130
8 Quantum Ratchets
˜
(˜ t)˜ (˜ t
)
=
D eff ( ˜
x)
τ s
τ c
3
k=1
D k
τ k
exp
−
t s
τ k
τ s
τ c
(8.43)
8.1.2.1 The Dimensionless Quantum Overdamped Langevin Equation
From Eqs. 8.37, 8.38, 8.46, we obtain
d x
d t
= − U ( x) +
F − 0.5 λ
U ( x) + ˜
˜
t
(8.44)
where we have used
U
(x) =
U
L 3
U ( x)
(8.45)
In Fig. 8.2 are represented the contributions, (blue curves) of the first term,
(Fig. 8.2a), and third term, (Fig. 8.2b). we can observe that this third term, quantum
correction, dominates the equation in this simulation.
In Fig. 8.3 we can see that the system behaves as a brownian motor, for the given
parameters.
8.1.3 The Quantum Underdamped Langevin Equation
In reference [12] Denisov et al. investigated the quantum ratchet effect under the
influence of weak dissipation. They predicted a ratchet current when all relevant
symmetries are violated (Fig. 8.4). in obtained these results they used a FloquetMarkov master equation approach. In this section we will use the quantum Langevin
equation to treat the same problem, namely (Fig. 8.5),
M
dv
dt
= −Mγ v − U
(x) − 0.5λU
(x) + F + (x, t)
(8.46)
We follow the parametrization of [12], namely,
1
k L
=
L
2π
, x =
1
k L
˜
x, λ =
1
k L
2
˜
λ, τ s =
1
k L
M
U 0
,
γ =
1
τ s
˜
γ ,
v =
U 0
M
˜
v,
dv
dt
= k L
U 0
M
d ˜
v
d ˜
t
,
¯
h =
1
k L
MU 0 ˜
¯
h,
τ c
τ s
= 0.5
U 0 (8.47)
Then the complete quantum Langevin equation is given by,
8 Quantum Ratchets
˜
(˜ t)˜ (˜ t
)
=
D eff ( ˜
x)
τ s
τ c
3
k=1
D k
τ k
exp
−
t s
τ k
τ s
τ c
(8.43)
8.1.2.1 The Dimensionless Quantum Overdamped Langevin Equation
From Eqs. 8.37, 8.38, 8.46, we obtain
d x
d t
= − U ( x) +
F − 0.5 λ
U ( x) + ˜
˜
t
(8.44)
where we have used
U
(x) =
U
L 3
U ( x)
(8.45)
In Fig. 8.2 are represented the contributions, (blue curves) of the first term,
(Fig. 8.2a), and third term, (Fig. 8.2b). we can observe that this third term, quantum
correction, dominates the equation in this simulation.
In Fig. 8.3 we can see that the system behaves as a brownian motor, for the given
parameters.
8.1.3 The Quantum Underdamped Langevin Equation
In reference [12] Denisov et al. investigated the quantum ratchet effect under the
influence of weak dissipation. They predicted a ratchet current when all relevant
symmetries are violated (Fig. 8.4). in obtained these results they used a FloquetMarkov master equation approach. In this section we will use the quantum Langevin
equation to treat the same problem, namely (Fig. 8.5),
M
dv
dt
= −Mγ v − U
(x) − 0.5λU
(x) + F + (x, t)
(8.46)
We follow the parametrization of [12], namely,
1
k L
=
L
2π
, x =
1
k L
˜
x, λ =
1
k L
2
˜
λ, τ s =
1
k L
M
U 0
,
γ =
1
τ s
˜
γ ,
v =
U 0
M
˜
v,
dv
dt
= k L
U 0
M
d ˜
v
d ˜
t
,
¯
h =
1
k L
MU 0 ˜
¯
h,
τ c
τ s
= 0.5
U 0 (8.47)
Then the complete quantum Langevin equation is given by,