Enoncé

◦12
y
|+zi
|+ >
z
D
E
z
y
x
z
A
B
R
A
B
|E >
|E >
I
x
|+zi |+xi,|−xi
R
z
D E A, B
R D, E PDPE
A

|+ziI|Ei
hE|Ei= 1 A
|EAiB
|EBi
A B
Hspin =C2
|Ei ∈ Henv Htot =Hspin ⊗ Henv
ψavant ∈ Htot I A, B
ψapres A, B
ψapres D, E
ρavant =|ψavantihψavant|
˜ρavant = Trenv (ρavant)|±zi Henv
˜ρapres = Trenv (ρapres)
hEA|EBiPDPE
hEA|EBi
hEA|EBi= 1 hEA|EBi= 0
˜ρapres ~ P R
˜ρapres =1
2Id + ~
P .ˆ ~σ
Hspin
Henv
˜ρ(t) = Trenv (ρ(t))
˜ρapres =1
20
01
2
τdecoh.
Tr (|aihb|) = ha|bi

I R
A B
Htot =C2⊗ Henv
|ψ(0)i=|S(0)i⊗|E(0)i
|S(0)i=|+zi=1
√2(|+xi+|−xi)∈C2,|E(0)i∈Henv.
ˆ
H=ˆ Sz⊗ˆ He,ˆ Sz=1/2 0
0−1/2|±xi
ˆ
HeHenv |E(0)iˆ He
|ψ(t)i=e−it ˆ
H|ψ(0)it
ρ(t) = |ψ(t)ihψ(t)|˜ρ=
Trenv (ρ(t)) |±xi
hE(0) |E(t)i=hE(0) |e−it ˆ
He|E(0)iN
N
Hˆ H:H → H
Henv =H ⊗ H ⊗ . . . ⊗ H
| {z }
N
ˆ
Henv =1
√N
N
X
i=1
ˆ
Hi
ˆ
Hi= Id ⊗. . . Id ⊗ˆ H
|{z}
position i ⊗Id . . . ⊗Id
i
|E(0)i=|e(0)i ⊗ . . . ⊗ |e(0)i,ke(0)k= 1.
he(0) |ˆ H|e(0)i= 0 σ2:= he(0) |ˆ H2|e(0)i
t N → ∞
hE(0) |E(t)i=e−t2σ2/2
˜ρ(t) = Trenv (ρ(t)) −→
N→∞
1
21e−t2σ2/2
e−t2σ2/21

˜ρ(t)|+zi=1
2(|+xi+|−xi)
|−zi=1
2(|+xi − |−xi)
S(˜ρ(t))
t→0t→ ∞ t
~
P(t) ˜ρ(t) = 1
2Id + ~
P(t).ˆ ~σ
τdecoh. =1
σ
ρ(0) = (|S(0)ihS(0) |)⊗ρenv
ρenv =ρe⊗. . . ⊗ρe
Tr ˆ Hρe= 0
Ei=
~ωi+1
2i∈N
T i Ei
pi=1
Z0e−Ei
kT =1
Z0e−~ω(i+1/2)
kT =1
Ze−βi
β=~ω
kT
1
Z=1 Z0e−~ω
2kT ZPipi= 1
Z β
S β
S β →0
β→ ∞