Nonlinear noninertial response of a quantum Brownian particle in a tilted periodic potential to a
strong ac force as applied to a point Josephson junction
William T. Coffey,
1
Yuri P. Kalmykov,
2
Serguey V. Titov,
3
and Liam Cleary
1
1
Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland
2
Laboratoire de Mathématiques, Physique et Systèmes, Université de Perpignan, 52, Avenue de Paul Alduy,
66860 Perpignan Cedex, France
3
Kotel’nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences, Vvedenskii Square 1,
Fryazino, 141190, Russian Federation
Received 4 November 2008; published 10 February 2009
The quantum Smoluchowski equation for the reduced Wigner function in configuration space pertaining to
the quantum Brownian motion of a particle in a tilted cosine potential in the high dissipation or noninertial
limit as applied to a model point Josephson junction namely, a resistively shunted junction in the presence of
noise and an arbitrarily large microwave ac driving current is considered. The solution of the resulting
recurrence relations for the Fourier amplitudes of the statistical moments describing the nonlinear dynamics of
the junction ignoring the capacitance is obtained using the matrix continued fractions previously developed
for the stationary ac field solution of the corresponding classical problem. Quantum effects in the nonlinear
response of the junction to an ac microwave current of arbitrary amplitude nonlinear microwave impedance,
frequency dependence of the dc current-voltage characteristic, etc. are estimated.
DOI: 10.1103/PhysRevB.79.054507 PACS numbers: 74.50.+r, 05.40.-a, 03.65.Yz
I. INTRODUCTION
Wigner’s phase-space formulation of quantum mechanics
in terms of quasiprobability distributions of the canonical
variables
1–7
as extended to open quantum systems see, e.g.,
Refs. 8–16 has recently been used
15,17
to derive a quantum
Smoluchowski equation QSE governing the time evolution
of the configuration-space distribution function Px , t for
particles with separable and additive Hamiltonians in the
overdamped or noninertial limit. In the present context,
pertaining to a quantum Brownian particle of mass m moving
along the x axis under the influence of a potential Vx, the
canonical variables are the position x and the momentum p
of the particle. The corresponding reduced single-particle
joint quasiprobability distribution function in phase space,
namely, the Wigner function Wx , p , t, represents the projec-
tion of all the other degrees of freedom of the system, com-
prised of the quantum Brownian particle and its heat bath,
onto the phase space x , p of that particle. The evolution of
Wx , p , t is governed by the master equation
8
W
t
+
p
m
W
x
-
1
i
V x +
i
2
p
- V x -
i
2
p
W
= M
ˆ
D
W, 1
where M
ˆ
D
is the collision kernel operator representing the
bath-particle interaction and is Planck’s constant. Here the
left-hand side is the single-particle Wigner equation
1,6,8
which is the quantum analog of the classical Liouville equa-
tion governing the evolution of the joint quasiprobability
distribution function for the closed system. The stationary
solution of this equation i.e., the master Eq. 1 with the
right-hand side equal to zero is the Wigner stationary distri-
bution W
0
x , p, which can be developed as a power series in
2
, viz.,
1
W
0
x, p = e
-x, p
1+
2
24m
V'
2
x - 3-
p
2
m
Vx
+ ¯
,
where x , p = p
2
/ 2m + Vx is the classical energy of the
particle, = kT
-1
, k is Boltzmann’s constant, and T is the
temperature. Regarding the general open system governed by
Eq. 1, various forms of the collision operator M
ˆ
D
have been
discussed in detail in Ref. 8. Now, on specializing to the
quantum Brownian motion in the high-temperature and
weak-coupling limits, the collision operator M
ˆ
D
can be rep-
resented just as in the classical theory by a Kramers-Moyal
expansion truncated at the second term. However, unlike the
classical theory in order that W
0
x , p should also render the
right-hand side of Eq. 1 zero, the coefficients of the trun-
cated Kramers-Moyal expansion must become functions of
the derivatives of the potential. The master Eq. 1 in phase
space then describes the relaxation of Wx , p , t to the sta-
tionary state given by W
0
x , p in the long time limit.
15,17
The
ansatz that the Wigner stationary distribution W
0
x , p ren-
ders the right-hand side of the master Eq. 1 zero whence
the diffusion coefficients must depend on the derivatives of
the potential may be used if the interactions between the
Brownian particle and the heat bath are small enough to al-
low one to use the weak-coupling limit, and if the correlation
time characterizing the bath is so short that one can regard
the stochastic process originating in the bath as Markovian.
For parameter ranges, where such an approximation is in-
valid e.g., throughout the very-low-temperature region,
where non-Markovian effects are substantial, other methods
should be used.
9
We remark that the imposition of W
0
x , p
as the stationary solution of Eq. 1 is exactly analogous to
the assumption of the Maxwell-Boltzmann distribution as the
PHYSICAL REVIEW B 79, 054507 2009
1098-0121/2009/795/0545079 ©2009 The American Physical Society 054507-1