Rabi oscillation at the quantum-classical boundary Generation of Schrödinger cat states

Rabi oscillation at the quantum-classical boundaryGeneration of Schrödinger cat states

Alexia Auffèves-Garnier

Soutenance de thèse29 juin 2004

About coherent superpositions

Any superposition of states is a possible state

| | |ia e b Young's hole experiment

Physical meaning of a "coherent superposition"?

a bP P P Addition of the probabilityamplitudes of each path

Interference fringes

Signature of a coherent superposition=interference fringes

Quantum phase of the superposition=phase of the fringes

| | |ia e b

Entanglement

(1)

S(2)

Coherent superposition for a bipartite system

"Entangled state" = non factorisable state

No system is in a definite state

Correlations in all the basis Quantum correlationsViolation of Bell's inequalities

H

V V

H

H

V

H

V

Anticorrelation

The two photonsform an EPR-pair

1| (| HV | VH )

2epr

1| (| H V | V H )

2epr

The Schrödinger-cat paradox

Entanglement of a microscopic system with a macroscopic one

A two-level atom and a cat in a box

| | alive

| | dead

e

g

Total correlation

The cat "measures" the atomic state

Linear evolution

The system form an EPR pair

Quantum correlations

Atom projected on |e>+|g>

Cat projected on |dead>+|alive>

Macroscopic state superposition

What enforces classicality: concept of complementarity

A superposition of states remains coherent No information about the state

A slit acts as a Which-Path detector

Two possible states for the slitcorrelated with the two particle's states

| |

| |a

b

a

b

The particle-slit system gets entangled

1| (| , | , )

2i

a ba e b

| | |a bC Contrast of the fringes

No interference if there exist a Which-Path information

Decoherence

A macroscopic object interacts with its environment and gets entangled with it

Coherent superposition (dead> and |alive>)

Statistical mixture (|dead> or |alive>)

No interference between macroscopic states

Time of decoherence all the faster as the two cat states are more different

Decoherence

Quantum correlations in all the basis

Classical correlations in the « natural » basis

The quantum-classical boundary

decohTintT

Microscopic object

Mesoscopic object

Environment

-3 parts-2 time scales

-Entanglement-« Schrödinger cat » states-Quantum behavior

Continuous parameter to explore the quantum-classical boundary?

Quantum world Classical world

int decohT T int decohT T-Continuous monitoring of the environment-No entanglement-Classical behavior

Tools of CQED

0 20T s

Rydberg states |e> and |g>High Q superconducting cavity

Vacuum Rabi period

Field relaxation

Strong coupling regime

30msatT

0 ,cav atT T T

1 mode of the electromagnetic field

A two level system

1 mscavT Long life time

51 (level e)

50 (level g)

51.1 GHz

From quantum to classical Rabi oscillation

Microscopic object

Mesoscopic object

Environment

Resonant interaction between the atom and the fieldRabi oscillation

Quantum Rabi oscillationClassical Rabi oscillation

intTdecohT

Rydberg atom

Coherent field in the cavity mode

Infinity of external modes

-Atom-field entanglement-Preparation of mesoscopic coherences of the field

-Field unsensitive to the atom-Factorized atom-field state

Outline

• Elements of theory– Results on Rabi oscillation – Rabi oscillation at the quantum-classical boundary: the atomic

point of view– Effect of the Rabi oscillation on the field

• Experimental study– The atom-field system– Preparation of mesoscopic states superpositions of the field:

experimental results– Evidence of correlation between atomic and field state– Coherence of the prepared superposition: where are the cats?

Basics on two-level systems

Y

Z

X

|e>

|g>

| cos( / 2) | sin( / 2) |ie e g

Two-level system {|e>;|g>}

General state

A vector in the Bloch sphere

Dynamics of the system

Precession around the eigenstates of the hamiltonian

|

|g>

|e>

|g>

|e>

Classical Rabi oscillation

|e>

Two-level system {|e>;|g>} interacting with a resonant field |g>

|e>

cos( )cl t

Rotating frameRotating wave approximation

Y

Z

X

|g>

|+>

| - >

Eigenstates of the hamiltonian1

| (| | )2

e g

1| ( | | )

2e g

Rabi oscillation at frequency cl1

1

0

Time (a.u.)

eP

Rabi oscillation as an interference

|+>

| - >

|e> |e>

Evo

lutio

nC

ha

nge

of b

asi

s

De

tect

ion

Ch

ang

e o

f ba

sis

// 2 21|| | ( )

2| i ti tet ee

Detection in {|e>,|g>} basis

Evolution

Classical Rabi oscillation: a quantum beat between two indistinguishable paths

1

1

0

Time (a.u.)

eP

Rabi oscillation in a quantized field

Two-level system {|e>;|g>} interacting with a resonantquantized field |n>

0 | | | |)2JCH a g e a e g

Jaynes-Cummings hamiltonian

Exchange of a quantum of energy

|e,n> |g,n+1>0 1n

|g>

|e>

|n-1>

|n>

|n+1>

|e,n>; |g,n+1> : two levels coupled and degenerate

Eigenstates of the hamiltonian:"Dressed states" 0 1n

| n

| n 1| (| , | , 1 )

2n e n g n

1| (| , | , 1 )

2n e n g n

Rabi oscillation between |e,n> and |g,n+1> at frequency

0 1n

The vacuum Rabi oscillation

0 30 60 900.0

0.2

0.4

0.6

0.8

Pe(t

)

time ( s)

Initial state |e,0> Rabi oscillation at0 Vacuum Rabi frequency

0 49kHz

0 0| , 0 cos | ,0 sin | ,12 2

t te e i g

Atom-field entangled state

Maximal entanglement at

0 2t

Formation of an EPR-pair

Rabi oscillation in a quantum field

entanglement

Rabi oscillation in a classical field

No entanglement

Continuous evolution?

Outline





Rabi oscillation in a mesoscopic coherent field

0t/T

eP (t)

0 1n n 1( ) ( ) cos( ) 1

2e nn

P t p n t with

Collapses and revivalsSpectrum of the frequencies

for a mesoscopic field

nDistribution width

Spectrum width

0 02

n

n nn

n

0

( . .)nP a u

n

0 n

0

Collapse when theside components arephase-shifted by

Revival when twosuccessive componentsrecover the same phase

0collT T 0

0

n

0revT T n

Classical limitSpectrum of the Rabi frequencies

( . .)nP a u

n

0 n

0

( . .)nP a u

n

In a coherent field In the classical limit

0

0

0

n

n

« Classical limit »

Atomic signal only dependson the energy of the field

Effect on the phase of the field?

Outline





Rabi oscillation in a mesoscopic field

| , | ,nn

e C e n Initial state:1

|| ,2

|n nn

n nn

CCe

| ( )t | ( )t Mesoscopic field

1n nC C ( )

2

n nn n o n n

n

2| | / 2

!

n

nC en

with

1n

0 16t n

| ( ) | ( ) | ( )att t t

21| ( ) | |

2at in it e e e g

( ) ( )| ( ) |in t i tt e e

Atomic superposition of quantum phase

0( )4

tt

n

( )t

( )tCoherent field of classical phase

Phase correlation

| ( ) | ( ) | ( )att t t

21| ( ) | |

2at in it e e e g Atomic superposition

of quantum phase ( )t

( )tCoherent field of classical phase

( ) ( )| ( ) |in t i tt e e

Rabi oscillation in a mesoscopic field

| ( ) | (1

| | ( ) | ( ),2

) atatt te t t

The atomic dipole and the field are phase-entangled

Generation of a Schrödinger-cat state

Classical limit

0 04

t

n

0

4 4

nt tn

, || |

/ / 22 |1

| |, |2

i ti t eee

Classical Rabi oscillation Field unchanged

No atom-field entanglement

Field « classical »

/ 2| |at i te

/ 2| |at i te

Geometrical representation

|

X

Y

Z

|+>X

Y

|+>

Equatorial planeof the Bloch sphere

Atomic state in the equatorial plane of the Bloch sphere

Coherent fieldin the Fresnel plane

Representation in the same plane

Phase correlation

Atomic dipole and field « aligned »

Evolution of the atom-field system

A microscopic objectleaves its imprint on a mesoscopic one

Schrödinger-cat situation

| , Initial state

|+>|

| ,

|->

| ( ), ( )at t t | ( ), ( )at t t

| ( )t

| ( )t

| ( ) | (1

| | ( ) | ( ),2

) atatt te t t

D

"Size" of the cat=D

02 sin4

tD n

n

2 ( )( ) | | |(( )) D ttC tt e

The field acts as a Which-Path detector

Contrast of the Rabi oscillation

New insights on collapse and revival

Time (a.u.)

2( )( ) | | |)( () D ttC tt e

Collapse as soon as the two components are well separated

024

tD n

n

00

2collT T

Maximal entanglement:"Schrödinger cat state"

"Size" of the cat=distance D

Factorization of the atomic state

11| | ( 1) |

2n

field i i -Unconditional mesoscopic states superposition-The field has a defined parity

Field states merge again

Revival of the Rabi oscillation0

4RT

n

02RT T n

| ( ) | (1

| | ( ) | ( ),2

) atatt te t t

A more realistic situation

Q function evolution in 20 photonsAtom initially in |g>Rabi oscillation in 20 photons

ref: J. Gea-Banacloche, PRA 44, 5913-5931(1991) V. Buzek et.al., PRA 45, 8190-8203(1992)

Outline





Selection rules: a two-level systemMicrowave transitionVery long lifetimeVery sensitive to small fields

Circular Rydberg atoms

Stable in a weak electric fieldComplex preparation (53 photons)

Stark tuning:

on e-g transition

Field ionisation detection

51 (level e)

50 (level g)

51.1 GHz

Atoms ofHigh principal quantum numberMaximal orbital and magnetic quantum numbers

85Rb

at 30 msT

0| | 1700egd qa

2255 kHz /(V/cm)

Two superconducting mirrors in Fabry-Pérot configurationcooled down to 0.6K

The superconducting cavity

High quality factor

High confinement of the field

Controlled potentiel between the mirrors

Small thermal field

Field per photon

Coupling with an external source

900Mode TEM

3cavV 769 mm

Recirculation ring

1n 2 modes: lift of degeneracy 86kHz

83.10 , 1 mscavQ T

51.099 GHz

0 02 . 50 kHzegd E

Strong coupling regime00 cav

E 1.58 mV/cm2 V

General scheme of the experiment

Oven« Circularising box » External source

Detection byionisation

External source

Atomic beam

Cavity mode

Lasers ofpreparation

Velocity selection

Outline





Field phase distribution measurement

SHow to measure a coherent field phase-shift?

Homodyne method

Injection of a coherent field

Second injection

Resulting field

A probe atom is sent in |g>

1gP -Field in the vacuum state

-Field in an excited state 1/ 2gP

Maximum displaced by Field phase-shifted by

| | Sie

| (1 )Sie

Back to the vacuum state 0S

S

= a signal to measure the field phase distribution( )g SP

Experimental field phase distribution

Maximum<1 (thermal field)

Width of the peak

1/ n

Detection of the atom in |e> or |g> Field projected on a superposition of and |

Phase splitting in quantum Rabi oscillation: timing of the experiment

SInjection of a coherent field |

A first atom is sent and interacts resonantly with the field

Injection of

A probe atom is sent in |g>

: two peaks corresponding to the vanishing of each component

Vanishing of |

Vanishing of |

|

|

S

S | Sie

( )g SP

|

Evidence of the phase splitting

v=335m/s

int 032 1.5T s T

Measured phase 23

Expected value 0 int 234

t

n

Experiment and theory in very good agreement

36n

Evolution of the phase distribution

int 0335 / , 1.5av m s t T

int 0200 / , 2.5bv m s t T

0 int

4

t

n

Various velocities

Various number of photons

exp 37

exp 19

30n

« Fast » atom

« Slow » atom

Experiment vs theory

0 int

4

t

n Measured phase vs theoretical phase

experimental points

theory (slope 1)

numerical simulations -second mode -thermal field -relaxation

Experiment andsimulations in very good

agreement

Outline





Atom-field correlations in the { , } basis

Atomic state with defined energyMesoscopic superposition

states of the field

Entangled state Correlations in all the basis

| |

Evidence of the superpositionEstimation of coherence (Wigner function)

Selective preparation of and

| ( ) | (1

| | ( ) | ( ),2

) atatt te t t

Correlation between | ( )at t | ( )t and

| ( )at t | ( )t

| ( 0) |at t

| ( 0) |at t

or between and

Beginning of Rabi oscillation

Selective preparation of Z

|g> |

|

|->

| |

Fast Stark pulse rotation around ZPreparation of / 2

Slow rotation in theequatorial plane

End of Rabi oscillation

| ,

Atom in its initial state Field in state |

|| , , at | ,| , at g

Phase distribution analysis

| | v=335m/s

30 photonsn

Outline





Coherence of the superposition

2

0

2

0

2

2

Preparation of a superposition of | | andCoherent superposition or statistical mixture?

A quasi-probability distribution: the Wigner function

Signature of coherence:

Interference fringes

Coherent superposition Statistical mixture

-Acts on the phase space of a harmonic oscillator

-Positive for a quasi-classical field

Experimental test based on induced revivals of the Rabi oscillationcf. Tristan Meunier’s thesis

Coherence of the superposition: where are the cats?

30n 2 50 photonsD

"Fast" atom "Slow" atom01.5at T 02.5bt T

Coherent superposition Statistical mixture

2

2 cavdecoh

TT

D

2 20 photonsD

0.7a

decoh

t

T 2.8b

decoh

t

T

Requirements for a cat living

Size of the cat

int

0

tD

T

0 int2 sin4

tD n

n

D independant of the

photon number

-The two components must be separated int

0

1t

T

-The superposition must remain coherent int decoht T

20

2 2 2int

2 2cav cavdecoh

T T TT

D t

20

int 2 2int

2 cavT Tt

twith

23 0int 2

2 cavT Tt

3

3int2

0 0

2 cavt T

T T

Condition to generate a Schrödinger cat:1

Requirements for a cat living

No catClassical field

Quantum behaviorof the field

No catClassical field

int

0

t

T

2 2 2maxD

With our setup

Preparation and decoherence at the same time2max 20D

int

0

1t

T

2min 1D

Observation of mesoscopic coherences2max 40D

2/3

2max

0

cavTD

T

32

0

22cavT

T

2max 80D

v=150m/s

8 cav vc aT T(reasonable hypothesis)

Efficient and promising method to generate cats

Conclusions-perspectives

Experimental study of the Rabi oscillation at the quantum-classical boundary

Field classical No entanglement with the atom

Generation of mesoscopic state superpositionsEstimation of their coherence

Experimental test of coherence (cf. Tristan Meunier’s thesis)

Continuous monitoring of the decoherence process-2 atoms experiment-Direct measurement of the Wigner function

Rabi oscillation at the quantum-classical boundary Generation of Schrödinger cat states

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