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Decoherence in quantum measurements Dominique Spehner

De
24 pages
Decoherence in quantum measurements Dominique Spehner Institut Fourier, Grenoble, France Reunion du GDR de physique quantique mesoscopique, Aussois, 20/03/2007 – p. 1

  • bath observables

  • single degrees

  • b? acting

  • quantum system

  • reunion du gdr de physique quantique

  • interaction hamiltonian

  • decoherence times


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Decoherence in quantum measurements
Dominique Spehner
Institut Fourier, Grenoble, France
Re´union du GDR de physique quantique me´soscopique, Aussois, 20/03/2007 – p. 1Outlines
• What is decoherence ?
• A model for a quantum measurement
• Entanglement & decoherence in q-measurements
• Decoherence times
• Conclusions
joint work with: F. Haake
Re´union du GDR de physique quantique me´soscopique, Aussois, 20/03/2007 – p. 2Outlines
• What is decoherence?
Re´union du GDR de physique quantique me´soscopique, Aussois, 20/03/2007 – p. 3Quantum superpositions “disappear”...
s
s s’
s’
t|s’−s|
1 1 1
ρ = (|ψi+|ϕi)(hψ|+hϕ|)−→ρ = |ψihψ|+ |ϕihϕ|
diag
2 2 2
Re´union du GDR de physique quantique me´soscopique, Aussois, 20/03/2007 – p. 4A simple model
[Braun, Haake & Strunz ’01]
H H
B
S
H
int
H = 0
S
SYSTEM
H = 0
B
BATH
• A small quantum system (‘S’) is coupled to a bath (‘B’).
The bath hasN 1 degrees of freedom.
• Interaction Hamiltonian: H =SB
int
S = observable of the system ‘S’
N
X
1

2
B =N B sum of independent bath observables
ν
ν=1
B acting on single degrees of freedomν ([B ,B ] = 0).
ν ν μ
Re´union du GDR de physique quantique me´soscopique, Aussois, 20/03/2007 – p. 5Pure vs mixed state (1)
• Initial state of ‘S+B’ = product pure state
|Φ i =|φ i⊗|Ψ i , |Ψ i =⊗|ψ i
0 S 0 0 ν
ν
X
|φ i = c|si with S|si =s|si.
S s
s
• The state of ‘S+B’ at timet> 0 is entangled,
X
−itSB −itsB
|Φi =e |Φ i = c|si⊗|Ψ i, |Ψ i =e |Ψ i
t 0 s s,t s,t 0
s
• Question: can one distinguish the entangled pure state
X
∗ 0
0
ρ (t) =|ΦihΦ| = c c |sihs|⊗|Ψ ihΨ |
0
SB t t s s,t s ,t
s
0
s,s
from the separable ‘diagonal’ mixed state
X
2
ρ (t) = |c| |sihs|⊗|Ψ ihΨ | ?
diag s s,t s,t
s
,→ in principle, YES, but...
Re´union du GDR de physique quantique me´soscopique, Aussois, 20/03/2007 – p. 66
Pure vs mixed state (2)
In practise, only complicated measurements on ‘S+B’ could
distinguishρ fromρ whent>τ (decoherence time).
SB diag dec
X
−itsB −1/2
|Ψ i =e ⊗|ψ i , B =N B .
s,t ν ν
ν
ν
Assume for simplicityhB i =hψ |B |ψ i = 0. Then
ν ν ν ν

Y
0
−it(s −s)B / N
ν
0
lnhΨ |Ψ i = ln hψ |e |ψ i
s,t ν ν
s ,t
ν
N

X
0 2 2 2 −3/2
= ln 1−(s −s) t hB i/(2N)+O(N )
ν
ν=1


2
2 −1/2
0
t 2hB i
s,s
0
hΨ |Ψ i' exp − , τ = .
s,t
s ,t 0
dec
s,s 0
|s −s|
τ
dec
Similarly, ifO acts trivially onM 1 bath degrees of freedom,
0
s,s
0 0
0
hs,Ψ |O|s,Ψ i' 0 ifs =s andtτ
s ,t s,t
dec
P
∗ 0
0
,→ tr(Oρ (t)) = c c hs,Ψ |O|s,Ψ i' tr(Oρ (t)).
0 0
SB s s,t
s ,t diag
s,s s
Re´union du GDR de physique quantique me´soscopique, Aussois, 20/03/2007 – p. 76
Reduced density matrix
If certain bath observables are “not measurable”,
everything happens as ifρ (t) =ρ (t) at timetτ .
SB diag dec
Disregard bath observables completely by tracing over ‘B’:
X
∗ 0
0
ρ (t) = tr(ρ (t)) = c c |sihs|hΨ |Ψ i
0
S SB s s ,t s,t
s
B
0
s,s

2
X
t
∗ 0
= c c |sihs|exp − .
0
s
0
s
s,s
τ
0
dec
s,s
,→ Decoherence is a loss of information about matrix
elements ofρ (t) between states which are
SB
nearly orthogonal in many bath subspaces.
Coherences of ‘S’ leak out in ‘B’.
0 −1 2 −1/2
,→ The decoherence timeτ ∝ ~|s −s| hB i is
dec
0
extremely short if|si,|si are macroscop. = states.
Re´union du GDR de physique quantique me´soscopique, Aussois, 20/03/2007 – p. 8Outlines
• What is decoherence ?
• A model for a quantum measurement
Re´union du GDR de physique quantique me´soscopique, Aussois, 20/03/2007 – p. 9H
Tripartite system (1)
• small quantum system ’S’
• macroscopic pointer ’P’
H
S
B
H
H
SP H PB
P
with 1 degree of freedom
POINTER
SYSTEM
• bath ‘B’ withN 1
(microscopic) (macroscopic)
BATH (N>>1)
degrees of freedom
MEASURING APPARATUS
Some information on the initial state of ‘S’ must be
transfered to the pointer→ ‘S’ and ‘P’ are strongly coupled
H =SP
SP
X
S = s|sihs| = observable of ‘S’ to be measured
s
,→ discrete non-degenerate spectrum
P = pointer momentum−→ translations of ‘P’ in position
[H ,S] = 0→ the coupling does not changeS.
PS
Re´union du GDR de physique quantique me´soscopique, Aussois, 20/03/2007 – p. 10