according to which the macroscopic system displaces the origin of the environmental oscillator α of environment E with the spring constant ω
2 α,E
by f
α,E
σ
z
, as it can be recognized from H
E
+ H
SE
= X
α
1 2
h p
2 α,E
+ ω
2 α,E
x
2 α,E
− f
α,E
σ
z 2
− hω
α,E
i 7
For simplicity, we assume that the interaction model is separable f
α,E
σ
z
= γ
α,E
f σ
z
, where γ
α,E
is the coupling strength and bilinear f σ
z
= σ
z
.
3. KINEMATICS
The shift in the system’s energy due to the perturbation H
SE
up to the second order is obtained as δE
n,E
≃
E
h0|hn|H
SE
|ni|0i
E
+ X
m6=n α6=0
E
hα|hm|H
ME
|ni|0i
E 2
E
n
− E
m
+ E
α,E
= 1
2 X
r
|σ
rn
|
2
Ω
rn
X
α
γ
2 α,E
ω
2 α,E
ω
αE
+ Ω
rn
8 where σ
mn
= hm|σ
z
|ni, Ω
mn
= E
m
− E
n
h. The state with the energy shifted to E
n
+ δE
n,E
due to the perturbation is not actually stationary and decays with a finite lifetime Γ
− 1
n,E
given by the Fermi’s golden rule as Γ
n,E
≃ 2π
h X
m6=n α6=0
E
h0|hn|H
SE
|ni|0i
E 2
δE
n
− E
m
+ E
α,E
= π
h X
r
|σ
rn
|
2
Ω
rn
X
α
γ
2 α,E
ω
2 α,E
δΩ
rn
− ω
α,E
9
4. DYNAMICS
We assume that the initial state of the whole system is |Ψ0i = |ψi|0i
A
|0i
B
10 where |ψi is an arbitrary state of the macroscopic system. The state of the whole system at time t is obtained by
|Ψti = e
− iH
th
U
I
t|Ψ0i 11
where we defined H = H
S
+ P
E
H
E
and U
I
t is the time evolution operator in the interaction picture. We expand U
I
t up to the second order with respect to the interaction Hamiltonian H
int
= P
E
H
SE
to find U
I
t ≃ 1 − i
h Z
t
dt
1
H
int
t
1
− 1
h
2
Z
t
dt
2
Z
t
2
dt
1
H
int
t
2
H
int
t
1
12 We first calculate
U
I
t|0i
A
|0i
B
≃ U
0A,0B
t|0i
A
|0i
B
+ U
αA,αB
t|αi
A
|αi
B
+ U
0A,αB
t|0i
A
|αi
B
+ U
αA,0B
t|αi
A
|0i
B
13 where
U
0A,0B
t =1 − i
2h X
E ,α
Z
t
dt
1
f
2 α,E
t
1
− 1
2h X
E ,α
ω
α,E
Z
t
dt
2
Z
t
2
dt
1
e
− it
2
− t
1
ω
α,E
f
α,E
t
2
f
α,E
t
1
− 1
4h
2
X
α
Z
t
dt
2
Z
t
2
dt
1
n f
2 α,A
t
2
, f
2 α,B
t
1
o U
αA,αB
t = − 1
2h X
α
ω
12 α,A
ω
12 α,B
Z
t
dt
2
Z
t
2
dt
1
n e
iω
α,A
t
2
f
α,A
t
2
, e
iω
α,B
t
1
f
α,B
t
1
o U
0A,αB
t = i
√ 2h
X
α
ω
12 α,B
Z
t
dt
1
e
iω
α,B
t
1
f
α,B
t
2
+ 1
√ 8h
3
X
α
ω
12 α,B
Z
t
dt
2
Z
t
2
dt
1
n f
2 α,A
t
2
, e
iω
α,B
t
1
f
α,B
t
1
o 14
to find |Ψti =
2
X
n=1
e
− iE
n
th
n |0i
A
|0i
B
hn|U
0A,0B
t|ψi + X
α
|0i
A
|αi
B
hn|U
0A,αB
t|ψi +
X
α
|αi
A
|0i
B
hn|U
αA,0B
t|ψi + X
α
|αi
A
|αi
B
hn|U
αA,αB
t|ψi o
15 where we defined f
α,E
t := ω
α,E
f
α,E
σ
z
t, σ
z
t := e
iH
S
th
σ
z
e
− iH
S
th
and {gt, ht
′
} := gtht
′
+ gt
′
ht. Note that U
αA,0B
t can be obtained by interconverting A ↔ B within U
0A,αB
t in 14. The problem is now reduced to the evaluation of matrix elements of interaction Hamiltonian in 15. The potential
U R being an even function, the energy eigenstates |ni have definite parity. Because U
0A,0B
t and U
αA,αB
t are even functions and U
0A,αB
t and U
αA,0B
t are odd functions, the following selection rules are identified hm|U
0A,0B
|ni = hm|U
αA,αB
|ni = 0 hn|U
0A,αB
|ni = hn|U
αA,0B
|ni = 0 16
The diagonal matrix elements of U
0A,0B
t are evaluated as hn|U
0A,0B
t|ni = 1 − it
h X
E
n δE
1 n,E
− 1
π X
m
σ
2 mn
Z
∞
dω
E
Jω
E
ω
E
+ Ω
mn
o 17
− 1
πh X
m
σ
2 mn
Z
∞
dω
E
Jω
E
1 − e
ıtω
E
+Ω
mn
ω
E
+ Ω
mn 2
− t
2
h
2
δE
1 n,A
δE
1 n,B
where Jω
E
is the spectral density of the environment E, corresponding to a continuous spectrum of environmental frequencies, ω
E
, defined as Jω
E
= π
2 X
α
γ
2 α,E
ω
3 α,E
δω
E
− ω
α,E
≡ J
E
ω
s E
e
− ω
E
Λ
E
18 where J
E
and Λ
E
are coupling strength and cut-off strength of environment E. The different types of environments are characterized by the value of parameter s as sub-ohmic 0 s 1, ohmic s = 1 and super-ohmic s 1.
The quantity embraced by the bracket on the right-hand side of 17 coincides with δE
n,E
in 8. Thus, the following expression is valid up to the second order
hn|U
0A,0B
t|ni = X
E
e
− itδE
n,E
h
1 − 1
πh X
m
σ
2 mn
Z
∞
dω
E
Jω
E
× 2 sin
2
{ω
E
+ Ω
mn
t2} ω
E
+ Ω
mn 2
− i sin{ω
E
+ Ω
mn
t} ω
E
+ Ω
mn 2
− t
2
h
2
δE
1 n,A
δE
1 n,B
− 1 19 We assume that the environmental cut-off frequency Λ
E
is much higher than the system’s characteristic frequency Ω, so that at times much higher than Ω
− 1
the first term of the integral in 19 can be approximated by a delta function δω
E
+ Ω
mn
. Obviously, the result of the corresponding integral would be Jω
E
+ Ω
mn
, which is zero for Ω
mn
≥ 0. The elements of U
0A,0B
t are then reduced to h1|U
0A,0B
t|1i = X
E
e
− itδE
1,E
h
1 + i
πh Z
∞
dω
E
Jω
E
sin{ω
E
+ ∆t} ω
E
+ ∆
2
− t
2
h
2
δE
1 1A
δE
1 1B
− 1 h2|U
0A,0B
t|2i = X
E
e
− itδE
2,E
h
1 − Γ
2,E
t 2
+ i
πh Z
∞
dω
E
Jω
E
sin{ω
E
− ∆t} ω
E
− ∆
2
− t
2
h
2
δE
1 2A
δE
1 2B
− 1 20
If the coupling between the system and each environment is weak, we have Γ
2,E
≪ Ω. At the temporal domain Ω
− 1
≪ t ≪ Γ
− 1
2,E
, we finally obtain h1|U
0A,0B
t|1i ≈ X
E
exp −
i h
tδE
1,E
− 1
π Z
∞
dω
E
Jω
E
sin{ω
E
+ ∆t} ω
E
+ ∆
2
− t
2
h
2
δE
1 1A
δE
1 1B
− 1 h2|U
0A,0Bt
|2i ≈ X
E
exp −
Γ
2,E
t 2
− i
h tδE
2,E
− 1
π Z
∞
dω
E
Jω
E
sin{ω
E
− ∆t} ω
E
− ∆
2
− t
2
h
2
δE
1 1A
δE
1 1B
− 1 21
The elements of U
αA,αB
t are obtained as hn|U
αA,αB
t|ni = − 1
2h Z
∞
Z
∞
dω
A
dω
B
Jω
A 12
Jω
B 12
× 1 − e
− iω
A
+ω
B
t
ω
A
+ ω
B
ω
B
+ −1
n+1
∆ +
e
− iω
A
+−1
n +1
∆t
− 1 ω
A
+ −1
n+1
∆ω
B
+ −1
n+1
∆ +
1 − e
− iω
A
+ω
B
t
ω
A
+ ω
B
ω
A
+ −1
n+1
∆ +
e
− iω
B
+−1
n +1
∆t
− 1 ω
A
+ −1
n+1
∆ω
B
+ −1
n+1
∆ 22
The elements of U
0A,αB
t are evaluated as hm|U
0A,αB
t|ni = 2
√ πh
i + t
h δE
1 n,A
Z
∞
dω
B
Jω
B
sin ω
B
+ −1
m
∆t2 ω
B
+ −1
m
∆ e
iω
B
+−1
m
∆t2
23 The elements of U
αA,0B
t are calculated by interconverting A ↔ B within hm|U
0A,αB
t|ni in 23. We suppose that the initial state of the system is the left-handed state |Li. The evolved state of the whole system
in the localized basis of the macroscopic system is written as |Ψti =
1 √
2 |χ
1
i − |χ
2
i |Li +
1 √
2 |χ
1
i + |χ
2
i |Ri
24 where we defined
|χ
n
ti = e
− iE
n
th
|0i
A
|0i
B
hn|U
0A,0B
t|Li + |0i
A
|αi
B
hn|U
0A,αB
t|Li + |αi
A
|0i
B
hn|U
αA,0B
t|Li + |αi
A
|αi
B
hn|U
αA,αB
t|Li 25
for n = 1, 2. We are interested in the probability of finding the macroscopic system in the right-handed state, i.e., P
R
t = |hR|Ψti|
2
= 1
2 hχ
1
t|χ
1
ti + hχ
2
t|χ
2
ti + Re hχ
1
t|χ
2
ti 26
4. ESTIMATION OF PARAMETERS
