Directed motion in a periodic potential of a quantum system coupled to a heat bath
driven by a colored noise
Satyabrata Bhattacharya,
1
Pinaki Chaudhury,
2
Sudip Chattopadhyay,
1,
*
and Jyotipratim Ray Chaudhuri
3,†
1
Department of Chemistry, Bengal Engineering and Science University, Shibpur, Howrah 711103, India
2
Department of Chemistry, University of Calcutta, Kolkata 700009, India
3
Department of Physics, Katwa College, Katwa, Burdwan 713130, India
Received 26 May 2008; published 18 August 2008
A system-reservoir nonlinear coupling model is proposed for a quantum system when the associated bath is
not in thermal equilibrium but is modulated by an external colored noise, to present a microscopic approach to
quantum state-dependent diffusion and multiplicative noise in terms of a quantum Langevin description.
Consequently, the Fokker-Planck equation in position space, valid in the overdamped limit, for multiplicative
colored noise is constructed to explore the possibility of observing a quantum current and dependence of the
current on various parameters of external noise is examined.
DOI: 10.1103/PhysRevE.78.021123 PACS numbers: 05.40.-a
I. INTRODUCTION
Thermal diffusion is an actively pursued area of research.
In a periodic potential it has an important role when studying
Josephson’s junctions 1, oscillators with noisy limit cycles
2, diffusion in crystal surfaces 3, and many others. In
contemporary research it has been followed with a lot of
interest while studying the transport properties of Brownian
particles moving in a periodic potential 4 with special
stress on giant diffusion and coherent transport 5.
The motivation for all these studies partly lies in an at-
tempt to understand how protein motors move in biological
systems 6. To understand such transport phenomena, vari-
ous models have been proposed such as the vibrational
ratchet 7, rocking ratchet 8, flashing ratchet 5, diffusion
ratchet 9,10, correlation ratchet 11, and others. These
models have large scale applications in nanoscopic systems
and biology 12 due to their effectiveness in understanding
experimental observations on biochemical motors active in
muscle contraction 13, directed transport in photovoltaic
and photoreflective materials 14, and others. The potential
in all these models is taken to be asymmetric in space. A
unidirectional current can also be obtained from a spatially
symmetric potential. In such nonequilibrium systems, time
asymmetric random forces or space-dependent diffusion is
required 15. The space-dependent friction coefficient or
space dependent temperature may lead to the emergence of
the space-dependent diffusion coefficient 16–18. In super-
lattice structures, semiconductors, or motion in porous me-
dia, frictional inhomogenities are common. Space-dependent
friction is experienced by particles moving close to a surface
19,20.
In 1987, Bütikker 18 had shown that a classical particle
experiences a net drift force resulting in the generation of
current if the particle is in a symmetric sinusoidal potential
field in the presence of sinusoidally modulated space-
dependent diffusion with the same periodicity. Bütikker had
shown this in the case of space-dependent friction in the
overdamped limit where a directional mass flow may be ob-
tained. The origin of this current is basically the phase dif-
ference between the symmetric periodic potential and space-
dependent diffusion. The current vanishes for phase
difference of zero and multiples of . van Kampen 21 had
come to similar conclusions in a latter work for systems in
overdamped condition with space-dependent temperature.
The problem of Langevin equation with multiplicative noise
and state-dependent dissipation for a thermodynamical
closed system has been well studied. The classical quantum-
mechanical system reservoir linear coupling model for mi-
croscopic description of additive noise and linear dissipation
which are related by the fluctuation dissipation relation
FDR is well known over many decades in several fields
22,23, the nature of nonlinear coupling and its conse-
quences have been explored with renewed interest only re-
cently. For example, the nonlinear coupling approach has
been extensively used by Tanimura and co-workers 24 in
explaining elastic and inelastic relaxation mechanisms along
with their effects on vibrational and Raman spectroscopy.
Without using the rotating wave approximation for the
system-bath coupling, recently they have developed 25 a
quantum dissipative equation with Gaussian-Markovian
noise that has applicability to low-temperature systems
strongly coupled to a harmonic bath. But in all such cases,
the corresponding Langevin equation has been considered
for a thermodynamical closed system. A Langevin equation
with state-dependent dissipation and multiplicative colored
noise processes for an open system has drawn little attention
apart from a few exceptions 26,27.
In the classical regime, the transport of macroscopic ob-
jects such as Brownian particles is well elaborated in litera-
ture 28, special interest has been devoted to transport in
ratchet systems also termed Brownian motor systems29.
In contrast, the quantum properties of directed transport are
only partially elaborated in such motor systems 30,31.
Challenges arise in the quantum region because the transport
can strongly depend on the mutual interplay of pure quantum
*
Author to whom correspondence should be addressed:
sudipchattopadhyay@rediffmail.com
†
Author to whom correspondence should be addressed:
jprc8@yahoo.com
PHYSICAL REVIEW E 78, 021123 2008
1539-3755/2008/782/02112315 ©2008 The American Physical Society 021123-1