Les Houches 2003 1

Cavity Quantum ElectrodynamicsLecture 2:

entanglement engineering with quantum gates

Michel BRUNE

DÉPARTEMENT DE PHYSIQUE DEL’ÉCOLE NORMALE SUPÉRIEURE

LABORATOIRE KASTLER BROSSEL

Les Houches 2003 2

CQED with Rydberg atoms and quantum information

• Vacuum Rabi oscillation: strongly couples two qubits→ achieves quantum gates:

Quantum phase gate, CNOT gate→ allows for step by step preparation of complex

entangled states:

|e>

|g>|0>

|1>

|2>

Atom qubit Field qubit

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Vacuum Rabi oscillation and quantum gates

g

eωge= ωcav

0 20 40 60 80 1000,0

0,2

0,4

0,6

0,8

1,0P g (

50 c

irc)

interaction time (µs)

e-

e

e,0

g,1

• EPR pair preparation

•Atom-field state exchange

• Phase gate, QND detection of a single photon

π/2

π

2π

Ω0=47 kHzTRabi=20µs

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outline

1. Cavity qubit as a quantum memory

2. Quantum phase gate (QPG) and CNOT gate

3. CNOT as QND detection of one photon

4. Step by step preparation of a GHZ triplet

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1. Cavity qubit as a quantum memory

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Vacuum Rabi oscillation: the field as a quantum memory

g

eωge= ωcav

0 20 40 60 80 1000,0

0,2

0,4

0,6

0,8

1,0P g (

50 c

irc)

interaction time (µs)

e-

e

e,0

g,1

• EPR pair preparation

•Atom-field state exchange:

Writing:

Reading:

π/2

π

Ω0=47 kHzTRabi=20µs

( ) ( )0 1 0e g e ggc e c g ic c⊗ −⊗+ +⇒

( ) ( )0 1 0e g e ggc e c g ic c⊗ −⊗+ +⇐

X. Maître et al. PRL 79, 769(1997)

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Quantum memory: experimental realization

• writing and reading the field state:

ωR≈ ωat

|e>

|g>

e

SR

π/2 pulse R1 g

π/2 pulse R2

Source atom:Writer

Probeatom:Reader

eg

•R1: preparation of arbitrary e-g superpositions•R2: analysis of arbitrary e-g superpositions

• Exp 1: no pulse R1 : writing and reading field energy• Exp 2: pulse R1 on, storing a superposition state

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Quantum memory: storing energy

0 1 2 3 4 50,0

0,2

0,4

0,6

0,8

1,0

cond

ition

al p

roba

bilit

y Πe2

/g1

T/Tr

Measurement of the cavity damping time with 1 injected photon!

Quantum memory, X. Maître et al. PRL 79, 769(1997)

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2. Quantum phase gate (QPG) and CNOT gate

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Single photon induced Rabi oscillation

g

eωge= ωcav

0 20 40 60 80 1000,0

0,2

0,4

0,6

0,8

1,0P g (

50 c

irc)

interaction time (µs)

e-

e

e,0

g,1

• EPR pair preparation

•Atom-field state exchange

• Phase gate, C-Not gate:

π/2

π

2π

Ω0=47 kHzTRabi=20µs

,0 ,0e e⇒ −A. Rauschenbeutel et al., PRL 83, 5166 (1999)

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Principle of the Quantum Phase Gate (QPG)

• uses three level atoms:e : 51c

ωcav

ωR

g : 50c

i : 49c

• qubit 1: atom qubit 1

0

g

i

=

=

e : ancillary level

• qubit 2: field qubit 1 : one photon

0 : cavity vacuum

• Truth table of a 2π pulse in C:

π phase shift if control and target =1

0,0 0,0

0,1 0,1

1,0 1,0

1,1 1,1ie π

⇒

⇒

⇒

⇒

,0 ,0

,1 ,1

,0 ,0

,1 ,1i

i i

i i

g g

g e gπ

⇒

⇒

⇒

⇒

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0

2

4

6

8

10

Pos

ition

(cm

)

From QPG to CNOT gate

2π

π/2

2π

D

π/2

Time

π/2

D

• QPG in C

• Classical π/2 pulse = Hadamar transform

• Detection

π/2

Atom π/2

π/2

Two π/2 pulses: Ramsey interferometer on the g-i transition

2π

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QPG as a CNOT

The field sate controls the phase of Ramsey fringes:

eN=1

N=0

0 2 4 6 8 100,0

0,2

0,4

0,6

0,8

1,0

φR

P i(ωR)

C-Not operation: the photon number controls the final atomic state.

0, 0,

0, 0,

1, 1,

1, 1,

i i

g g

i g

g i

⇒

⇒

⇒

⇒

Ramsey fringes:

π/2

π/2

2π

g

g

g

i

i

i

For a proper choice of the phase of the interferometer:

Ramsey interferometer

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CNOT as QND detection of one photon

The field sate controls the phase of Ramsey fringes:

eN=1

N=0

0 2 4 6 8 100,0

0,2

0,4

0,6

0,8

1,0

φR

P i(ωR)

Ramsey fringes:

π/2

π/2

2π

g

g

g

i

i

i

For a proper choice of the phase of the interferometer

the atom state is perfectly correlatedto the photon number 0 or 1

Ramsey interferometer1

0

, 1

0,

,

, g

g

g

i⇒

⇒ QND detection of one photon

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3. CNOT as QND detection of one photon

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Input-output characterization of the QND measurement

QNDmeasurement

Signal: atomdetected in g or i

Input:0 ou 1photon

Output:0 ou 1photon

1

3

2

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0

2

4

6

8

10

Pos

ition

(cm

)

QND detection of one photon: experimental timing

π 2π

π/2

D D

π/2

Atom # 1

Atom # 2

• Initial field state: N=0, prepared with a bunch of atoms in g. (Cavity “Cooling”)• atom # 1 prepares one photon• atom # 2 performs a QND detectionHow good is the fidelity of the QND measurement?

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QND detection of one photon:Ramsey fringes signal

G. Nogues et al., Nature 400, 239 (1999)A. Rauschenbeutel et al., PRL 83, 5166 (1999)

0 10 20 30 40 50 600,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

0 photon

1 photon

Pro

babi

lity

Pi 2

frequency (ω R-ω gi)/2π (kHz)

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Input-output characterization of the QND measurement

QNDmeasurement

Signal: atomdetected in g or i

Input:0 ou 1photon

Output:0 ou 1photon

1

3

2

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0

2

4

6

8

10

Pos

ition

(cm

)

QND detection: the photon is still there

π2π

π/2

D D

π/2

Atom # 1: g

Atom # 2: g

• Atom # 1 detects a small blackbody field in C: nth=0.25 photon• Atom # 2 prepared in g checks the result by absorbing the field

→ lock at two atom correlation to check if the QND detected photon is still there

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QND detection: Checking the result

0 10 20 30 40 50 60

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

Frequency ωR (kHz)

Prob

abilit

y

P(e2 if i1)

The absorption rate of atom 2 is modulated depending whetherDetection of atom 1 in i1 corresponds to 0 or 1 photon in C

→ The photon is still here!

i1 → 0 photon

i1→ 1 photon

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Input-output characterization of the QND measurement

QNDmeasurement

Signal: atomdetected in g or i

Input:0 ou 1photon

Output:0 ou 1photon

1

3

2

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0

2

4

6

8

10

Pos

ition

(cm

)

QND measurement: field input-output correlation

π/2 π2π

π/2

D DD

π/2

Time

Atom # 1

Atom # 2

Atom # 3

• Depending on the detected state of atom # 1, 0 or 1 photon is prepared in C• atom #2 performs the QND measurement• atom #3 checks the final state of the field

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0

2

4

6

8

10

Pos

ition

(cm

)

Input-output correlation without QND meter

π/2 π

DD

Time

Atom # 12π

π/2

D

π/2

Atom # 2

Atom # 3

• Depending on the detected state of atom # 1, 0 or 1 photon is prepared in C• atom #2 is not prepared• atom #3 checks the final state of the field

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Input-output correlation without QND meter

• Atom#1 - atom#3 correlation

0

0.1

0.2

0.3

0.4

0.5

atom #1 and #3 coincidence

Prob

abili

ty

no atom #2

g1g3 g1e3

e1g3

e1e3

1 photon 0 photon

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Input-output correlation with QND meter

• Atom#1 - atom#3 correlation

1 photon 0 photon

0

0.1

0.2

0.3

0.4

0.5

atom #1 and #3 coincidence

Prob

abili

ty

no atom #2

g1g3 g1e3

e1g3

e1e3

0

atom #2 detected in gatom #2 detected in i

g2

g2

i2

i2

Easy quantitative analysis of performances:• absorption rate of atom 2: 10%• fidelity of QND measurement: 80%

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Factor of merit of the QND detection

0,0 0,2 0,4 0,6 0,8 1,00,0

0,2

0,4

0,6

0,8

1,0

dete

ctio

n fid

elity

η

absoption rate ε

QND

classical

Ideal QND

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4. Step by step preparation of a GHZ triplet

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0

2

4

6

8

10

Pos

ition

(cm

)

Three qubits entanglement : experimental sequenceD

Time

Atom # 1

e-

V

field

D

Atom # 2

enta

ngle

men

t

π/2

• Atome # 1 π/2 ( )1 1 11, ,0 0 ,2

1e e g⇒ + EPR Pair preparation

π/2

• Atome #2 π/2 ( )2 2 212

g i g⇒ +

2π

2π ( ) ( )12 2 21 20, ,2

11 e gi g i g+ − ⇒ + C-Notgate

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The "GHZ" state"

• prepared state:

• In term of qubits:

• In term of spin 1/2:

" GHZ triplet " (Greenberger Horne Zeilinger)

( )1 0,0,0 1,1,12

+

( )2 21 1 ,,2

,1 ,c c−+++ −+ −

( ) ( )2 2 2 21 11 0, 1,2

e gi g i g−++

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Caracterization of the prepared state

• ideal case:

• Density matrix of the prepared state:

* . . . . . .. * . . . . . .. . * . . . . .. . . * . . . .. . . . * . . .. . . . . * . .. . . . . . * .

. . . . . .* *

*

tripletρ

=

.....................+ + + − − −

performed measurements:* measurement of σz1. σz2. σz3

* measurement of σx1. σx2. σx3

( )1 12 21 , ,2

, ,trip clet cψ + −+ −++ −=

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Measurement of σz1. σz2. σz3 : practical realization

• Step 1: transfer of the field state to a third atomperforming a π absorption pulse in C:

• step 2: detection of each atom for measuring σz1. σz2. σz3- atoms 1 et 3 : direct measurement of energy- atome 2: measurement of energy after applucation of

an external π/2 pulse:

( ) ( )2 321 2 211 ,0 12

, g i g ge g i+ − + ⊗

( ) ( )1 13 32 2 2 2 012

g i gge g ei+ − ⇒ + ⊗

( )321 1 2 3, ,1 , ,2

+ − −++ −+

( )( )

2 2 2

2 2

1 2

1 2

g i i

g i g

+ →

− →π/2 ( )1 2 3 1 2 3

1 , , , ,2

e i g g g e⇒ +

Les Houches 2003 35

0

2

4

6

8

10

Pos

ition

(cm

)

Full set of operations for measurement of σz1. σz2. σz3

π/2 π2π

π/2

D DD

π/2

Timeθ

π/2

D

• Rabi oscillation in C

• Classical p/2 pulse

• Detection

π/2

Atom # 1

Atom # 2

Atom # 3

State before detection:

( )1 2 3 1 2 31 , , , ,2

e i g g g e⇒ +

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Measurement results:

• measurement of σz1. σz2. σz3

Plong =Peig + Pgge = 0.58 (0.02)

Pgi

g

Pgi

e

Pgg

g

Pgg

e

Pei

g

Pei

e

Peg

g

Peg

e

0.4

0.3

0.2

0.1

0

|+1,+2,+3⟩

|-1,-2,-3⟩

Rauschenbeutel et al., Science 288, 2024 (2000)

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0

2

4

6

8

10

Pos

ition

(cm

)

Full set of operations for measurement of σx1. σx2. σx3

π/2 π2π

π/2

D DD

π/2 π/2

Timeθ

π/2

D

• Rabi oscillation in C• Classical p/2 pulses, phase of detection pulses adjusted to measure σx1 and sx3

• Detection

π/2

Atom # 1

Atom # 2

Atom # 3

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Fidelity of preparation of the GHZ state

• fidelity:

F > 0.3 garanties non-separabilitysee also: Sacket et al. Science 288, 2024 (2000)

preparation of a 4 ions GHZ state in one step

0.54 (0.03)triplet tripletF ψ ρ ψ= =

• measurement of σz1. σz2. σz3

Plong =Peig + Pgge = 0.58 (0.02)• measurement of σx1. σx2. σx3

A= ⟨ σx1. σx2. σx3 ⟩ = -0.28 (0.03)

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Engineered versus "spontaneous" entanglement• first experiments on Bell inequalities:

use of "spontaneous" entanglement relying on a symetry of the system

• cavity QED: provide many tools for step by step entanglement engineering

• Where is the limit? answer not clear…. ….but there are still interesting things to do!

Non-linearcristal

σ−

σ−σ+

σ+

• atomic cascadeAspect, Grangier

• Parametric down conversion: Mandel, Zeilinger Gisin …

0

2

4

6

8

10

Pos

ition

(cm

)

π/2 π2π

π/2

D DD

π/2 π/2

Time

Atom # 1

Atom # 2

Atom # 3

the most complex sequence of gatesapplied on individually addressable

and measurable qubits

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References (1)• QND measurement in microwave CQED experiments:

M. Brune, S. Haroche, V. Lefevre-Seguin, J.M. Raimond and N. Zagury: "Quantum non-demolition measurement of small photon numbers by Rydberg-atom phase sensitive detection", Phys. Rev. Lett. 65, 976 (1990).M.Brune, S. Haroche, J.M. Raimond,L. Davidovich and N. Zagury. "Manipulation of photons in a cavity by dispersive atom-field coupling: QND measurement and generation of "Schrödinger cat"states". Phys Rev A45, 5193, (1992).S. Haroche, M. Brune and J.M. Raimond. "Manipulation of optical fields by atomic interferometry: quantum variations on a theme by Young".Appl. Phys. B, 54, 355, (1992).S. Haroche, M. Brune and J.M. Raimond. "Measuring photon numbers in a cavity by atomic interferometry: optimizing the convergence procedure". Journal de Physique II

Les Houches 2003 41

References (2)• Gates: QPG or C-Not, algorithm:

M. Brune et al., Phys. Rev. Lett, 72, 3339(1994).Q.A. Turchette et al., Phys. Rev. Lett. 75, 4710 (1995).C. Monroe et al., Phys. Rev. Lett. 75, 4714 (1995).A. Reuschenbeutel et al., PRL. G. Nogues et al. Nature 400, 239 (1999). S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune, J.M. Raimond and S. Haroche, Phys. Rev. Lett. 87, 037902 (2001)F. Yamaguchi, P. Milman, M. Brune, J-M. Raimond, S. Haroche: "Quantum search with two-atom collisions in cavity QED", PRA 66, 010302 (2002).

• Q. memory: X. Maître et al., Phys. Rev. Lett. 79, 769 (1997).

• Atom EPR pairs:CQED: E. Hagley et al., Phys. Rev. Lett. 79, 1 (1997). Ions: Q.A. Turchette et al., Phys. Rev. Lett. 81, 3631 (1998).

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4. Non-resonant gate

entanglement directly generated by a two atom "collision" catalyzed by the cavity

application to a non-resonant phase gate

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Two atoms and one mode

• Non-resonant coupling

RΩ

, ,1g g

, , 0g g

cavνatν

One photon coupling

, ,0g e, , 0e g

cav atδ ν ν= −

, , 0g e, ,0e g

•Atoms can exchange energy by virtually emitting a photon in C.•Two atom EPR state preparation for a π/2 "Raman" pulse:

( )1 , , 02EPR e g g eψ = + ⊗

Zheng et al PRL 85 2392 (2000)

20

2R δΩ

Ω =

Les Houches 2003 45

• effect of cavity damping:projection on |g,g,0>Full loss of entanglement• probability of error:

• Resonant case:

• error rate reduced as:

efficient with slower atoms

Advantage of non-resonant method of entanglement:Sensitivity to cavity damping

cavδ Γ

RΩ , , 0g e, ,0e g

, ,1g g

, , 0g g

2

int.colerr cavP T

δΩ ≈ Γ

int. 2R T πΩ =

int.res res

err cavP T≈ Γint

. 2resT πΩ =

colerrres

err

PP δ

Ω≈

20

2R δΩ

Ω =

Les Houches 2003 46

Advantage of non-resonant method of entanglement:Sensitivity to blackbody radiation

• coupling in the presence of N photons:

RΩ, ,e g N

, ,g e N

, , 1g g N +

, , 1e e N −

NΩ

1NΩ +

Due to destructive interference between two probability amplitudes,

the effective coupling is to first order independent of N:

The method works even in the presence of blackbody radiationSimilar to "hot" gate for ions:

Moelmer et al PRL 82 1835 (2000)

( )2 2 20 0 0. 1 .

2 2 2R

N Nδ δ δ

Ω + Ω ΩΩ ≈ − =

Les Houches 2003 47

Principle of the experiment

• Two atoms with different velocity "collide" in C.

Posi

tion

cavity centeratom #1

atom

#2

δ,θ

enta

ngle

men

tTime

Les Houches 2003 48

Cavity assisted "collision": experimental signal

0 1 2 3 40,0

0,2

0,4

0,6

0,8

1,0

Pe1-g2

Pg1-e2

Pro

babi

lity

η (x10-6)

π/2 pulse: preparation of:

Osnaghi et al., PRL 87, 037902 (2001)

• Solid line: second order coupling• doted line: numerical integration

( )1 , , 02EPR e g g eψ = + ⊗

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Measuring the EPR entanglement:

-1 0 1 2 3-1,0

-0,5

0,0

0,5

1,0

Tranverse EPR correlation

⟨σ1,

xσ 2,φ ⟩

φ/π

Fidelity of the EPR state: 0,79

Les Houches 2003 50

• Definition of logical qubits:

• result of a 2π two atom "Raman" pulse (taking into account light shift of levels:

Quantum phase gate using a cavity assisted collision

e : 51cωcav

ωgi

g : 50c

i : 49c

atom 1 atom 2

1

10

0

0,0 0,0

0,1 0,1

1,0 1,0

1,1 1,1ie π

⇒

⇒

⇒

⇒

The cavity field is not affected

, ,

, ,

, ,

, ,i

i g i g

i i i i

e g i

e

g

e i e iπ

→

→

→

→

Application to Grover algorithm: F. Yamaguchi, et al.

"Quantum search with two-atom collisions in cavity QED", PRA 66, 010302 (2002)