dynamics of the macroscopic system, especially the phenomenon of macroscopic quantum tunneling, can be studied in the typical double-well potential. At sufficiently low temperatures, the system’s Hamiltonian can be expanded by
two first states of energy. When the two-level system couples linearly to an oscillator environment, the result is the renowned Spin-Boson model [7, 32]. Due to decoherence, then, a mixture of localized states of the system appears.
Here, we examine the effective dynamics of a macroscopic quantum system, in interaction with a non-equilibrium environment. The system is conveniently modeled by the motion of a particle in a macroscopic double-well potential,
and the non-equilibrium environment is minimally described as two independent equilibrium environments at different temperatures. We analyze the dynamics of the system induced by such environment.
The paper is organized as follows. In the next section, we describe the physical model of the whole system, consisting of the macroscopic quantum system and surrounding environments. We examine the kinematics and dynamics of the
system in sections 3 and 4, respectively. The parameters of the model are estimated in section 5. The results are discussed in section 6. Our concluding remarks are presented in the last section.
2. MODEL
The Hamiltonian of the whole system composed of the macroscopic quantum system S and the environments A and B is conveniently defined as
H = H
S
+ X
E
H
E
+ X
E
H
SE
1 where E = A, B. The system is modeled as a particle in a symmetric double-well potential. Such a potential can be
represented by a quartic potential as U R = U
h R
R
2
− 1 i
2
, U
= M Ω
2
R
2
8 2
where M is the effective mass of the particle, Ω is the harmonic frequency at the bottom of each well, and R is
the distance between two minima from the origin. To quantify the extent to which the system exhibits quantum coherence, we incorporate the dimensionless form of the model. The potential has the characteristic energy U
and the characteristic length R
which we adopt as the units of energy and length, respectively. The corresponding characteristic time can be defined as τ
= R U
M
12
which we consider as the unit of time. Likewise, the unit of momentum is taken as P
= M U
12
. We then define the dynamical variables, x and p, as RR and PP
◦
, respectively. The corresponding commutation relation is defined as [x, p] = ıh, where the Planck constant is redefined
as h = ~R P
= ~U τ
, which we call “reduced Planck constant”. In the limit E
th
≪ Ωτ h ≪ 1 with E
th
= k
B
T U as thermal energy, k
B
as Boltzmann constant, and T as temperature, the states of the system are confined in the two-dimensional Hilbert space spanned by first two states
of energy |1i and |2i. The states corresponding to the particle localized in the left and right wells, |Li and |Ri, can be expanded as the maximal superpositions of energy states, |1i and |2i. Accordingly, the effective Hamiltonian of the
system in the localized basis would be
H
S
= −∆σ
x
3 where the tunneling frequency is identified as ∆ = hΩτ
4. For an isolated system, the probability of the tunnelling from the left state to the right one is given by
P
L→R
= sin
2
∆t 2
4 A frequently employed model for an environment is a collection of harmonic oscillators. The α-th harmonic oscillator
in the environment E is characterized by its natural frequency, ω
α,E
, and position and momentum operators, x
α,E
and, p
α,E
, respectively, according to the Hamiltonian H
E
= X
α
1 2
p
2 α,E
+ ω
2 α,E
x
2 α,E
− hω
α,E
5 The last term, which merely displaces the origin of energy, is introduced for later convenience. For the environment
E, we define |0i
E
as the vacuum eigenstate and |αi
E
as the excited eigenstate with energy E
α,E
. The interaction between the macroscopic quantum system and the environment E has the form [33]
H
SE
= − X
α
ω
2 α,E
f
α,E
σ
z
x
α,E
+ 1
2 ω
2 α,E
f
2 α,E
σ
z
6
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
