)A�L},�@�

@t

�(a)A�L =

1

i~

h� ~

MXl=�M

�a(t)~�

ly01e

ikazl�i!at + b(t)~�ly12e

�ikbzl�i!bt + h:c:�; �i;

(B.40)

and various components in�@�@t

�(a)A�L are

Tr{E(�)Xl

eikazl~�ly01�} =Xl

eikazlh� (i�5)2

@

@(i�5)+ (i�5)(i�6)

@

@(i�6)� (i�5)(i�7)

@

@(i�7)+ (i�5)(i�8)

@

@(i�8)

�(i�5)(i�9)@

@(i�9)� i�5

@

@(i�11)� i�5

@

@(i�12)� 2i�5

@

@(i�13)+ i�5Nz � i�7

@

@(i�6)

�i�9@

@(i�8)+ i�16e

i�13@

@(i�15)+ i�18e

i�13@

@(i�17)+ ei�13

@

@(i�19)

il�;

Tr{E(�)Xl

e�ikazl~�l01�} =Xl

e�ikazlh @

@(i�5)

il�;

Tr{E(�)Xl

�eikazl~�ly01} =Xl

eikazlh @

@(i�19)

il�;

Tr{E(�)Xl

�e�ikazl~�l01} = Tr{E(�)Xl

eikazl~�ly01�}��5$��19;��6$��18;��7$��17;��8$��16;��9$��15;��11$��11;��12$��12;��13$��13

(B.41)

where a correspondence that we denote as C later is ��5 $ ��19; ��6 $ ��18; ��7 $

��17; ��8 $ ��16; ��9 $ ��15; ��11 $ ��11; ��12 $ ��12; ��13 $ ��13, can be observed

to help calculate the characteristic function. Also

123

Tr{E(�)Xl

e�ikbzl~�ly12�} =Xl

e�ikbzlh� (i�6)2

@

@(i�6)� (i�6)(i�8)

@

@(i�8)� (i�6)(i�7)

@

@(i�7)(B.42)

�(i�6)(i�10)@

@(i�10)+ (i�7)(i�6)

@

@(i�7)+ i�6

@

@(i�13)� i�6

@

@(i�12)+ i�7

@

@(i�5)

+(i�8)(i�10)(@

@(i�12)� @

@(i�11))� i�8(i�10)

2 @

@(i�10)� i�8e

i�12�i�11 @

@(i�14)

+i�10ei�11�i�13 @

@(i�16)+ ((i�10)(i�14)e

i�11�i�13 + ei�12�i�13)@

@(i�18)

il�;

Tr{E(�)Xl

eikbzl~�l12�} =Xl

eikbzlh @

@(i�6)+ i�5

@

@(i�7)

il�;

Tr{E(�)Xl

�e�ikbzl~�ly12} = Tr{E(�)Xl

eikbzl~�l12�}�C ;

Tr{E(�)Xl

�eikbzl~�l12} = Tr{E(�)Xl

e�ikbzl~�ly12�}�C ; (B.43)

and the atom-�eld interaction characteristic function for quantum �elds, which we

denote as part (b), is

�@�@t

�(b)A�L = Tr{E(�)

�@�@t

�(b)A�L},�@�

@t

�(b)A�L =

1

i~

h� ~

MXl=�M

�gsp2M + 1~�ly32as;le

�ikszl

+gip2M + 1~�ly03ai;le

ikizl + h:c:�; �i: (B.44)

For the part of �elds only,

Tr{E(�)as;l�} = [@

@(i�l3)]�, Tr{E(�)�as;l} = [

@

@(i�l3)+ i�l4]�;

Tr{E(�)ays;l�} = [@

@(i�l4)+ i�l3]�, Tr{E(�)�a

ys;l} = [

@

@(i�l4)]�;

Tr{E(�)ai;l�} = [@

@(i�l1)]�, Tr{E(�)�ai;l} = [

@

@(i�l1)+ i�l2]�;

Tr{E(�)ayi;l�} = [@

@(i�l2)+ i�l1]�, Tr{E(�)�a

yi;l} = [

@

@(i�l2)]�; (B.45)

124

and for the part of atomic operators associated with signal �eld,

Tr{E(�)~�ly32�} =hi�6

@

@(i�8)+ i�7

@

@(i�9)� (i�10)2

@

@(i�10)

+i�10(@

@(i�11)� @

@(i�12)) + ei�12�i�11

@

@(i�14)

il�;

Tr{E(�)~�l32�} =hi�8

@

@(i�6)+

@

@(i�10)+ i�9

@

@(i�7)

il�;

Tr{E(�)�~�ly32} = Tr{E(�)~�l32�}�C ;

Tr{E(�)�~�l32} = Tr{E(�)~�ly32�}�C ; (B.46)

and for the part of atomic operators associated with idler �eld,

Tr{E(�)~�ly03�}

=h(i�6)(i�8)

2 @

@(i�8)+ (i�5)(i�6)(i�8)

@

@(i�6)� (i�5)(i�8)(

@

@(i�13)� i�9

@

@(i�9)

+(i�10)@

@(i�10)� @

@(i�11)) + (i�5)(i�6)

@

@(i�10)� i�5e

i�11�i�13 @

@(i�16)

�(i�5)(i�14)ei�11�i�13@

@(i�18)� (i�5)(i�9)

@

@(i�5)� (i�7)(i�8)

@

@(i�6)� i�7

@

@(i�10)

�(i�7)(i�9)@

@(i�7)+ i�9Nz � (i�9)2

@

@(i�9)+ i�8[�(i�9)

@

@(i�8)+ i�16e

i�13@

@(i�15)

+i�18ei�13

@

@(i�17)+ ei�13

@

@(i�19)] + ei�11

@

@(i�15)+ i�9[�

@

@(i�13)� i�10

@

@(i�10)

� @

@(i�12)+ 2i�10

@

@(i�10)� 2 @

@(i�11)] + i�14e

i�11@

@(i�17)

il�;

Tr{E(�)~�l03�} =h @

@(i�9)

il�;

Tr{E(�)�~�ly03} = Tr{E(�)~�l03�}

�C ;

Tr{E(�)�~�l03} = Tr{E(�)~�ly03�}

�C : (B.47)

B.2.5 Characteristic function - dissipation part

We calculate the characteristic function from�@�@t

�spup to the second order of various

��s (where we denote (2)) that account for drift and di¤usion terms in Fokker-Planck

equation. Below we drop the summation over spatial slices l; which we will retrieve

125

later,

Tr{E(�)�01��y01}

(2)

= Tr{[(i�8)(i�10)�12 + (i�6)(i�14)�13 + (i�8)(i�15)�10 +

(i�8)(i�16)�11 + (i�6)(i�17)�10 + e�i�13�11 � i�19e�i�13�10

+(i�6)(i�18)�11 � i�8e�i�11�13 � i�6e

�i�12�12]E(�)�}(2) (B.48)

where various properties of tracing can be found in previous sections, and the one we

did not have before is (up to �rst order)

Tr{�13E(�)�}(1)

= [i�15@

@(i�19)� i�16(

@

@(i�11)� @

@(i�13))� i�18

@

@(i�14)

+ei�11�i�13@

@(i�8)+ i�10

@

@(i�6)]�: (B.49)

Put everything together, and for the dissipation of �rst laser transition we have

01Tr{E(�)[�01��y01 �

1

2�11��

1

2��11]}

(2) =

01[�i�52

@

@(i�5)� i�19

2

@

@(i�19)� i�6

2

@

@(i�6)� i�18

2

@

@(i�18)� i�8

2

@

@(i�8)

�i�162

@

@(i�16)� i�13

@

@(i�13)+ (i�13)(i�18)

@

@(i�18)+ (i�13)(i�16)

@

@(i�6)

+(i�13)(i�16)@

@(i�16)+ (i�13)(i�8)

@

@(i�8)+ (i�8)(i�18)

@

@(i�14)+ (i�6)(i�16)

@

@(i�10)

+(i�8)(i�16)@

@(i�11)+ (i�6)(i�18)

@

@(i�12)+(i�13)

2

2

@

@(i�13)]�: (B.50)

And for the second laser,

Tr{E(�)�12��y12}

(2) = Tr{h(i�14)(i�15)�20 + (i�16)(i�14)�21 + ei�13�i�12

��22 � i�17�20

�i�14�23 � i�18�21 + (i�19)(i�18)�20

�iE(�)�}(2): (B.51)

126

The above requires

Trf�20E(�)�g = [@

@(i�17)]�:

Then we have

12Tr{E(�)[�12��y12 �

1

2�22��

1

2��22]}

(2) =

12[�i�62

@

@(i�6)� i�18

2

@

@(i�18)� i�7

2

@

@(i�7)� i�17

2

@

@(i�17)� i�10

2

@

@(i�10)

�i�142

@

@(i�14)+ (i�13 � i�12)

@

@(i�12)+ (i�5)(i�19)

@

@(i�12)+(i�13 � i�12)

2

2

@

@(i�12)]�:

(B.52)

And the dissipation for the signal transition,

Tr{E(�)�32��y32}

(2) =

Tr{[(i�8)(i�16)�22 + (i�9)(i�15)�22 + (i�14)(i�15)�20 + (i�14)(i�16)�21

+ei�11�i�12(�i�17�20 � i�14�23 + �22 � i�18�21 + (i�19)(i�18)�20)]E(�)�},

(B.53)

so we have

32Tr{E(�)[�32��y32 �

1

2�22��

1

2��22]}

(2) =

32[�i�62

@

@(i�6)� i�18

2

@

@(i�18)� i�7

2

@

@(i�7)� i�17

2

@

@(i�17)� i�10

2

@

@(i�10)

�i�142

@

@(i�14)+ (i�11 � i�12)

@

@(i�12)+ (i�5)(i�19)

@

@(i�12)+ (i�8)(i�16)

@

@(i�12)

+(i�11 � i�12)

2

2

@

@(i�12)]�: (B.54)

And for idler transition,

127

Tr{E(�)�03��y03}

(2) =

Tr{[(i�10)(i�18)�31 + (i�10)(i�14)�33 + (i�10)(i�17)�30 � i�10e�i�12�32

e�i�11(�33 � i�16�31 + (i�16)(i�19)�30 � i�15�30]E(�)�}(2): (B.55)

The above needs

Trf�31E(�)�g = [@

@(i�16)+ i�19

@

@(i�15)]�;

then we have

03Tr{E(�)[�03��y03 �

1

2�33��

1

2��33]}

(2)

= 03[�i�82

@

@(i�8)� i�16

2

@

@(i�16)� i�9

2

@

@(i�9)� i�15

2

@

@(i�15)� i�10

2

@

@(i�10)

�i�142

@

@(i�14)� i�11

@

@(i�11)+ (i�11)(i�14)

@

@(i�14)+ (i�10)(i�11)

@

@(i�10)

+(i�10)(i�14)@

@(i�12)+(i�11)

2

2

@

@(i�11)]�: (B.56)

B.3 Stochastic Di¤erential Equation

A distribution function can be found by Fourier transforming the characteristic func-

tions,

f(~�) =1

(2�)n

Z:::

Ze�i~��

~��(~�)d�1:::d�n; (B.57)

then

@f

@t=

1

(2�)n

Z:::

Ze�i~��

~�@�

@td�1:::d�n: (B.58)

If @�@t= i��

@�@(i� )

, use integration by parts and neglect the boundary terms, we

have @f@t= � @

@(��)� f where a minus sign is from i��. Correspondingly, if

@�@t= ei�� ,

we have @f@t= e

� @@(��) .

128

B.3.1 Fokker-Planck equation

Let@f

@t= Lf =

Xl;l0

[LA�ll0 + LL + L(a)A�L�ll0 + L(b)A�L�ll0 + Lsp�ll0 ]f; (B.59)

then we have for the atomic part,

LA = �i!1[@

@�l5(��l5)�

@

@�l6(��l6)�

@

@�l8(��l8)]

�i!2[@

@�l6(��l6) +

@

@�l7(��l7) +

@

@�l10(��l10)]

�i!3[@

@�l8(��l8) +

@

@�l9(��l9)�

@

@�l10(��l10)] + (c:c: with C 0) (B.60)

where C 0 is ��5 $ �19; ��6 $ �18; �

�7 $ �17; �

�8 $ �16; �

�9 $ �15; �

�10 $ �14; �

�11 $ �11;

��12 $ �12; ��13 $ �13; �

�1 $ �2; �

�3 $ �4; and c:c: is complex conjugation. Also for

the �eld part,

LL = [i!s@

@�l3�l3 � i!s

@

@�l4�l4 + i!i

@

@�l1�l1 � i!i

@

@�l2�l2]�ll0

+i!l0l@

@�l3�l

0

3 � i!l0l@

@�l04

�l4 + i!ll0@

@�l1�l

0

1 � i!ll0@

@�l02

�l2: (B.61)

129

The atom-�eld interaction part (a) is

L(a)A�L =

iaeikazl�i!at[� @2

@�l5@�l5

(�l5) +@2

@�l5@�l6

(�l6)�@2

@�l5@�l7

(�l7) +@2

@�l5@�l8

(�l8)

� @2

@�l5@�l9

(�l9)�@

@�l5(��l11 � �l12 � 2�l13 +Nz)�

@

@�l7(��l6)�

@

@�l9(��l8)

+@

@�l16e� @

@�l13 (��l15) +@

@�l18e� @

@�l13 (��l17) + e� @

@�l13 (�l19)]� iaeikazl�i!at(�l19)

+ibe�ikbzl�i!bt[� @2

@�l6@�l6

(�l6)�@2

@�l6@�l8

(�l8)�@2

@�l6@�l10

(�l10)

+@

@�l6(��l13 + �l12) +

@

@�l7(��l5) +

@2

@�l8@�l10

(�l12 � �l11)�@3

@�l8@�l10@�

l10

(�l10)

� @

@�l8e� @

@�l12

+ @

@�l11 (��l14) +@

@�l10e� @

@�l11

+ @

@�l13 (��l16) + (@2

@�l10@�l14

e� @

@�l11

+ @

@�l13

+e� @

@�l12

+ @

@�l13 )(�l18)]� ibe�ikbzl�i!bt[�l18 +

@

@�l19(��l17)] + (c:c: with C 0);(B.62)

and let L(b)A�L = L(b)A�L;S + L(b)A�L;I ; which are the terms for signal (S) and idler (I)

parts,

L(b)A�L;S = igsp2M + 1e�ikszl [

@

@�l6(��l8) +

@

@�l7(��l9)�

@2

@�l10@�l10

(�l10)

+@

@�l10(��l11 + �l12) + e

� @

@�l12

+ @

@�l11 (�l14)]�l3 + ig�s

p2M + 1eikszl

[@

@�l8(��l6) + �l10 +

@

@�l9(��l7)](�l4 �

@

@�l3) + (c:c: with C 0) (B.63)

and

130

L(b)A�L;I = igip2M + 1eikizl [

@3

@�l5@�l8@�

l8

(��l8) +@3

@�l5@�l6@�

l8

(��l6) +@2

@�l5@�l6

(�l10)

� @2

@�l5@�l9

(�l5)�@2

@�l5@�l8

(�l13 +@

@�l9�l9 �

@

@�l10�l10 � �l11)

� @

@�l5e� @

@�l11

+ @

@�l13 (��l16)�@2

@�l5@�l14

e� @

@�l11

+ @

@�l13 (�l18)�@2

@�l7@�l8

(�l6)

� @

@�l7(��l10)�

@2

@�l7@�l9

(�l7)�@2

@�l9@�l9

(�l9)�Nz@

@�l9� @

@�l8

�� @

@�l9(��l8) +

@

@�l16e� @

@�l13 (��l15) +@

@�l18e� @

@�l13 (��l17) + e� @

@�l13�l19�

+e� @

@�l11�l15 �@

@�l9

�� l13 +

@

@�l10(�l10)� �l12 + 2

@

@�l10(��l10)

�2�l11�+

@

@�l14e� @

@�l11 (��l17)]�l1

+ig�ip2M + 1e�ikizl(�l9)(�

l2 �

@

@�l1) + (c:c: with C 0): (B.64)

The dissipation part Lsp can be derived accordingly and the above equation, which

involves higher order derivatives (third order and higher), is neglected. The validity

of truncation to second order is due to the expansion in the small parameter 1=Nz.

If the Fokker-Planck equation is

@f

@t= � @

@�A�f �

@

@�A�f +

1

2(@2

@�@�+

@2

@�@�)D��f (B.65)

where A and D are drift and di¤usion terms then we have a corresponding classical

Langevin equation

@�

@t= A� + ��,

@�

@t= A� + �� (B.66)

with a correlation function h��i = �(t� t0)D��. So we have according to various

131

L�s,

_�l5 = (�i!1 � 012)�l5 + iae

ikazl�i!at(�l0 � �l13) + i�beikbzl+i!bt�l7

�igip2M + 1eikizl�l16�

l1 + �

l5;

_�l6 = i(!1 � !2 + i 01 + 22

)�l6 � i�ae�ikazl+i!at�l7 + ibe

�ikbzl�i!bt(�l13 � �l12)

+igsp2M + 1e�ikszl�l8�

l3 + �

l6;

_�l7 = (�i!2 � 22)�l7 � iae

ikazl�i!at�l6 + ibe�ikbzl�i!bt�l5

+igsp2M + 1e�ikszl�l9�

l3 � igi

p2M + 1eikizl�l10�

l1 + �

l7;

_�l13 = � 01�l13 + 12�l12 + iae

ikazl�i!at�l19 � i�ae�ikazl+i!at�l5

�ibe�ikbzl�i!bt�l18 + i�beikbzl+i!bt�l6 + �

l13;

_�l12 = � 2�l12 + ibe�ikbzl�i!bt�l18 � i�be

ikbzl+i!bt�l6 + igsp2M + 1e�ikszl�l14�

l3

�ig�sp2M + 1eikszl�l10�

l4 + �

l12;

_�l11 = � 03�l11 + 32�l12 � igs

p2M + 1e�ikszl�l14�

l3 + igs

p2M + 1eikszl�l10�

l4

+igip2M + 1eikizl�l15�

l1 � ig�i

p2M + 1e�ikizl�l9�

l2 + �

l11;

_�l8 = i(!1 � !3 + i 01 + 03

2)�l8 � i�ae

�ikazl+i!at�l9 � ibe�ikbzl�i!bt�l14

+ig�sp2M + 1eikszl�l6�

l4 + igi


l1 + �

l8;

_�l9 = (�i!3 � 032)�l9 � iae

ikazl�i!at�l8 + ig�sp2M + 1eikszl�l7�

l4

+igip2M + 1eikizl(�l0 � �l11)�

l1 + �

l9;

_�l14 = i(!2 � !3 + i 03 + 22

)�l14 � i�beikbzl+i!bt�l8

+ig�sp2M + 1eikszl(�l12 � �l11)�

l4 + igi


l1 + �

l14;

_�l4 = i!s�l4 + i

Xl0

!ll0�l0

4 � igsp2M + 1e�ikszl�l14 + �

l4;

_�l1 = �i!i�l1 � iXl0

!ll0�l0

1 + ig�ip2M + 1e�ikizl�l9 + �

l1; (B.67)

where 2 = 12 + 32: We postpone the derivations of di¤usion coe¢ cients after the

scaling is made in the next section, and note that the complete equations of motion

are found by making complex conjugate of the above with correspondence C 0 and

132

changing Langevin noises correspondingly, say ��5 ! �19:

B.3.2 Slowly varying envelopes and scaled equations of motion

Here we introduce the slowly varying envelopes and de�ne our cross-grained collective

atomic and �eld observables, then �nally transform the equations in a dimensionless

form for later numerical simulations. We note that

iXl0

!ll0�l0

4 = cd

dzl�l4, � i

Xl0

!ll0�l0

1 = �c@

@zl�l1; (B.68)

and �l0 = Nz � �l13 � �l12 � �l11. De�ne slow varying observables that

e�5(z; t) � 1

Nz�l5e

�ikazl+i!at; e�6(z; t) � 1

Nz�l6e

ikbzl+i!bt;

e�7(z; t) � 1

Nz�l7e

�ikazl+ikbzl+i!bt+i!at; e�8(z; t) � 1

Nz�l8e

�i!at+i!3t+ikazl�ikizl ;

e�9(z; t) � 1

Nz�l9e

�ikizl+i!3t; e�11(z; t) � 1

Nz�l11;

e�12(z; t) � 1

Nz�l12; e�13(z; t) � 1

Nz�l13;

e�14(z; t) � 1

Nz�l14e

�i(!23+�2)teikazl�ikbzl�ikizl (B.69)

where ei�kz = eikazl�ikbzl�ikizl+ikszl. Also for the �eld variables,

E�s (z; t) �g�sdi=~

p2M + 1�l4e

�i!st; E+i (z; t) �gidi=~

p2M + 1�l1e

i!it; (B.70)

where we use the idler dipole moment in signal �eld scaling for the purpose of scale-

free atomic equation of motions, so we need to keep in mind that in calculating signal

intensity or correlation function, an extra factor of (di=ds)2 needs to be taken care of.

We choose the central frequency of signal and idler as !s = !23 + �2; !i = !3

where �1 = !a � !1 and �2 = !a + !b � !2. With a scaling of Arecchi-Courtens

cooperation length [115], we set up the units of �eld strength, time, and length in the

following,

133

EcTc=N jgij2di=~

; Lc = cTc;1

Tc=

sd2in!i2~�0

; Ec =

rn~!i2�0

=1

Tc

1

di=~: (B.71)

Compared with optical density and superradiant time scale, we have (in terms of

single atomic decay rate )

N jgij2 = NL=c

; N = N3

8�

�2

A ; n =

N

V: (B.72)

Now the slowly varying and dimensionless equations of motion with Langevin

noises in Ito�s form are

@

@te�5 = (i�1 �

012)e�5 + ia(e�0 � e�13) + i�be�7 � ie�16E+i + F5;

@

@te�6 = i(�2 ��1 + i

01 + 22

)e�6 � i�ae�7 + ib(e�13 � e�12) + ie�8E+s e�i�kz + F6;@

@te�7 = (i�2 �

22)e�7 � iae�6 + ibe�5 + ie�9E+s e�i�kz � ie�10E+i + F7;

@

@te�13 = � 01e�13 + 12e�12 + iae�19 � i�ae�5 � ibe�18 + i�be�6 + F13;

@

@te�12 = � 2e�12 + ibe�18 � i�be�6 + ie�14E+s e�i�kz � ie�10E�s ei�kz + F12;

@

@te�11 = � 03e�11 + 32e�12 � ie�14E+s e�i�kz + ie�10E�s ei�kz + ie�15E+i � ie�9E�i + F11;@

@te�8 = �(i�1 +

01 + 032

)e�8 � i�ae�9 � ibe�14 + ie�6E�s ei�kz + ie�19E+i + F8;@

@te�9 = � 03

2e�9 � iae�8 + ie�7E�s ei�kz + i(e�0 � e�11)E+i + F9;

@

@te�14 = �(i�2 +

03 + 22

)e�14 � i�be�8 + i(e�12 � e�11)E�s ei�kz + ie�17E+i + F14;(B.73)

and �eld propagation equations are

(@

@t� @

@z)E�s = �ie�14e�i�kz jgsj2jgij2

+ F4;

(@

@t+

@

@z)E+i = ie�9 + F1; (B.74)

134

where jgsj2jgij2 is a unit transformation factor from the signal �eld strength to the idler

one. For a recognizable format of the above equations used in the text of Chapter 4,

we change the labels in the below,

e�5 $ �01; e�6 $ �12; e�7 $ �02; e�8 $ �13; e�9 $ �03; e�10 $ �32; e�11 $ �33;

e�12 $ �22; e�13 $ �11; e�14 $ �y32; e�15 $ �y03; e�16 $ �y13; e�17 $ �y02;

e�18 $ �y12; e�19 $ �y01; (B.75)

where �ij is the stochastic variable that corresponds to the atomic populations of

state jii when i = j and to atomic coherence when i 6= j. Note that the associated

c-number Langevin noises are changed accordingly.

The Langevin noises are de�ned as

F5(z; t) =1

Nz�l5e

�ikazl+i!at;F6(z; t) =1

Nz�l6e

ikbzl+i!bt;

F7(z; t) =1

Nz�l7e

�ikazl+ikbzl+i!bt+i!at;F13(z; t) =1

Nz�l13;F12(z; t) =

1

Nz�l12;

F11(z; t) =1

Nz�l11;F8(z; t) =

1

Nz�l8e

�i!at+i!3t+ikazl�ikizl ;

F9(z; t) =1

Nz�l9e

�ikizl+i!3t;F14(z; t) =1

Nz�l14e

�i(!23+�2)teikazl�ikbzl�ikizl ;

F4(z; t) =g�sdi=~

p2M + 1e�i!st�l4;F1(z; t) =

gidi=~

p2M + 1ei!it�l1 (B.76)

where other Langevin noises can be found by using the correspondence similar to C 0,

for example, F�5 $ F19.

Before we proceed to formulate the di¤usion coe¢ cients, we need to be careful

about the scaling factor for the transformation to continuous variables when numerical

135

simulation is applied. Take hF6F5i for example,

hF6(z; t)F5(z0; t0)i

=1

N2z

eikbzl+i!bte�ikazl0+i!at0D�l6�

l0

5

E=

1

N2z

eikbzl+i!bte�ikazl+i!at[iaeikazl�i!at�l6 + igi


l1]�(t� t0)�ll0

=1

Nz

�iaTcTce�6 + i

di~Ece�10E+i

Ec

��(t� t0)�(z � z0)

L

2M + 1

=�i(aTc)e�6 + ie�10(E+i =Ec)� 1T 2c �(t� t0)Tc�(z � z0)Lc

L

Lc

NzN

1

Nz

=1

Nc

�i(aTc)e�6 + ie�10(E+i =Ec)� 1T 2c �(t� t0)Tc�(z � z0)Lc (B.77)

where we have used limM!12M+1L

�ll0 = �(z� z0), 2M +1 = NNz; and Nc = NLc

Lis the

cooperation number. Then we have the dimensionless form of di¤usion coe¢ cients.

T 2c hF6(z; t)F5(z0; t0)i =1

NcD6;5�(t� t0)�(z � z0) (B.78)

D6;5 =�iae�6 + ie�10E+i � : (B.79)

The dimensionless di¤usion coe¢ cients Dij are

136

(i)D5;5 = �i2ae�5; D5;6 = i(ae�6 + e�10E+i ); D5;7 = �iae�7;D5;8 = i(ae�8 + (e�11 � e�13)E+i ); D5;9 = �i(ae�9 + e�5E+i );D5;11 = �ie�16E+i ; D5;13 = ie�16E+i ; D5;14 = �ie�18E+i ; D5;19 = 12e�12;

(ii)D6;6 = �i2be�6; D6;8 = �ibe�8; D6;10 = �ibe�10;D6;13 = �i�ae�7 + 01e�6; D6;16 = �ie�7E�i + 01e�10; D6;18 = 01e�12;

(iii)D7;8 = �ie�6E+i ; D7;9 = �ie�7E+i ;(iv)D8;9 = �ie�8E+i ; D8;10 = ib(e�12 � e�11); D8;11 = ibe�14;D8;12 = �ibe�14;D8;13 = �i�ae�9 + ie�19E+i + 01e�8;D8;16 = ie�15E+i � ie�9E�i + 01e�11 + 32e�12; D8;18 = ie�17E+i + 01e�14;

(v)D9;9 = �i2e�9E+i ; D9;10 = ie�10E+i ; D9;15 = 32e�12;(vi)D10;10 = �i2e�10E+s e�i�kz; D10;11 = i(be�16 � e�7E�i ) + 03e�10;

D10;13 = �ibe�16;D10;14 = ibe�18 � i�be�6 + 03e�12; D10;19 = ie�6E�i ;(vii)D11;11 = ie�14E+s e�i�kz � ie�10E�s ei�kz + ie�15E+i � ie�9E�i + 32e�12 + 03e�11;

D11;12 = ie�10E�s ei�kz � ie�14E+s e�i�kz � 32e�12;(viii)D12;12 = ibe�18 � i�be�6 � ie�10E�s ei�kz + ie�14E+s e�i�kz + 2e�12;

D12;13 = �ibe�18 + i�be�6 � 12e�12;(ix)D13;13 = iae�19 � i�ae�5 + ibe�18 � i�be�6 + 01e�13 + 12e�12;(x)D3;8 =

jgsj2jgij2

ie�6ei�kz; D3;9 =jgsj2jgij2

ie�7ei�kz: (B.80)

B.3.3 Alternative method to derive di¤usion coe¢ cients by Einstein re-lations

Before going further to set up the stochastic di¤erential equation, we show here

how we derive the di¤usion coe¢ cients from the Heisenberg-Langevin approach with

Einstein relations, and it provides the important check for Fokker-Planck equations.

137

We note here that a symmetric property of the di¤usion coe¢ cients is within Fokker-

Planck equation, whereas the quantum di¤usion coe¢ cients in quantum Langevin

equation do not have symmetric property simply because the quantum operators do

not necessarily commute with each other.

The approach involves a quantum-classical correspondence in deriving c-number

Langevin equations and requires a chosen normal ordering of quantum operators. We

use the same ordering as we use for deriving Fokker-Planck equations in Eq. (B.30),

~�y01; ~�y12; ~�

y02; ~�

y13; ~�

y03; ~�

y32; ~�11; ~�22; ~�33; ~�32; ~�03; ~�13; ~�02; ~�12; ~�01; a

ys; as; a

yi ; ai

and its classical correspondence is e�19;e�18;:::e�1:We take D8;13 = D13;8 for a demonstration. We �rst calculate the quantum

di¤usion coe¢ cient, D13;8; using Einstein relations where we attach the hat to it, and

then we can �nd �D13;8; a classical di¤usion coe¢ cient, which is reviewed in Chapter 2.

Note that in calculating the quantum coe¢ cients, we take advantage of Eq. (B.73)

where the drift terms are directly corresponded to quantum Langevin equations. For

clarity, D13;8 = D~�11;~�13 with e�13 ! ~�11; e�8 ! ~�13 representing a correspondence to

quantum Langevin equations. The index in the classical variables e� represents theordering we choose as de�ned above, and in various quantum operators ~�, the index

represents the atomic levels for atomic coherences or populations. We should �nd

�D13;8 = D13;8 = D8;13; and the proof is illustrated below by the Einstein�s relation,

Eq. (2.8),

138

DD13;8

E=

�Dh� 01~�11 + 12~�22 + ia~�

y01 � i�a~�01 � ib~�

y12 + i�b~�12

i~�13

E��~�11

��(i�1 +

01 + 032

)~�13 � i�a~�03 � ib~�y32 + i~�12E

�s e

i�kz + i~�y01E+i

��+@

@th~�11~�13i

= h 01~�13i ; (B.81)

where the term @@th~�11~�13i = @

@th~�13i is the drift term of the quantum Langevin

equation that can be found from Eq. (B.73),

@

@t~�13 = �(i�1 +

01 + 032

)~�13 � i�a~�03 � ib~�y32 + i~�12E

�s e

i�kz + i~�y01E+i : (B.82)

From Eq. (2.15), we have�D13;8

�=D

D13;8E+nDh

� 01~�11 + 12~�22 + ia~�y01 � i�a~�01 � ib~�

y12 + i�b~�12

i~�13

E+�

~�11

��(i�1 +

01 + 032

)~�13 � i�a~�03 � ib~�y32 + i~�12E

�s e

i�kz + i~�y01E+i

�� classical counterpart

o= 01e�8 � i�ae�9 + ie�19E+i � ; (B.83)

where classical counterpart represents the last two terms of Eq. (2.15). We have

used the commutation relations for non-normal correlation functions that

[~�01; ~�13] = ~�03;

[~�12; ~�13] = 0;h~�11; ~�

y32

i= 0;h

~�11; ~�y01

i= ~�y01; (B.84)

139

and use the correspondence ~�03 ! e�9 and ~�y01 ! e�19 in �D13;8: The rest of the di¤usioncoe¢ cients are con�rmed by the method of Einstein relations illustrated above.

B.3.4 Ito and Stratonovich stochastic di¤erential equations

The c-number Langevin equations derived from Fokker-Planck equations have a di-

rect correspondence to Ito-type stochastic di¤erential equations. In stochastic sim-

ulations, it is important to �nd the expressions of Langevin noises from di¤usion

coe¢ cients.

For any symmetric di¤usion matrix D(�), it can always be factorized into

D(�) = B(�)BT (�) (B.85)

where B ! BS (an orthogonal matrixS that SST = I) preserves the di¤usion

matrix so B is not unique. The matrix B is in terms of the Langevin noises where

�idt = dW it (Wiener process) and

�i(t)�j(t

0)�= �ij�(t � t0) and the �i below is just

a random number in Gaussian distribution with zero mean and unit variance.

In numerical simulation, we use the semi-implicit algorithm that guarantees the

stability and convergence in the integration of stochastic di¤erential equations. So a

transformation from Ito to Stratonovich-type stochastic di¤erential equation is nec-

essary,

dxit = Ai(t;�!xt )dt+

Xj

Bij(t;�!xt )dW j

t (Ito) (B.86)

dxit = [Ai(t;�!xt )�

1

2

Xj

Xk

Bjk(t;�!xt )

@

@xjBik(t;

�!xt )]dt

+Xj

Bij(t;�!xt )dW j

t (Stratonovich) (B.87)

where a correction in drift term appears due to the transformation.

Here we have the full equations with 19 variables in the positive-P representation,

64 di¤usion matrix elements, and 117 noise terms (random number generators). A

140

correction in drift term is underlined and we have (S for Stratonovich)

@

@�e�5 = (

ia2)| {z }+(i�1 �

012)e�5 + ia(e�0 � e�13) + i�be�7 � ie�16E+i + F5; (S)

@

@�e�19 = (

�i�a2)| {z }+(�i�1 �

012)e�19 � i�a(e�0 � e�13)� ibe�17 + ie�8E�i + F19;

@

@�e�6 = (ib)+| {z } i(�2 ��1 + i

01 + 22

)e�6 � i�ae�7 + ib(e�13 � e�12)+ie�8E+s e�i�kz + F6;

@

@�e�18 = (�i�b)| {z }�i(�2 ��1 � i

01 + 22

)e�18 + iae�17 � i�b(e�13 � e�12)�ie�16E�s ei�kz + F18;

@

@�e�7 = (i�2 �

22)e�7 � iae�6 + ibe�5 + ie�9E+s e�i�kz � ie�10E+i + F7;

@

@�e�17 = (�i�2 �

22)e�17 + i�ae�18 � i�be�19 � ie�15E�s ei�kz + ie�14E�i + F17;

@

@�e�8 = (�i�1 �

01 + 032

)e�8 � i�ae�9 � ibe�14 + ie�6E�s ei�kz + ie�19E+i + F8;@

@�e�16 = (i�1 �

01 + 032

)e�16 + iae�15 + i�be�10 � ie�18E+s e�i�kz � ie�5E�i + F16;@

@�e�9 = (iE+i )| {z }� 032 e�9 � iae�8 + ie�7E�s + i(e�0 � e�11)E+i + F9;

@

@�e�15 = (�iE�i )| {z }� 032 e�15 + i�ae�16 � ie�17E+s � i(e�0 � e�11)E�i + F15;

@

@�e�10 = (

i

2E+s )| {z }+(i�2 �

03 + 22

)e�10 + ibe�16 � i(e�12 � e�11)E+s e�i�kz�ie�7E�i + F10;

@

@�e�14 = (� i

2E�s )| {z }+(�i�2 �

03 + 22

)e�14 � i�be�8 + i(e�12 � e�11)E�s ei�kz+ie�17E+i + F14;

@

@�e�13 =

�5 01 + 124| {z }� 01e�13 + 12e�12 + iae�19 � i�ae�5 � ibe�18

+i�be�6 + F13;@

@�e�12 = � 2

4|{z}� 2e�12 + ibe�18 � i�be�6 + ie�14E+s e�i�kz � ie�10E�s ei�kz + F12;@

@�e�11 =

�3 03 + 324| {z }� 03e�11 + 32e�12 � ie�14E+s e�i�kz + ie�10E�s ei�kz + ie�15E+i

�ie�9E�i + F11;141

� @

@zE+s = ie�10 jgsj2jgij2

+ F3;

� @

@zE�s = �ie�14 jgsj2jgij2

+ F4;

@

@zE+i = ie�9 + F1;

@

@zE�i = �ie�15 + F2: (B.88)

The Langevin noises are formulated as a non-square form [35, 84]

F1 = F2 = 0;

F5 =pD5;5�1 +

rD5;19

2(�12 + i�13) +

rD5;6

2(�14 + i�15) +

rD5;7

2(�16 + i�17)

+

rD5;8

2(�18 + i�19) +

rD5;9

2(�20 + i�21) +

rD5;14

2(�22 + i�23)

+

rD5;13

2(�24 + i�25) +

rD5;11

2(�26 + i�27);

F19 =

rD5;19

2(�12 � i�13) +

pD19;19�2 +

rD19;18

2(�28 + i�29) +

rD19;17

2(�30 + i�31)

+

rD19;16

2(�32 + i�33) +

rD19;15

2(�34 + i�35) +

rD19;10

2(�36 + i�37)

+

rD19;13

2(�38 + i�39) +

rD19;11

2(�40 + i�41);

F6 =

rD5;6

2(�14 � i�15) +

pD6;6�3 +

rD6;18

2(�42 + i�43) +

rD6;8

2(�44 + i�45)

+

rD6;16

2(�46 + i�47) +

rD6;10

2(�48 + i�49) +

rD6;13

2(�50 + i�51);

F18 =

rD19;18

2(�28 � i�29) +

rD6;18

2(�42 � i�43) +

pD18;18�4 +

rD18;8

2(�52 + i�53)

+

rD18;16

2(�54 + i�55) +

rD18;14

2(�56 + i�57) +

rD18;13

2(�58 + i�59);

F7 =

rD5;7

2(�16 � i�17) +

rD7;8

2(�60 + i�61) +

rD7;9

2(�62 + i�63);

F17 =

rD19;17

2(�30 � i�31) +

rD17;16

2(�64 + i�65) +

rD17;15

2(�66 + i�67);

142

F8 =

rD5;8

2(�18 � i�19) +

rD6;8

2(�44 � i�45) +

rD18;8

2(�52 � i�53)

+

rD7;8

2(�60 � i�61) +

rD8;16

2(�68 + i�69) +

rD8;9

2(�70 + i�71)

+

rD8;10

2(�72 + i�73) +

rD8;13

2(�74 + i�75) +

rD8;12

2(�76 + i�77)

+

rD8;11

2(�78 + i�79) +

rD8;3

2(�80 + i�81);

F16 =

rD19;16

2(�32 � i�33) +

rD6;16

2(�46 � i�47) +

rD18;16

2(�54 � i�55)

+

rD17;16

2(�64 � i�65) +

rD8;16

2(�68 � i�69) +

rD16;15

2(�82 + i�83)

+

rD16;14

2(�84 + i�85) +

rD16;13

2(�86 + i�87) +

rD16;12

2(�88 + i�89)

+

rD16;11

2(�90 + i�91) +

rD16;4

2(�92 + i�93);

F9 =

rD5;9

2(�20 � i�21) +

rD7;9

2(�62 � i�63) +

rD8;9

2(�70 � i�71)

+pD9;9�5 +

rD9;15

2(�94 + i�95) +

rD9;10

2(�96 + i�97) +

rD9;3

2(�98 + i�99);

F15 =

rD19;15

2(�34 � i�35) +

rD17;15

2(�66 � i�67) +

rD16;15

2(�82 � i�83)

+

rD9;15

2(�94 � i�95) +

pD15;15�6 +

rD15;14

2(�100 + i�101)

+

rD15;4

2(�102 + i�103);

143

F10 =

rD19;10

2(�36 � i�37) +

rD6;10

2(�48 � i�49) +

rD8;10

2(�72 � i�73)

+

rD9;10

2(�96 � i�97) +

pD10;10�7 +

rD10;14

2(�104 + i�105)

+

rD10;13

2(�106 + i�107) +

rD10;11

2(�108 + i�109);

F14 =

rD5;14

2(�22 � i�23) +

rD18;14

2(�56 � i�57) +

rD16;14

2(�84 � i�85)

+

rD15;14

2(�100 � i�101) +

rD10;14

2(�104 � i�105) +

pD14;14�8

+

rD14;13

2(�110 + i�111) +

rD14;11

2(�112 + i�113);

F13 =

rD5;13

2(�24 � i�25) +

rD19;13

2(�38 � i�39) +

rD6;13

2(�50 � i�51)

+

rD18;13

2(�58 � i�59) +

rD8;13

2(�74 � i�75) +

rD16;13

2(�86 � i�87)

+

rD10;13

2(�106 � i�107) +

rD14;13

2(�110 � i�111) +

pD13;13�9

+

rD12;13

2(�114 + i�115);

F12 =

rD8;12

2(�76 � i�77) +

rD16;12

2(�88 � i�89) +

rD12;13

2(�114 � i�115)

+pD12;12�10 +

rD11;12

2(�116 + i�117);

F11 =

rD5;11

2(�26 � i�27) +

rD19;11

2(�40 � i�41) +

rD8;11

2(�78 � i�79)

+

rD16;11

2(�90 � i�91) +

rD10;11

2(�108 � i�109) +

rD14;11

2(�112 � i�113)

+

rD11;12

2(�116 � i�117) +

pD11;11�11;

F3 =

rD8;3

2(�80 � i�81) +

rD9;3

2(�98 � i�99);

F4 =

rD16;4

2(�92 � i�93) +

rD15;4

2(�102 � i�103): (B.89)

In numerical simulations, we have a factor 1pNc�t�z

for Langevin noises F and

1Nc�t�z

for correction terms.

144

APPENDIX C

MULTIMODE DESCRIPTION OF CORRELATED

TWO-PHOTON STATE

In this Appendix, we introduce a general model for quantum detection e¢ ciency

for multimode analysis in various quantum communication scheme. Based on this

detection model with the spectral description of correlated two-photon state, we derive

the e¤ective density matrix conditioning on the detection events of entanglement

swapping, polarization maximally entangled (PME) state projection, and quantum

teleportation.

C.1 Quantum E¢ ciency of Detector

To account for quantum e¢ ciency of detector and the a¤ect of its own spectrum

�ltering, we introduce an extra beam splitter (B.S.) with a transmissivity �(!; !0)

[117] before the detection event. � models the quantum e¢ ciency of the detectors

in the microscopic level (response at frequency !0) and the macroscopic level (time-

integrated detection). One example of conditioning on the single click of the detector,

the output density operator becomes

�out =

Z 1

�1d!0�1Trref

�UBS �inU

yBS

��1 (C.1)

�1 �Z 1

�1d!j!ih!j (C.2)

UBS �

0B@ p1� �

p�

p� �

p1� �

1CA (C.3)

145

Figure C.1: Model of quantum e¢ ciency of detector.

where Trref is the trace over the re�ected modes my3; and the �at spectrum projection

operator �1 (only photon number is projected and no frequency resolution) is con-

sidered in the measurement process [76]. In Figure C.1, my1 is the incoming photon

operator before the detection, my3 is the re�ected mode, and m

y4 is now the detec-

tion mode with a modelling of spectral quantum e¢ ciency and an e¤ective quantum

e¢ ciency is de�ned as

Z 1

�1�(!; !0)d!0 = �eff (!): (C.4)

C.2 Multimode Description of Entanglement Swapping

From Eq. (5.4), we use single mode �(!) for Raman photon and a multimode de-

scription f(!s; !i) for cascade photons and rewrite the e¤ective state. Note that a

symmetric setup is considered so the mode description is the same for both sides A

and B in the scheme of entanglement swapping.

146

jieff = �1(1� �2)�Zf(!s; !i)a

y;As (!s)a

y;Ai (!i)d!sd!i

Zf(!0s; !

0i)a

y;Bs (!0s)a

y;Bi (!0i)d!

0sd!

0ij0i+

�2(1� �1)

Z�(!)d!ay;Ar (!)SyA

Z�(!0)d!0ay;Br (!0)SyBj0i+

p�1(1� �1)�p

�2(1� �2)

Zf(!s; !i)d!sd!i � ay;As (!s)a

y;Ai (!i)

Z�(!0)d!0ay;Br (!0)SyBj0i+p

�1�2(1� �1)(1� �2)

Z�(!)d!ay;Ar (!)SyA

Zf(!0s; !

0i)a

y;Bs (!0s)a

y;Bi (!0i)d!

0sd!

0ij0i:

(C.5)

With the B.S., we have ay;Ai =my1+m

y2p

2, ay;Bi =

ny1+ny2p

2, ay;Ar =

my1�m

y2p

2, ay;Br =

ny1�ny2p

2,

where ayi is the creation operator for idler photon and ayr is for Raman photon. The

input density operator is �in = jieffhj and conditioning on the pair of single

click (my1;2; n

y1;2), we are able to generate maximally entangled singlet or triplet state

jiDLCZ = SyA�SyBp

2j0iA;B. Without loss of generality, we consider a triplet state along

with a pair of clicks (my1; n

y1) and use the model of quantum e¢ ciency in Eq. (C.1)

with tracing over the detection modes (my4; n

y4). Note that m

y1 =

p1� �my

3 +p�my

4

and ny1 =p1� �ny3 +

p�ny4 as we model the quantum e¢ ciency in the previous

Section.

�out =

Z 1

�1d!0Trm4;n4

�Trm3;n3

�UBBSU

ABS �inU

y;ABS U

y;BBS

�M4;4

(C.6)

M4;4 � (Iym4 � j0im4h0j) j0im2h0j (Iyn4 � j0in4h0j) j0in2h0j (C.7)

where the unitary B.S. operator is denoted by both sides (A and B) and NRPD

projection operators are used [99]. These operators project the state with single

click of the detected mode without resolving the number of photons. I is identity

operator. The un-normalized output density operator after tracing out these modes

becomes

147

�out =�21(1� �2)

2

4�Z

d!id!0i�eff (!i)�eff (!

0i)� Z

f(!s; !i)ay;As (!s)d!s

Zf(!0s; !

0i)a

y;Bs (!0s)d!

0s

�j0ih0j

� Zf �(!00s ; !i)a

As (!

00s)d!

00s

Zf �(!000s ; !

0i)a

Bs (!

000s )d!

000s

�+�1�2(1� �1)(1� �2)

4

�Zd!i�eff (!i)

h Zf(!s; !i)dws

Zf �(!0s; !i)d!

0sZ

j�(!)j2�eff (!)d!i�ay;As (!s)S

yBj0ih0jSBaAs (!0s) +

ay;Bs (!s)SyAj0ih0jSAaBs (!0s)

�+

Z Zf(!s; !i)d!s�

�(!i)�eff (!i)d!i �Z Zf �(!0s; !

0i)d!

0s�(!

0i)�eff (!

0i)d!

0i

�ay;As (!s)S

yBj0ih0jSAaBs (!0s) +

ay;Bs (!s)SyAj0ih0jSBaAs (!0s)

��+ �0out (C.8)

where �eff (!) is introduced after integration of !0; and we denote it as an e¤ective

quantum e¢ ciency for idler �eld !i or Raman photon at frequency ! (wavelength

780 nm for D2 line of Rb atom). �0out includes the terms that won�t survive after

the interference of telecom photons in the middle B.S. (conditioning on a single click

of detector). They involve operators like ay;As ay;Bs j0ih0jaAs SB, ay;As ay;Bs j0ih0jSASB and

Sy;ASy;Bj0ih0jSASB.

The normalization factor is derived by tracing over the atomic degree of freedom.

Tr(�out) � N =

�21(1� �2)2

4

Zd!sd!i�eff (!i)jf(!s; !i)j2

Zd!0sd!

0i�eff (!

0i)jf(!0s; !0i)j2 +

�1�2(1� �1)(1� �2)

2

Zd!sd!i�eff (!i)jf(!s; !i)j2

Zj�j2(!)�eff (!)d! +

�22(1� �1)2

4

Zj�j2(!)�eff (!)d!

Zj�j2(!0)�eff (!0)d!0 (C.9)

which will be put back when we calculate the heralding and success probabilities.

Next we interfere telecom photons with B.S. that ay;As =cy1+c

y2p2, ay;Bs =

cy1�cy2p2;

and again a quantum e¢ ciency �(!; !0) for telecom photon is introduced. Use

148

cy1 =p1� �cy3 +

p�cy4 and trace over the re�ected mode c

y3 conditioning on the click

of cy4 from NRPD. The e¤ective density matrix becomes

�(2)out =

Z 1

�1d!0Trc4

�Trc3

�UCBS �inU

y;CBS

�M4

�

Z 1

�1d!0�

(2)out(!0); (C.10)

�(2)out(!0) � Trc4

��(2)in (!0)

(C.11)

M4 � (Iyc4 � j0ic4h0j) j0ic2h0j; (C.12)

�(2)in (!0) =

�21(1� �2)2

16

Zd!id!

0i�eff (!i)�eff (!

0i)

�Zd!s(1� �(!s))f(s; i)f

�(s; i0)

Zd!0sf(s

0; i0)p�(!0s)c

y4(!

0s)j0ih0j �Z

d!00s c4(!00s)p�(!00s)f

�(s00; i) +

Zd!s(1� �(!s))f(s; i)f

�(s; i)�Zd!0sf(s

0; i0)p�(!0s)c

y4(!

0s)j0ih0j

Zd!000s c4(!

000s )p�(!000s )f

�(s000; i0) +Zd!0s(1� �(!0s))f(s

0; i0)f �(s0; i0)

Zd!sf(s; i)

p�(!s)c

y4(!s)j0ih0j �Z

d!00s c4(!00s)p�(!00s)f

�(s00; i) +

Zd!0s(1� �(!0s))f(s

0; i0)f �(s0; i)�Zd!sf(s; i)

p�(!s)c

y4(!s)j0ih0j

Zd!000s c4(!

000s )p�(!000s )f

�(s000; i0) +Zd!0s

p�(!0s)f(s

0; i0)

Zd!s

p�(!s)f(s; i)c

y4(!s)c

y4(!

0s)j0ih0j �Z

d!00sp�(!00s)f

�(s00; i)

Zd!000s

p�(!000s )f

�(s000; i0)c4(!00s)c4(!

000s )

�+

�1�2(1� �1)(1� �2)

8

�Zd!i�eff (!i)

Zf(s; i)d!s

Zf �(s0; i)d!0s �Z

d!j�(!)j2�eff (!)p�(!s)c

y4(!s)


��

c4(!0s)p�(!0s)

Z Zf(s; i)d!s�

�(!i)�eff (!i)d!i �Z Zf �(s0; i0)d!0s�(!

0i)�eff (!

0i)d!

0i

p�(!s)c

y4(!s)��

SyBj0ih0jSA + SyAj0ih0jSB�c4(!

0s)p�(!0s)

�(C.13)

149

where a brief notation for spectrum f(s; i) � f(!s; !i) and quantum e¢ ciency �(!) �

�(!; !0). This quantum e¢ ciency refers to the telecom photon. We proceed to trace

over the detected modes and the density matrix can be simpli�ed by interchange of

variables in integration.

�(2)out(!0) =

�21(1� �2)2

8

Zd!id!

0i�eff (!i)�eff (!

0i)

�Zd!s(1� �(!s; !0))f(!s; !i)f

�(!s; !0i)

Zd!0sf(!

0s; !

0i)f

�(!0s; !i)�(!0s; !0) +Z

d!s(1� �(!s; !0))jf(!s; !i)j2Zd!0sjf(!0s; !0i)j2�(!0s; !0) +

1

2

Zd!0s�(!

0s; !0)jf(!0s; !0i)j2

Zd!s�(!s; !0)jf(!s; !i)j2 +

1

2�Z

d!0s�(!0s; !0)f(!

0s; !

0i)f

�(!0s; !i)

Zd!s�(!s; !0)f(!s; !i)f

�(!s; !0i)

�j0ih0j

+�1�2(1� �1)(1� �2)

8

�Zd!i�eff (!i)

Z�(!s; !0)jf(!s; !i)j2d!s �Z

d!j�(!)j2�eff (!)�SyBj0ih0jSB + SyAj0ih0jSA

�+Z Z

�(!s; !0)f(!s; !i)d!s��(!i)�eff (!i)d!i

Zf �(!s; !

0i)�(!

0i)�eff (!

0i)d!

0i ��


(C.14)

where the trace over two photon states requires the commutation relation of photon

operators.

Tr[my4(!s)m

y4(!

0s)j0ih0jm4(!

00s)m4(!

000s )]

= h0jm4(!00s)[�(!s; !

000s ) + my

4(!s)m4(!000s )]m

y4(!

0s)j0i

= �(!s; !000s )�(!

00s ; !

0s) + �(!s; !

00s)�(!

0s; !

000s ): (C.15)

The above is the general formulation for the un-normalized density matrix condi-

tioning on three clicks of NRPD�s. We�ve included spectral quantum e¢ ciency of the

detector either for near-infrared (�eff) or telecom wavelength (�t �R1�1 �(!; !0)d!0)

150

To proceed, we assume a �at and �nite spectrum response (�eff (!) = �eff , �t(!) =

�t) with the range !0 2 [ � �; + �] centered at (near-infrared or telecom)

and ! 2 [!0 � �; !0 + �]. The widths 2� and 2� are large enough compared to

our source bandwidth so these detection events do not give us any information of

spectrum for our source. A perfect e¢ ciency also means no photon loss during

detection. Note that the integral involves multiplication of two telecom photon

e¢ ciencyR1�1 �(!; !0)�(!

0; !0)d!0 = �2t (!) that is valid if the source bandwidth is

smaller than detector�s.

After the integration of !0, we have

�(2)out =

�21(1� �2)2

8�2eff

Zd!id!

0i

�(1� �t)�t

Zd!sf(!s; !i)f

�(!s; !0i)�Z

d!0sf(!0s; !

0i)f

�(!0s; !i) + (1� �t)�t

Zd!sjf(!s; !i)j2

Zd!0sjf(!0s; !0i)j2 +

�2t2

Zd!0sjf(!0s; !0i)j2

Zd!sjf(!s; !i)j2 +

�2t2

Zd!0sf(!

0s; !

0i)f

�(!0s; !i)�Zd!sf(!s; !i)f

�(!s; !0i)

�j0ih0j+ �1�2(1� �1)(1� �2)

8�t�

2eff ��Z

d!i

Zjf(!s; !i)j2d!s

Zd!j�(!)j2


�+Z Z

f(!s; !i)d!s��(!i)d!i

Zf �(!s; !

0i)�(!

0i)d!

0i�


: (C.16)

C.3 Density Matrix of PME Projection and Quantum Tele-portation

In Chapter 5.4, we have the normalized density operator �(2);ABout;n of the DLCZ entan-

gled state through entanglement swapping. With another pair of DLCZ entangled

state, �(2);CDout;n , the joint density operator for these two pairs constructs the polarization

maximally entangled state (PME) projection and is interpreted as

151

�(2);ABout;n �

(2);CDout;n =

1

(a+ b)2

�a2j0ih0j+ ab

2

hj0iABh0j

�SyC j0ih0jSC + SyDj0ih0jSD

+�1SyC j0ih0jSD + �1S

yDj0ih0jSC

�+ j0iCDh0j


+�1SyBj0ih0jSA + �1S

yAj0ih0jSB

�i+b2

4

�SyC j0ih0jSC + SyDj0ih0jSD

+�1SyC j0ih0jSD + �1S

yDj0ih0jSC

��SyBj0ih0jSB + SyAj0ih0jSA

+�1SyBj0ih0jSA + �1S

yAj0ih0jSB

��; (C.17)

which is used to calculate the success probability after post measurement [a click from

each side, the side of (A or C) and (B or D)]. a = �r(2� �)�1 +

Pj �

2j

�; b = 4, and

�r = �1=�2, � = �t, �j is Schmidt number that is used to decompose the two-photon

source from the cascade transition.

In DLCZ protocol, quantum teleportation uses the similar setup in PME projec-

tion and combines with the desired teleported state, j�i = (d0SyI1+d1SyI2)j0i, which is

represented by two other atomic ensembles I1 and I2. The requirement of normaliza-

tion of the state is d0j2+ jd1j2 = 1, and the density operator of quantum teleportation

is �QT = j�ih�j �(2);ABout;n �

(2);CDout;n . Conditioning on clicks of DI1 and DI2, the

e¤ective density matrix for quantum teleportation is (using SyI1 = (DI1 + DA)=p2;

SyI2 = (DI2 + DC)=p2 for the e¤ect of beam splitter)

152

�QT;eff =h jd0j22(Dy

I1j0ih0jDI1) +

jd1j22(Dy

I2j0ih0jDI2) +

d0d�1

2(Dy

I1j0ih0jDI2)

+d�0d12(Dy

I2j0ih0jDI1)

i 1

(a+ b)2

�a2j0ih0j+ ab

2

hj0iABh0j�Dy

I2j0ih0jDI2

2+ SyDj0ih0jSD + �1

DyI2p2j0ih0jSD + �1S

yDj0ih0j

DI2p2

�+j0iCDh0j

�SyBj0ih0jSB +

DyI1j0ih0jDI1

2+ �1S

yBj0ih0j

DI1p2+ �1

DyI2p2j0ih0jSB

�i+b2

4

�DyI2j0ih0jDI2

2+ SyDj0ih0jSD + �1

DyI2p2j0ih0jSD + �1S

yDj0ih0j

DI2p2

��

SyBj0ih0jSB +DyI1j0ih0jDI1

2+ �1S

yBj0ih0j

DI1p2+ �1

DyI2p2j0ih0jSB

��; (C.18)

which is used to calculate the success probability for teleported state.

153

APPENDIX D

HAMILTONIAN AND EQUATION OF MOTION FOR

FREQUENCY CONVERSION IN A DIAMOND TYPE

ATOMIC ENSEMBLE

In this appendix, we derive the Hamiltonian and the Maxwell-Bloch equation for

frequency conversion in ladder-type transition. The steady state solutions for atoms

are solved, and the solution to the �eld equations are discussed in Chapter 6. Similar

to the derivation in Appendix B where the cascade emissions are investigated, the

conversion scheme here also involves four-wave mixing with two classical driving lasers

and two quantum �elds, signal and idler. The driving lasers are applied in a way

that signal or idler is converted only when an idler or signal is put into interaction

with the atoms (see Figure 6.1). We will use the same quantization procedure for

electromagnetic �elds as discussed in Appendix B.2.1.

D.1 Hamiltonian and Maxwell-Bloch Equation

To derive the coupled Maxwell-Bloch equations it is convenient to employ a quan-

tized description of the electromagnetic �eld [29] and use Heisenberg-Langevin equa-

tion methods, and then invoke a standard semiclassical factorization assumption. The

propagation length L is discretized into 2M+1 elements. The positive frequency com-

ponent of the electric �eld operator is given by E+(z) =PM

n=�M

q~!s;n2�0V

ei(ks+kn)z cn

where [cn; cyn0 ] = �nn0, kn = 2�n

L; !s;n = !s + knc ; n = �M; :::;M and !s = ksc is

the central frequency. De�ne the local boson operators al = 1p2M+1

PMn=�M cne

iknzl

where [al; ayl0 ] = �ll0. Similar de�nitions hold for the signal, s, and idler �eld, i, which

carry an additional index in the following.

155

The Hamiltonian for the interacting system, HI = �~d � ~E, depicted in Figure 6.1

is given by, (we ignore the interactions responsible for atomic spontaneous emission

for the moment)

H = H0 + HI ; (D.1)

where

H0 =3Xi=1

MXl=�M

~!i�lii + ~!sMX

l=�M

ays;las;l + ~Xl;l0

!ll0 ays;las;l0

+~!iMX

l=�M

ayi;lai;l + ~Xl;l0

!ll0 ayi;lai;l0 ; (D.2)

and

HI = �~MX

l=�M

na(t)�

ly01e

ikazl�i!at + b(t)�ly32e

�ikbzl�i!bt

+ gsp2M + 1�ly12as;le

�ikszl + gip2M + 1�ly03ai;le

ikizl + h:c:o

(D.3)

where �lmn �PNz

� ��;lmn =PNz

� jmi�hnj��r�=zl

; the Rabi frequencies a;(b)(t) =

fa;(b)(t)d10;(23)E(ka;(b))=(2~) is half the standard de�nition, and fa;(b) is a slowly vary-

ing temporal pro�le without spatial dependence (ensemble scale much less than pulse

length). The dipole matrix element dmn � hmjdjni, coupling strength gs;(i) �

d21;(30)E(ks;(i))=~; E(k) =p~!=2�0V ; and zp = pL

2M+1; p = �M; :::;M . The ma-

trix !ll0 �PM

n=�M kneikn(zl�zl0 )=(2M + 1) accounts for �eld propagation by coupling

the local mode operators.

The dynamical equations including dissipation due to spontaneous emission may

be treated by standard Langevin-Heisenberg equation methods [30], and we de�ne ij

as the natural transition rate from jji ! jii: Since we are interested in a semiclassical

description, we replace the �eld operators by c-numbers in the Langevin equations,

156

and drop the zero-mean Langevin noise sources. All atomic spin operators are also

replaced by their expectation values. Finally, in the co-moving frame coordinates z

and � = t� z=c the atomic equations are

@

@�~�01 = (i�1 �

012)~�01 + ia(~�00 � ~�11) + ig�s ~�02E

�s � igi~�

y13E

+i ;

@

@�~�12 = (i�!s �

01 + 22

)~�12 � i�a~�02 + igs(~�11 � ~�22)E+s + iP �b~�13;

@

@�~�02 = (i�2 �

22)~�02 � i~�12a + igs~�01E

+s + iP �~�03b � iP �gi~�32E

+i ;

@

@�~�11 = � 01~�11 + 12~�22 + ia~�

y01 � i�a~�01 � igs~�

y12E

+s + ig�s ~�12E

�s ;

@

@�~�22 = � 2~�22 + igs~�

y12E

+s � ig�s ~�12E

�s + ib~�

y32 � i�b~�32;

@

@�~�33 = � 03~�33 + 32~�22 � ib~�

y32 + i�b~�32 + igi~�

y03E

+i � ig�i ~�03E

�i ;

@

@�~�13 = (i�!i � i�1 �

01 + 032

)~�13 � i�a~�03 � iPgs~�y32E

+s + iP�b~�12

+igi~�y01E

+i ;

@

@�~�03 = (i�!i �

032)~�03 � ia~�13 + iP�b~�02 + igi(~�00 � ~�33)E+i ;

@

@�~�y32 = (�i�b �

03 + 22

)~�y32 � iP �g�s ~�13E�s + i�b(~�22 � ~�33) + iP �gi~�

y02E

+i

(D.4)

where 2 = 12 + 32; P � ei�kz�i�!t; the four-wave mixing mismatch wavevector

�k = ka� ks+ kb� ki; the frequency mismatch �! = !a+!s�!b�!i = �1��b+

�!s � �!i; and various detunings are de�ned as �!i = !i � !3, �!s = !s � !12;

�1 = !a � !1; �2 = !a + !s � !2 = �1 + �!s , �b = !b � !23: The slow-varying

atomic operators are de�ned

~�01 � 1

Nz�l01e

�ikazl+i!at; ~�12 �1

Nz�l12e

ikszl+i!st; ~�02 �1

Nz�l02e

�ikazl+ikszl+i!st+i!at;

~�13 � 1

Nz�l13e

�i!at+i!it+ikazl�ikizl ; ~�03 �1

Nz�l03e

�ikizl+i!it; ~�y32 �1

Nz�ly32e

�i!bte�ikbzl ;

~�22 � 1

Nz~�l22; ~�33 �

1

Nz~�l33; ~�11 �

1

Nz~�l11 (D.5)

157

where Nz(2M + 1) = N .

The �eld equations are

@

@zE+s =

iNg�sc~�12; (D.6)

@

@zE+i =

iNg�ic~�03 (D.7)

where the �eld operators are de�ned as

E�s (z; t) �p2M + 1ays;le

�i!st; E+i (z; t) �p2M + 1ai;le

i!it: (D.8)

Langevin noises are not concerned here for we are interested in the normally-

ordered quantity, frequency conversion e¢ ciency, of input �eld and additional quan-

tum noise corrections vanish as the j2i ! j3i transition driven by pump laser b has

vanishing populations and atomic coherence. For energy and momentum conserva-

tion (P = 1), and in the weak �eld limit, we solve atomic operators in steady state

after linearizing with respect to the probe �elds

T01~�01 = ia(1� 2~�11 � ~�22 � ~�33);

T �32~�y32 = i�b(~�22 � ~�33);

T02~�02 = �ia~�12 + igs~�01E+s + ib~�03 � igi~�32E

+i ;

T13~�13 = �i�a~�03 � igs~�y32E

+s + i�b~�12 + igi~�

y01E

+i ;

T12~�12 = �i�a~�02 + igs(~�11 � ~�22)E+s + i~�13b;

T03~�03 = �ia~�13 + i~�02�b + igi(~�00 � ~�33)E+i (D.9)

where T01 = 012� i�1; T

�32 =

03+ 22

+ i�b; T02 = 22� i�2; T13 =

01+ 032

+ i�1� i�!i;

T12 = 01+ 2

2� i�!s; T03 = 03

2� i�!i; and note that ~�02; ~�13; ~�12; ~�03 are expressed

in �rst order of �elds and ~�01; ~�y32 in zeroth order. For population operators, we

solve them in the zeroth order of �elds and the nonzero steady states of population

and coherence operator are (s denotes steady state solution)

158

~�11;s =jaj2

�21 +

2014+ 2jaj2

; ~�00;s = 1� ~�11;s; ~�01;s =ia

012� i�1

(1� 2~�11;s): (D.10)

Substitute the above back into Eq. (D.9) and solve for ~�12 and ~�03: The parametric

coupling equations for the signal and idler �elds become

@

@zE+s = �sE

+s + �sE

+i

@

@zE+i = �iE

+s + �iE

+i (D.11)

where

�s =�N jgsj2cD

[~�11;s(T03 +jaj2T13

+jbj2T02

)� i�a~�01;sT02

(T03 +jaj2 � jbj2

T13)];

(D.12)

�s =�Ngig�scD

[~�00;s(�abT02

+�abT13

) +ib~�

y01;s

T13(T03 +

jbj2 � jaj2T02

)]; (D.13)

�i =�Ngsg�icD

[~�11;s(a

�b

T02+a

�b

T13) +

i�b~�01;sT02

(T12 +jbj2 � jaj2

T13)]; (D.14)

�i =�N jgij2cD

[~�00;s(T12 +jaj2T02

+jbj2T13

)�ia~�

y01;s

T13(T12 +

jaj2 � jbj2T02

)];

(D.15)

D � T12T03 + T12(jaj2T13

+jbj2T02

) + T03(jaj2T02

+jbj2T13

) +(jaj2 � jbj2)2

T02T13:

(D.16)

The absorption coe¢ cient for idler �eld is the real part of (��i) and phase velocity

vp = !=k(!) = c=n(!) that k(!) = n(!)!=c: The wavevector is related to coe¢ cient

�i that k(!) = Im(�i) + !=c so n(!) = 1+ Im(�i)=(!=c). The group velocity

is vg = d!=dk(!) = c=(n + !dn=d!) where n � 1; and it is this steep slope of

refractive index that makes a large group delay (dn=d! > 0 inside EIT window).

As an example in Figure D.1, we demonstrate the real and imaginary parts of self-

coupling coe¢ cient �i with the optical depth (opd) ��L = 150 (see Chapter 6 for

159

Figure D.1: Self-coupling coe¢ cient �i: A dimensionless quantity �iL is plotted withreal (solid blue) and imaginery (dashed red) parts as a dependence of idler detuning�!i showing a normal dispersion inside the EIT window.

more details on other parameters): The dispersion curve (Im(�iL)) inside the left

parametric coupling window bounded by two absorption peaks (Re(�iL)) shows a

normal dispersion indicating a group delay at the center of the window (see Figure

6.3 for complete parametric coupling windows). Note that we plot out unitless �iL

where L is in the order of millimeter for regular cold atomic ensemble, and see Sec.

II and III for detail discussion of various coupling coe¢ cients in Eq. (D.11) and

e¢ ciency dependence on optical depth.

160

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