EDOM Orsay 21-1-04 1 Entanglement, quantum measurement and quantum information with Rydberg atoms and cavities Michel BRUNE DÉPARTEMENT DE PHYSIQUE DE L’ÉCOLE NORMALE SUPÉRIEURE LABORATOIRE KASTLER BROSSEL Paris France Permanents: S. Haroche J.-M. Raimond G. Nogues Post-Doc: S. Kuhr PhD: P. Maioli A. Auffeves T. Meunier S. Gleyzes P. Hyafil J. Mozley
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EDOM Orsay 21-1-04 1
Entanglement, quantum measurement and quantum information with Rydberg atoms and cavities
Michel BRUNE
DÉPARTEMENT DE PHYSIQUE DEL’ÉCOLE NORMALE SUPÉRIEURE
LABORATOIRE KASTLER BROSSELParis France
Permanents: S. HarocheJ.-M. RaimondG. Nogues
Post-Doc: S. Kuhr
PhD: P. MaioliA. AuffevesT. MeunierS. GleyzesP. HyafilJ. Mozley
Dresden 10-05-04 2
Quantum and classical interacting systems
Coupling→ entanglement
State preparation
Measurement:• randomness• EPR correlation
|ψA,B⟩≠ |ψA⟩⊗|ψB⟩
|ψA⟩
"Rest of the world"
isolated system
Clas
sica
l in f
orm
a tio
n
|ψB⟩
Dresden 10-05-04 3
Quantum entanglement and Cavity QED
• Principle of cavity QED experiments:
•Two level atoms interacting with a single mode of a high Q
Illustrating Bohr-Einstein dialog central concept: complementarity
A B C
Light slits: recoil of the slit monitors which path
informationNo interferences: mater behave as particles
Massive slits: insensitive to collisions with single particlesInterferences: mater behave
as wavesExperiment performed with photons, electrons, atoms,
molecules.
Dresden 10-05-04 5
The"Schrödinger cat"
• Elementary formulation of the problem: Superposition principle in quantum mechanics:→ Any superposition state is a possible state→ Schödinger: this is obviously absurd when applied to
macroscopic objects such as a cat !!
Up to which scale does the superposition principle applies?
Dresden 10-05-04 6
Schrödinger cat and quantum theory of measurement
• Hamiltonian evolution of a microscopic system coupled to a measurement apparatus:
...12at chat+ +Ψ = +
→ Entangled atom-meter state Problem: real meters provide one of the possible results not a
superposition of the two → too much entanglement in QM?
Need to add something?NO! "decoherence" does the work
Dresden 10-05-04 7
Entanglement and quantum informationCan one do something "useful" from the strangeness of quantum logic?• Yes:
2. Rabi oscillation in vacuum:entanglement and complementarity at work
A B C
Dresden 10-05-04 19
Complementarity in the Ramsey interferometer
Classical π/2 pulses Ramsey fringes signalwith two classical
pulses: results from a two path
interference
-1 0 1 20,0
0,2
0,4
0,6
0,8
1,0
φ/π
P g(φ)
Atomic beam
Classical microwave fields acts as "beam-splitters"
for the internal atomic state
e
gS
De
g
e
S
e
Dresden 10-05-04 20
From classical to quantum beam splitters: • π/2 pulse in a large coherent state:
( )12
e e gα α⊗ → + ⊗
When α is large enough, one more photon in the field does not make any difference on the field state
NO which path information stored in the field: "classical beam splitter"
n n
0 2 0 4 0 6 0 8 0 10 0 12 0
P(n)
n
nPoisson law
e gα α α≈ ≈
R1
e e
g
α
n
( ),0 ,112
0 e ge +⊗ →R1
,0e
,1g
,0eAtom-field EPR pair:Hagley et al. PRL 79,1 (1997)
The photon number is a perfect label of the atomic stateFULL which path information stored in the field: "quantum beam splitter"
• π/2 pulse in vacuum: α=0
Dresden 10-05-04 21
Resonant interaction with a coherent field as "beam-splitter" in a Ramsey interferometer
Classical π/2 pulses
e
gS
De
g
e
S
e
π/2 pulse in C
π/2
entangled EPR pair
-1 0 1 20,0
0,2
0,4
0,6
0,8
φ/π
0,2
0,4
0,6
0,8
0,2
0,4
0,6
0,8
n=12.8
n=2
n=0.31
α=3.6
α=1.41
α=0.56
0,2
0,4
0,6
0,8
1,0
n=0α=0
Pg
Pg
( )12 gee gα α⊗ + ⊗
Dresden 10-05-04 22
Quantitative interpretation in term of atom-cavity entanglement
• Variation of fringe Visibility V: R1 R2
e α⊗
( )12 gee gα α⊗ + ⊗
.e gV α α η=
η: saturated contrast at large n
Reduced atom density matrix:
*11
2 1
e gat
e g
α αρ
α α
=
0 2 4 6 8 10 12 14 160,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
experimenttheory
V
Average photon number <n>
η=0.73
EDOM Orsay 21-1-04 23
4. Rabi oscillation a mesoscopic field:Schrödinger cat state and decoherence
Dresden 10-05-04 24
Coherent field states
• Number state: |N⟩• Quasi-classical state:
2 2 ;!
Ni
N
e N eN
α αα α α− Φ= =∑
( ) 2;!
NNP e NN
NN α−= =
•Photon number distribution
∆N = 1/|α|
∆N . ∆Φ > 1
∆Φ=1/|α|
1
0 1 2 3 4 5 60,0
0,1
0,2
0,3
0,4Poisson distribution
N=2
P(N
)
Photon number N
|α|
• Phase space representation
Φ
Re(α)
Im(α)
Dresden 10-05-04 25
Rabi oscillation in a classical field
Rabi oscillation results from a quantum interference between probability amplitudes of the two atomic "eigenstates"
( )
( ) ( ) ( )
( ). 2 . 2, ,
0
cos . 2 . .sin . 2
1 . .2
R R
at
at R R
i t i tat at
e
t t e i t e
e e
ψ
ψ
ψ ψ− ++ −
Ω Ω
=
→ = Ω − Ω
= +
( )( )
,
,
1 2
1 2
at
at
e g
e g
ψ
ψ
+
− =
+=
−
Dresden 10-05-04 26
Rabi oscillation in a mesoscopic field
( )
( ) ( ) ( ) ( ) ( )( )
( ) ( )( )
. 2 . 2, ,
. 2 . 2
0
1 . .2
1 . .2
R R
R R
i t i tat at
i t i t
e
t e t t e t t
e t e t
ψ α
ψ α ψ α ψ
ψ ψ
− ++ + − −
− ++
Ω Ω
Ω Ω−
= ⊗
→ ≈ ⊗ + ⊗
= +
Rabi oscillation frequency:2
0 1 where R N N αΩ = Ω + =
Dresden 10-05-04 27
Graphical representation of the atom-field state in the complex plane:
( ) ( ) ( ),att t tψ α ψ+ + +≈ ⊗
Re(α)
Im(α)
φ
( ) ( )
( ) ( )( ),
.
12
i t
at
ti
e
e
t
t e g
α α
ψ
− Φ
−+
Φ
+ =
= +
( ) 0. 4t t NΦ = Ω
( )( ),at
t
t
α
ψ+
+
Dresden 10-05-04 28
Graphical representation of the atom-field state in the complex plane:
( ) ( ) ( ),att t tψ α ψ− − −≈ ⊗
Re(α)
Im(α)
−φ
( ) ( )
( ) ( )( ),
.
12
i t
at
ti
e
e
t
t e g
α α
ψ
+ Φ−
+ Φ− = −
=
( )( ),at
t
t
α
ψ+
+
( )( ),at
t
t
α
ψ−
−
( ) 0. 4t t NΦ = Ω
Dresden 10-05-04 29
Graphical representation of the atom-field state in the complex plane:
Re(α)
Im(α)
−φ
( )( ),at
t
t
α
ψ+
+
( )( ),at
t
t
α
ψ−
−
φ
( ) ( ) ( )( ). 2 . 21 . .2
R Ri t i tt e t e tψ ψ ψ− ++
Ω−
Ω≈ +
( ) 0. 4t t NΦ = Ω
• Entangled atom-field state:
A Schrödinger cat state:Field phase "measures" the atomic state
Dresden 10-05-04 30
Measured phase distribution of the "Schrödinger cat" field state
- Coherent field containing 10 to 40 photons- atom 1: prepares the "cat state"- atom 2: measures the phase distribution of the field
S
|g> Pg ?Atom 1
-200 -150 -100 -50 0 50 100 150
0,5
0,6
0,7
0,8
Pg
Phase θ (°)
33 injected photons no atom 1 atom 1: 335m/s atom 1: 200m/s
Dresden 10-05-04 31
Wigner of the field state
15 injected photonsV=335 m/sTcav=800 µs
-6
-4
-2
0
2
4
6
-6-4
-2
0
2
4
6
-0,2
-0,1
0,0
0,1
0,2
0,3
0,4
0,5
Im(α)
Re(α)
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
6
X Axis Title
Y Ax
is T
itle
-0,2500
-0,1563
-0,06250
0,03125
0,1250
0,2188
0,3125
0,4063
0,5000
Wigner function can be measured: B. Englert et al., Opt. Comm. 100, 526 (1993), Lutterbach and Davidovich, PRL 78 2547 (1997)P. Bertet et al., PRL 89, 200402 (2002)
Dresden 10-05-04 32
Rabi oscillation entanglement and complementarity
Rabi oscillation results from a quantum interference between andNo oscillations as soon as: complementarity again!
( ) ( ) ( ) ( ) ( )( ). 2 . 2, ,
1 . .2
R Ri t i tat att e t t e t tψ α ψ α ψ− +
+ + −Ω Ω
−≈ ⊗ + ⊗
,atψ + ,atψ −
( ) ( ) 0t tα α+ − ≈
Preparation of a phase catField states coincide again: revival of Rabi oscillation signature of the coherence of the "cat" state
Collapse of Rabi oscillation10 20 30 40 50
0.2
0.4
0.6
0.8
1Pe(t)
Ω0t/2π
Dresden 10-05-04 33
Demonstration of coherence by induced revival
Atom-field entanglement
π rotation of the atomic state
Field phase evolution is reversed
Recombinaison ot the two field components:
Revival of Rabi oscillation
Morigi et al PRA 65, 040102
Dresden 10-05-04 34
Induced revival signal
-20 -10 0 10 20 30 40 50 60
0,0
0,2
0,4
0,6
0,8
1,0
Tra
nfe
r
Interaction time
18.5 µs
-20 -10 0 10 20 30 40 50 60
0,0
0,2
0,4
0,6
0,8
1,0
Tra
nfe
r
Interaction time
-20 -10 0 10 20 30 40 50 60
0,0
0,2
0,4
0,6
0,8
1,0
Tra
nsf
ert
Interaction time
Π Pulse
22 µs
-20 -10 0 10 20 30 40 50 60
0,0
0,2
0,4
0,6
0,8
1,0
Tra
nsf
er
Interaction time
23.5 µs
Dresden 10-05-04 35
Perspectives
• Rydberg atoms and superconducting cavitiesA two cavity experiment
EPR pair of Schrödinger cat states (non-locality at the mesoscopic scale)