rabi oscillation at the quantum-classical boundary generation of schrödinger cat states
DESCRIPTION
Rabi oscillation at the quantum-classical boundary Generation of Schrödinger cat states. Alexia Auffèves-Garnier. Soutenance de thèse 29 juin 2004. About coherent superpositions. Any superposition of states is a possible state. Physical meaning of a "coherent superposition"?. - PowerPoint PPT PresentationTRANSCRIPT
Rabi oscillation at the quantum-classical boundaryGeneration of Schrödinger cat states
Alexia Auffèves-Garnier
Soutenance de thèse29 juin 2004
About coherent superpositions
Any superposition of states is a possible state
| | |ia e b Young's hole experiment
Physical meaning of a "coherent superposition"?
a bP P P Addition of the probabilityamplitudes of each path
Interference fringes
Signature of a coherent superposition=interference fringes
Quantum phase of the superposition=phase of the fringes
| | |ia e b
Entanglement
(1)
S(2)
Coherent superposition for a bipartite system
"Entangled state" = non factorisable state
No system is in a definite state
Correlations in all the basis Quantum correlationsViolation of Bell's inequalities
H
V V
H
H
V
H
V
Anticorrelation
The two photonsform an EPR-pair
1| (| HV | VH )
2epr
1| (| H V | V H )
2epr
The Schrödinger-cat paradox
Entanglement of a microscopic system with a macroscopic one
A two-level atom and a cat in a box
| | alive
| | dead
e
g
Total correlation
The cat "measures" the atomic state
Linear evolution
The system form an EPR pair
Quantum correlations
Atom projected on |e>+|g>
Cat projected on |dead>+|alive>
Macroscopic state superposition
What enforces classicality: concept of complementarity
A superposition of states remains coherent No information about the state
A slit acts as a Which-Path detector
Two possible states for the slitcorrelated with the two particle's states
| |
| |a
b
a
b
The particle-slit system gets entangled
1| (| , | , )
2i
a ba e b
| | |a bC Contrast of the fringes
No interference if there exist a Which-Path information
Decoherence
A macroscopic object interacts with its environment and gets entangled with it
Coherent superposition (dead> and |alive>)
Statistical mixture (|dead> or |alive>)
No interference between macroscopic states
Time of decoherence all the faster as the two cat states are more different
Decoherence
Quantum correlations in all the basis
Classical correlations in the « natural » basis
The quantum-classical boundary
decohTintT
Microscopic object
Mesoscopic object
Environment
-3 parts-2 time scales
-Entanglement-« Schrödinger cat » states-Quantum behavior
Continuous parameter to explore the quantum-classical boundary?
Quantum world Classical world
int decohT T int decohT T-Continuous monitoring of the environment-No entanglement-Classical behavior
Tools of CQED
0 20T s
Rydberg states |e> and |g>High Q superconducting cavity
Vacuum Rabi period
Field relaxation
Strong coupling regime
30msatT
0 ,cav atT T T
1 mode of the electromagnetic field
A two level system
1 mscavT Long life time
51 (level e)
50 (level g)
51.1 GHz
From quantum to classical Rabi oscillation
Microscopic object
Mesoscopic object
Environment
Resonant interaction between the atom and the fieldRabi oscillation
Quantum Rabi oscillationClassical Rabi oscillation
intTdecohT
Rydberg atom
Coherent field in the cavity mode
Infinity of external modes
-Atom-field entanglement-Preparation of mesoscopic coherences of the field
-Field unsensitive to the atom-Factorized atom-field state
Outline
• Elements of theory– Results on Rabi oscillation – Rabi oscillation at the quantum-classical boundary: the atomic
point of view– Effect of the Rabi oscillation on the field
• Experimental study– The atom-field system– Preparation of mesoscopic states superpositions of the field:
experimental results– Evidence of correlation between atomic and field state– Coherence of the prepared superposition: where are the cats?
Outline
• Elements of theory– Results on Rabi oscillation – Rabi oscillation at the quantum-classical boundary: the atomic
point of view– Effect of the Rabi oscillation on the field
• Experimental study– The atom-field system– Preparation of mesoscopic states superpositions of the field:
experimental results– Evidence of correlation between atomic and field state– Coherence of the prepared superposition: where are the cats?
Basics on two-level systems
Y
Z
X
|e>
|g>
| cos( / 2) | sin( / 2) |ie e g
Two-level system {|e>;|g>}
General state
A vector in the Bloch sphere
Dynamics of the system
Precession around the eigenstates of the hamiltonian
|
|g>
|e>
|g>
|e>
Classical Rabi oscillation
|e>
Two-level system {|e>;|g>} interacting with a resonant field |g>
|e>
cos( )cl t
Rotating frameRotating wave approximation
Y
Z
X
|g>
|+>
| - >
Eigenstates of the hamiltonian1
| (| | )2
e g
1| ( | | )
2e g
Rabi oscillation at frequency cl1
1
0
Time (a.u.)
eP
Rabi oscillation as an interference
|+>
| - >
|e> |e>
Evo
lutio
nC
ha
nge
of b
asi
s
De
tect
ion
Ch
ang
e o
f ba
sis
// 2 21|| | ( )
2| i ti tet ee
Detection in {|e>,|g>} basis
Evolution
Classical Rabi oscillation: a quantum beat between two indistinguishable paths
1
1
0
Time (a.u.)
eP
Rabi oscillation in a quantized field
Two-level system {|e>;|g>} interacting with a resonantquantized field |n>
0 | | | |)2JCH a g e a e g
Jaynes-Cummings hamiltonian
Exchange of a quantum of energy
|e,n> |g,n+1>0 1n
|g>
|e>
|n-1>
|n>
|n+1>
|e,n>; |g,n+1> : two levels coupled and degenerate
Eigenstates of the hamiltonian:"Dressed states" 0 1n
| n
| n 1| (| , | , 1 )
2n e n g n
1| (| , | , 1 )
2n e n g n
Rabi oscillation between |e,n> and |g,n+1> at frequency
0 1n
The vacuum Rabi oscillation
0 30 60 900.0
0.2
0.4
0.6
0.8
Pe(t
)
time ( s)
Initial state |e,0> Rabi oscillation at0 Vacuum Rabi frequency
0 49kHz
0 0| , 0 cos | ,0 sin | ,12 2
t te e i g
Atom-field entangled state
Maximal entanglement at
0 2t
Formation of an EPR-pair
Rabi oscillation in a quantum field
entanglement
Rabi oscillation in a classical field
No entanglement
Continuous evolution?
Outline
• Elements of theory– Results on Rabi oscillation – Rabi oscillation at the quantum-classical boundary: the atomic
point of view– Effect of the Rabi oscillation on the field
• Experimental study– The atom-field system– Preparation of mesoscopic states superpositions of the field:
experimental results– Evidence of correlation between atomic and field state– Coherence of the prepared superposition: where are the cats?
Coherent states of the field
Re( )
Im( )
n
Field radiated by a classical source in the mode
2| | / 2| |!
n
n
e nn
2 | |
( )!
n
p n en
Poissonian distribution of the photon number
Representation in the complex plane
| | n n 1
| |
n
n
"Quantum" fieldBig fluctuations
| | 1 "Classical" fieldSmall fluctuations
| | : a continuous parameterto explore the quantum
classical boundary
| | 1
Rabi oscillation in a mesoscopic coherent field
0t/T
eP (t)
0 1n n 1( ) ( ) cos( ) 1
2e nn
P t p n t with
Collapses and revivalsSpectrum of the frequencies
for a mesoscopic field
nDistribution width
Spectrum width
0 02
n
n nn
n
0
( . .)nP a u
n
0 n
0
Collapse when theside components arephase-shifted by
Revival when twosuccessive componentsrecover the same phase
0collT T 0
0
n
0revT T n
Classical limitSpectrum of the Rabi frequencies
( . .)nP a u
n
0 n
0
( . .)nP a u
n
In a coherent field In the classical limit
0
0
0
n
n
« Classical limit »
Atomic signal only dependson the energy of the field
Effect on the phase of the field?
Outline
• Elements of theory– Results on Rabi oscillation – Rabi oscillation at the quantum-classical boundary: the atomic
point of view– Effect of the Rabi oscillation on the field
• Experimental study– The atom-field system– Preparation of mesoscopic states superpositions of the field:
experimental results– Evidence of correlation between atomic and field state– Coherence of the prepared superposition: where are the cats?
Rabi oscillation in a mesoscopic field
| , | ,nn
e C e n Initial state:1
|| ,2
|n nn
n nn
CCe
| ( )t | ( )t Mesoscopic field
1n nC C ( )
2
n nn n o n n
n
2| | / 2
!
n
nC en
with
1n
0 16t n
| ( ) | ( ) | ( )att t t
21| ( ) | |
2at in it e e e g
( ) ( )| ( ) |in t i tt e e
Atomic superposition of quantum phase
0( )4
tt
n
( )t
( )tCoherent field of classical phase
Phase correlation
| ( ) | ( ) | ( )att t t
21| ( ) | |
2at in it e e e g Atomic superposition
of quantum phase ( )t
( )tCoherent field of classical phase
( ) ( )| ( ) |in t i tt e e
Rabi oscillation in a mesoscopic field
| ( ) | (1
| | ( ) | ( ),2
) atatt te t t
The atomic dipole and the field are phase-entangled
Generation of a Schrödinger-cat state
Classical limit
0 04
t
n
0
4 4
nt tn
, || |
/ / 22 |1
| |, |2
i ti t eee
Classical Rabi oscillation Field unchanged
No atom-field entanglement
Field « classical »
/ 2| |at i te
/ 2| |at i te
Geometrical representation
|
X
Y
Z
|+>X
Y
|+>
Equatorial planeof the Bloch sphere
Atomic state in the equatorial plane of the Bloch sphere
Coherent fieldin the Fresnel plane
Representation in the same plane
Phase correlation
Atomic dipole and field « aligned »
Evolution of the atom-field system
A microscopic objectleaves its imprint on a mesoscopic one
Schrödinger-cat situation
| , Initial state
|+>|
| ,
|->
| ( ), ( )at t t | ( ), ( )at t t
| ( )t
| ( )t
| ( ) | (1
| | ( ) | ( ),2
) atatt te t t
D
"Size" of the cat=D
02 sin4
tD n
n
2 ( )( ) | | |(( )) D ttC tt e
The field acts as a Which-Path detector
Contrast of the Rabi oscillation
New insights on collapse and revival
Time (a.u.)
2( )( ) | | |)( () D ttC tt e
Collapse as soon as the two components are well separated
024
tD n
n
00
2collT T
Maximal entanglement:"Schrödinger cat state"
"Size" of the cat=distance D
Factorization of the atomic state
11| | ( 1) |
2n
field i i -Unconditional mesoscopic states superposition-The field has a defined parity
Field states merge again
Revival of the Rabi oscillation0
4RT
n
02RT T n
| ( ) | (1
| | ( ) | ( ),2
) atatt te t t
A more realistic situation
Q function evolution in 20 photonsAtom initially in |g>Rabi oscillation in 20 photons
ref: J. Gea-Banacloche, PRA 44, 5913-5931(1991) V. Buzek et.al., PRA 45, 8190-8203(1992)
Outline
• Elements of theory– Results on Rabi oscillation – Rabi oscillation at the quantum-classical boundary: the atomic
point of view– Effect of the Rabi oscillation on the field
• Experimental study– The atom-field system– Preparation of mesoscopic states superpositions of the field:
experimental results– Evidence of correlation between atomic and field state– Coherence of the prepared superposition: where are the cats?
Selection rules: a two-level systemMicrowave transitionVery long lifetimeVery sensitive to small fields
Circular Rydberg atoms
Stable in a weak electric fieldComplex preparation (53 photons)
Stark tuning:
on e-g transition
Field ionisation detection
51 (level e)
50 (level g)
51.1 GHz
Atoms ofHigh principal quantum numberMaximal orbital and magnetic quantum numbers
85Rb
at 30 msT
0| | 1700egd qa
2255 kHz /(V/cm)
Two superconducting mirrors in Fabry-Pérot configurationcooled down to 0.6K
The superconducting cavity
High quality factor
High confinement of the field
Controlled potentiel between the mirrors
Small thermal field
Field per photon
Coupling with an external source
900Mode TEM
3cavV 769 mm
Recirculation ring
1n 2 modes: lift of degeneracy 86kHz
83.10 , 1 mscavQ T
51.099 GHz
0 02 . 50 kHzegd E
Strong coupling regime00 cav
E 1.58 mV/cm2 V
General scheme of the experiment
Oven« Circularising box » External source
Detection byionisation
External source
Atomic beam
Cavity mode
Lasers ofpreparation
Velocity selection
Outline
• Elements of theory– Results on Rabi oscillation – Rabi oscillation at the quantum-classical boundary: the atomic
point of view– Effect of the Rabi oscillation on the field
• Experimental study– The atom-field system– Preparation of mesoscopic states superpositions of the field:
experimental results– Evidence of correlation between atomic and field state– Coherence of the prepared superposition: where are the cats?
Field phase distribution measurement
SHow to measure a coherent field phase-shift?
Homodyne method
Injection of a coherent field
Second injection
Resulting field
A probe atom is sent in |g>
1gP -Field in the vacuum state
-Field in an excited state 1/ 2gP
Maximum displaced by Field phase-shifted by
| | Sie
| (1 )Sie
Back to the vacuum state 0S
S
= a signal to measure the field phase distribution( )g SP
Experimental field phase distribution
Maximum<1 (thermal field)
Width of the peak
1/ n
Detection of the atom in |e> or |g> Field projected on a superposition of and |
Phase splitting in quantum Rabi oscillation: timing of the experiment
SInjection of a coherent field |
A first atom is sent and interacts resonantly with the field
Injection of
A probe atom is sent in |g>
: two peaks corresponding to the vanishing of each component
Vanishing of |
Vanishing of |
|
|
S
S | Sie
( )g SP
|
Evidence of the phase splitting
v=335m/s
int 032 1.5T s T
Measured phase 23
Expected value 0 int 234
t
n
Experiment and theory in very good agreement
36n
Evolution of the phase distribution
int 0335 / , 1.5av m s t T
int 0200 / , 2.5bv m s t T
0 int
4
t
n
Various velocities
Various number of photons
exp 37
exp 19
30n
« Fast » atom
« Slow » atom
Experiment vs theory
0 int
4
t
n Measured phase vs theoretical phase
experimental points
theory (slope 1)
numerical simulations -second mode -thermal field -relaxation
Experiment andsimulations in very good
agreement
Outline
• Elements of theory– Results on Rabi oscillation – Rabi oscillation at the quantum-classical boundary: the atomic
point of view– Effect of the Rabi oscillation on the field
• Experimental study– The atom-field system– Preparation of mesoscopic states superpositions of the field:
experimental results– Evidence of correlation between atomic and field state– Coherence of the prepared superposition: where are the cats?
Atom-field correlations in the { , } basis
Atomic state with defined energyMesoscopic superposition
states of the field
Entangled state Correlations in all the basis
| |
Evidence of the superpositionEstimation of coherence (Wigner function)
Selective preparation of and
| ( ) | (1
| | ( ) | ( ),2
) atatt te t t
Correlation between | ( )at t | ( )t and
| ( )at t | ( )t
| ( 0) |at t
| ( 0) |at t
or between and
Beginning of Rabi oscillation
Selective preparation of Z
|g> |
|
|->
| |
Fast Stark pulse rotation around ZPreparation of / 2
Slow rotation in theequatorial plane
End of Rabi oscillation
| ,
Atom in its initial state Field in state |
|| , , at | ,| , at g
Phase distribution analysis
| | v=335m/s
30 photonsn
Outline
• Elements of theory– Results on Rabi oscillation – Rabi oscillation at the quantum-classical boundary: the atomic
point of view– Effect of the Rabi oscillation on the field
• Experimental study– The atom-field system– Preparation of mesoscopic states superpositions of the field:
experimental results– Evidence of correlation between atomic and field state– Coherence of the prepared superposition: where are the cats?
Coherence of the superposition
2
0
2
0
2
2
Preparation of a superposition of | | andCoherent superposition or statistical mixture?
A quasi-probability distribution: the Wigner function
Signature of coherence:
Interference fringes
Coherent superposition Statistical mixture
-Acts on the phase space of a harmonic oscillator
-Positive for a quasi-classical field
Experimental test based on induced revivals of the Rabi oscillationcf. Tristan Meunier’s thesis
Coherence of the superposition: where are the cats?
30n 2 50 photonsD
"Fast" atom "Slow" atom01.5at T 02.5bt T
Coherent superposition Statistical mixture
2
2 cavdecoh
TT
D
2 20 photonsD
0.7a
decoh
t
T 2.8b
decoh
t
T
Requirements for a cat living
Size of the cat
int
0
tD
T
0 int2 sin4
tD n
n
D independant of the
photon number
-The two components must be separated int
0
1t
T
-The superposition must remain coherent int decoht T
20
2 2 2int
2 2cav cavdecoh
T T TT
D t
20
int 2 2int
2 cavT Tt
twith
23 0int 2
2 cavT Tt
3
3int2
0 0
2 cavt T
T T
Condition to generate a Schrödinger cat:1
Requirements for a cat living
No catClassical field
Quantum behaviorof the field
No catClassical field
int
0
t
T
2 2 2maxD
With our setup
Preparation and decoherence at the same time2max 20D
int
0
1t
T
2min 1D
Observation of mesoscopic coherences2max 40D
2/3
2max
0
cavTD
T
32
0
22cavT
T
2max 80D
v=150m/s
8 cav vc aT T(reasonable hypothesis)
Efficient and promising method to generate cats
Conclusions-perspectives
Experimental study of the Rabi oscillation at the quantum-classical boundary
Field classical No entanglement with the atom
Generation of mesoscopic state superpositionsEstimation of their coherence
Experimental test of coherence (cf. Tristan Meunier’s thesis)
Continuous monitoring of the decoherence process-2 atoms experiment-Direct measurement of the Wigner function