1

Course outline

1.

A reminder about concepts and an overview of experiments: how to entangle atoms and photons and

realise quantum gates.

2.

Tests of complementarity and exploration of the quantum/classical boundary with coherent states of

radiation

2

2.Tests of complementarity and exploration of the

quantum/classical boundary with coherent states of radiation

Entangle a qubit with a mesoscopic system: how to encode information in a large object

When is a coherent field “quantum” or “classical”?

How to prepare large Schrödinger cats with a resonant atom/field interaction?

3

Outline of lecture

2.1. A complementarity experiment at the quantum/classical boundary

Realization of a thought experiment based on Rabi oscillation and Ramsey interferometry

2.2. Single atom/mesoscopic field entanglement: how a coherent field evolves from quantum to classical .

– An unexpected aspect of Rab Oscillationi– A new tool to prepare and study Schrödinger cats

4

2.1. A complementarity experiment at the Quantum/Classical boundary

5

The “strangeness” of the quantum

• Superposition principle and quantum interferences

– Feynman: Young’s slits experiment contains all the mysteries of the quantum

Shimizu et al 1992

6

The “strangeness” of the quantum:a thought experiment about complementarity

(Bohr-Einstein debate, Solvay 1927)

• Microscopic slit: set in motion when deflecting particle. Which path information and no fringes

• Macroscopic slit: insensitive to interfering particle. No which path information: fringes are observed.

• Wave and particle are complementary aspects of the quantum object.

Particle/slit entanglement

7

A “modern” version of Bohr’s proposal

• Mach Zehnder interferometer

φ

φ

D

•Interference between two well-separated paths.• Getting a which-path information?

8

A “modern” version of Bohr’s proposal

φ

φ

D

• Mach-Zehnder interferometer with a moving beam splitter

• Massive beam splitter: negligible motion, no which- path information, fringes• Microscopic beam splitter: which path information and no fringes

9

• A more general analyzis of Bohr’s experiment

– Initial beam-splitter state

– Final state for path b

– Particle/beam-splitter state

– Particle/beam-splitter entanglement– (an EPR pair if states orthogonal)

– Final fringes signal• Small mass, large kick

NO FRINGES– Large mass, small kick

FRINGES

0

α

0a b αΨ = Ψ + Ψ

0a b αΨ Ψ

0 0α =

0 1α =

φ

O

B1

B2

M

M'

a

b

P

Dφ

Complementarity and entanglement

10

Entanglement and complementarity

Entanglement with another system destroys interference• explicit detector (beam-splitter/ external)• uncontrolled measurement by the environment (decoherence)

φ

φ

D

Complementarity, decoherence and entanglement intimately linked

11

A more realistic system: Ramsey interferometry

• Two resonant π/2 classical pulses on an atomic transition e/g

φ B2

B1

M

M'

a

b

D

Which path information?Atom emits one photon in R1 or R2

Ordinary macroscopic fields(heavy beam-splitter)Field state not appreciably affected. No "which path" information

FRINGESMesoscopic Ramsey field

(light beam-splitter)Addition of one photon changes the field. "which path" info

NO FRINGES

R1 R2

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

Pg

Fréquence relative (kHz)

12

Coherent states of the field: a system evolving from quantum to classical

2| | / 2| |!

n

n

e nn

α αα −>= >∑2 | |( )

!

n

p n en

α α−=

Field radiated by a classical source in the mode

Poissonian distribution of the photon number

Representation in the complex plane

1| |

nn α

∆=

ϕ

Re( )α

Im( )α

n

| | n nα = = ∆

| | 1α ≈| | 1α

| |α

"Quantum" fieldBig fluctuations"Classical" field

Small fluctuations

: a continuous parameterto explore the quantum

classical boundary

13

Experimental requirements

• Ramsey interferometry– Long atomic lifetimes– Millimeter-wave transitions

• Circular Rydberg atoms

• π/2 pulses in mesoscopic fields– Very strong atom-field coupling

• Circular Rydberg atoms

• Field coherent over atom/field interaction• Superconducting millimeter-wave cavities

14

General scheme of the experiments

Rev. Mod. Phys. 73, 565 (2001)

15

Resonant atom-cavity interaction: Rabi oscillation in vacuum

Initial state |e,0>

Oscillatory Spontaneous emission and strong coupling regime.

Atom Cavi ty

e>g> | 0>

| 1>| 2>

| 3>

Atom Cavi ty

| e>| g> | 0>

| 1>| 2>

| 3>

Ω

0 30 60 900.0

0.2

0.4

0.6

0.8

Pe(t)

time ( µ s)

π/2 pulseVacuum Rabi

frequency

Ω = 50 kHz

In a large coherent field, Rabi frequency becomes Ω √ n

µ

16

Bohr’s experiment with a Ramsey interferometer

• Illustrating complementarity: Store one Ramsey field in a cavity

– Initial cavity state– Intermediate atom-cavity state

• Ramsey fringes contrast– Large field

• FRINGES

– Small field• NO FRINGE

D

S

R1 R2

e

g

C

φφ

α( )1 , ,

2 e ge gα αΨ = +

e gα α

e gα α α≈ ≈

0 , 1e gα α= =

Atom-cavity interaction timeTuned for π/2 pulse

Possible even if C empty

From quantum to

classical classical

17

Quantum/classical limit for an interferometer

Fringes contrast versus photon number N in first Ramsey field

Fringes vanish for quantum field

photon number plays the role of the beam-splitter's "mass"

Also an illustration of the ∆N∆Φ uncertainty relation :

• Ramsey fringes reveal field pulses phase correlations.

• Small quantum field: large phase uncertainty and low fringe contrast

Nature, 411, 166 (2001)

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

Frin

ges

cont

rast

N

18

An elementary quantum eraser

• Another thought experiment

φ

φ

D

Two interactions with the same beamsplitter assembly erase the which path informationand restore the interference fringes

19

Ramsey “quantum eraser”

• A second interaction with the mode erases the atom-cavity entanglement

• Ramsey fringes without fields !– Quantum interference fringes without external field– A good tool for quantum manipulations

φ, 0e

( )1 ,0 ,12e g+

, 0e

10 12 14 16 18 20 22 240.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pe|g,1>

Atom found in g: one photon in C whatever the path:no info and fringes

20

Entanglement between a mesoscopic coherent field and a single atom

The Ramsey interference experiment shows that, during a π/2 pulse, the atom and the field do not get entangled when n >>1:

NO ENTANGLEMENT during time t π/2 = π / 2 Ω √n

Atom and field get however ENTANGLED if they are coupled for a longer time, of the order of 2π/Ω:

|e> | α > → | Ψ +atom > | α + > + | Ψ −atom > | α

− >t > 2π/Ω

Atom dipole states

α +

α −

Coherent field split into two components:

Mesoscopic superposition of coherent states with

opposite phases

Rabi oscillation collapse and

revivals revisited

21

To be classical a field in a cavity must be coherent and contain many photons on average.

The interaction with an atom, which can emit or absorb at most one photon, is expected to leave a « large » field practically « unperturbed »

and the « atom + field system » unentangled:

| α (0)> | Ψatom(0)> → | α (t)> | Ψ(α) atom(t)>

How large must the photon number be for this classical limit to be valid?

a

φ

Correspondance principle: a coherent field with many photons has small relative fluctuations and behaves asymptotically classically

It depends on how long the interaction lasts…A large field exhibits quantum features if the interaction with the atom has enough time to create

entanglement….and if there is no decoherence

Mesoscopic physics in Quantum Optics

22

2.2Single atom/mesoscopic field entanglement:

how a coherent field evolves from quantum to classical

23

Classical Rabi oscillation

|g>

|e>ωTwo-level system {|e>;|g>}

interacting with a resonant field cos( )cl tωΩ

Rotating frameRotating wave approximation

|e>

Y

Z

X

|g>

|+>

| - >

Eigenstates of the hamiltonian1| (| | )2e g+ > = > + >

1| ( | | )2

e g− > = − > + >

1−

1

0

Time (a.u.)

eP

Rabi oscillation at frequency clΩ

24

Classical Rabi oscillation: an interference effect

( )// 2 21 || | ( )2

| i ti tet ee ψ − Ω Ω>→ >= ++ > − >Evolution

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

| - >Evol

utio

nC

hang

e of

bas

is

Det

ectio

nC

hang

e of

bas

is Classical Rabi oscillation:

a quantum beat betweentwo indistinguishable paths

|e> |+> |e>

1−

1

0

Time (a.u.)

eP

25

Rabi oscillation in a quantized field

|g>

|e>

|n-1>

|n>

|n+1>

ωωTwo-level system {|e>;|g>}

interacting with a resonantquantized field |n>

( )0 | | | |)2JCH a g e a e g+Ω= >< + ><

Jaynes-Cummings hamiltonian

Exchange of a quantum of energy

|e,n> |g,n+1>0 1nΩ +0 1nΩ +

| n+ >|e,n>; |g,n+1> : two levelscoupled and degenerated

Eigenstates of the hamiltonian:"Dressed states"

| n− >1| (| , | , 1 )2ne n g n+ >= > + + >

1| (| , | , 1 )2ne n g n− >= > − + >

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

0 1nΩ +

26

The vacuum Rabi oscillation

0 30 60 900.0

0.2

0.4

0.6

0.8

Pe(t)

time ( µ s)

Initial state |e,0> Rabi oscillation at0Ω

0 49kHzΩ ∼

0 0| , 0 cos | , 0 sin | ,12 2t te e i gΩ Ω >→ > − >

Maximal entanglement at

0 2t πΩ =

Formation of an EPR-pair

Rabi oscillation in a quantum field

Vacuum Rabi frequency

Atom-field entangled state

entanglement

Rabi oscillation in a classical field

No entanglement

Continuous evolution?

27

Rabi oscillation in a mesoscopic coherent field

0t/T

eP (t)0 1n nΩ = Ω +( )

1( ) ( ) cos( ) 12e nn

P t p n t= Ω +∑

Collapses and revivalsSpectrum of the frequencies

for a mesoscopic field

nDistribution width

Spectrum width

0 0 2nn nnn

−Ω ≈ Ω + Ω

0Ω

( . .)nP a u

nΩ

0 nΩ

0Ω

Collapse when theside components arephase-shifted by π

Revival when twosuccessive componentsrecover the same phase

0collT T∼ 0∆Ω Ω∼

0

nδ ΩΩ ∼

0revT T n∼

δΩwith

28

Collapse and revival in cavity QED

Rabi oscillation in a 0.85 photons coherent field (Brune et al PRL 76, 1800)

Also observed for in • closed cavities (Rempe, PRL 58, 353)• ion traps (Meekhof, PRL 76, 1796)

What about larger n’s?What about the field evolution in this complex Rabi oscillation process ?

1n∝

FFT

0 50 100 150Frequency (kHz)

1 2 3 4P

e (t)

0.0

0.5

1.0

Time ( µs)

0 30 60 90

Direct proof of fieldquantization

(photon graininess)

29

Classical limit

Spectrum of the Rabi frequencies

( . .)nP a u

nΩ

0 nΩ

0Ω

In a coherent field

0

0

0n

n

→ ∞Ω →

Ω → Ω

« Classical limit »

In the classical limit

( . .)nP a u

nΩ

Ω

Atomic signal only dependson the energy of the field

Effect on the phase of the field?

30

Rabi oscillation in a mesoscopic field

| , | ,nn

e C e nα >= >∑Initial state:1 || ,2

|n nn

n nn

CCe α >= + − >+

>∑ ∑2| | / 2

!

n

nC e nα α−=with

| ( )tψ + > | ( )tψ − >Mesoscopic field

1n nC C +≈2( )

2n nn n o n nn

−≈ + + −

1n0 16t nΩ

| ( ) | ( ) | ( )att t tψ ψ α+ + +>= > >

( )21| ( ) | |2

at in it e e e gϕ ϕψ − −+ >= > + >

( ) ( )| ( ) |in t i tt e eϕ ϕα α −+ >= >

Atomic superposition of quantum phase

0( )4ttn

ϕ Ω=

( )tϕ−

( )tϕ−Coherent field ofclassical phase

Phase correlation

31

Rabi oscillation in a mesoscopic field

( )| ( ) | (1| | ( ) | ( ),2

) atatt te t tαα ψα ψ+ + − −> > >>→ + >

The atomic dipole and the field are phase-entangledGeneration of a Schrödinger-cat state

0

0

0n

n

→ ∞Ω →

Ω → Ω

Classical limit0 0

4tn

ϕ Ω= →

0

4 4nt tnϕ Ω Ω= →

, || | α αα+ −> >→ >/ 2| |i teψ − Ω+ >→ + >/ 2| |i teψ Ω− >→ − >

( )/ / 22 |1| |, |2

i ti t eee α α− ΩΩ>→ + − >+ > > No atom-field entanglementField « classical »

Field unchangedClassical Rabi oscillation

32

Geometrical representation

|α >

X

Y

Z

|+>X

Y

|+>

Atomic state in the equatorial plane of the Bloch sphere

Coherent fieldin the Fresnel planeRepresentation in the same plane

Equatorial planeof the Bloch sphere

Phase correlation

Atomic dipole and field « aligned »

33

Evolution of the atom-field system

A microscopic objectleaves its imprinton a mesoscopic one

ϕ

| ,α+ >Initial state

|+> |α >

| ,α− >

|-> ϕ

| ( ), ( )at t tψ α+ +→ >| ( ), ( )at t tψ α− −→ >

| ( )tα+ >

( )| ( ) | (1| | ( ) | ( ),2

) atatt te t tαα ψα ψ+ + − −> > >>→ + >

D 02 sin4tD nn

Ω =

Schrödinger-cat situation

"Size" of the cat=D

The field acts as a Which-Path detector| ( )tα− >

2 ( )( ) | | |(( )) D ttC tt eαα −−

+= < > =Contrast of the Rabi oscillation

34

New insights on collapse and revival

2( )( ) | | |)( () D ttC tt eαα −−

+= < > =

( )| ( ) | (1| | ( ) | ( ),2

) atatt te t tαα ψα ψ+ + − −> > >>→ + >

Time (a.u.)Collapse as soon as the twocomponents are well separated

024tD nn

Ω∼ 00

2collT T∝Ω∼

Maximal entanglement:"Schrödinger cat state"

"Size" of the cat=distance D

Factorization of the atomic state

( )11| | ( 1) |2

nfield i iψ α α

+>= > + − − >-Unconditional mesoscopic states superposition-The field has a defined parity

Field states merge againRevival of the Rabi oscillation

0

4RTn

ϕ πΩ= ≈02RT T n≈

35

An exact calculation

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

36

Field phase distribution measurement

S

Injection of a coherent fieldSecond injection

1gP ≈-Field in the vacuum state-Field in an excited state 1/ 2gP ≈

Maximum displaced by φ∆Field phase-shifted by φ∆

|α >| Sie φα− >

| (1 )Sie φα − >

Back to the vacuum state 0Sφ =

Sφ

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

How to measure a coherent field phase-shift?

Homodyne method

Resulting field

A probe atom is sent in |g>

37

Experimental field phase distribution

Maximum

38

Phase splitting in quantum Rabi oscillation: timing of theexperiment

SInjection of a coherent field |α >

Detection of the atomField projected on

( )1|2

| |field α αψ −+ > +>= >

Injection of | ie φα− >

: two peaks corresponding to the vanishing of each component( )gP φ

ϕ

ϕ−|α+ >

α− >

|α+ >

|α− >

Sφ ϕ=

Sφ ϕ= −

A first atom is sent and interactsresonantly with the field

Vanishing of

Vanishing of |A probe atom is sent in |g>

39

Evidence of the phase splitting

v=335m/s

int 032 1.5T s Tµ= ≈

36n =

Measured phase 23ϕ = °

Expected value 0 int 234tn

ϕ Ω= = °Experiment and theory in

very good agreement

40

Evolution of the phase distribution

0 int

4tn

ϕ Ω=2 velocities

Various numberof photons

30n =

int 0200 / , 2.5bv m s t T= ≈

exp 37ϕ = °int 0335 / , 1.5av m s t T= ≈

exp 19ϕ = °

41

Experiment vs theory

0 int

4tn

θ Ω=

numerical simulations-second mode-thermal field-relaxation

Experiment andsimulations in very good

agreement

theory (slope 1)

experimental points

Measured phase vs theoretical phase

42

Test of coherence: induced quantum revivals

Initial Rabi rotation,

Collapse

And slow phase rotation

Stark pulse (duration short compared to phase rotation).

Equivalent to a Z rotation by πReverse phase rotation

Recombine field components and resume Rabi oscillation

A spin echo experiment

Echo experiments in Cavity QED to study decoherenceproposed by G.Morigi, E.Solano, B.G.Englert andH.Walther, Phys.Rev.A 65, 0401202(R),2002.

43

0,45

0,60

0,75

0,45

0,60

0,75

-80 -40 0 40 80

0,45

0,60

0,75

Sg(φ)

φ (degrees)

βx

-6

-4

-2

0

-4 -2 0 2 4-6

-4

-2

0

βy

-6

-4

-2

0

0,00

0,25

0,50

0,75

0,00

0,25

0,50

0,75

0 20 40 600,00

0,25

0,50

0,75

Stark pulse after 20µs

Pg

No stark pulse

Interaction time (µs)

Stark pulse after 25µs

Separation and recombination of fieldcomponents by Stark switching

Rabi oscillation revivals

44

Conclusions and perspectives

Larger and longer lived cats (n in the hundreds) withbetter cavities

Prepare and detect | α , 0> + | 0, α > (similar to |n,0> + |0,n > « high noon states »)

Non local field states in two cavities

-4

-2

0

2

4-2

-10

12 -2

-1

0

1

2

-4

-2

0

2

4

-2

-1

0

1

2

Wigner function measurements anddecoherence studies of cat states

45

Alexia Auffeves

M.BruneJ.M.Raimond

G.Nogues

S.GleyzesP.Maioli

T.Meunier

Support:JST (ICORP, Japan), EC, CNRS, UMPC, IUF, CdF

2.Tests of complementarity and exploration of the quantum/classical boundary with coherent states of radiationOutline of lectureThe “strangeness” of the quantumThe “strangeness” of the quantum: a thought experiment about complementarityA “modern” version of Bohr’s proposalA “modern” version of Bohr’s proposalComplementarity and entanglementEntanglement and complementarityA more realistic system: Ramsey interferometryCoherent states of the field: a system evolving from quantum to classicalExperimental requirementsGeneral scheme of the experimentsResonant atom-cavity interaction: Rabi oscillation in vacuumBohr’s experiment with a Ramsey interferometerQuantum/classical limit for an interferometerAn elementary quantum eraserRamsey “quantum eraser”Classical Rabi oscillationClassical Rabi oscillation: an interference effectRabi oscillation in a quantized fieldThe vacuum Rabi oscillationRabi oscillation in a mesoscopic coherent fieldCollapse and revival in cavity QEDClassical limitRabi oscillation in a mesoscopic fieldRabi oscillation in a mesoscopic fieldGeometrical representationEvolution of the atom-field systemNew insights on collapse and revivalAn exact calculationField phase distribution measurementExperimental field phase distributionPhase splitting in quantum Rabi oscillation: timing of the experimentEvidence of the phase splittingEvolution of the phase distributionExperiment vs theoryTest of coherence: induced quantum revivals