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 , 0⊗ 1 √ 2 (|i 2 + |g 2 )+ |g 1 , 1⊗ 1 √ 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 ) «
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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
Lecture 4: future directions
Gilles NOGUES
Laboratoire Kastler Brossel
Ecole normale supérieure, Paris
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
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
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
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
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>
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
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
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>)
The first step of the QND measurement
Measurement of the Wigner distribution
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
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
− +
= =
∫
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
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)
RESULTATS EXPERIMENTAUXRESULTATS EXPERIMENTAUX
Smithey et al., PRL 70, 1244 (1993)
Comprimé Vide
Breitenbach et al, Nature 387, 471 (1997)
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)
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= − −
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
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
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
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
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>
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
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
…….
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
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
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
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
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.
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
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
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
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)
Implementation of 3 qubit error correction
S S
Error
Dé
tect
ion
Dé
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
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
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:
Bell inequalities violation
Optimum Bell signal versus γ
A compromise between violation amplitude and decoherence: γ²=2
Probing the Wigner functionA second atom to read out both cavities (same scheme as for single mode Wigner function)
A difficult but feasible experiment
Bell signal versus time Tc=30 and 300 ms
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