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Transcript of DECOHERENCE AND QUANTUM INFORMATION JUAN PABLO PAZ Departamento de Fisica, FCEyN Universidad de...
![Page 1: DECOHERENCE AND QUANTUM INFORMATION JUAN PABLO PAZ Departamento de Fisica, FCEyN Universidad de Buenos Aires, Argentina paz@df.uba.ar Paraty August 2007.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649ee55503460f94bf4e7c/html5/thumbnails/1.jpg)
DECOHERENCE AND QUANTUM INFORMATION
JUAN PABLO PAZDepartamento de Fisica, FCEyN
Universidad de Buenos Aires, Argentina
Paraty
August 2007
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Lecture 1 & 2: Decoherence and the quantum origin of the classical world
Lecture 2 & 3: Decoherence and quantum information processing, noise characterization (process tomography)
Colaborations with: W. Zurek (LANL), M. Saraceno (CNEA), D. Mazzitelli (UBA), D. Dalvit (LANL), J. Anglin (MIT), R. Laflamme (IQC), D. Cory (MIT),
G. Morigi (UAB), S. Fernandez-Vidal (UAB), F. Cucchietti (LANL),
• 1 & 2. Decoherence, an overview. – Hadamard-Phase-Hadamard and the origin of the classical world!
Information transfer from the system to the environment.
– Bosonic environment. Complex environments. Spin environments (some new results)
• 2 &3 . Decoherence in quantum information. How to characterize it?
– Quantum process tomography (some new ideas)
Current/former students: D. Monteoliva, C. Miquel (UBA), P. Bianucci (UBA, UT), L. Davila (UEA, UK), C. Lopez (UBA, MIT), A. Roncaglia (UBA), C. Cormick (UBA), A. Benderski (UBA), F. Pastawski (UNC),
C. Schmiegelow (UNLP, UBA)
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HOW DOES THE WIGNER FUNCTION OF A QUANTUM STATE LOOK LIKE?: SUPERPOSITION OF TWO GAUSSIAN STATES
€
˙ W = H0,W{ }MB
+ D∂ 2pp W +L
OSCILLATIONS IN WIGNER FUNCTION: THE SIGNAL OF
QUANTUM INTERFERENCE. HOW DOES DECOHERENCE AFFECTS THIS
STATE?
€
Distance L
Wavelength λ p =h
L
DECOHERENCE IN QUANTUM BROWNIAN MOTION (VI)
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€
˙ W = H0,W{ }MB
+ D∂ 2pp W +L
€
Distance L
Wavelength λ p =h
L
€
˙ W = H0,W{ }MB
+ D∂ 2pp W +L
Wosc ≈ A(t)cos(kp p)⇒ A(t) ≈ exp(−Γt)
Γ = Dkp2
DECOHERENCE RATE: MUCH LARGER THAN
RELAXATION RATE
€
Γ=DL2 /h2, D = 2 mγ kBT, λ DB =h / 2m kBT
⇒ Γ =γ L /λ DB( )2
≈1040 γ, m =1gr, T = 300K, L =1cm
DECOHERENCE IN QUANTUM BROWNIAN MOTION (VII)
MASTER EQUATION CAN BE REWRITTEN FOR THE WIGNER FUNCTION: IT HAS THE FORM OF A FOKER-PLANCK EQUATION
€
˙ W = H0,W{ }MB
+ D∂ 2pp W +L
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A DIGRESSION ON WIGNER FUNCTIONS
€
0
€
ρB
€
U€
H
€
X = Re TrB ρ BU( )( )
Y = −Im TrB ρ BU( )( )
Know : Use circuit to learn about (“spectrometer”)
Know : Use circuit to learn about (“tomographer”)
€
ρ
€
U
€
U
€
ρAPLICATION:
DAVIDOVICH-LUTTERBACH SCHEME FOR MEASURING WIGNER FUNCTION
REMEMBER THE SIMPLE SCHEME TO USE ONE QUBIT TO LEARN ABOUT EITHER THE QUANTUM STATE OF ANOTHER SYSTEM OR ABOUT THE QUBIT-SYSTEM INTERACTION
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• REMEMBER THE DEFINITION OF WIGNER FUNCION
€
A(x, p) =1
πhD(x, p) R D+(x, p) =
dλ dζ
(2πh)2exp(−i(xζ − pλ ) /h) D(λ ,ζ )∫
€
W (x, p) =dy
2πhe ipy / h∫ < x −
y
2| ρ | x +
y
2> = Tr(ρ A(x, p) )
€
A(x, p) : phase space point operators
• PROPERTIES OF WIGNERS ARE CONSEQUENCES OF
Integral along lines give a projection operator:
€
Tr(A(x, p) A( ′ x , ′ p ) =1
2πhδ(x − ′ x ) δ(p − ′ p )
€
A(x, p) = A+(x, p)
€
A(x, p) form a complete orthonormal set
€
dx dp∫ A(x, p) = Ψ Ψ
€
ax + bp = c
€
(a ˆ X + b ˆ P ) Ψ = c Ψ
phase space displacement operator
R: reflection operator
€
D(λ ,ζ ) = exp(−i(λ ˆ P −ζ ˆ X ) /h)
R x = −x
A DIGRESSION ON WIGNER FUNCTIONS
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TOMOGRAPHY OF (CONTINUOUS) WIGNER FUNCTION ALLA DAVIDOVICH-LUTTERBACH
This quantum circuit describes the ENS-experiment to measure Wigner function
Qbit: 2-level atom; System: em-field inside the cavity
Displacement operator: inject classical field in cavity; Reflection: dispersive interaction between field & atom
APLICATION II: CAN THIS TYPE OF TOMOGRAPHY BE GENERALIZED TO OTHER SYSTEMS? DISCRETE WIGNER FUNCTIONS
A DIGRESSION ON WIGNER FUNCTIONS
€
0
€
ρ
€
R€
H
€
Z = Prob(0) − Prob(1)
Z = Tr D+(x, p)ρD(x, p)R( ) = Tr ρD(x, p)RD+(x, p)( )
Z = πh W (x, p)
€
D(x, p)€
H
€
W (x, p) = Tr(ρ A(x, p) )
ρ = W (x, p)x,p
∑ A(x, p)
Wigner function is nothing but the matrix element of the density matrix in a special operator basis
€
ρ x, x '( ) = Tr(ρ x x ' )
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• A SYSTEM WITH A FINITE DIMENSIONAL (N) HILBERT SPACE
€
{ n , n =1,K ,N}
{ k , k =1,K ,N}
Position basis (arbitrary)
Momentum basis
€
k =1
Nexp(i2π nk /N)
n=1
N
∑ n
• PHASE SPACE DISPLACEMENT OPERATORS ARE EASY TO CONSTRUCT
€
U n = n +1, mod(N)
V k = k +1, mod(N)
Position shift
Momentum shift
€
D(q, p) = U qV − p exp(iπ qp /N)
• PHASE SPACE POINT OPERATORS ARE NOT SO EASY TO CONSTRUCT
€
A(q, p) =1
N 2n=1
N
∑ exp(i2π (np + mp) /N) D(n,m)n=1
N
∑Naive attempt
Does not have all the right properties (for all values of N)
A DIGRESSION ON WIGNER FUNCTIONS
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0 1 2 3 4 5 6 7
0
1
3
4
2
5
6
7
q+p=0N=8
0 1 2 3 4 5 6 7
0
1
3
4
2
5
6
7
2q+2p=0
Lines may have moremore than N pointsq & p are integers, arithmetic mod N
WHY??: {1,2,…,N} is NOT a FINITE FIELD: 2 x 4 = 0 (mod 8)
(it is a field if N is a prime number GF(N))
A DIGRESSION ON WIGNER FUNCTIONS
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• SOLUTION (M. Berry, 1980; U. Leonhardt,…)
• DISCRETE WIGNER FUNCTION
€
A(q, p) =1
2N n=1
2N
∑ exp(i2π (np + mp) /2N) D(n,m)n=1
2N
∑
= U qRV − p exp(iπ qp /N)
Phase space operators:
DFT of displacements
In a grid of 2N by 2N
• SAME PROPERTIES THAN IN THE CONTINUOUS CASE:
W(q,p) is real
Sum over lines is always possitive!
€
Nq,p=1
2N
∑ W1(q, p) W2(q, p) = Tr(ρ1 ρ 2)
€
W (q, p)q,p=1
2N
∑ = ϕ ρ ϕ , D(b,a) ϕ = exp(i2πc /N) ϕ
€
ax − bp = c
€
W (q, p) =1
2NTr(ρ A(q, p)), 0 ≤ q, p ≤ 2N −1
A DIGRESSION ON WIGNER FUNCTIONS
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• Discrete Wigner function: How does it look like?
Position eigenstate Gaussian wavepacket
positive oscillatory oscillatory
osci
llato
ry
OSCILLATIONS DUE TO PERIODIC BOUNDARY CONDITIONS
A DIGRESSION ON WIGNER FUNCTIONS
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DISCRETE WIGNER FUNCTIONS IN QUANTUM COMPUTATION; see “Quantum computers in phase space”, C. Miquel, J.P.P & M. Saraceno, PRA 65 (2002), 062309
TOMOGRAPHY OF (DISCRETE) WIGNER FUNCTION
€
0
€
ρ
€
A(x, p)€
H
€
X = Tr ρA(x, p)( )
X = N W (x, p)
EFFICIENT QUANTUM CIRCUIT TO IMPLEMENT ‘CONTROLLED-A(x,p) EXISTS
A DIGRESSION ON WIGNER FUNCTIONS
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MEASUREMENT OF DISCRETE WIGNER FUNCTION OF FOUR COMPUTATIONAL STATES IN A TCE-SAMPLE IN A 500Mhz NMR SPECTROMETER (C. Miquel, J.P.P, M. Saraceno, E. Knill, R. Laflamme, C.
Negrevergne, Nature 418, 59 (2002)
A DIGRESSION ON WIGNER FUNCTIONS
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€
˙ W = H0,W{ }MB
+ D∂ 2pp W +L
€
Distance L
Wavelength λ p =h
L
€
˙ W = H0,W{ }MB
+ D∂ 2pp W +L
Wosc ≈ A(t)cos(kp p)⇒ A(t) ≈ exp(−Γt)
Γ = Dkp2
DECOHERENCE RATE: MUCH LARGER THAN
RELAXATION RATE
€
Γ=DL2 /h2, D = 2 mγ kBT, λ DB =h / 2m kBT
⇒ Γ =γ L /λ DB( )2
≈1040 γ, m =1gr, T = 300K, L =1cm
DECOHERENCE IN QUANTUM BROWNIAN MOTION (VII)
MASTER EQUATION CAN BE REWRITTEN FOR THE WIGNER FUNCTION: IT HAS THE FORM OF A FOKER-PLANCK EQUATION
€
˙ W = H0,W{ }MB
+ D∂ 2pp W +L
![Page 15: DECOHERENCE AND QUANTUM INFORMATION JUAN PABLO PAZ Departamento de Fisica, FCEyN Universidad de Buenos Aires, Argentina paz@df.uba.ar Paraty August 2007.](https://reader036.fdocuments.in/reader036/viewer/2022062422/56649ee55503460f94bf4e7c/html5/thumbnails/15.jpg)
EVOLUTION OF WIGNER FUNCTION
€
ν
NOTICE: NOT ALL STATES ARE AFFECTED BY THE ENVIRONMENT IN THE SAME WAY (SOME
SUPERPOSITIONS LAST LONGER THAN OTHERS)
POINTER STATES, DECOHERENCE TIMESCALE (I)
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WARNING: DECOHERENCE TIMESCALE OBTAINED IN THE HIGH TEMPERATURE LIMIT FOR INITIAL “CAT STATE” IS ONLY AN APPROXIMATION!
POINTER STATES, DECOHERENCE TIMESCALE (II)
€
˙ W = H0,W{ }MB
+ D∂ 2pp W +L
Wosc ≈ A(t)cos(kp p)⇒ A(t) ≈ exp(−Γt)
Γ = Dkp2
EXAMPLE 1: EVOLUTION OF FRINGE VISIBILITY FACTOR IN AN ENVIRONMENT AT ZERO TEMPERATURE FOR QUANTUM BROWNIAN MOTION (NON-EXPONENTIAL
DECAY: CAN BE UNDERSTOOD FROM TIME-DEPENDENCE OF COEFFICIENTS OF MASTER EQUATION)
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POINTER STATES, DECOHERENCE TIMESCALE (II’)
Measure degradation of system’s state with entropy (von Neuman) or purity decay
€
SVN (t) = −Tr ρ(t)ln ρ(t)( )( ), ζ (t) = Tr ρ 2(t)( )
These quantities depend on time AND on the initial state
USE MASTER EQUATION TO ESTIMATE PURITY DECAY OR ENTROPY GROWTH
€
˙ ρ =− i HR +m
2δω2(t)x 2,ρ
⎡ ⎣ ⎢
⎤ ⎦ ⎥− iγ(t) x, p,ρ{ }[ ]− D(t) x, x,ρ[ ][ ]− f (t) x, p,ρ[ ][ ]
€
˙ ζ = 2Tr ˙ ρ ρ( ) = 2γζ − 2DTr x,ρ[ ]2
( ) + 2 fTr x,ρ[ ] p,ρ[ ]( )
System & Environment become entangled
(large environment: irreversible process)
€
Ψ(0) Ψ(0)
€
ρ(t)
PURE STATE MIXED STATE
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WARNING: DECOHERENCE TIMESCALE OBTAINED IN THE HIGH TEMPERATURE LIMIT FOR INITIAL “CAT STATE” IS ONLY AN APPROXIMATION!
POINTER STATES, DECOHERENCE TIMESCALE (II)
€
˙ W = H0,W{ }MB
+ D∂ 2pp W +L
Wosc ≈ A(t)cos(kp p)⇒ A(t) ≈ exp(−Γt)
Γ = Dkp2
EXAMPLE 1: EVOLUTION OF FRINGE VISIBILITY FACTOR IN AN ENVIRONMENT AT ZERO TEMPERATURE FOR QUANTUM BROWNIAN MOTION (NON-EXPONENTIAL
DECAY: CAN BE UNDERSTOOD FROM TIME-DEPENDENCE OF COEFFICIENTS OF MASTER EQUATION)
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CLASSICAL QUANTUM
WIGNER FUNCTION DEVELOPS OSCILLATORY STRUCTURE AT SUB-PLANCK SCALES
W.H. Zurek, Nature 412, 712 (2001); D. Monteoliva & J.P. Paz, PRE 64, 056238 (2002)
€
λX ≈h /P, λ P ≈h /L ⇒ A ≈λ X λ P ≈hh
LP<< h
NONTRIVIAL TIMESCALE: HARMONICALLY DRIVEN DOUBLE WELL (CHAOS)
€
V (x) = −a x 2 + b x 4 + cx cos(ω t)
POINTER STATES, DECOHERENCE TIMESCALE (III)
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DECOHERENCE DESTROYS THE FRINGES THAT ARE DYNAMICALLY PRODUCED
CLASSICAL + NOISE QUANTUM + DECOHERENCE
DECOHERENCE IMPLIES INFORMATION TRANSFER INTO CORRELATIONS BETWEEN SYSTEM AND ENVIRONMENT. WHAT IS THE RATE?:
Lyapunov regime: Above a certain threshold, rate of entropy production fixed by the Lyapunov exponent of the system (W. Zurek & J.P.P., 1994)
POINTER STATES, DECOHERENCE TIMESCALE (IV)
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INTUITIVE EXPLANATION: WHY IS THERE A REGIME OF ENTROPY GROWTH FIXED BY THE LYAPUNOV EXPONENT?
€
SVN = −Tr ρ log ρ( )( ), SLIN = −log Tr ρ 2( )( ), Tr ρ 2
( ) = 2πh dx dpW 2(x, p)∫
STRETCHING
+ FOLDING
€
t ≈1
λ
DECOHERENCE
€
t ≈ tDECO ≈1
Γ<<
1
λ
€
S = 0 (pure)
€
S = 0 (pure)
€
S ≈1 (mixed)
DIGRESSION: DECOHERENCE FOR CLASSICALLY CHAOTIC SYSTEMS
€
eλt
€
e−λ t€
eλt
€
σ c = 2D /λ
€
Area ≈ σ c eλt
S ≈ log(Area) ≈ λ t
STRETCHING DECOHERENCE
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NUMERICAL EVIDENCE IS RATHER STRONG
(D. Monteoliva and J.P.P., PRL (2000), PRE (2002), (2006))
DIGRESSION: DECOHERENCE FOR CLASSICALLY CHAOTIC SYSTEMS
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time time
Log(D)
Log(1/a)
€
MC ≈aexp(−λ t)
€
a ≈ 1/D
€
MC ≈bexp(−Γt)
€
Γ ≈Dkp2(t)
D. Monteoliva and J.P. P. (2006)
DIGRESSION: DECOHERENCE FOR CLASSICALLY CHAOTIC SYSTEMS
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POINTER STATES, DECOHERENCE TIMESCALE (V)
NOT ALL STATES ARE AFFECTED BY DECOHERENCE IN THE SAME WAY
QUESTION: WHAT ARE THE STATES WHICH ARE MOST ROBUST UNDER DECOHERENCE? (STATES WHICH ARE LESS SUSCEPTIBLE TO BECOME ENTANGLED
WITH THE ENVIRONMENT). THE STATES THAT EXIST “OBJECTIVELY”
POINTER STATES: STATES WHICH ARE MINIMALLY AFFECTED BY THE INTERACTION WITH THE ENVIRONMENT (MOST ROBUST STATES OF THE SYSTEM)
Information initially ‘stored’ in the system flows into correlations with the environment
€
SVN (t) = −Tr ρ(t)ln ρ(t)( )( ), ζ (t) = Tr ρ 2(t)( )
Measure information loss by entropy growth (or purity decay)
AN OPERATIONAL DEFINITION OF POINTER STATES:
“PREDICTABILITY SIEVE”
€
Ψ(0) Ψ(0)
€
ρ(t)
Initial state of the system (pure) State of system at time t (mixed)
t
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POINTER STATES, DECOHERENCE TIMESCALE (VI)
Measure degradation of system’s state with entropy (von Neuman) or purity decay
€
SVN (t) = −Tr ρ(t)ln ρ(t)( )( ), ζ (t) = Tr ρ 2(t)( )€
Ψ(0) Ψ(0)
€
ρ(t)
These quantities depend on time AND on the initial state
PREDICTABILITY SIEVE: FIND THE INITIAL STATES SUCH THAT THESE QUANTITIES ARE MINIMIZED (FOR A DYNAMICAL RANGE OF TIMES)
PREDICTABILITY SIEVE IN A PHYSICALLY INTERESTING CASE?
ANALIZE QUANTUM BROWNIAN MOTION
USE MASTER EQUATION TO ESTIMATE PURITY DECAY OR ENTROPY GROWTH
€
˙ ρ =− i HR +m
2δω2(t)x 2,ρ
⎡ ⎣ ⎢
⎤ ⎦ ⎥− iγ(t) x, p,ρ{ }[ ]− D(t) x, x,ρ[ ][ ]− f (t) x, p,ρ[ ][ ]
€
˙ ζ = 2Tr ˙ ρ ρ( ) = 2γζ − 2DTr x,ρ[ ]2
( ) + 2 fTr x,ρ[ ] p,ρ[ ]( )
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Minimize over initial state: Pointer states for QBM are minimally uncertainty coherent states! W.Zurek, J.P.P & S. Habib, PRL 70, 1187 (1993)
A SIMPLE SOLUTION FROM THE PREDICTABILITY SIEVE CRITERION
€
˙ ζ = 2Tr ˙ ρ ρ( ) = 2γζ − 2DTr x,ρ[ ]2
( ) + 2 fTr x,ρ[ ] p,ρ[ ]( )
Approximations I: Neglect friction, use asymptotic form of coefficients, assume initial state is pure and apply perturbation theory:
€
ζ (T) −ζ (0) = 2D dt0
T
∫ Tr x(t),ρ[ ]2
( )
POINTER STATES, DECOHERENCE TIMESCALE (VI)
€
ζ (T) = ζ (0) − 2D Δx 2 +1
m2ω2Δp2 ⎛
⎝ ⎜
⎞
⎠ ⎟
Approximations II: State remains approximately pure, average over oscillation period
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Minimize over initial state: Pointer states for QBM are minimally uncertainty coherent states! W.Zurek, J.P.P & S. Habib, PRL 70, 1187 (1993)
A SIMPLE SOLUTION FROM THE PREDICTABILITY SIEVE CRITERION
€
˙ ζ = 2Tr ˙ ρ ρ( ) = 2γζ − 2DTr x,ρ[ ]2
( ) + 2 fTr x,ρ[ ] p,ρ[ ]( )
Approximations I: Neglect friction, use asymptotic form of coefficients, assume initial state is pure and apply perturbation theory:
€
ζ (T) −ζ (0) = 2D dt0
T
∫ Tr x(t),ρ[ ]2
( )
POINTER STATES, DECOHERENCE TIMESCALE (VII)
€
ζ (T) = ζ (0) − 2D Δx 2 +1
m2ω2Δp2 ⎛
⎝ ⎜
⎞
⎠ ⎟
Approximations II: State remains approximately pure, average over oscillation period
€
ζ (t)
“momentum eigenstate”
“position eigenstate”
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1) Dynamical regime (QBM): Pointer basis results from interplay between system and environment
3) “Slow environment” regime: The evolution of the environment is very “slow” (adiabatic environment): Pointer states are eigenstates of the Hamiltonian of the system! The
environment only “learns” about properties of system which are non-vanishing when averaged in time. J.P. Paz & W.Zurek, PRL 82, 5181 (1999)
€
ζ (t)
2) “Slow system” regime: (Quantum Measurement): System’s evolution is negligible, Pointer basis is determined by the interaction Hamiltonian (position in QBM)
POINTER STATES, DECOHERENCE TIMESCALE (VIII)
WARNING: DIFFERENT POINTER STATES IN DIFFERENT REGIMES!
TAILOR MADE POINTER STATES? Environmental engeneering…
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DECOHERENCE: AN OVERVIEW
SUMMARY: SOME BASIC POINTS ON DECOHERENCE
•POINTER STATES: W.Zurek, S. Habib & J.P. Paz, PRL 70, 1187 (1993), J.P. Paz & W. Zurek, PRL 82, 5181 (1999)
•TIMESCALES: J.P. Paz, S. Habib & W. Zurek, PRD 47, 488 (1993), J. Anglin, J.P. Paz & W. Zurek, PRA 55, 4041 (1997)
•CONTROLLED DECOHERENCE EXPERIMENTS: Zeillinger et al (Vienna) PRL 90 160401 (2003), Haroche et al (ENS) PRL 77, 4887 (1997), Wineland et al (NIST), Nature 403, 269 (2000).
• DECOHERENCE AND THE QUANTUM-CLASSICAL TRANSITION:YES: HILBERT SPACE IS HUGE, BUT MOST STATES ARE UNSTABLE!! (DECAY VERY FAST INTO MIXTURES)
CLASSICAL STATES: A (VERY!) SMALL SUBSET. THEY ARE THE POINTER STATES OF THE SYSTEM DYNAMICALLY CHOSEN BY THE ENVIRONMENT
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DECOHERENCE: AN OVERVIEW
LAST DECADE: MANY QUESTIONS ON DECOHERENCE WERE ANSWERED
• NATURE OF POINTER STATES: QUANTUM SUPERPOSITIONS DECAY INTO MIXTURES WHEN QUANTUM INTERFERENCE IS SUPRESSED. WHAT ARE THE STATES SELECTED BY THE INTERACTION? POINTER STATES: THE MOST STABLE STATES OF THE SYSTEM, DYNAMICALLY
SELECTED BY THE ENVIRONMENT: W.Zurek, S. Habib & J.P. Paz, PRL 70, 1187 (1993), J.P. Paz & W. Zurek, PRL 82, 5181 (1999)
• TIMESCALES: HOW FAST DOES DECOHERENCE OCCURS? J.P. Paz, S. Habib & W. Zurek, PRD 47, 488 (1993), J. Anglin, J.P. Paz & W. Zurek, PRA 55, 4041 (1997)
•INITIAL CORRELATIONS: THEIR ROLE, THEIR IMPLICTIONS L. Davila Romero & J.P. Paz, Phys Rev A 54, 2868 (1997), MORE REALISTIC PREPARATION OF INITIAL STATE (FINITE TIME
PREPARATION, ETC) J. Anglin, J.P. Paz & W. Zurek, PRA 55, 4041 (1997)
• DECOHERENCE FOR CLASSICALLY CHAOTIC SYSTEMS: W. Zurek & J.P. Paz, PRL 72, 2508 (1994), D. Monteoliva & J.P. Paz, PRL 85, 3373 (2000).
• CONTROLLED DECOHERENCE EXPERIMENTS: S. Haroche et al (ENS) PRL 77, 4887 (1997), D. Wineland et al (NIST), Nature 403, 269 (2000), A. Zeillinger et al (Vienna) PRL 90 160401 (2003),
• ENVIRONMENT ENGENEERING: Cirac & Zoller, etc; see Nature 412, 869 (2001) (review of PRL work published by UFRJ group on environment engeneering with ions)
•SPIN ENVIRONMENTS: ASK CECILIA CORMICK (POSTER NEXT WEEK).