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
To examine the dynamics of open macroscopic quantum system, we first estimate the parameters relevant to our analysis. The two-level approximation requires that E
th
≪ 1, so considering the thermal energy in the range of temperature variation 1K − 100K as k
B
T ∼ 10
− 23
J − 10
− 21
J, we estimate the typical value of characteristic energy as U
∼ 10
− 20
J. Since the harmonic frequency at the bottom of each well Ω coincides naturally with the characteristic frequency of the system τ
− 1
, we have ∆ ≈ h. The reduced Planck constant h quantifies the macroscopicity of the system in question. The system in which h ≪ 1 is called the quasi-classical system. The typical range of h for a
macroscopic quantum system is 0.01 − 0.1 [33]. As for the environment, we assume that two environments are dilute. A dilute environment cannot exactly be
modeled as a collection of oscillators. In fact, the fluctuating force produced by the dilute environment on the system is not Gaussian as it would be for oscillators. Nevertheless, if every collision is sufficiently weak, then the force
could still approximately considered as Gaussian, when averaged over longer time scales, including many collisions. In order to analyze the dynamics explicitly, we should specify the spectral density of the environments. As Harris and
Stodolsky pointed out, in the dilute environment, the dynamics of the system samples the velocity distribution of the environmental particles, which is strongly temperature-dependent [34, 35]. Starting from a microscopic model of the
collisions with the environmental particles, one can drive the corresponding spectral density. The interaction process envisaged here is a sequence of collisions between light environmental particles and a heavier macroscopic system. We
also assume that the collision does not lead to any internal transitions of the macroscopic system. The details of the derivation can be found in [36, 37]. For a Gaussian interaction potential, the spectral density is obtained as
J
E
ω; T = JT
E
ω
1 2
E
e
− ω
E
ΛT
E
27 The temperature dependence of coupling strength and cut-off strength are respectively as JT
E
= cT
E
K
− 34
and ΛT
E
≈ 0.026R
2
T
E
K
12
, where c is a dimensionless constant, depending on the details of the environmental interactions, and R is the range of intermolecular interactions. At temporal domain Γ
2,E
≪ Ω, the best range for the constant is c 10. Also, since the environment is dilute we assume that R 1.
5. RESULTS AND DISCUSSION
