EDOM Orsay 21-1-04 1

Entanglement, quantum measurement and quantum information with Rydberg atoms and cavities

Michel BRUNE

DÉPARTEMENT DE PHYSIQUE DEL’ÉCOLE NORMALE SUPÉRIEURE

LABORATOIRE KASTLER BROSSELParis France

Permanents: S. HarocheJ.-M. RaimondG. Nogues

Post-Doc: S. Kuhr

PhD: P. MaioliA. AuffevesT. MeunierS. GleyzesP. HyafilJ. Mozley

Dresden 10-05-04 2

Quantum and classical interacting systems

Coupling→ entanglement

State preparation

Measurement:• randomness• EPR correlation

|ψA,B⟩≠ |ψA⟩⊗|ψB⟩

|ψA⟩

"Rest of the world"

isolated system

Clas

sica

l in f

orm

a tio

n

|ψB⟩

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Quantum entanglement and Cavity QED

• Principle of cavity QED experiments:

•Two level atoms interacting with a single mode of a high Q

cavity

•The "strong coupling" regime: coupling >> dissipation

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Probing EPR entanglement and complementarity

Illustrating Bohr-Einstein dialog central concept: complementarity

A B C

Light slits: recoil of the slit monitors which path

informationNo interferences: mater behave as particles

Massive slits: insensitive to collisions with single particlesInterferences: mater behave

as wavesExperiment performed with photons, electrons, atoms,

molecules.

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The"Schrödinger cat"

• Elementary formulation of the problem: Superposition principle in quantum mechanics:→ Any superposition state is a possible state→ Schödinger: this is obviously absurd when applied to

macroscopic objects such as a cat !!

Up to which scale does the superposition principle applies?

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Schrödinger cat and quantum theory of measurement

• Hamiltonian evolution of a microscopic system coupled to a measurement apparatus:

...12at chat+ +Ψ = +

→ Entangled atom-meter state Problem: real meters provide one of the possible results not a

superposition of the two → too much entanglement in QM?

Need to add something?NO! "decoherence" does the work

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Entanglement and quantum informationCan one do something "useful" from the strangeness of quantum logic?• Yes:

Quantum cryptography: deos workQuantum communication: teleportationQuantum algorithms:

Search problems (Grover)Factorization (Shor)

• How? by manipulating qubits with a quantum computer

• Practically: extreeeeemely difficult to realize: a quantum computer manipulates huge Schrödinger cat states.

Anyway interesting for understanding the essence and the limits of quantum logic

Qubit: lwo level system0 and 1Classical bit:

0 ou 1 1 0 12

a b+

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outline1. Rydberg atoms in a cavity:

achieving the strong coupling regime

2. Rabi oscillation in vacuum: entanglement and complementarity at work

4. Rabi oscillation a mesoscopic field:Schrödinger cat state and decoherence

A B C

EDOM Orsay 21-1-04 9

1. Rydberg atoms in a cavity: achieving the strong coupling regime

One photon and one atomin a box:

Photon box: superconducting microwave cavity

“circular" Rydberg atoms

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The cavity

- one mode- a few trapped

photons

a "photon box":- superconducting Niobium mirrors- microwave photons:λ=6 mm, νcav=51GHz- photon lifetime:

Tcav=1ms ( Q = 3.108 )

0 2 4 6 8 10 12 14 16 1818

27

36

45

Measurement of cavity damping time

τcav= 1 msQ = 3.3 108

ν= 51.0989 GHz

Mic

row

ave

Pow

er(d

B)

time (ms)

5 cm

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The "circular" Rydberg atoms

e

g

n=51

n=50

νat=51.1 GHzνcav

"Circular Rydberg atoms":l=|m|=n-1

- radiative lifetime: 30 ms- dipôle: d= 1500 u.a.- ideal closed two levelsystem

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Laser velocity selection

Circular atompreparation:- 53 photons process- pulsed preparation0.1 to 10 atoms/pulse e-

State selectivedetector

One atom = one click

Cryogenic environmentT=0.6 to 1.3 K

weak blackbody radiation

85Rb

Experimental set-up

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Experimental setup

Atom preparation

detection

atomicbeam

lasers

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Resonant atom-field coupling

→ Coherent Rabi oscillation

Ω0/2π=50 kHzTrabi=20 µs

22

ge

0

1ge

0

1

e,0 g,1

0 0, 0 cos ,0 sin ,12 2

t te e i gΩ Ω → ⋅ − ⋅

Electricdipole

couplingΩ0

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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-

eΩ0=47 kHzTRabi=20µse,0

g,1

Coherent Rabi oscillation replaces irreversible damping

by spontaneous emission

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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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0

2

4

6

8

10

Pos

ition

(cm

)

Tailored preparation of of tree entangled qubits

π/2 π2π

π/2

θ

D DD

π/2 π/2 π/2

Time

π/2

D

• Rabi oscilation in C

• Classical p/2 pulse

• Detection

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

−π/2

Atom # 1

Atom # 2

Atom # 3

EDOM Orsay 21-1-04 18

2. Rabi oscillation in vacuum:entanglement and complementarity at work

A B C

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Complementarity in the Ramsey interferometer

Classical π/2 pulses Ramsey fringes signalwith two classical

pulses: results from a two path

interference

-1 0 1 20,0

0,2

0,4

0,6

0,8

1,0

φ/π

P g(φ)

Atomic beam

Classical microwave fields acts as "beam-splitters"

for the internal atomic state

e

gS

De

g

e

S

e

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From classical to quantum beam splitters: • π/2 pulse in a large coherent state:

( )12

e e gα α⊗ → + ⊗

When α is large enough, one more photon in the field does not make any difference on the field state

NO which path information stored in the field: "classical beam splitter"

n n

0 2 0 4 0 6 0 8 0 10 0 12 0

P(n)

n

nPoisson law

e gα α α≈ ≈

R1

e e

g

α

n

( ),0 ,112

0 e ge +⊗ →R1

,0e

,1g

,0eAtom-field EPR pair:Hagley et al. PRL 79,1 (1997)

The photon number is a perfect label of the atomic stateFULL which path information stored in the field: "quantum beam splitter"

• π/2 pulse in vacuum: α=0

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Resonant interaction with a coherent field as "beam-splitter" in a Ramsey interferometer

Classical π/2 pulses

e

gS

De

g

e

S

e

π/2 pulse in C

π/2

entangled EPR pair

-1 0 1 20,0

0,2

0,4

0,6

0,8

φ/π

0,2

0,4

0,6

0,8

0,2

0,4

0,6

0,8

n=12.8

n=2

n=0.31

α=3.6

α=1.41

α=0.56

0,2

0,4

0,6

0,8

1,0

n=0α=0

Pg

Pg

( )12 gee gα α⊗ + ⊗

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Quantitative interpretation in term of atom-cavity entanglement

• Variation of fringe Visibility V: R1 R2

e α⊗

( )12 gee gα α⊗ + ⊗

.e gV α α η=

η: saturated contrast at large n

Reduced atom density matrix:

*11

2 1

e gat

e g

α αρ

α α

=

0 2 4 6 8 10 12 14 160,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

experimenttheory

V

Average photon number <n>

η=0.73

EDOM Orsay 21-1-04 23

4. Rabi oscillation a mesoscopic field:Schrödinger cat state and decoherence

Dresden 10-05-04 24

Coherent field states

• Number state: |N⟩• Quasi-classical state:

2 2 ;!

Ni

N

e N eN

α αα α α− Φ= =∑

( ) 2;!

NNP e NN

NN α−= =

•Photon number distribution

∆N = 1/|α|

∆N . ∆Φ > 1

∆Φ=1/|α|

1

0 1 2 3 4 5 60,0

0,1

0,2

0,3

0,4Poisson distribution

N=2

P(N

)

Photon number N

|α|

• Phase space representation

Φ

Re(α)

Im(α)

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Rabi oscillation in a classical field

Rabi oscillation results from a quantum interference between probability amplitudes of the two atomic "eigenstates"

( )

( ) ( ) ( )

( ). 2 . 2, ,

0

cos . 2 . .sin . 2

1 . .2

R R

at

at R R

i t i tat at

e

t t e i t e

e e

ψ

ψ

ψ ψ− ++ −

Ω Ω

=

→ = Ω − Ω

= +

( )( )

,

,

1 2

1 2

at

at

e g

e g

ψ

ψ

+

− =

+=

−

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Rabi oscillation in a mesoscopic field

( )

( ) ( ) ( ) ( ) ( )( )

( ) ( )( )

. 2 . 2, ,

. 2 . 2

0

1 . .2

1 . .2

R R

R R

i t i tat at

i t i t

e

t e t t e t t

e t e t

ψ α

ψ α ψ α ψ

ψ ψ

− ++ + − −

− ++

Ω Ω

Ω Ω−

= ⊗

→ ≈ ⊗ + ⊗

= +

Rabi oscillation frequency:2

0 1 where R N N αΩ = Ω + =

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Graphical representation of the atom-field state in the complex plane:

( ) ( ) ( ),att t tψ α ψ+ + +≈ ⊗

Re(α)

Im(α)

φ

( ) ( )

( ) ( )( ),

.

12

i t

at

ti

e

e

t

t e g

α α

ψ

− Φ

−+

Φ

+ =

= +

( ) 0. 4t t NΦ = Ω

( )( ),at

t

t

α

ψ+

+

Dresden 10-05-04 28

Graphical representation of the atom-field state in the complex plane:

( ) ( ) ( ),att t tψ α ψ− − −≈ ⊗

Re(α)

Im(α)

−φ

( ) ( )

( ) ( )( ),

.

12

i t

at

ti

e

e

t

t e g

α α

ψ

+ Φ−

+ Φ− = −

=

( )( ),at

t

t

α

ψ+

+

( )( ),at

t

t

α

ψ−

−

( ) 0. 4t t NΦ = Ω

Dresden 10-05-04 29

Graphical representation of the atom-field state in the complex plane:

Re(α)

Im(α)

−φ

( )( ),at

t

t

α

ψ+

+

( )( ),at

t

t

α

ψ−

−

φ

( ) ( ) ( )( ). 2 . 21 . .2

R Ri t i tt e t e tψ ψ ψ− ++

Ω−

Ω≈ +

( ) 0. 4t t NΦ = Ω

• Entangled atom-field state:

A Schrödinger cat state:Field phase "measures" the atomic state

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Measured phase distribution of the "Schrödinger cat" field state

- Coherent field containing 10 to 40 photons- atom 1: prepares the "cat state"- atom 2: measures the phase distribution of the field

S

|g> Pg ?Atom 1

-200 -150 -100 -50 0 50 100 150

0,5

0,6

0,7

0,8

Pg

Phase θ (°)

33 injected photons no atom 1 atom 1: 335m/s atom 1: 200m/s

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Wigner of the field state

15 injected photonsV=335 m/sTcav=800 µs

-6

-4

-2

0

2

4

6

-6-4

-2

0

2

4

6

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

Im(α)

Re(α)

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

X Axis Title

Y Ax

is T

itle

-0,2500

-0,1563

-0,06250

0,03125

0,1250

0,2188

0,3125

0,4063

0,5000

Wigner function can be measured: B. Englert et al., Opt. Comm. 100, 526 (1993), Lutterbach and Davidovich, PRL 78 2547 (1997)P. Bertet et al., PRL 89, 200402 (2002)

Dresden 10-05-04 32

Rabi oscillation entanglement and complementarity

Rabi oscillation results from a quantum interference between andNo oscillations as soon as: complementarity again!

( ) ( ) ( ) ( ) ( )( ). 2 . 2, ,

1 . .2

R Ri t i tat att e t t e t tψ α ψ α ψ− +

+ + −Ω Ω

−≈ ⊗ + ⊗

,atψ + ,atψ −

( ) ( ) 0t tα α+ − ≈

Preparation of a phase catField states coincide again: revival of Rabi oscillation signature of the coherence of the "cat" state

Collapse of Rabi oscillation10 20 30 40 50

0.2

0.4

0.6

0.8

1Pe(t)

Ω0t/2π

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Demonstration of coherence by induced revival

Atom-field entanglement

π rotation of the atomic state

Field phase evolution is reversed

Recombinaison ot the two field components:

Revival of Rabi oscillation

Morigi et al PRA 65, 040102

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Induced revival signal

-20 -10 0 10 20 30 40 50 60

0,0

0,2

0,4

0,6

0,8

1,0

Tra

nfe

r

Interaction time

18.5 µs

-20 -10 0 10 20 30 40 50 60

0,0

0,2

0,4

0,6

0,8

1,0

Tra

nfe

r

Interaction time

-20 -10 0 10 20 30 40 50 60

0,0

0,2

0,4

0,6

0,8

1,0

Tra

nsf

ert

Interaction time

Π Pulse

22 µs

-20 -10 0 10 20 30 40 50 60

0,0

0,2

0,4

0,6

0,8

1,0

Tra

nsf

er

Interaction time

23.5 µs

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Perspectives

• Rydberg atoms and superconducting cavitiesA two cavity experiment

EPR pair of Schrödinger cat states (non-locality at the mesoscopic scale)

– Non-locality and decoherence

Complex manipulation of quantum information– Quantum feedback– Elementary algorithm– Error correction code

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The teamPhD

• Frédérick .Bernardot• Paulo Nussenzweig• Abdelhamid Maali• Jochen Dreyer• Xavier Maître• Gilles Nogues• Arno Rauschenbeutel• Patrice Bertet• Stefano Osnaghi• Alexia Auffeves• Paolo Maioli• Tristan Meunier• Philippe Hyafil• Sébastien Gleyzes• Jack Mozley

Post doc• Ferdinand Schmidt-Kaler• Edward Hagley• Christof Wunderlich• Perola Milman

Colaboration• Luiz Davidovich• Nicim Zagury• Wojtek Gawlik

Permanent• Gilles Nogues• Michel Brune• Jean-Michel Raimond• Serge Haroche

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Les bizarrerie quantiques:intrication, mesure et décohérence

• Limite classique-quantiquePas de superpositions à notre échelleLe “chat de Schrödinger"

Nous n’observons qu’une toute petite fraction des états possibles

• Décohérence

Couplage inévitable d’un système macroscopique à un environnement

Quelques états stables (base préférée)Les superpositions quantiques de ces états sont très rapidement détruites

Environement

( )12

+ ⇔

Dresden 10-05-04 38

Rabi oscillation in a small coherent field

0 20 40 60 800,0

0,5

1,0

interaction time (µs)

P g(t) exp

fitS

e

e-

e

2 0.85 photonα =

α

Dresden 10-05-04 39

Rabi oscillation in a small coherent field:observing discrete Rabi frequencies

Fourier transform of the Rabi oscillation signal

0 25 50 75 100 125 150

Frequency (kHz)

FFT

(arb

. u.)

Discrete peaks corresponding to

discrete photon numbers

0Ω0 2Ω

0 3Ω

Direct observation of field quantization

in a "box"

Dresden 10-05-04 40

Rabi oscillation in a small coherent field:Measuring the photon number distribution

( ) ( )( )0/1 1 co( ) s 1 .

2t

gN

P t P et NN τ−− Ω += ∑Fit of P(n) on the Rabi oscillation signal:

0 1 2 3 4 50,0

0,1

0,2

0,3

0,4

0,5

Photon number

P(n)

Measured P(n) Poisson law

accurate field statistics measurement

0.85N =

Dresden 10-05-04 41

Rabi oscillation in small coherent fields

0,0

0,5

1,0

Four

ier

Tran

sfor

m A

mpl

itude

(β)

(γ)

(δ)

(α)

(d)

(c)

(b)

(a)

(D)

(C)

(B)

(A)

Probability P(n)

e to

g t

rans

fer r

ate

0,0

0,5

0,0

0,5

0 30 60 90

0,0

0,5

Time (µs)0 50 100 150

Frequency (kHz)

0,00,5

1,00,0

0,50,0

0,5

0 1 2 3 4 5

0,00,3

n

<n>=0.06

<n>=0.4

<n>=0.85

<n>=1.77

Phys. Rev. Lett. 76, 1800 (1996)

Dresden 10-05-04 42

1. Préparation du champ à mesurer:Un état cohérent |α> par exemple2. Addition d'un champ de même

amplitude et de phase variable θ

3. Un atome absorbe le champ résiduel: mesure P(0)

Observation de la distribution de phase du champ:“detection homodyne”

α

( ). ie π θα −

θ

signal d'absorption par un atome S

|g> Pg ?

-200 -150 -100 -50 0 50 100 1500,4

0,5

0,6

0,7

0,8

Pg

Phase θ (°)

33 photons

Interférence destructive pour θ=0.

Dresden 10-05-04 43

Détection des deux composantes de phase

20 25 30 35 40 45 50 55

-40

-20

0

20

40

60

Fiel

d ph

ase

shift

(°)

Atomes "lents"N petit

Atomes rapidesN grand

0 int. 4t NΦ = Ω