128
8 Quantum Ratchets
D eff (x) =
1
ββ 0
1 − λβU
(x)
−1
(8.33)
where the prime denotes the derivative with respect to the coordinate x. The
quantum correction parameter λ describes quantum fluctuations in position given
by
λ =
¯
h
ππ 0
γ +
1 +
τ c 0
Mπ
,
β =
1
k B T
(8.34)
Here, (z) is the digamma function, γ 0.5772 the Euler-Mascheroni constant.
Note that for k B T ¯
hh 0 /M, λ becomes
λ =
¯
h
ππ 0
γ + ln
τ c 0
Mπ
,
(8.35)
where, for a moderate damping, [9],
0 = MM 0 with h 0 = 2π
2
U 0 k B T
L 2 M
1/2
(8.36)
where the dimensionless U 0 is the amplitude of the ratchet potential and M is the
mass of the particle.
8.1.1.1 Dimensionless Parameters
Defining
x = L ˜
x, λ = L
2 ˜
λ, ˜
β = /k B T ,
U( ˜
x) = U(L ˜
x)//U ,
F = (U/L)
F , τ s = L
2 //U
(8.37)
we obtain the rescaled dimensionless diffusion coefficient
D eff ( ˜
x) =
1 − ˜
λ ˜
β
U
( ˜
x)
−1
(8.38)
we have used in 8.33:
U
(x) =
U
L 2
U ( x)
(8.39)
The barrier height U is the difference between the maximal and minimal values
of the unbiased potential.
8 Quantum Ratchets
D eff (x) =
1
ββ 0
1 − λβU
(x)
−1
(8.33)
where the prime denotes the derivative with respect to the coordinate x. The
quantum correction parameter λ describes quantum fluctuations in position given
by
λ =
¯
h
ππ 0
γ +
1 +
τ c 0
Mπ
,
β =
1
k B T
(8.34)
Here, (z) is the digamma function, γ 0.5772 the Euler-Mascheroni constant.
Note that for k B T ¯
hh 0 /M, λ becomes
λ =
¯
h
ππ 0
γ + ln
τ c 0
Mπ
,
(8.35)
where, for a moderate damping, [9],
0 = MM 0 with h 0 = 2π
2
U 0 k B T
L 2 M
1/2
(8.36)
where the dimensionless U 0 is the amplitude of the ratchet potential and M is the
mass of the particle.
8.1.1.1 Dimensionless Parameters
Defining
x = L ˜
x, λ = L
2 ˜
λ, ˜
β = /k B T ,
U( ˜
x) = U(L ˜
x)//U ,
F = (U/L)
F , τ s = L
2 //U
(8.37)
we obtain the rescaled dimensionless diffusion coefficient
D eff ( ˜
x) =
1 − ˜
λ ˜
β
U
( ˜
x)
−1
(8.38)
we have used in 8.33:
U
(x) =
U
L 2
U ( x)
(8.39)
The barrier height U is the difference between the maximal and minimal values
of the unbiased potential.