Quantum Communication, Quantum Entanglement and All That Jazz
Entanglement, quantum measurement and quantum information ...rydphy04/lectures/Brune_coll.pdf ·...
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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⟩
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Quantum entanglement and Cavity QED
• Principle of cavity QED experiments:
•Two level atoms interacting with a single mode of a high Q
cavity
•The "strong coupling" regime: coupling >> dissipation
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Probing EPR entanglement and complementarity
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.
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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?
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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
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Entanglement and quantum informationCan one do something "useful" from the strangeness of quantum logic?• Yes:
Quantum cryptography: deos workQuantum communication: teleportationQuantum algorithms:
Search problems (Grover)Factorization (Shor)
• How? by manipulating qubits with a quantum computer
• Practically: extreeeeemely difficult to realize: a quantum computer manipulates huge Schrödinger cat states.
Anyway interesting for understanding the essence and the limits of quantum logic
Qubit: lwo level system0 and 1Classical bit:
0 ou 1 1 0 12
a b+
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outline1. Rydberg atoms in a cavity:
achieving the strong coupling regime
2. Rabi oscillation in vacuum: entanglement and complementarity at work
4. Rabi oscillation a mesoscopic field:Schrödinger cat state and decoherence
A B C
EDOM Orsay 21-1-04 9
1. Rydberg atoms in a cavity: achieving the strong coupling regime
One photon and one atomin a box:
Photon box: superconducting microwave cavity
“circular" Rydberg atoms
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The cavity
- one mode- a few trapped
photons
a "photon box":- superconducting Niobium mirrors- microwave photons:λ=6 mm, νcav=51GHz- photon lifetime:
Tcav=1ms ( Q = 3.108 )
0 2 4 6 8 10 12 14 16 1818
27
36
45
Measurement of cavity damping time
τcav= 1 msQ = 3.3 108
ν= 51.0989 GHz
Mic
row
ave
Pow
er(d
B)
time (ms)
5 cm
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The "circular" Rydberg atoms
e
g
n=51
n=50
νat=51.1 GHzνcav
"Circular Rydberg atoms":l=|m|=n-1
- radiative lifetime: 30 ms- dipôle: d= 1500 u.a.- ideal closed two levelsystem
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Laser velocity selection
Circular atompreparation:- 53 photons process- pulsed preparation0.1 to 10 atoms/pulse e-
State selectivedetector
One atom = one click
Cryogenic environmentT=0.6 to 1.3 K
weak blackbody radiation
85Rb
Experimental set-up
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Experimental setup
Atom preparation
detection
atomicbeam
lasers
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Resonant atom-field coupling
→ Coherent Rabi oscillation
Ω0/2π=50 kHzTrabi=20 µs
22
ge
0
1ge
0
1
e,0 g,1
0 0, 0 cos ,0 sin ,12 2
t te e i gΩ Ω → ⋅ − ⋅
Electricdipole
couplingΩ0
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Single photon induced Rabi oscillation
g
eωge= ωcav
0 20 40 60 80 1000,0
0,2
0,4
0,6
0,8
1,0P g (
50 c
irc)
interaction time (µs)
e-
eΩ0=47 kHzTRabi=20µse,0
g,1
Coherent Rabi oscillation replaces irreversible damping
by spontaneous emission
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Vacuum Rabi oscillation and quantum gates
g
eωge= ωcav
0 20 40 60 80 1000,0
0,2
0,4
0,6
0,8
1,0P g (
50 c
irc)
interaction time (µs)
e-
e
e,0
g,1
• EPR pair preparation
•Atom-field state exchange
• Phase gate, QND detection of a single photon
π/2
π
2π
Ω0=47 kHzTRabi=20µs
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0
2
4
6
8
10
Pos
ition
(cm
)
Tailored preparation of of tree entangled qubits
π/2 π2π
π/2
θ
D DD
π/2 π/2 π/2
Time
π/2
D
• Rabi oscilation in C
• Classical p/2 pulse
• Detection
( )1 0,0,0 1,1,12
−π/2
Atom # 1
Atom # 2
Atom # 3
EDOM Orsay 21-1-04 18
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
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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
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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⊗ + ⊗
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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)
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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
+
− =
+=
−
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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 + =
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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+
+
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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
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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
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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)
– Non-locality and decoherence
Complex manipulation of quantum information– Quantum feedback– Elementary algorithm– Error correction code
Dresden 10-05-04 36
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
Post doc• Ferdinand Schmidt-Kaler• Edward Hagley• Christof Wunderlich• Perola Milman
Colaboration• Luiz Davidovich• Nicim Zagury• Wojtek Gawlik
Permanent• Gilles Nogues• Michel Brune• Jean-Michel Raimond• Serge Haroche
Dresden 10-05-04 37
Les bizarrerie quantiques:intrication, mesure et décohérence
• Limite classique-quantiquePas de superpositions à notre échelleLe “chat de Schrödinger"
Nous n’observons qu’une toute petite fraction des états possibles
• Décohérence
Couplage inévitable d’un système macroscopique à un environnement
Quelques états stables (base préférée)Les superpositions quantiques de ces états sont très rapidement détruites
Environement
( )12
+ ⇔
Dresden 10-05-04 38
Rabi oscillation in a small coherent field
0 20 40 60 800,0
0,5
1,0
interaction time (µs)
P g(t) exp
fitS
e
e-
e
2 0.85 photonα =
α
Dresden 10-05-04 39
Rabi oscillation in a small coherent field:observing discrete Rabi frequencies
Fourier transform of the Rabi oscillation signal
0 25 50 75 100 125 150
Frequency (kHz)
FFT
(arb
. u.)
Discrete peaks corresponding to
discrete photon numbers
0Ω0 2Ω
0 3Ω
Direct observation of field quantization
in a "box"
Dresden 10-05-04 40
Rabi oscillation in a small coherent field:Measuring the photon number distribution
( ) ( )( )0/1 1 co( ) s 1 .
2t
gN
P t P et NN τ−− Ω += ∑Fit of P(n) on the Rabi oscillation signal:
0 1 2 3 4 50,0
0,1
0,2
0,3
0,4
0,5
Photon number
P(n)
Measured P(n) Poisson law
accurate field statistics measurement
0.85N =
Dresden 10-05-04 41
Rabi oscillation in small coherent fields
0,0
0,5
1,0
Four
ier
Tran
sfor
m A
mpl
itude
(β)
(γ)
(δ)
(α)
(d)
(c)
(b)
(a)
(D)
(C)
(B)
(A)
Probability P(n)
e to
g t
rans
fer r
ate
0,0
0,5
0,0
0,5
0 30 60 90
0,0
0,5
Time (µs)0 50 100 150
Frequency (kHz)
0,00,5
1,00,0
0,50,0
0,5
0 1 2 3 4 5
0,00,3
n
<n>=0.06
<n>=0.4
<n>=0.85
<n>=1.77
Phys. Rev. Lett. 76, 1800 (1996)
Dresden 10-05-04 42
1. Préparation du champ à mesurer:Un état cohérent |α> par exemple2. Addition d'un champ de même
amplitude et de phase variable θ
3. Un atome absorbe le champ résiduel: mesure P(0)
Observation de la distribution de phase du champ:“detection homodyne”
α
( ). ie π θα −
θ
signal d'absorption par un atome S
|g> Pg ?
-200 -150 -100 -50 0 50 100 1500,4
0,5
0,6
0,7
0,8
Pg
Phase θ (°)
33 photons
Interférence destructive pour θ=0.
Dresden 10-05-04 43
Détection des deux composantes de phase
20 25 30 35 40 45 50 55
-40
-20
0
20
40
60
Fiel
d ph
ase
shift
(°)
Atomes "lents"N petit
Atomes rapidesN grand
0 int. 4t NΦ = Ω