Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles...

109
Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing entanglement Entanglement and measurement Summary: preparation of a GHZ state Preparation of a GHZ state Experimental realization A first atom in |e 1 performs a π/2 pulse |ψ 1 = 1 2 (|e 1 , 0 + |g 1 , 1) A second atom in 1/ 2(|i 2 + |g 2 ) performs a QPG gate without affecting the field state (QND) |ψ 2 = 1 2 |e 1 , 01 2 (|i 2 + |g 2 )+ |g 1 , 11 2 (|i 2 -|g 2 ) « An atom-field-atom GHZ state A third atom in |g 3 can perform a π pulse in order to read the field state |ψ 3 = 1 2 |e 1 , g 3 1 2 (|i 2 + |g 2 )+ |g 1 , e 3 1 2 (|i 2 -|g 2 ) «

Transcript of Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles...

Page 1: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Cavity QED

Gilles Nogues

cavity QED withRydberg atoms

Basic atom-fieldinteractions:producingentanglement

Entanglement andmeasurement

Summary:preparation of aGHZ state

Preparation of a GHZ state

Experimental realizationI A first atom in |e1〉 performs a π/2 pulse

|ψ1〉 =1√

2(|e1, 0〉+ |g1, 1〉)

I A second atom in 1/√

2(|i2〉+ |g2〉) performs a QPGgate without affecting the field state (QND)

|ψ2〉 =1√

2

„|e1, 0〉 ⊗

1√

2(|i2〉+ |g2〉) + |g1, 1〉 ⊗

1√

2(|i2〉 − |g2〉)

«

An atom-field-atom GHZ stateI A third atom in |g3〉 can perform a π pulse in order to

read the field state

|ψ3〉 =1√

2

„|e1, g3〉 ⊗

1√

2(|i2〉+ |g2〉) + |g1, e3〉 ⊗

1√

2(|i2〉 − |g2〉)

«

Page 2: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Cavity QED

Gilles Nogues

cavity QED withRydberg atoms

Basic atom-fieldinteractions:producingentanglement

Entanglement andmeasurement

Summary:preparation of aGHZ state

Detection of the GHZ state

2nd Ramsey pulse for atom 2 at frequency ν0It corresponds to the QND detection conditions:

1√2(|i2〉+ |g2〉) → |i2〉

1√2(|i2〉 − |g2〉) → |g2〉

Page 3: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Cavity QED

Gilles Nogues

cavity QED withRydberg atoms

Basic atom-fieldinteractions:producingentanglement

Entanglement andmeasurement

Summary:preparation of aGHZ state

Detection of the GHZ state

2nd Ramsey pulse for atom 2 at frequency ν0It corresponds to the QND detection conditions:

1√2(|i2〉+ |g2〉) → |i2〉

1√2(|i2〉 − |g2〉) → |g2〉

Results

0

0.1

0.2

0.3

0.4

0.5

Coincidences between atom #1 and #3

Pro

bab

ility

no atom #2

G1G3 G1E3

E1G3

E1E3

Page 4: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Cavity QED

Gilles Nogues

cavity QED withRydberg atoms

Basic atom-fieldinteractions:producingentanglement

Entanglement andmeasurement

Summary:preparation of aGHZ state

Detection of the GHZ state

2nd Ramsey pulse for atom 2 at frequency ν0It corresponds to the QND detection conditions:

1√2(|i2〉+ |g2〉) → |i2〉

1√2(|i2〉 − |g2〉) → |g2〉

Results

0

0.1

0.2

0.3

0.4

0.5

Coincidences between atom #1 and #3

Pro

bab

ility

no atom #2

G1G3 G1E3

E1G3

E1E3

atom #2 detected in gatom #2 detected in i

Page 5: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Cavity QED

Gilles Nogues

cavity QED withRydberg atoms

Basic atom-fieldinteractions:producingentanglement

Entanglement andmeasurement

Summary:preparation of aGHZ state

Correlation in another basis

EPR correlation for atoms 1 and 3Measurement of 〈σx ,1σφ,3〉 as in previous experiments.

-3 -2 -1 0 1 2 3

-0,2

-0,1

0,0

0,1

0,2

0,3 Two atoms signal

Frequency (kHz)of The e-i two photon transition

Pe1e3- Pe1g3- Pg1e3+ Pg1g3

signal if i 2 signal if g2 signal if no at 2

QPG action of atom 2 changes the sign of the correlationof atoms 1 and 3.

Page 6: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Cavity QED

Gilles Nogues

cavity QED withRydberg atoms

Basic atom-fieldinteractions:producingentanglement

Entanglement andmeasurement

Summary:preparation of aGHZ state

General conclusion

I Simple quatum gates demonstratedI most complicated algorithm uses up to 4 qubits and

entangles 3 of themI complete measurement of the density matrix not

realised (see ion trap experiment)I early experimentI data acquisition time too long due to the very low

probability for detecting coincidences (20h for the 3atom experiment)

Page 7: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Decoherence in Quantum MechanicsA cavity QED approach

Gilles NOGUES

Laboratoire Kastler Brossel

Ecole normale supérieure, Paris

Page 8: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

A century of quantum mechanics

And yet an intriguing theory which defies our « classical intuition »

Many paradoxes as soon as one tries to interpret its results Entanglement: after interaction 2 subsystems,

although separated in space, cannot be considered as independant

Superposition principle: Any linear combination of physical state is a physical state

Page 9: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

The Schrödinger cat

A gedanken experiment

Three related questions Why do we need the

box? Why do we found the

cat dead or alive? Why can’t we predict

the outcome of a single realization?

Environment

Preferred basis

?

A cat in a box with an excited atom

If the atom desexcite, a setup kills the cat

What is the state of the system after a half-lifetime of the atom?

Page 10: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Outline

A cavity QED experiment How to create and destroy and restore a coherent

superposition The decoherence in quantum mechanics

Effect of the size of the system and of the environment Monitoring the decoherence

What can be an optical Schrödinger cat? Observing the decoherence

Beyond decoherence Pushing the limits?

Page 11: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

A cavity QED experiment on complementarity

How to wash out fringes in an interferometer by gaining a « which-path » information?

How to restore the interference by manipulating the information

Page 12: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

The “strangeness” of the quantum Superposition principle and quantum interferences

The sum of quantum states is yet another possible state A system “suspended” between two different classical realities

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

1 2Ψ = Ψ + Ψ

0 1 22 ReI I= + Ψ Ψ

Da

dλ=

d

D

a

Page 13: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Interferometers for photons and atoms

Mach-Zender

2 optical path separated by beamsplitters

Ramsey:

2 energy path separated by interaction with classical field

R1 R2

B2

B1

M

M'

a

b

D

0 10 20 30 40 50 600.0

0.2

0.4

0.6

0.8

1.0

Pg

Fréquence relative (kHz)

Page 14: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Destroying the fringes

By gaining a “which-path” information on the system

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

Macroscopic slit: impervious to interfering particle. No which path information and fringes

Wave and particle are complementary aspects of the quantum object.

(From Einstein-Bohr at the 1927 Solvay congress)

One slit is mounted on springs

Page 15: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

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 FRINGES

α( )1

, ,2

e ge gα αΨ = +

e gα α

e gα α α≈ ≈

0 , 1e gα α= =

Atom-cavity interaction time

Tuned for π/2 pulsePossible even if C

empty

D

S

R1 R2

e

g

C

φφ

Page 16: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Interferences versus field size

A microscopic field is strongly affected by the interaction Stores an information

about the atomic state For larger fields the

« which path » information is lost Fringes restored

0

0.5

1

0.5

0

0.5

0

0.5

1

N=0

N=0.31

N=2

N=12.8 Nature, 411, 166 (2001)

Page 17: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Interference versus field size (2)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16Mean photon number

Maximum contrast 72% final contrast due to external imperfections

(velocity dispersion, 2 atom events…)

•Photon number acts as the mass of the slit in the Young’s experiment

•Classical behaviour observed for about 10 photons

Page 18: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Important remarks

No need to measure the field state The mere fact that the information exists destroy

the interferences Close link with entanglement

Atom and field no longer separable Local operation on the atom cannot recover the

whole information Fringes are recoverable

If one operates on the atom and the field… … in order to erase the information

Page 19: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

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

( )1,0 ,1

2e g+

,0e ,1g

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

S

e

C

D

R1R2

g

φ

Page 20: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Intermediate conclusions

Any coherent superposition can be created and probed in a interferometer experiment

The fringe contrast is a direct measurement of how much « which-path » information one has about the system

It is in principle possible to manipulate this amount of information to restore the coherence

Cavity QED is an appropriate tool to manipulate those concepts in the case of an atom coupled to a few tens of photons

Page 21: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

A general frame for decoherence

The environment as an unavoidable « which-path » meter

Effect of the size of the system Link with the measurement theory

Page 22: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Why the box?

Hide the cat Physically: prevent

diffusion of light on the system

Isolate from its environment

Coupling with the outside world Metallic box?

Blackbody radiation Surrounding gaz

Collision: brownian motion

Vibration: noise All those interactions

provide a « which-path » information

Page 23: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Why the cat? What is the difference between a photon in an

interferometer and a cat in a box Strength of the coupling with the environment Distance (in Hilbert space) between states

Tdec=Trel/Distance

How to evaluate the distance? Depends on the nature of the coupling

Diffusion of photons: distance = (distance in space) / wavelength

Not a trivial relaxation mechanism But is explained by relaxation theory for simple models

Final state Trace over the environment: statistical mixture

Page 24: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Link with measurement theory What is a good meter?

Coupled to the observed system

Gives an answer that you can see with your eye

Open, macroscopic system

A two step process System-meter coupling

May be unitary leads to an entangled

state « which – path »

information gathered by environment

Coherence is lost And unrecoverable if

the environment is huge

Quantum system

0+-

∆x

Environment

Page 25: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Pointer states

Only a small fraction of the Hilbert space is observed Example: or , but no

superpositions This « preferred basis » correspond to meter

state which do not get entangled with the environment Pointer states

0 +-0 +-

Page 26: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

What is the final state of the cat

The coherence leaks into the environment One has to account for the lack of knowledge

about the environment Density matrix description… …with a trace over the environment

ρfin=1/2(| >< |+| >< |)0 +-

0 +-0 +-

0 +-

Statistical mixture of two pointer state

The meter is either pointing one way or the other

Page 27: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

A short summary: what is important in decoherence

An open system whose interaction with the environment selects a preferred basis

Pointer states which can be discriminated by a classical experience Large distance in Hilbert space The larger the distance, the faster the process

This is the case of every large system Demonstration:

One never observes a cat dead and alive at the same time

Page 28: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Monitoring the decoherence

Requirements to observe the phenomenon A system as isolated as possible

Long relaxation time A mesoscopic system

Whose « size » can be varied between macroscopic and microscopic dimensions

An easy way to prepare and analyse coherent superpositions of the system

A small coherent field in a cavity is a perfect example

Page 29: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Preparing and observing a Schrödinger kitten

Decoherence for a superposition of coherent states of light

How to prepare such a state?

An interferometric experiment to test the coherence.

Page 30: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Cavity relaxation

Due to diffraction on the surface Not perfectly spherical

over a few mm

Interaction Hamiltonian:

H=Σ gi(abi+ + a+bi)

State at time t: At T=0K:

|αe-γ t/2⟩ Π|βi(t)⟩

A coherent state remains disentangled

Pointer state Energy conservation

Σ |βi(t)|2= |α|2(1-e-γ t)

= n(1-e-γ t)

Mode i

Page 31: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Decoherence of a coherent state superposition Initial state

1/N(|0⟩ +|α⟩ ) Π |0⟩ iN=√2+⟨ 0|α⟩ =√2+e-|α|2/2

≈ √2 if |α|>>1

Page 32: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Decoherence of a coherent state superposition

Entangled system Which-path information

has leaked to the environment

Contrast:

C=Π i⟨ 0|βi(t)⟩

=Π e-|βi(t)|2/2= e-Σ|βi(t)|2/2

=e-|α|2 (1-e-γ t)/2 ≈e-|α|2γt/2

Decoherence time

Tdec= 2 Tr / |α|2

At time t:

|0⟩ Π |0⟩ i +|αe-γ t/2⟩ Π|βi(t)⟩

Page 33: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

A single atom in a coherent field

Non resonant interaction No exchange of energy ω0 ω

ω

|g⟩

|e⟩

|0⟩

|1⟩

|2⟩

-Ω2(n+1)/4δ

-Ω2n/4δ

-Ω2/4δ

ω

ω

ω+δω

ω+δω

|g,0⟩

|e,0⟩

|g,1⟩

|e,n-1⟩

|g,n⟩

|e,n⟩

|g,n+1⟩

δ

Uncoupled Coupled •If δ>>Ω:

|+,n⟩ ≈|g,n⟩ and ∆E+,n=-Ω2n/4δ

Cavity frequency

Atom position incavity mode

ω

Atom acts as a dielectric that phase shifts the field in the cavity…

ωg

… depending on its state

ωe

Page 34: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Another experiment on complementarity

Cavity as an external detector in the Ramsey interferometer

Cavity contains initially a coherent field

Non-resonant atom-field interaction:Atom modifies the cavity field phase

(index of refraction effect)

Phase shift α 1/ δ ( δ:atom-cavity detuning)

"Which path" information:

Small phase shift (large δ)(smaller than quantum phase noise)

field phase almost unchanged No which path information Standard Ramsey fringes

Large phase shift (small δ)(larger than quantum phase noise)

Cavity fields associated to the two paths distinguishable

Unambiguous which path information

No Ramsey fringes

D

S

R1R2

e

g

C

φ

α1

e e

g g

Page 35: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Fringes and field state

State transformationsR1 R2

C

Before R1

Before C

After C

After R2

Detection probabilities

Ramsey fringes signal multiplied by

Complementarity

PRL 77, 4887 (96)

( )1

2e e g→ +

( )

( )

1

21

2

i

i

e e e g

g e e g

ϕ

ϕ−

→ +

→ − +

, , , ,i i ie e e e g g eα α α αΦ Φ − Φ→ →

,e α

( )1

2e g α+

( )1, ,

2i i ie e e g eα αΦ Φ − Φ+

( )

( ) ( )

1

21

2

i i i i

i i i i

e e e e e

g e e e e

ϕ

ϕ ϕ

α α

α α

Φ Φ − +Φ − Φ

+Φ Φ − +Φ − Φ

+ +

( ),

11 Re

2i i i

g eP e e eϕ α α− +Φ Φ − Φ = ±

i ie eα αΦ − Φ0 2 4 6 8 10

Ram

sey

Fri

nge

Sig

nal

104 kHz

ν (kHz)

0.0

0.5

1.0

347 kHz

712 kHz,

9.5 photons

0.0

0.5

1.0

712 kHz

Vacuum

Page 36: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Signal analysis

Fringe signal multiplied by

Modulus

Contrast reduction

Phase

Phase shift corresponding to cavity light shifts

Phase leads to a precise (and QND) measurement of the average photon number

i ie eα αΦ − Φ

2 22 sin / 2n De e− Φ −=

Fringes contrast and phase

Excellent agreement with theoretical predictions.

Not a trivial fringes washing out effect

Calibration of the cavity field 9.5 (0.1) photons

D

2 sinn Φφ (radians)

0.0 0.2 0.4 0.6 0.80

20

40

60

Fri

nge

Con

tras

t (%

)

0.0 0.2 0.40

2

4

6

φ (radians)

Fringe S

hift (rd)

n=9.5 (0.1)

Page 37: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Probing the coherence Field state after atomic

detection

A coherent superposition of two 'classical' states.

Decoherence will transform this superposition into a statistical mixture Is it possible to study the

dynamics of this phenomenon

Requires to perform an interferometry experiment with the cavity state

A second atom to probe the field

First atom

Second atom

Two indistinguishable quantum paths to the same final state e1g2 and e2g1

Quantum interferences

( )1

2+ Φ

−ΦD

−2Φ

Page 38: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Atomic correlations A correlation signal

Independent of Ramsey interferometer frequency

when Φ is neither 0 nor π/2

0.5 for a quantum superposition

0 for a statistical mixture

0 for an empty cavity

Principle of the experiment Send a first atom to

prepare the cat Wait for a delay τ Send a second probe

atom Measure η versus τ

Raw correlation signals

1Re

2η α α=

, ,

,,

, , , ,

e e g e

g ee e

e e e g g e g g

PP

P P P P

η = Π − Π

= −+ +

0 2 4 6 8 10 12

-0.1

0.0

0.1

0.2

0.3

corr

elat

ion

sign

al

ν (kHz)

15000 coincidences

τ=40 µs

Page 39: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Seeing the decoherence over time

0 1 2

0.0

0.1

0.2

n=3.3 δ/2π =70 and 170 kHz

Tw

o-A

tom

Cor

rela

tion

Sig

na

l

τ/Tr

0 1 20

4

8

12

16

20

n=3.3

t /Tr

corr

ela

tion

sig

na

l

0 1 20

4

8

12

16

20

n=5.5

δ/2π =70 kHz

corr

ela

tion

sig

nal

t /Tr

PRL 77, 4887 (1996)

Effect of atomic phase shift Effect of field amplitude

Page 40: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

What can we say from those results

It is possible to prepare mesocopic superpositions of states Schrödinger kitten

However the coherence is lost very very very very fast The larger the cat, the faster the phenomenon Experimental data in good agreement with

theory Tdec=Trel/D In our case D goes up to 9 photons

Page 41: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

How to obtain larger cat states?

Increase cavity lifetime (i.e. T rel) Increase coupling

Dispersive interaction ∝Ω2/ δ with δ>>Ω Resonant interaction ∝Ω

Increase interaction time (lower atomic velocity)

What is the effect of a resonant atom on a mesoscopic coherent field?

Page 42: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Rabi oscillation in a mesoscopic field

Intermediate regime of a few tens of photons. A first insight

A simple theoretical problem

Page 43: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Collapse and revival Collapse: dispersion of field amplitudes due to

dispersion of photon number

Revival: rephasing of amplitudes at a finite time such that oscillations corresponding to n and n+1 come back in phase

Revival is a genuinely quantum effect

Page 44: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Oscillations in a small coherent field

Brune et al., PRL 76, 1800 (1996)

Initial state |e,α⟩ |α|2=<n>

Observation of the collapse and revival A fourier transform reveals

the different frequencies Ω(n)=Ω0√n

Open questions How is the field affected? What is happening if <n>

increases (classical limit)

0 20 40 60 800,0

0,2

0,4

0,6

0,8

1,0

<n>=1,77

Interaction time (µs)

0 20 40 60 800,0

0,2

0,4

0,6

0,8

1,0

<n>=0,85

Pe

0 20 40 60 800,0

0,2

0,4

0,6

0,8

1,0

<n>=0,4

Page 45: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

All results

N=0

0.4

0.85

1.77

0.0

0.5

1.0

nth

= 0.06

FF

T A

mpl

itude

P

(n)

Pe (

t)

0.0

0.5

1.0

0.0

0.5

1.0

0 30 60 90

0.0

0.5

1.0

Time ( µs)

0 50 100 150

Frequency (kHz)

0.0

0.5

1.0

0.0

0.5

ncoh

= 1.77

ncoh

= 0.85

ncoh

= 0.4

0.0

0.5

0 1 2 3 4 5

0.0

0.3

n

Page 46: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Remark at short time

time necessary for doing a π/2 pulse Complementarity experiment shows that the

field is not affected

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16Mean photon number

Maximum contrast 72%

Page 47: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

A more detailed analysis of Rabi oscillation

Valid in the case of mesoscopic fields ∆n<<n Expansion of |Ψ⟩ around n=<n> (Gea Banacloche PRL 65, 3385, Buzek et al PRA

45, 8190)

First non-trivial order:

Atomic states slowly rotating in the equatorial plane of the Bloch sphere (<n> times slower than Rabi oscillation)

A slowly rotating field state in the Fresnel plane Graphical representation of the joint atom-field evolution in a plane

t=0: both field states coincide with original coherent state Atomic states are the classical eigenstates

1( ) ( ) ( ) ( ) ( )

2a c a ct t t t t+ + − − Ψ = Ψ Ψ + Ψ Ψ

0 / 21

2i nt i

a e e e i g± Ω± ± Φ Ψ = 0

4

t

n

ΩΦ =

0 / 4i nt ic e eαΩ± ± ΦΨ =

Page 48: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Atom-field states evolution

c+Ψ c

−Ψ a−Ψa

Initial state: a coherent superposition of two states

Page 49: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Atom-field states evolution

c+Ψ c

−Ψ a−Ψa

•At most times: ⟨ Ψc-|Ψc

+⟩ =0 an atom-field entangled state

•In spite of large photon number: considerable reaction of the atom on the field

Page 50: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

10 20 30 40 50

0.2

0.4

0.6

0.8

1P

e(t)

0t/ 2

Link with collapses and revival

Rabi oscillation: interference between |Ψa+⟩ and |Ψa

-⟩

Contrast vanishes when ⟨ Ψc-|Ψc

+⟩

A direct link between Rabi collapse and complementarity

Page 51: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

10 20 30 40 50

0.2

0.4

0.6

0.8

1P

e(t)

0t/ 2

Link with collapses and revival

Rabi oscillation: interference between |Ψa+⟩ and |Ψa

-⟩

Contrast vanishes when ⟨ Ψc-|Ψc

+⟩

A direct link between Rabi collapse and complementarity

Loss of contrast as the field gets which-path information

Page 52: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

10 20 30 40 50

0.2

0.4

0.6

0.8

1P

e(t)

0t/ 2

Link with collapses and revival

Rabi oscillation: interference between |Ψa+⟩ and |Ψa

-⟩

Contrast vanishes when ⟨ Ψc-|Ψc

+⟩

A direct link between Rabi collapse and complementarity

trev/2: atom disentangled from a field in a Schrödinger cat state

Page 53: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

10 20 30 40 50

0.2

0.4

0.6

0.8

1P

e(t)

0t/ 2

Link with collapses and revival

Rabi oscillation: interference between |Ψa+⟩ and |Ψa

-⟩

Contrast vanishes when ⟨ Ψc-|Ψc

+⟩

A direct link between Rabi collapse and complementarity

revival: field phase information is erased

Page 54: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Phase distribution measurement

Heterodyning a coherent fieldS

•Inject a coherent field |α>

•Add a coherent amplitude –αeiφ

•Resulting field (within global phase) |α(1-eiφ)>•Zero final amplitude for φ=0

•Probe final field amplitude with atom in g•Pg=1 for a zero amplitude

•Pg=1/2 for a large amplitude

•More generally: Pg(φ) reveals field phase distribution

•In technical terms, Pg(φ)=Q distribution

Atom in |g>

Page 55: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Phase distribution measurement

Heterodyning a coherent fieldS

•Inject a coherent field |α>

•Add a coherent amplitude –αeiφ

•Resulting field (within global phase) |α(1-eiφ)>•Zero final amplitude for φ=0

•Probe final field amplitude with atom in g•Pg=1 for a zero amplitude

•Pg=1/2 for a large amplitude

•More generally: Pg(φ) reveals field phase distribution

•In technical terms, Pg(φ)=Q distribution

Atom in |g>

Page 56: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Phase distribution measurement

Heterodyning a coherent fieldS

•Inject a coherent field |α>

•Add a coherent amplitude –αeiφ

•Resulting field (within global phase) |α(1-eiφ)>•Zero final amplitude for φ=0

•Probe final field amplitude with atom in g•Pg=1 for a zero amplitude

•Pg=1/2 for a large amplitude

•More generally: Pg(φ) reveals field phase distribution

•In technical terms, Pg(φ)=Q distribution

Atom in |g>

Page 57: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Phase distribution measurement

Heterodyning a coherent fieldS

•Inject a coherent field |α>

•Add a coherent amplitude –αeiφ

•Resulting field (within global phase) |α(1-eiφ)>•Zero final amplitude for φ=0

•Probe final field amplitude with atom in g•Pg=1 for a zero amplitude

•Pg=1/2 for a large amplitude

•More generally: Pg(φ) reveals field phase distribution

•In technical terms, 2xPg(φ)-1=Q(α) distribution

Page 58: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Coherent field phase distribution

One can apply the same phase measurement after interaction with an atom Detection of states |Ψc

+⟩ and |Ψc-⟩

-100 -50 0 50 100

0,4

0,5

0,6

0,7

0,8

0,9

Pg

Phase (°)

MicrowaveAttenuation injection

76dB 73dB 70dB 67dB

0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1

10

15

20

25

30

35

40

45

64 dB - 23 ph

76 dB - 2.1 ph

wid

th (

in d

egre

as)

n

1

Homodyning signal Width versus <n>

Page 59: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Phase splitting in quantum Rabi oscillation

Timing

S

•Inject a coherent field

•Send a first atom: Rabi oscillation and phase shift

•Inject a phase tunable coherent amplitude

•Send a second atom in g: final amplitude read out

1st atom2nd atom

Page 60: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Phase splitting of a mesoscopic field

…versus <n> and interaction time335 m/s 200 m/s

(degrees)

n

Sg(φ)

0 150 15

20

25

30

35

40

-150 0 150

20

25

30

35

40

0,50,6

0,7

(degrees)

0

4

t

n

ΩΦ =

A. Auffèves et al., Phys. Rev. Lett. 91, 230405 (2003)

Page 61: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Phase splitting in quantum Rabi oscillationObserved phase versus theoretical phase

Large Shrödinger cat states (up to 40 photons separation)

φ+ (degrees)

Pha

se (

degr

ees)

15 20 25 30 35 40 45 50 55 60-60

-40

-20

0

20

40

60

0 2 4 6 8-4

-2

0

2

4

x

y

0

4

t

n

ΩΦ =

Numerical simulation of field master equation

Page 62: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

How to test that one has a coherent superposition for the field state

Interference between the two states Waiting for the revival will do the trick

If atomic populations oscillates again then it proves the coherence

But interaction time is limited Maximum interaction time 100µs for atoms at

100 m/s Revival time for n=25: 300µs

Page 63: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Test of coherence: induced quantum revivalsInitial Rabi rotation,

Collapse

And slow phase rotation

Stark pulse (duration short compared to phase rotation).

Equivalent to a Z rotation by π

Reverse phase rotation

Recombine field components and resume Rabi oscillation

Page 64: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Test of coherence: induced quantum revivalsInitial Rabi rotation,

Collapse

And slow phase rotation

Stark pulse (duration short

compared to phase rotation).

Equivalent to a Z rotation by π

Reverse phase rotation

Recombine field components and resume Rabi oscillation

Page 65: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Test of coherence: induced quantum revivalsInitial Rabi rotation,

Collapse

And slow phase rotation

Reverse phase rotation

Recombine field components and resume Rabi oscillation

Stark pulse (duration short

compared to phase rotation).

Equivalent to a Z rotation by π

Page 66: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Test of coherence: induced quantum revivalsInitial Rabi rotation,

Collapse

And slow phase rotation

Reverse phase rotation

Recombine field components and resume Rabi oscillation

Stark pulse (duration short

compared to phase rotation).

Equivalent to a Z rotation by π

Page 67: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Induced quantum revivals (I) Effect on the field observed by homodyning

Two components clearly resolved Separated in phase space by a distance of 9 photons Associated decoherence time 100 µs

t=015 photons injected

-100 -50 0 50 100-100 -50 0 50 100-100 -50 0 50 100

0,50

0,60

0,70

0,80

phase (°)

sign

al

t=Tπ=22 µs t=2Tπ

Page 68: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Induced quantum revivals (II)

Atomic population as a function of π pulse delay

Contrast decreases when Tπ increases simulation of the experimental data Main limitation due to inhomogeneous effect (velocity

dispersion) Additional decoherence term

-20 -10 0 10 20 30 40 50 60

0,0

0,2

0,4

0,6

0,8

1,0Pg

Interaction time Interaction time-20 -10 0 10 20 30 40 50 60

0,0

0,2

0,4

0,6

0,8

1,0Tπ=18.5 µs Tπ=23.5 µs

Page 69: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Conclusion

Study of decoherence in the frame of cavity QED A Schödinger cat for a « trapped field »

In good agreement with theory Effect of the size of the cat

Close relation with the theory of measurement in quantum physics Possible application and testground for basic

Quantum information processing experiments

Page 70: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Lecture 4: future directions

Gilles NOGUES

Laboratoire Kastler Brossel

Ecole normale supérieure, Paris

Page 71: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

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 *

Christine Guerlin

Thomas Nirrengarten*

Post doc

Ferdinand Schmidt-Kaler

Edward Hagley

Christof Wunderlich

Perola Milman

Stefan Kuhr

Angie Quarry*

Collaboration

Luiz Davidovich

Nicim Zagury

Wojtek Gawlik

Daniel Estève

Permanent

Gilles Nogues *

Michel Brune

Jean-Michel Raimond

Serge Haroche

*: atom chip team

Page 72: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Outline

More about the fieldQND detection of more than one atom

Measurement of the Wigner distribution

More about the atomQND detection of the atomic state

A Schrödinger cat state for a mesoscopic ensemble of atoms

New experimental toolsA two-cavity setup

Page 73: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Outline

More about the fieldQND detection of more than one atom

Measurement of the Wigner distribution

More about the atomQND detection of the atomic state

A Schrödinger cat state for a mesoscopic ensemble of atoms

New experimental toolsA two-cavity setup

Page 74: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

QND detection experiment

Only works for |0> and |1>If one has 2 photons in the cavity

Rabi frequency Ω0√2≈1.5xΩ0

A 3π pulse is performed |g,2>→|e,1>

One photon is absorbed and atom is neither in g or i

2 main problemsA measured qubit can only provide one bit of informationDispersive interaction is required to prevent energy exchange for all photon number

Page 75: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Dispersive interaction

But : light shift

)1n(δ4

E2

ne, +Ω=∆

n4δ

E2

ng, Ω−=∆

Empty cavity |0> Phase shift

of Ramsey fringes

on the e-g transition

1

0

P(e

)

φ

∆φ(n)

Fock state |n>

1( )

2e g+ ( )1

( )2

i ne e g∆Φ+ ∆Φ(n)=Φ0n

Φ0=Ω2/ δTint

2/ωωδ cavat Ω> >−=No energy exchange

ω ωcav at

δDispersive regime :

|g>

|e>

Page 76: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Parity Measurement

φ

Vacuum |0>

State |1>

State ρ

Φ0=πFor φ=φ* :

If N even, detection in e

If N odd, detection in gC

C(g)P-(e)PP(n))1(P **

n

n^==−=∑ φφ

P=+1 for state |2n>

P=-1 for state |2n+1>

1

0

P(e

)

φ∗

N even

N odd

The measurement of the final atomic state gives the parity operator value

P=eia+a

Page 77: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Photon number decimation

0 1 2 3 4 5 6 7 8n

P(n)

Pg(n)

0 1 2 3 4 5 6 7 8n

P(n)

0 1 2 3 4 5 6 7 8n

P(n)

|e> |g>

Ramsey interferometer set on a bright fringe for |0>

∆Φ(n)=Φ0n=π.n

Initial distribution

Final conditionalprobabilities

Page 78: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

A complete QND measurement

0 1 2 3 4 5 6 7n

P(n)

D étection dans |e> D étection dans |g>

0 1 2 3 4 5 6 7n

P(n)

0 1 2 3 4 5 6 7n

P(n)

D étection dans |e> Détection dans |g>

0 1 2 3 4 5 6 7n

P(n)

0 1 2 3 4 5 6 7n

P(n)

Détection dans |e> D étection dans |g>

0 1 2 3 4 5 6 7n

P(n)

0 1 2 3 4 5 6 7n

P(n)

Pg(n)8

8Pg(n)

Pg(n)8

8

Φ0=π

Φ0=π/ 2

Φ0=π/ 4

Adapt the phase shift per photon and the interferometer phase at each step

Number of steps:

Ns=Log2(<n>)

Page 79: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

The first step of the QND measurement

Measurement of the Wigner distribution

Page 80: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Wigner function: an insight into a quantum state

A quasi-probability distribution in phase space.

Characterizes completely the quantum state

Negative for non-classical states.

Describes the motion of a particle or

a quantum single mode field q

p

Re(α)

Im(α)

Motion

of a particle

Electromagnetic

field

Page 81: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Properties

Definition

By inverse Fourier transform

In particular

The probability distribution of x is obtained by integrating W over p

This property should obviously be invariant by rotation in phase space

All elements of density matrix derived from W: contains all possible information on quantum state.

( )( ) ( ) ( )†

cos sin , sin cos

ˆ ˆˆ

W q p q p dp

P q q U U q

θ θ θ θ θ

θ

θ θ θ θ

θ ρ θ

− +

= =

Page 82: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Examples of Wigner functions

-20

2

-2

0

2-2

-1

0

1

2

-2

-1

0

1

2

Vacuum |0>

-2 0 2

-2

0

2

-2

-1

0

1

2

-2

-1

0

1

2

Fock state |1>

-4 -2 0 2 4

-4

-2

0

24-2

-1

0

1

2

-2

-1

0

1

2

Thermal field nth=1

-4 -2 0 2 4

-4

-2

0

24-2

-1

0

1

2

-2

-1

0

1

2

Coherent state |β>(β=1.5+1.5i)

42

02

4

42

02 4

0.3

0

42

02

4

42

02 4

5 photons Fock state

Page 83: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

How to measure W for the electromagnetic field ?

Propagating fields : « Tomographic » methods

Principle : - Homodyning measures marginal distributions P(qθ) for different

θ - inverse Radon transform allows reconstruction of W(q,p)

( medical tomography)

Refs : - Coherent and squeezed states : - Smithey et al., PRL 70, 1244 (1993)

- Breitenbach et al., Nature 387, 471 (1997)

- One-photon Fock state : Lvovsky et al., PRL 87, 050402 (2001)

- α|0>+β|1> : Lvovsky et al., PRL 88, 250401-1 (2002)

Page 84: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

RESULTATS EXPERIMENTAUXRESULTATS EXPERIMENTAUX

Smithey et al., PRL 70, 1244 (1993)

Comprimé Vide

Breitenbach et al, Nature 387, 471 (1997)

Page 85: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

MESURE COMPLETE DE LA DISTRIBUTION DE WIGNER MESURE COMPLETE DE LA DISTRIBUTION DE WIGNER POUR UN PHOTONPOUR UN PHOTON

Lvovsky et al, PRL 87, 050402 (2001)

Page 86: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Other methods

Use the link between W and parity operator

Displace the field and measure parity by determination of photon number probability

Direct counting (Banaszek et al for coherent states)

Quantum Rabi oscillations for an ion in a trap (Wineland)

A demanding method. Much more information than the mere average parity needed

^ ^ ˆW(α) 2Tr(D( α)ρ D(α) )( 1)N= − −

Page 87: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Wigner distribution for a trapped ion

D. Liebfried et al, PRL 77, 4281 (1996), NIST, Boulder

Etat nombre 1n = ( )10 1

2+

Matrice densité

• Same outcome for trapped neutral atom:- G.Drobny and V. Buzek, PRA 65 053410 (2002)

From the data of C. Salomon et I. Bouchoule

Page 88: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Our approach

))1()α(Dρ)α(D(Tr2)α(Wˆ^^

−−= N

- Proposed by Lutterbach and Davidovich (Lutterbach et al.PRL 78 (1997) 2547)

- Based on :

W is the expectation value of the Parity operator in the displaced state ρ( −α)

)1( N−

A) Apply D(-α) Inject –α in cavity mode OK

B) Parity measurement directly gives )1( N−

ρ(-α)

D(-α)

ρ(-α)

ρ

« parity » operator

ˆ( 1)N n− =

n+ if n=2k

n− if n=2k+1

Page 89: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

p osi

ti on

Atomicfrequency

νcav

Cavity mode

time

D(-α)

π/2∆φ

π/2Dispersive interaction

e-g detection

δ

|g,0>R1

R2

Testing the method: vacuum state Wigner function

•Use Stark effect to tune interferometer phase•No phase information in cavity field: injected field phase irrelevant•Finite intrinsic contrast of the Ramsey interferometer

Page 90: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Wigner function of the "vacuum"

0.2

0.4

0.6

0.2

0.2

0.4

0.4

0.6

0.6

-1 0 1 2 3φ/π

πW(α)

0

1

2

0 21 α

P(e

)P

(e)

P(e

)

Nphot

0 1 20

0.5

1 0.83

0.120.05

(norm.)

α=0

α=0.6

α=1.25

Page 91: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Single photon Wigner function measurement

p osi

ti on

Atomicfrequency

νcav

Cavity mode

time

D(-α)

π/2∆φ

π/2

Wigner function measurement scheme

Dispersive interaction

e-g detection

δ

Preparation

of cavity state

R1

R2

π|e,0>

|g,1>

Page 92: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Wigner function of a "one-photon" Fock state

0,0 0,5 1,0 1,5 2,0

-1,0

-0,5

0,0

0,5

1,0

απW(α

)

Nphot

0.25

0.71

0.040

0.5

1

0 1 2

(norm.)

Φ/ π

Φ/ π

P(e

)P

(e)

P(e

)

0 1 2 30,3

0,4

0,5

0,6

0,7

0 1 2 30,3

0,4

0,5

0,6

0,7

0 1 2 30,3

0,4

0,5

0,6

0,7

α=0

α=0.3

α=0.81

Φ/ π

Page 93: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Towards other states- Cavity QED setup : direct measurement of the field

- Next improvements : - better isolation

- better detectors

- In the future : « movie » of the decoherence of a Schrödinger cat

-20

2

-2

0

2

-2

-1

0

1

2-2

02

2)/10( +More complex states : ex

-4

-2

0

2

4-2

-10

12 -2

-1

0

1

2

-4

-2

0

2

4

-2

-1

0

1

2

-4

-2

0

2

4-2

-10

12 -2

-1

0

1

2

-4

-2

0

2

4

-2

-1

0

1

2

…….

Page 94: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Outline

More about the fieldQND detection of more than one atom

Measurement of the Wigner distribution

More about the atomQND detection of the atomic state

New experimental toolsA two-cavity setup

Page 95: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Phase shift with dispersive atom-field interaction

Non resonant atom: no energy exchange but cavity mode frequency shift (atomic index of refraction effect).

Phase shift of the cavity field (slower than in the resonant case)

-150 -100 -50 0 50 100 150

0,45

0,50

0,55

0,60

0,65

0,70

Atom in e

-150 -100 -50 0 50 100 150

0,45

0,50

0,55

0,60

0,65

0,70

0,75

No atom

-150 -100 -50 0 50 100 150

0,45

0,50

0,55

0,60

0,65

0,70

Atom in g

Atom in g

No atom

Atom in e

Opposite values for e and g

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

0,45

0,50

0,55

0,60

0,65

0,70

0,75

No atom 1 atom in g 2 atoms in g

Phase (°)

Proportional to atom number

Page 96: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Absolute measurement of atomic detection efficiency

Histogram of field phase reveals exact atom count

Comparison with detected atom counts provides field ionization detectors efficiency in a precise and absolute way

0.4 atoms samples:

70-90 % detection efficiency

Page 97: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Inject a very large coherent field in the cavity

Send an atomic sample

Different phase shifts for e, g or no atom

Inject homodyning amplitude

Zero amplitude for e. Larger for no atom.

Still larger for g

Read final field amplitude by sending a large number of atoms in g

Final number of atoms in e proportional to photon number

Towards a 100% efficiency atomic detection

Re(α)

Im(α)

−φ

g

φ

e

e

g

Page 98: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Preliminary experimental results

detection efficiency: 87%

error probability: 0 atom detected as 1: 10% (main present limitation)

e in g: 1.6%

g in e: 3%

100% detection efficiency within reach with slower atoms: v=150 m/s ….experiment in progress.

0 10 20 30 40

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

Atom 1 prepared in g no atom 1 atom in g

Atom1 prepared in e no atom 1 atom in e

Pro

babi

lity

Total number of excited atoms

Experimental conditions:• 75 photons initially• v=200 m/s• δ=50 kHz• 70 absorber atoms

Page 99: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Outline

More about the fieldQND detection of more than one atom

Measurement of the Wigner distribution

More about the atomQND detection of the atomic state

New experimental toolsA two-cavity setup

Page 100: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

A two-cavity experimentRydberg atoms and superconducting cavities:

Towards a two-cavity experiment

Creation of non-local mesoscopic Schrödinger cat statesNon-locality and decoherence (real time monitoring of W function)

Complex quantum information manipulationsQuantum feedback

Simple algorithms

Three-qubit quantum error correction code

Page 101: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Teleportation of an atomic state

A

C

B

Da

Db

Dc

R1

R2

R3

Pb

Pa

C1

C2

beam 3

3'beam 1

03

2

EPR pair

beam 2

1

1'

Davidovich et al,PHYS REV A 50 R895 (1994)

• This scheme works for massive particles• Detection of the 4 Bell states and application of the "correction" to the target is possible using a C-Not gate (beam 2 and 3)• The scheme can be compacted to 1 cavity and 1 atomic beam

Page 102: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Entangling two modes of the radiation field

Principle:

First atom

Initial state

π/2 pulse in Ma

π pulse in Mb

Second atom:

probes field states

Final transfer rate modulated versus the delay at the beat note between modes

Single photon beats

(b)As Ap

π

ππ/2

π/2DD

Ma

Mb

t0 π/2Ω 3π/2Ω Τ+π/ΩT

0

−δ

, 0,0e

( )1,0,0 ,1,0

2e g+

( )10,1 1,0

2g +

48 50 52 54 56 580.0

0.5

1.0

200 202 204 206 2080.0

0.5

1.0

400 402 404 406 4080.0

0.5

1.0

698 700 702 704 7060.0

0.5

1.0(d)

(c)

(b)

(a)

T(µs)

Pe(T)

Page 103: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Implementation of 3 qubit error correction

S S

Error

tect

ion

tect

ion

Cor

rect

ion

all the tools exist!

Cor

rect

ion

encoding decoding

R

R

R

R R’

R’

R’

R’0 1α β+

0

0

Error?

σz2

σz3

Ramsey π/2 pulses

encoding and decoding: preparation of a GHz triplet

0

error detection

Page 104: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

New non-locality explorations

Use a single atom to entangle two mesoscopic fields in the cavity

A non-local Schrödinger cat or a mesoscopic EPR pair

Easily prepared via dispersive atom-cavity interaction

Page 105: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Mesoscopic Bell inequalities

A Bell inequality form adapted to this situation

Here, Π is the parity operator average. Dichotomic variable for which the Bell inequalities argument can be used (transforms the continuous variable problem in a spin-like problem)

Maximum violation for parity entangled states:

Page 106: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Bell inequalities violation

Optimum Bell signal versus γ

A compromise between violation amplitude and decoherence: γ²=2

Page 107: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Probing the Wigner functionA second atom to read out both cavities (same scheme as for single mode Wigner function)

Page 108: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

A difficult but feasible experiment

Bell signal versus time Tc=30 and 300 ms

Page 109: Preparation of a GHZ state Cavity QED Gilles Nogues ...issqui05/Nogues_all.pdf · Cavity QED Gilles Nogues cavity QED with Rydberg atoms Basic atom-field interactions: producing

Outline

More about the fieldQND detection of more than one atom

Measurement of the Wigner distribution

More about the atomQND detection of the atomic state

New experimental toolsA two-cavity setup