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Quantum Suppression of Ratchet Rectification in a Brownian System
Driven by a Biharmonic Force
Akihito Kato
†
and Yoshitaka Tanimura*
,†,‡
†
Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto606-8502, Japan
‡
Universität Augsburg, Institut für Physik, Universitätsstrasse 1, 86135 Augsburg, Germany
ABSTRACT: We rigorously investigate the quantum dis-
sipative dynamics of a ratchet system described by a periodic
potential model based on the Caldeira−Leggett Hamiltonian
with a biharmonic force. In this model, we use the reduced
hierarchy equations of motion in the Wigner space
representation. These equations represent a generalization of
the Gaussian−Markovian quantum Fokker−Planck equation
introduced by Tanimura and Wolynes (1991), which was
formulated to study non-Markovian and nonperturbative
thermal effects at finite temperature. This formalism allows
us to treat both the classical limit and the tunneling regimes,
and it is helpful for identifying purely quantum mechanical
effects through the time evolution of the Wigner distribution.
We carried out extensive calculations of the classical and quantum currents for various temperatures, coupling strengths, and
barrier heights. Our results reveal that at low temperature, while the quantum current is larger than the classical current in the
case of a high barrier, the opposite is true in the case of a low barrier. We find that this behavior results from the fact that the
tunneling enhances the current in the case of a high barrier, while it suppresses the current in the case of a low barrier. This is
because the effect of the ratchet potential is weak in the case of a low barrier due to the large dispersion of the distribution
introduced by tunneling. This causes the spatiotemporal asymmetry, which is necessary for ratchet current, to be weak, and as a
result, the net current is suppressed.
1. INTRODUCTION
A system that is able to rectify thermal or mechanical
fluctuations through a periodic potential is called a ratchet.
1−8
With recent advances in microscopy and nanotechnology,
several new mechanisms of rectification of fluctuations resulting
in transport have been identified through theoretical works
9−19
and experimental works
20−30
in biology, physics, and chemistry.
Well-known examples include asymmetric quantum dots,
20−22
vortices in superconductors,
23−25
cold atoms in asymmetric
optical lattices,
26−28
and molecular motors in biological
systems.
29−31
A ratchet system can produce a directed current
from fluctuations only when the system is in a nonequilibrium
state because the extraction of work from unbiased fluctuations
is not allowed by the second law of thermodynamics. Moreover,
to realize the ratchet effects, it is necessary to break the
spatiotemporal symmetries because otherwise the contributions
from positive and negative currents would cancel.
32−36
Quantum mechanical effects may also play a role because
ratchet systems must be of microscopic size to rectify the
fluctuating motion.
37−47
A Fokker−Planck approach
48
and a Langevin approach
49
have been used in classical studies of ratchets, while varieties of
equation of motion approaches have been employed for
quantum mechanical studies.
37−47
In the quantum mechanical
case, dissipative systems are commonly modeled as potential
systems coupled to heat-bath degrees of freedom at finite
temperature. This coupling gives rise to thermal fluctuations
and dissipation that drive the systems toward the thermal
equilibrium state. The heat-bath degrees of freedom are then
reduced using such methods as the projection operator method
and the path integral method, for example. The quantum
Langevin equation
37
and the quantum Fokker−Planck
equation
42
have been used for the purpose of understanding
the quantum aspects of ratchet dynamics in a heat bath.
Although these equations are analogous to the classical kinetic
equations, which have proved to be useful in the treatment of
classical transport problems, such equations cannot be derived
in a quantum mechanical framework without significant
approximations or assumptions. For example, in dealing with
the quantum Langevin equation expressed in operator form, it
is generally assumed that the antisymmetric correlation
function of the noise is positive, but this is valid only for
slow non-Markovian modulation at high temperature, as we
demonstrate in Section 2. Similarly, the quantum Fokker−
Planck equation can be derived from the Caldeira−Leggett
Special Issue: Peter G. Wolynes Festschrift
Received: March 28, 2013
Revised: April 23, 2013
Published: May 2, 2013
Article
pubs.acs.org/JPCB
© 2013 American Chemical Society 13132 dx.doi.org/10.1021/jp403056h |J. Phys. Chem. B 2013, 117, 13132−13144
Hamiltonian under a Markovian approximation,
51−55
but for
this to be possible, the heat bath must be at a sufficiently high
temperature, in which case quantum tunneling processes play a
minor role. The quantum master equation expressed in terms
of Floquet states provides a more rigorous treatment of
quantum dissipative dynamics than the methods mentioned
above,
46
but it can only be applied to systems possessing weak
interactions.
50
The methods discussed to this point represent
the main approaches used in the treatment of ratchet systems
carried out to this time. However, as we have pointed out, none
of them provide fully quantum mechanical descriptions of
broad validity.
We employ the reduced hierarchy equations of motion
(HEOM) in the Wigner space representation,
56
which are a
generalization of the Gaussian−Markovian quantum Fokker−
Planck equation introduced by Tanimura and Wolynes
57,58
to
study non-Markovian and nonperturbative thermal effects at
finite temperature.
59−63
Because the HEOM are derived from
the full system-bath Hamiltonian with no approximation, the
entire system approaches a thermal equilibrium state at finite
temperature when no external perturbation is applied.
56,64
This
means that the second law of thermodynamics holds within the
HEOM approach, and hence there can be no finite current in
the absence of a driving force, which is the essential behavior to
investigate the ratchet rectification of thermal fluctuations. The
HEOM have been used to study a variety of phenomena and
systems, including exciton transfer,
65−69
electron transfer,
70−73
quantum information,
74,75
and resonant tunneling diodes.
62,63
The HEOM are ideal for studying quantum transport systems
when implemented using the Wigner representation because
they allow the treatment of continuous systems utilizing open
boundary conditions and periodic boundary conditions.
76
Elucidation of the dynamical behavior of the system through
the time evolution of the Wigner distribution functions is also
possible. The classical hierarchy equations of motion can be
obtained easily by taking the classical limit of the HEOM, and
utilizing the Wigner representation, we can easily compare
quantum and classical distribution functions.
58,61
Considering
both classical and quantum mechanical cases, we report the
results of numerical calculations of the ratchet current, which is
the induced net current resulting from ratchet effects, in a
dissipative environment for various temperatures and system-
bath coupling strengths. We then clarify the roles of
fluctuations, dissipation, and the biharmonic force in both the
classical and quantum cases.
The organization of the paper is as follows. In Section 2, we
introduce the HEOM approach in the Wigner representation.
In Section 3, we discuss ratchet systems from the point of view
of space-time symmetries. In Section 4, the numerical results
obtained for the ratchet current in classical and quantum
mechanical cases are presented. Section 5 is devoted to
conclusions.
2. REDUCED HIERARCHY EQUATIONS OF MOTION
FORMALISM
Ratchet systems are often modeled by a Brownian particle in a
periodic potential under an ac or dc driving force. We take this
approach and employ a model based on the Caldeira−Leggett
(or Brownian) Hamiltonian
77
∑
ω
ω
̂=̂+̂
+̂+̂−̂
⎡
⎣
⎢
⎢
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎤
⎦
⎥
⎥
H
p
mUq t
p
m
mxaV q
m
2(; )
22
()
j
j
j
jj
j
j
jj
tot
2
22
2
2
(1)
Here m,p̂,q̂, and U(q̂,t) are the mass, momentum, position,
and potential of the particle, and mj,p̂j,x̂j, and ωjare the mass,
momentum, position, and frequency variables of the jth bath
oscillator mode. The quantities ajare coefficients that depend
on the nature of the system-bath coupling. From eq 1, it is seen
that the interaction part of the Hamiltonian is assumed to take
the form ĤI=−V(q̂)X̂, where V(q̂) is an arbitrary function of q̂
and X̂≡Σjajx̂jis the interaction coordinate. We introduced the
counter term Σjaj
2V(q̂)2/2mjωj
2to maintain the translational
symmetry of the Hamiltonian for U(q̂;0) = 0. Here we consider
areflection-symmetric potential system driven by a biharmonic
force.
14−19,45−49
This force prevents the system from reaching
equilibrium and breaks the spatiotemporal symmetry. We write
the potential as U(q̂;t)=U(q̂)−q̂F(t), where U(q̂) is static,
reflection-symmetric potential, and −q̂F(t) is the dynamic
potential. Note that the time average of F(t) is zero. The total
system obeys the von-Neumann equation (the quantum
Liouville equation).
After the bath degrees of freedom are traced out, the reduced
density matrix elements of the system are obtained in the path
integral form as
∫∫∫
ρ
ρ
ρ
′= ′ ′ ′
×′′ ℏ′
×−′ℏ
qq t q q Dq Dq q q
qq t q q iS qt Fqq t
iS q t
(, ; ) d d ( , )
( , , ; , )exp( [ , ]/ ) [ , ; ]
exp( [ , ]/ )
0000
CS 00A
A(2)
where SA[q;t] is the action corresponding to the system’s
Hamiltonian, ĤA=q̂2/2m+U(q̂;t), ρ(q0,q′0) is the initial state
of the system at time t0, and ρCS(q,q′,t;q0,q′0) is the initial
correlation function between the system and the heat bath.
78
The effect of the bath is incorporated into the Feynman−
Vernon influence functional, which is written
∫
∫
∫
′= −
ℏ
×∂
∂
ℏΨ− °
+−
×
×
⎧
⎨
⎩
⎡
⎣
⎢
⎤
⎦
⎥
⎫
⎬
⎭
Fq q t sV s
suisuVu
uCs uV u
[, ; ] exp 1d()
d2()()
d( )()
t
t
t
s
t
s
2
0
0
0(3)
where V×(t)≡V(q(t)) −V(q′(t)) and V°(t)≡V(q(t)) +
V(q′(t)). Although the method we employ here can be used
with any form of V(q),
59−61
in this paper, we consider the
linear−linear system bath coupling case, defined by V(q)=q.
The interaction coordinate X̂≡Σjajx̂jis regarded as a driving
force through the interaction −
̂̂
q
X
. The canonical and
symmetrized correlation functions, respectively, are then
expressed as Ψ(t)≡β⟨X̂;X̂(t)⟩Band C(t)≡(1/
2)⟨{X̂(t),X̂(0)}⟩B, where β=1/kBTis the inverse temperature,
X̂(t)isX̂in the Heisenberg representation, and ⟨···⟩Brepresents
the thermal average over the bath modes.
56,64
Using the
spectral density J(ω)=Σjaj
2δ(ω−ωj)/2mjωj, we can rewrite
these functions as
The Journal of Physical Chemistry B Article
dx.doi.org/10.1021/jp403056h |J. Phys. Chem. B 2013, 117, 13132−1314413133
∫ωω
ωω
Ψ
=∞
tJt() 2 d ()
cos( )
0(4)
and
∫ωω ω βω
=ℏ ℏ
∞
⎜⎟
⎛
⎝
⎞
⎠
Ct J t() d ( )cos( )coth 2
0(5)
The function C(t) is analogous to the classical correlation
function of X(t) and corresponds to the correlation function of
the bath-induced noise (fluctuations), whereas Ψ(t) corre-
sponds to dissipation. The function C(t) is related to Ψ(t)
through the quantum version of the fluctuation−dissipation
theorem, C[ω]=ℏωcoth(βℏω/2)/2Ψ[ω], which ensures that
the system evolves toward the thermal equilibrium state for
finite temperatures, trB{exp[−βĤtot]}, in the case that there is
no driving force.
79
As shown in Figure 1, the noise correlation C(t) becomes
negative at low temperature due to the contribution of the
Matsubara frequency terms with ν=2π/βℏin the region of
small t. This behavior is characteristic of quantum noise. The
fact that the noise correlation takes negative values introduces
problems when the quantum Langevin equation is applied to
quantum tunneling at low temperature. We note that the
characteristic time scale over which we have C(t)<0is
determined by the temperature and is not influenced by the
spectral distribution J(ω). Thus, the validity of the Markovian
(or δ(t)-correlated) noise assumption is limited in the quantum
case to the high-temperature regime. Approaches employing
the Markovian master equation and the Redfield equation,
which are usually applied to systems possessing discretized
energy states, ignore or simplify such tunneling contributions,
often through use of the rotating wave approximation (RWA).
The main problem created by the RWA is that a system treated
under this approximation will not satisfy the fluctuation−
dissipation theorem, and thus it may introduce significant error
in the evolution of the system toward equilibrium. In the
classical limit, with ℏtending to zero, C(t) is always positive.
We assume that spectral density J(ω) has an Ohmic form
with a Lorentzian cutoff, that is
ωζ
π
γω
γω
=+
Jm
()
2
22 (6)
where ζis the system-bath coupling strength, which represents
the magnitude of damping, and γis the width of the spectral
density of the bath mode. The canonical and symmetrized
correlation functions then become
ζγ
Ψ
=γ−||
tm() e t(7)
and
∑
ζγ β γ
β
ν
γν
=ℏℏ
−ℏ−
γν−||
=
−||
⎜⎟
⎡
⎣
⎢
⎢
⎛
⎝
⎞
⎠
⎤
⎦
⎥
⎥
Ct m
() 2cot 2e14e
t
k
k
k
t
2
1
22
k
(8)
where νk=2πk/βℏis the kth Matsubara frequency.
80
Note that
in the high- temperature limit, βℏγ≪1, the noise correlation
function reduces to C(t)≈mζγe−γ|t|/β. This indicates that the
heat bath oscillators interact with the system in the form of
Gaussian−Markovian noise.
The reduced HEOM can be obtained by considering the
time derivative of the reduced density matrix elements given in
eq 2 in the Wigner representation,
56
as a generalization of the
Gaussian−Markovian quantum Fokker−Planck equation in-
troduced by Tanimura and Wolynes.
57,58
If we choose Kso as
to satisfy νK=2πK/(βℏ)≫ωc, where ωcis the characteristic
frequency of the system, the factor e−νk|t|in eq 8 can be replaced
with the Dirac delta function, using the approximation νke−νk|t|/
2≃δ(t) (for k≥K+ 1), with negligible error. For the kernel
Figure 1. Symmetric correlation C(t)defined by eq 8 is depicted as a function of the dimensionless time tfor (a) the slow modulation case, γ=1,
and (b) the fast modulation case, γ= 10. The constants mζγ2/2 and ℏare set to unity. Note that γ→∞corresponds to the Ohmic (Markovian)
limit, as can be seen from eq 6. In each of the Figures, the inverse temperatures are, from top to bottom, β= 0.2, 0.5, 1.0, and 5. The noise
correlation C(t) becomes negative for (b), the fast modulation case, at low temperature (large β) due to the contribution of Matsubara frequency
terms.
The Journal of Physical Chemistry B Article
dx.doi.org/10.1021/jp403056h |J. Phys. Chem. B 2013, 117, 13132−1314413134
eqs 7 and 8, the number of hierarchy elements for γis denoted
by n, and the number of kth Matsubara frequencies is denoted
by jk.
56,81
Then, the HEOM can be expressed as
61−63
∑
∑
∑
γν
γ
ν
∂
∂=− +Ξ
̂++
×+Φ
̂
++Θ
̂
+Θ
̂
···
=
··· ···
+
=
··· + ··· ···
−
=
··· − ···
tWqpt n j
Wqpt Wqpt
WqptnWqpt
jW qpt
(, ; ) [ ]
(, ; ) [ (, ; )
(, ; )] (, ; )
(, ; ),
jj
n
k
K
kk
jj
n
jj
n
k
K
jj j
n
jj
n
k
K
kkk j j j
n
,,
() QM
1
,,
() ,,
(1)
1
,, 1,,
() 0,,
(1)
1
,, 1,,
()
K
KK
kK K
kK
1
11
11
1
3
(9)
where the quantum Liouvillian in the Wigner representation is
expressed as
76
∫
π
−=−
∂
∂
−′
ℏ−′ ′
−∞
∞
Wqp p
mq
Wqp
pUqp ptWqp
(, ) (, )
d
2(, ; ) (, )
W
QM
2
3
(10)
with UW(q,p;t)=2∫0
∞drsin(pr/ℏ)[U(q+r/2;t)−U(q−r/
2;t)]. The other operators are defined as
Φ
̂≡∂
∂p(11)
ζγβγ
Θ̂≡+
ℏℏ
∂
∂
⎜⎟
⎡
⎣
⎢⎛
⎝
⎞
⎠
⎤
⎦
⎥
pm
p2cot 2
0(12)
γζ
βν γ
Θ̂≡−
∂
∂
m
p
2
()
k
k
2
22
(13)
for k> 0 and
∑
ζ
β
βγ βγ γ
νγ
Ξ
̂≡− − ℏℏ
−−
∂
∂
=
⎜⎟
⎡
⎣
⎢
⎢
⎛
⎝
⎞
⎠
⎤
⎦
⎥
⎥
m
p
12cot 2
2
k
K
k
1
2
22
2
2
(14)
Note that the zeroth element is identical to the Wigner
function W0,0,···,0
(0) (q,p;t)≡W(q,p;t), and the other elements are
introduced in the numerical calculations in order to treat the
nonperturbative, non-Markovian system-bath interaction.
Although these elements do not have direct physical meaning,
they allow us to take into account the quantum coherence and
entanglement between the system and the bath.
74,75
The
importance of the system-bath coherence was pointed out in
the context of correlated initial conditions,
78
and it was shown
with a nonlinear response theory approach that the system-bath
coherence plays an essential role in the case that the system is
driven by a time-dependent external force.
56
The HEOM consist of an infinite number of equations, but
they can be truncated at finite order with negligible error.
57,58
Essentially, the condition necessary for the error introduced by
the truncation to be negligibly small is that the total number of
hierarchy elements or the total number of Matsubara
frequencies retained be sufficiently large. Explicitly, it can be
shown that the condition N≡n+Σk=1
Kjk≫ωc/min(γ,ν1)is
sufficient for this purpose.
81
For the Caldeira−Leggett
Hamiltonian, the equations of motion are then truncated by
using the “terminators”, expressed in the Wigner representation
as
61−63
∂
∂=− +Ξ
̂
··· ···
tWqpt Wqpt(, ; ) ( ) (, ; )
jj
n
jj
n
,,
() QM , ,
()
KK11
3
(15)
with n+Σk=1
Kjk=N. The HEOM formalism can be used to
treat a strong system-bath coupling nonperturbatively. It is ideal
for studying quantum transport systems when employing the
Wigner representation because it allows the treatment of
continuous systems utilizing open boundary conditions and
periodic boundary conditions.
76
In addition, the formalism can
accommodate the inclusion of an arbitrary time-dependent
external field while still accounting for the system-bath
coherence. Note that such coherence cannot be described if
we assume that the state of the total system takes a factorized
system-bath form.
56
Such system-bath coupling features are
necessary to properly treat quantum ratchet systems. In the
white noise (or Markovian) limit, γ→∞, which is taken after
imposing the high-temperature approximation, βℏγ≪1, the
quantum Fokker−Planck equation can be derived in a similar
form as the Kramers equation,
51,52
which is identical to the
quantum master equation without RWA.
82
Because we assume
βℏγ≪1 with γ→∞, this equation cannot be applied to low-
temperature systems, where quantum effects play a major role.
The Wigner distribution function reduces to the classical one
in the limit ℏ→0, and hence we can directly compare the
quantum results with the classical results. The classical HEOM,
which is derived from eq 9 by setting ℏ→0, is given by
57,58
γ
γζ β
∂
∂=− +
+∂
∂
++
∂
∂
+
−
⎛
⎝
⎜⎞
⎠
⎟
tWqpt nWqpt
pWqpt
np m
pWqpt
(, ; ) ( ) (, ; )
(, ; )
(, ; )
nn
n
n
() CL ()
(1)
(1)
3
(16)
and the classical terminator is
γ
ζβ
γζ β
∂
∂=− +
+∂
∂+∂
∂
++
∂
∂
−
⎛
⎝
⎜⎞
⎠
⎟
⎛
⎝
⎜⎞
⎠
⎟
tWqpt NWqpt
ppm
pWqpt
Np m
pWqpt
(, ; ) ( ) (, ; )
(, ; )
(, ; )
NN
N
N
() CL ()
()
(1)
3
(17)
The classical Liouvillian is defined by
−=−
∂
∂+∂
∂
∂
∂
Wqp p
mq
Wqp Uq t
qp
Wqp(, ) (, ) (; ) (, )
CL
3
(18)
The classical equation of motion is helpful because knowing the
classical limit allows us to identify the purely quantum
mechanical effects.
58,61
Note that the above equations of
motion can also be derived from the classical Langevin
equation, where the classical fluctuation dissipation theorem
is hold.
57
Then, in the white noise (or Markovian) limit, the
above equations reduce to the Kramers equation.
In Section 4, we use eqs 9−15 and eqs 16−18 to calculate
quantum and classical ratchet currents, respectively. Because we
are dealing with the distribution function for both quantum and
classical cases, there is no need to sample kinetic trajectories,
which is necessary in approaches employing the Langevin
equation and in some kinetic approaches. This is convenient if
The Journal of Physical Chemistry B Article
dx.doi.org/10.1021/jp403056h |J. Phys. Chem. B 2013, 117, 13132−1314413135
we wish to calculate the current at low temperature, even in the
classical case, because this sampling becomes inefficient due to
the localization of the particle trajectories trapped in the
potential. Another important aspect of the present method-
ology is that it allows elucidation of the dynamical behavior of
the system through the time evolution of the distribution
function in the phase space. In addition to the dependence of
the current on the temperature, the barrier height, and the
system-bath coupling, we analyze the mechanism of ratchet
rectification using the Wigner and classical distribution
functions.
3. POTENTIAL, FORCE, AND SYMMETRY
In the present study, we consider the 1D symmetric potential
(see Figure 2)
κ=
U
qU q() cos( )
02(19)
and the biharmonic force
θ=Ω+ Ω+Ft F t F t( ) cos( ) cos(2 )
12
(20)
where U0is the barrier height, κis the wavenumber, and F1,F2,
Ω, and θdenote the amplitudes, frequency, and the phase
difference of the biharmonic forces, respectively.
14−19,45−49
Experimentally, such a situation has been realized for fluxons in
long Josephson junctions
25
and for cold rubidium and cesium
atoms in optical lattices.
26−28
Before reporting the results of our calculations of the ratchet
current, it is important to mention two symmetry conditions
under which the current vanishes.
32−36
The first one is the
time-shift symmetry defined by the transformation (q,p,t)→
(−q,−p,t+T/2), where Tis the period of the external force,
and the second one is the time-reversal symmetry defined by
the transformation (q,p,t)→(q,−p,−t). When the equation of
motion of the system is invariant under either of these two
symmetry transformations, the current vanishes because each of
these transformations reverses the sign of the momentum. For
the external force given by eq 20 and for the quantum and
classical HEOM appearing in eqs 9−15 and eqs 16−18,
respectively, time-shift symmetry exists if F1=0orF2=0.
Time-reversal symmetry is broken if θ≠0orπin the
Hamiltonian case, that is, if ζ= 0, while, in the presence of
dissipation, this symmetry is always broken, which implies that
directed motion can exist for arbitrary θ. For the classical
overdamped Langevin equation with white noise, the system
possesses time-reversal symmetry if θ=π/2 or 3π/2. The
overdamped (Smoluchowski) case can be studied within the
HEOM formalism by choosing a large value for ζwith γ≫Ω
in the classical case and by choosing a large value of ζwith γ≫
Ωand βℏγ≪1 (i.e., the high-temperature limit) in the
quantum case. In such cases, we expect the calculated current to
vanish if θ=π/2 or 3π/2.
4. NUMERICAL RESULTS
We numerically integrated the HEOM in the form of finite
difference equations using the fourth-order Runge−Kutta
method. We also imposed the periodical boundary condition
W(p,q)=W(p,q+2π/κ). The spatial derivative of the kinetic
term in the Liouville operator, −(p/m)∂W(p,q)/∂q,was
approximated using a third-order left-handed or right-handed
difference scheme, depending on the sign of the momentum,
while other derivatives with respect to pwere approximated
using a third-order center difference scheme.
63
We employed
the following units of the length, momentum, and frequency: qr
=1/κ,pr=ℏκ, and ωr=ℏκ2/m. Note that with these units we
have m=ℏ=κ= 1. The dimensionless position and
momentum then are given by q̅=q/qrand p̅=p/pr. The mesh
sizes for the position and the momentum were chosen in the
ranges 0.0524 < Δq̅< 0.1047 and 0.04 < Δp̅< 0.10. The depth
of the hierarchy and the number of Matsubara frequencies were
chosen so as to satisfy N∈{5 −10} and K∈{1 −2}. In this
study, we fixed the inverse of the noise correlation time to γ=
1.0ωrand the amplitude and frequency of the biharmonic force
to F1=F2= 0.20ℏωr/qrand Ω= 1.0ωr, respectively. Three
values were used for the height of potential barrier, U0=ℏωr,
2ℏωr, and 4ℏωr(see Figure 2).
Figure 2. U0cos 2(q̅) potential for the cases of (a) a low barrier, U0=
1, (b) an intermediate barrier, U0= 2, and (c) a high barrier, U0=4.In
each case, eigenenergies and eigenfunctions with periodic boundary
conditions are plotted as the black dashed lines and the red curves,
respectively.
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Employing the biharmonic force given in eq 20, we
integrated the classical and quantum HEOM until the
distributions reached the steady state. Then, the classical and
quantum directed currents (or the ratchet current) were
evaluated from the distribution function as
∫
≡′⟨′
⟩
→∞
+
JTtptlim 1d()
tt
tT
(21)
where
∫
⟨
⟩=pt q ppWq p t() d d ( , ; ) (22)
We repeated the calculation for fixed physical conditions, while
varying θin the range 0 < θ<2π. From these calculations, we
find that as a function of θthe current roughly takes the
following form:
26,48
θθθ=−JJ() sin( )
max 0(23)
where Jmaxand θ0are the maximum value of the current and the
phase lag evaluated from the numerical simulations, respec-
tively. Below we investigate the dependence of the current on
the temperature, the barrier height, and the bath coupling
strength through the dependences of Jmax and θ0on these
quantities. To gain further insight into the mechanism of
ratchet rectification, we also present plots of the Wigner and
classical distribution functions where helpful.
4.1. Temperature Effects: the Classical Case. Figure 3
displays (a) the maximum value, Jmax, and (b) the phase lag, θ0,
of the current as functions of the inverse temperature, β, for
three values of the barrier height, U0, evaluated using the
classical HEOM given in eqs 16−18. Here we chose the weak
coupling strength ζ= 0.10ωr. In all three cases depicted in
Figure 3a, it is seen that the maximum current realizes a
maximum value at some intermediate value of the inverse
temperature. The value of βat which this maximum is realized
is found to increase as U0decreases. As shown in Figure 3b, a
similar profile is found for the phase lag θ0as a function of β,
although the peak positions are different.
The behavior depicted in Figure 3 can be understood in
terms of the thermal activation of a particle. To illustrate this, in
Figure 4, we plot snapshots of the classical distribution function
for (a) low-, (b) intermediate-, and (c) high-temperature cases
at five different values of the time for one cycle. In Figure 4a-i,b-
i, it is seen that when the temperature takes a low or
intermediate value the distribution function is concentrated in
the potential wells. The distributions are periodically
accelerated by the biharmonic force and rotate clockwise
around the bottom of each potential well. Figure 4a−ii,b-ii
reveals that when the distribution approaches the left side of
the barrier a small part of the distribution with positive
momentum is transferred to the right potential well by crossing
the barrier. This can be seen as the elongation of the
distribution on the right side. In Figures 4a-iii,b-iii, it is seen
that while the distribution begins to flow in the positive
direction the sign of the biharmonic force changes, and the
main part of the distribution moves away from the barrier. The
part of the distribution that crosses the barrier flows into the
right potential and rotates clockwise in a spiral form while
losing energy due to dissipation. Figure 4a-iv,b-iv shows the
flow stops shortly after the biharmonic force change direction,
with a small time delay. In Figure 4a-v,b-v, it is seen that when
the distribution approaches the left-hand side of the barrier, a
part of the distribution with negative momentum also flows in
the opposite direction, but the effect of the acceleration caused
by the biharmonic force is smaller in this case than in the
positive case due to the asymmetric influence of the biharmonic
force on the system dynamics with the phase lag θ0.
Figure 3. Maximum current in (a) the classical and (a′) the quantum case and the corresponding phase lag in (b) the classical and (b′) the quantum
case as functions of the inverse temperature, β, for three values of the barrier height U0. The coupling strength here is ζ= 0.10ωr.
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In the intermediate temperature case depicted in Figure 4b,
the distribution is broadened compared with Figure 4a due to
thermal effects. The current is greater than in the low-
temperature case because the weight of the distribution at
energies above the barrier under the biharmonic force is
greater. The response of the distribution to the external force is
also delayed due to the broadening of the distribution in the p
direction. Because the phase lag, θ0, changes in accordance with
the timing of the excitation created by F1cos(Ωt)qand F2
cos(2Ωt+θ−θ0)qthrough the gradients of the potential, this
delay may be the reason that θ0increases as βdecreases up to
the maximum point.
In the high-temperature case considered in Figure 4c, the
distribution is populated even at the top of barrier. This causes
the spatiotemporal asymmetry induced by biharmonic
perturbation, which is necessary for ratchet current to be
weak, and as a result, the current decreases as the temperature
increases. Because the activation energy increases as the barrier
increases, the maxima of the peaks shift toward higher
temperature as U0increases. The phase lag, θ0, decreases as
the temperature increases because the distribution can flow to
the neighboring potentials in both the positive and negative
directions with only a small delay. This can be understood from
Figure 4c-ii, where it is seen that the current flows quickly
following the movement of the distribution.
4.2. Temperature Effects: the Quantum Case. Next, we
consider the quantum mechanical case under the same physical
conditions as in the classical case considered above. Figure 3a′
displays the maximum value, Jmax, and Figure 3b′displays the
phase lags, θ0, of the current as functions of the inverse
Figure 4. Snapshots of the classical distribution for θ= 0.7πin the case of intermediate barrier height, U0= 2, for three values of the inverse
temperature: (a) low temperature, β= 3; (b) intermediate temperature, β= 2; and (c) high temperature, β= 1. The snapshots correspond to the
following times: t= (i) 0, (ii) 0.4π, (iii) 0.8π, (iv) 1.2π, and (v) 1.6π(1/Ω).
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temperature, β, for three values of the barrier height, U0,
calculated using the quantum mechanical HEOM given in eqs
9−15. It is seen that the profiles of Jmax and θ0differ
significantly from those in the classical case. In the intermediate
and high barrier cases, The Figure also shows that the quantum
mechanical Jmax is significantly larger than the classical one for
the cases of an intermediate and high barrier in the low-
temperature regime (β> 2), while they exhibit similar behavior
in the high-temperature regime (β< 1).
To see a role of quantum effects, we consider the crossover
temperature at which the classical thermal-activated regime
crosses over to the quantum tunneling regime with regard to
chemical reaction rates
83
for the bath spectral density, J(ω),
given in eq 6 with the potential heights U0= 1, 2, and 4. The
evaluated crossover temperatures are at βc= 4.5, 3.2, and 2.2,
respectively. These values of βcare much smaller than those at
which we observed the difference between the quantum and
classical cases. This suggests that our analysis based on the
reaction rate may not apply directly, because here we
considered Jmax under the external driving force while altering
the phase.
A distinctive feature of the quantum mechanical results is
found in the low barrier case (the red curve) in Figure 3a′. One
may expect that the ratchet current in the quantum mechanical
case is larger than that in the classical case, especially at low
temperatures due to tunneling. However, our results in the low
barrier case presented in Figure 3a′reveal that in fact this is not
the case.
Figure 5. Snapshots of the quantum distribution for θ= 0.4πin the case of low barrier height, U0= 1, calculated using the HEOM for three values of
the inverse temperature: (a) low temperature, β= 3; (b) intermediate temperature, β= 2; and (c) high temperature, β= 1. These snapshots
correspond to the following times: t= (i) 0, (ii) 0.4π, (iii) 0.8π, (iv) 1.2π, and (v) 1.6π(1/Ω).
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To elucidate the reason for the behavior depicted in Figure
3a′for the case U0= 1, we consider the Wigner distribution in
the low and intermediate barrier cases for three values of the
temperature in Figures 5 and 6, respectively. In both cases, the
Wigner distribution function is dispersed over the potential,
and it is negative in some regions (the regions inside the white
and black loops) at low temperature (β≥2). The Wigner
function for the quantum system is not definitely positive, and
in cases that the tunneling process is important, the region in
which it takes negative values is larger.
84
This indicates that
delocalization of the distribution in the low barrier case
depicted in Figure 5a arises from tunneling. This can be
understood from Figure 2a, where the ground state in the low
potential case is seen to have a large population under the
barrier. The tunneling contribution induced by biharmonic
perturbation may be suppressed at high temperature, because
the temperature enters the theory through βℏ, and hence the
classical limit, ℏ→0, is equivalent to the high-temperature
limit, β→0. In this case, however, thermal activation causing
flow over the potential barrier is significant. For this reason, the
distribution is dispersed for all temperatures. As shown in the
high-temperature classical case depicted in Figure 4b, the effect
of the ratchet potential is weak when the distribution function
is dispersed. This is the reason that the ratchet current is
smaller in the quantum case than the classical case if the barrier
is low.
In the cases of intermediate and high barriers, depicted in
Figures 3a′,b′, tunneling also plays a significant role. As shown
in Figure 6, the Wigner distribution is not dispersed when the
barrier is high. As a result, the ratchet mechanism is effective
Figure 6. Snapshots of the quantum distribution for θ= 0.6πin the case of intermediate barrier height, U0= 2, for the following values of β: (a) low
temperature, β= 3; (b) intermediate temperature, β= 2; and (c) high temperature, β= 1. These snapshots correspond to the following times: t= (i)
0, (ii) 0.4π, (iii) 0.8π, (iv) 1.2π, and (v) 1.6π(1/Ω).
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with the help of tunneling, and thus the quantum mechanical
current is larger than the classical current. However, when the
temperature becomes very high, the quantum coherence is lost
due to the thermal noise, and as a result, the quantum value
approaches the classical one.
As shown in Figure 3b′, the phase lag has a critical point in
the high-temperature region (β< 1.5) as in the classical case
depicted in Figure 3b. However, in the low-temperature region
(β> 2.5), the phase is seen to increase monotonically in the
quantum mechanical case, while it decreases monotonically in
the classical case. We carefully investigated the dynamics of the
Wigner distribution in the low-temperature region by studying
snapshots of it and the corresponding current using fine time
slices. We thus found that the transfer of the distribution
through tunneling takes a longer time at low temperature. This
is because thermal transitions between the ground and excited
states, which are the cause of tunneling transition induced by
biharmonic perturbation, may be suppressed at low temper-
ature. Because the net current flows slowly in the low-
temperature case, the phase lag that controls the timing of the
biharmonic force increases when the temperature decreases.
For higher barriers, the tunneling plays a minor role, and thus
the increase in the phase is smaller than in the low barrier case.
Note that as discussed in ref 85 quantum diffusion may be
suppressed by quantum reflection. In the cases we studied,
however, the curvature at the top of the barrier and in the
potential well are equal, and the force that creates the difference
between the curvatures is small.
86
Thus, in the present case,
quantum reflection is not the cause of the suppression.
4.3. Varying the Coupling Strength. We next investigate
the change of the ratchet current as the system-bath coupling
strength ζis varied by taking advantage of the fact that the
HEOM constitute a nonperturbative theory. In Figure 7, we
plot the classical (red curve) and quantum (green curve)
ratchet current for (a) the low barrier case, U0= 1, and (b) the
intermediate barrier case, U0= 2, with β= 1. In both cases, the
classical and quantum currents seem to approach the same
value for large ζ, although the quantum values reach the large ζ
limit more rapidly. To compare the two cases, we plot the
quantum and classical distribution functions in Figure 8. It is
seen that in the low barrier case the quantum distribution is
dispersed, as in the case depicted in Figure 5. Moreover, in the
quantum mechanical case, the distribution in the pdirection
becomes a symmetric Gaussian for ζ> 0.3, while there is some
deviation from Gaussian in the classical case, due to the external
perturbation. This Gaussian feature arises from quantum
tunneling, as can be observed more clearly in the low barrier
case. It is interesting that the situation in which there exists a
symmetric Gaussian distribution in momentum space is similar
to that described by the classical Smoluchowski equation, which
can be derived in the overdamped limit from the Kramers
equation. In the overdamped limit, the distribution of the
momentum direction is Gaussian. In the presently considered
case, we find that such a situation arises due to the effect of
tunneling even if the system-bath coupling is weak. Therefore,
it is natural to call this situation a tunneling-induced
Smoluchowski limit.
The phase approaches π/2 as ζincreases, as depicted in the
inset of Figure 7i). This is similar to the situation for the
classical Smoluchowski equation. For U0= 2, the quantum and
classical cases exhibit similar ζdependence. This is due to the
fact that the dispersion of the distribution through tunneling
does not occur in the higher barrier case, as depicted in Figure
6. The quantum values approach the classical values for large ζ
because in that case the quantum coherence is destroyed by the
system-bath coupling.
5. CONCLUDING REMARKS
The classical and quantum mechanical ratchet currents in a
system driven by a biharmonic force were calculated in a
rigorous manner for the first time using the reduced hierarchy
equations of motion (HEOM) in the Wigner representation
over a wide range of values of the temperature, the system-bath
coupling strength, and the potential height. The roles of
fluctuations, dissipation, and the biharmonic force on the
ratchet current were investigated in classical and quantum
mechanical cases by studying the dependence of the ratchet
current on these parameters.
In the classical case, we found that over the temperature
range we studied the current realizes a maximum at an
intermediate value and falls offfor both high and low
temperatures. This decrease on the high temperature side is a
result of thermal activation, which induces dispersion of the
distribution over the barrier in the high-temperature region. In
the quantum mechanical case, that tunneling enhances the
current in the high barrier case, while it suppresses the current
in the low barrier case. This is because the effect of the ratchet
potential is weak in the case of a low barrier due to the
dispersion of the distribution that arises from tunneling.
Figure 7. Maximum ratchet current in the classical (red curve) and
quantum mechanical (green curve) cases for (a) a low barrier, U0=1,
and (b) an intermediate barrier, U0= 2, with β= 1. The insets depict
the phase lag θ0as a function of ζ.
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Moreover, in the quantum mechanical case, there is a classical
Smoluchowski-like regime induced by tunneling effects. We
also found that the phase lag of the biharmonic force is related
to the response time of the system to the external perturbation.
As we demonstrated, the dynamical behavior of the system is
clearly and readily elucidated by the time evolution of the
Wigner distribution function. Although we must introduce
hierarchies of Wigner functions described by a discretized
mesh, a modern personal computer is sufficiently powerful to
solve the equations of motion in a reasonable amount of time.
Numerical techniques, such as the optimization of the
hierarchy,
87−90
the utilization of a graphic processing unit
(GPU),
91
and memory allocation for massive parallel
computing,
92
have been developed for the HEOM approach
to accelerate numerical calculations. As an approximated
method, it may be possible to adapt a multiconfiguration
time-dependent Hartree (MCTDH) approach, which is shown
to be an efficient method to solve Schrödinger equations of
motion.
93−95
With these developments, the extension of the
present study to the variety of multidimensional potential
systems with non-Drude-type spectral distribution functions
may be possible.
■AUTHOR INFORMATION
Corresponding Author
*E-mail: tanimura@kuchem.kyoto-u.ac.jp. Tel:+81-75-753-
4017.
Notes
The authors declare no competing financial interest.
■ACKNOWLEDGMENTS
Y.T. is grateful for stimulating discussions with Professor Peter
Hänggi and the hospitality of him and his group members
during his stay at the University of Augsburg, made possible by
the Humboldt Foundation. A.K. acknowledges a research
fellowship from Kyoto University. We are grateful for useful
comments on the finite difference expressions of the HEOM
with Dr. Atsunori Sakurai. This research is supported by a
Grant-in-Aid for Scientific Research (B2235006) from the
Japan Society for the Promotion of Science.
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Figure 8. Snapshots of the classical (upper panel) and quantum mechanical (lower panel) distribution functions at the fixed time t=2π(1/Ω) for
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