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ournal of Statistical Mechanics:An IOP and SISSA journalJ Theory and Experiment

The quantum rate of escape from ametastable state non-linearlycoupled to a heat bath driven byexternal colored noise*

Pulak Kumar Ghosh1 and Jyotipratim Ray Chaudhuri2

1 Indian Association for the Cultivation of Science, Jadavpur,Kolkata 700 032, India2 Department of Physics, Katwa College, Katwa, Burdwan 713130,West Bengal, IndiaE-mail: gpulakchem@gamil.com and jprc 8@yahoo.com

Received 5 October 2007Accepted 22 January 2008Published 28 February 2008

Online at stacks.iop.org/JSTAT/2008/P02014doi:10.1088/1742-5468/2008/02/P02014

Abstract. We have studied the quantum stochastic dynamics of a system whoseinteraction with a reservoir is non-linear in system coordinates. In addition, thebath particles are driven by external noise with finite correlation. The effects ofthe space dependent friction and correlation of the external noise on the rate ofdecay of a particle from a metastable well have been examined. At a critical valueof the correlation, the variation of the quantum decay rate versus the externalnoise strength exhibits a resonance.

Keywords: driven diffusive systems (theory), stochastic processes

* This paper is dedicated to Professor S P Bhattacharyya on the occasion of his sixtieth birthday.

c©2008 IOP Publishing Ltd and SISSA 1742-5468/08/P02014+21$30.00

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Contents

1. Introduction 2

2. Quantum stochastic dynamics of a noise driven bath in the presence ofnon-linear coupling 3

3. Overdamped limit 7

4. Quantum rate of decay from a metastable state 8

5. Conclusion 18

Acknowledgment 18

Appendix. Quantum correction terms 19

References 21

1. Introduction

The paradigm of Brownian motion from a microscopic point of view is essentially a modelcomprising of a system coupled to its environment considered to be a reservoir [1]–[4].The complete dynamics is described, in principle, by the equations of motion of thesystem plus an infinite number of reservoir elements. By appropriate elimination of thereservoir degrees of freedom one sets up an equation of motion for the system which, ineffect, includes, first, the drive of a random force which originates from the uncertainty ofinitial conditions of coordinates and momenta of the large number of bath variables and,second, a dissipative term. The stochastic force is related to dissipation by the fluctuation-dissipation relation which ensures that the overall system is thermodynamically closed sothat the principle of detailed balance holds good. In the early 1940s Kramers, on the basisof the theory of Brownian motion in phase space, proposed a diffusion model for chemicalreaction. Since then the model and several of its variants have been widely employed inmany areas in physics, chemistry and biology for understanding the nature of activatedprocesses in classical, quantum and semiclassical systems, in general. These have becomethe subject of several reviews and monographs in the recent past [4]–[9].

In the majority of these treatments one is essentially concerned with additive noise ofan equilibrium heat bath at a finite temperature. In this paper, we address the problemof a Langevin equation with multiplicative noise and state dependent diffusion for athermodynamically open system in a quantum mechanical context. The physical situationthat we consider here is the following: at t = 0, the reservoir is in thermal equilibriumin the presence of the system. At t = 0+, an external noise agency is switched on tomodulate the heat bath [10, 11]. This modulation makes the system thermodynamicallyopen [20]. Using a standard method, we then construct the operator Langevin equationfor a general non-linear system–reservoir coupling.

To put the discussions into an appropriate perspective we begin with the followingnote. In a previous work Ray Chaudhuri et al [11] studied the dynamics of a metastablestate non-linearly coupled to a heat bath driven by external noise in the classical limit.

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The authors [11] have derived the generalized Langevin equation with space dependentfriction and multiplicative noise, and the corresponding Fokker–Planck equation, to studythe rate of escape from a metastable state in the moderate to large damping regime.In this paper we focus on the dynamics of the corresponding situation in a quantummechanical context. Although the quantum mechanical system–reservoir linear couplingmodel for microscopic description of additive noise and linear dissipation which are relatedby a fluctuation-dissipation relation for a closed system has become well known overmany decades in several fields, the nature of the non-linear coupling and its consequencesfor closed systems have been explored with renewed interest only recently [13, 14]. Forexample, it has been shown that quantum dissipation can reduce the appearance ofmetastable states and barrier drift in a double-well potential [12]; the study of elasticand inelastic relaxation mechanisms and their interplay in the Raman and infraredspectra [3] and quantum transport as a consequence of state dependent diffusion havebeen investigated [13] in a periodic potential, etc. Furthermore the system may beassociated with both thermal and non-thermal environments. The effect of a non-thermalenvironment is important in different noise-induced phenomena such as noise-inducedquantum transport [15]–[19], stochastic resonance [21]–[23], resonant activation [24]–[27],noise-induced transitions [23, 28], stochastic localization [29, 30], to name but a few.

The aim of the present paper is thus to explore the associated quantum effects inthe rate of decay of a metastable state of a particle in a correlated noise driven bath inthe presence of non-linear coupling. The layout of the paper is as follows. In section 2we have formulated the quantum stochastic dynamics of a noise driven bath with finitecorrelation and non-linear coupling. We have derived the quantum Langevin equation inthe overdamped limit in section 3. In section 4 we have calculated the rate of decay froma metastable state under the influence of inhomogeneous diffusion, non-linear dissipationand an external drive with finite correlation. This paper is concluded in section 5.

2. Quantum stochastic dynamics of a noise driven bath in the presence ofnon-linear coupling

We consider a particle of mass M non-linearly coupled to a medium comprised of a setof harmonic oscillators with frequency ωj. The bath oscillators are additionally drivenby an external colored noise. The total Hamiltonian of such a composite system can bedescribed by the following system–bath Hamiltonian [3, 4, 10, 12]:

H =p2

2M+ V (q) +

∑

j

[p2

j

2+

1

2ω2

j (xj − cjf(q))2

]+

∑

j

kjxjε(t) (2.1)

where q and p are the coordinate and momentum operators of the particle and the {xj , pj}are the set of coordinate and momentum operators for the bath oscillators with unit mass.The system particle is coupled to the bath oscillators non-linearly through the generalcoupling terms cjf(q). cj is the coupling strength. The coordinate and momentumoperators follow the usual commutation relations [q, p] = i� and [xj , pk] = i�δjk. Thepotential V (q) is due to external force field for the system particle. The last term ofthe Hamiltonian (2.1) represents the interaction between the heat bath and the externalclassical noise ε(t). ε(t) is a stationary Gaussian noise with the following characteristics:

〈ε(t)〉 = 0 and 〈ε(t)ε(t′)〉 = 2Qχ(t − t′) (2.2)

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where Q is the strength of the noise and χ(t) is some decaying memory kernel. Theclassical counterpart [31] of (2.1) for linear coupling and in the absence of interactionbetween bath oscillators and external noise has been known for many years. The modelin a quantum system in the absence of interaction between bath oscillators and externalnoise has also been studied on a few occasions [3, 4, 12].

Eliminating the bath degrees of freedom in the usual way we obtain the operatorLangevin equation [4, 13] for the particle

˙q(t) = p(t) (2.3)

˙p(t) = −V ′(q(t)) − f ′(q(t))

∫ t

0

f ′(q(t′))γ(t − t′)p(t′) dt′ + f ′(q(t)){η(t) + π(t)I} (2.4)

where the noise operator η(t) and the memory kernel γ(t) are given by

η(t) =∑

j

[{xj(0) − cjf(q(0))} cjω

2j cos ωjt +

pj(0)

ωjsin ωjt

](2.5)

γ(t) =∑

j

c2jω

2j cos ωjt (2.6)

π(t) = −∫ t

0

dt ′φ(t − t′)ε(t′) (2.7)

φ(t) =∑

j

κjcjωj sin{ωjt}. (2.8)

I is the unit operator.It is clear from the operator Langevin equation (equations (2.3) and (2.4)) for the

system that the noise operator is multiplicative and the dissipative term is non-linearwith respect to system coordinates due to the non-linear coupling term in the system–bath Hamiltonian. In the case of linear coupling, i.e., f(q) = q (equations (2.3) and (2.4)),this reduces to a quantum generalized Langevin equation [4] in which the noise term isadditive and the dissipative term is linear.

In the Markovian limit [4, 6, 7, 9] the generalized quantum Langevin equations (2.3)and (2.4) reduce to the form

˙q(t) = p(t) (2.9a)

˙p(t) = −V ′(q(t)) − γ[f ′(q(t))]2p(t) + f ′(q(t)){η(t) + π(t)I} (2.9b)

where γ is a dissipation constant in the Markovian limit.Following [4] we then carry out a quantum mechanical averaging 〈· · ·〉 over the product

separable bath modes with coherent states and system mode with an arbitrary state att = 0 in equations (2.9a) and (2.9b) to obtain a generalized Langevin equation as

q = p (2.10a)

p = −V ′(q) + QV − γ[f ′(q)]2p + Q1 + f ′(q) [η(t) + π(t)] + Q2 (2.10b)

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with the quantum mechanical mean values of the position and momentum operators 〈q〉 =q and 〈p〉 = p. QV = V ′(q)−〈V ′(q)〉 represents quantum correction due to non-linearity ofthe system potential. Q1 = γ[[f ′(q)]2p−〈[f ′(q)]2p〉] and Q2 = {η(t)+π(t)}[〈f ′(q)〉−f ′(q)]are also quantum correction terms, due to non-linearity of the system–bath couplingfunction.

Furthermore the quantum mechanical mean Langevin force is given by

η(t) =∑

j

cjω2j

[{〈xj(0)〉 − cj〈f(q(0))〉} cos ωjt +

〈pj(0)〉ωj

〈pj(0)〉 sinωjt

]. (2.11)

To realize η(t) as an effective c-number noise we now introduce the ansatz [4, 20, 23, 27, 32]that the momentum 〈pj(0)〉, and the shifted coordinates {〈xj(0)〉 − cj〈f(q(0))〉} of thebath oscillators are distributed according to a Wigner canonical thermal distribution ofGaussian form as

Pj (〈xj(0)〉 − cj〈f(q(0))〉, 〈pj(0)〉) = N exp

{−〈pj(0)〉2 + {〈xj(0)〉 − cj〈f(q(0))〉}2

2�ωj (nj(ωj) + 1/2)

}

(2.12)

so that for any quantum mechanical mean value, Oj({〈xj(0)〉−cj〈f(q(0))〉}, 〈pj(0)〉) whichis a function of mean value of the bath operators 〈xj(0)〉 and 〈pj(0)〉; its statistical average〈· · ·〉S is

〈Oj〉S =

∫OjPj d〈pj(0)〉 d{〈xj(0)〉 − cj〈f(q(0))〉}. (2.13)

Here nj(ωj) indicates the average thermal photon number of the jth oscillator at thetemperature T and nj(ωj) = [exp(�ωj/kBT )− 1]−1, and N is the normalization constant.

The distribution Pj (equation (2.12)) and the definition of statistical average(equation (2.13)) imply that the c-number noise η(t) must satisfy

〈η(t)〉S = 0 (2.14)

〈η(t)η(t′)〉S =1

2

∑

j

c2jω

2j �ωj

(coth

�ωj

2kBT

)cos ωj(t − t′). (2.15)

In the Markovian limit the noise correlation becomes [33, 34]

〈η(t)η(t′)〉S = 2D0δ(t − t′) (2.16a)

D0 = 12γ�ω0

(n(ω0) + 1

2

)(2.16b)

where ω0 is the average bath frequency and the spectral density function is considered inthe Ohmic limit. The use of the Gaussian form (equation (2.12)) for a bath distributionneeds some elaboration. It follows from a simple calculation of the partition function ofnon-linearly coupled harmonic oscillators that a Gaussian form of ansatz is not valid ingeneral. In the present case we emphasize that one is concerned here with a system–reservoir non-linear coupling where the coupling is non-linear in system coordinates butstill linear in bath coordinates, so that the use of a coherent state basis for harmonicoscillators enables us to factor out the dependence of the non-linear system coordinate in

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our calculation. This is also reflected in the fluctuation-dissipation relation (2.15) whichretains its usual form, in spite of the non-linear system–reservoir interaction. At this pointwe emphasize that the overdamped situation in our treatment refers to large γ rather thanγ[f ′(x)]2. For better validity of the theory the non-linearity should not be too strong.

We now define an effective noise, ξ(t) = π(t) + η(t), which is characterized by

〈ξ(t)〉S = 0 (2.17a)

〈ξ(t)ξ(t′)〉S = D0 + 2Q

∫ t

0

dt′′∫ t′

0

dt′′′φ(t − t′′)φ(t′ − t′′′)χ(t′′ − t′′′). (2.17b)

Let us consider the external noise to be a stationary Gaussian Ornstein–Uhlenbeck process,which can be characterized by the following equation:

〈ε(t)ε(t′)〉S =De

τeexp

(|t − t′|

τR

)(2.18)

where De and τe denote the strength and correlation time of the external noise respectively.The effective noise ξ(t) is also Gaussian with the following form of two-time correlation:

〈ξ(t)ξ(t′)〉S =DR

τRexp

(|t − t′|

τR

)(2.19)

where

DR =γ

2

[�ω0

(n(ω0) +

1

2

)+ 2Deκ

20

]and τR =

γDeκ20

DR

τe. (2.20)

k0 is a measure of the extent of the coupling between the external noise and bath particle.Now, simplifying the quantum correction terms in the equations (2.10a) and (2.10b), theLangevin equation can be rewritten as follows:

q = p (2.21)

p = −R′(q) + h(q)p + g(q)ξ(t) (2.22)

where

R′(q) = V ′(q) − QV + 2γf ′(q)Q4 + γQ5 (2.23)

h(q) = [f ′(q)]2 + 2f ′(q)Qf + Q3 (2.24)

and

g(q) = f ′(q) + Qf . (2.25)

The above equation is characterized by a force term, R′, and non-linear dissipation andmultiplicative noise. The force term, R′(q), in equation (2.23) contains a classical forceterm, V ′(q), as well as its correction QV due to non-linearity of the system potential.The corrections for non-linear coupling function also contribute. The terms containingh(q) are non-linear dissipative terms where Qf , Q3, Q4 and Q5 are due to associatedquantum contributions in addition to the classical non-linear dissipative term γ[f ′(q)]2p.The explicit expressions for QV , Qf , Q3, Q4 and Q5 are given in the appendix. The lastterm in the above equation (2.22) contains an effective multiplicative noise term. For in

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the absence of external noise, the classical limit of the above equation was derived earlierby Lindenberg and Seshadri [31]. Furthermore quantum dispersions due to potential andcoupling terms in the Hamiltonian are entangled with non-linearity. The quantum noisedue to the heat bath on the other hand is expressed in terms of the fluctuation-dissipationrelation (in the absence of an external driving force).

Using van Kampen’s method, the probabilistic description corresponding equa-tions (2.21) and (2.22) is given by a Fokker–Planck equation of the following form:

∂P (q, v, t)

∂t= − ∂

∂q(vP ) +

∂

∂v{γh(q)v + V ′(q) − QV − 2g(q)g′(q)τRDR}P (q, v, , t)

+[[g(q)]2DR − γh(q)g2(q)τRDR − vg(q)g′(q)τRDR

] ∂2P (q, v, t)

∂v2

+ g2(q)τRDR∂2P (q, v, t)

∂v∂q(2.26)

where we have neglected the transient correction terms Q4 and Q5 which is indeed a validassumption on the timescale 1/γ [13]. This equation works for small correlation time,i.e., when τc � 1/γ. For small correlation time, we neglect the small non-Markoviancontributions g2(q)τRDR(∂2P/∂v∂q) and also the term vg(q)g′(q)τRDR in comparison toother terms to get the approximate evolution equation for the probability density P (q, v, t)as

∂P (q, v, t)

∂t= − ∂

∂q(vP ) +

∂

∂v{γh(q)v + V ′(q) − QV − 2g(q)g′(q)τRDR}P (q, v, , t)

+[[g(q)]2DR − γh(q)g2(q)τRDR

] ∂2P (q, v, t)

∂v2. (2.27)

The above equation is equivalent to the multiplicative Langevin equation with spacedependent dissipation:

q = p (2.28)

p = −V ′(q) + QV − h(q)p + G(q)ζ(t) (2.29)

where G(q) = g(q)√

1 − γg2(q)τR and

〈ζ(t)〉s = 0 and 〈ζ(t)ζ(t′)〉s = 2DRδ(t − t′). (2.30)

3. Overdamped limit

In general, when the fluctuation is position/state dependent or equivalently when the noiseis multiplicative with respect to system variables the conventional adiabatic reduction offast variables does not work correctly. To obtain a correct equilibrium distribution Sanchoet al [7] had proposed an alternative approach to the Langevin equation in the case ofa multiplicative noise system. By carrying out a systematic expansion of the relevantvariables in powers of γ−1 and neglecting terms smaller than O(γ−1) they obtained aLangevin equation corresponding to a Fokker–Planck equation in position space. Thisdescription leads to the correct stationary probability distribution of the system withcoordinate dependent friction.

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Following the method of Sancho et al [7], we obtain the Fokker–Planck equation inposition space corresponding to the Langevin equation (equation (2.28))

∂P (q, t)

∂t=

∂

∂q

[1

γh(q)

{V ′(q) − QV +

DR

γ

∂

∂q

G2(q)

h(q)

}P (q, t)

]. (3.1)

The Stratonovich prescription leads to the corresponding Langevin equation as given by

q = −V ′(q) − QV

γh(q)− DR

G′(q)G(q)

γ(h(q))2+

G(q)

γh(q)ζ(t). (3.2)

Equation (3.2) is a quantum Langevin equation for multiplicative noise with positiondependent friction in the overdamped limit (i.e., corrected up to O(1/γ)).

4. Quantum rate of decay from a metastable state

We carry out a quantum mechanical study of the barrier crossing dynamics of anoverdamped particle in a noise driven bath in the presence of non-linear system–bathcoupling. The particle coordinate q (which in our case is the quantum mechanical meanposition) corresponds to the reaction coordinate and its values at the minima of V (q)denote the reactant and product states separated by a finite barrier, the top being ametastable state representing the transition state. We choose the potential of the formV (q) = (1/2)aq2 − (1/3)bq3. a and b are the potential parameters and the maximum andthe minimum of the potential are at the points q = qb = a/b and q = qa = 0, respectively.

In our present study we have assumed f(q) = b1q + 12b2q

2. So from equation (2.1)the linear–linear coupling term (

∑cjb1q) is proportional to b1 while the square–linear

coupling term (∑

cjb2q2) is proportional to b2. The linear–linear coupling model is very

commonly used for studying quantum dissipative dynamics, e.g., energy dissipation froma vibrational system mode to the heat bath modes during population decay. In additionto energy dissipation due to linear–linear coupling, one also encounters fluctuation in thesystem potential due to non-linear coupling. Since this fluctuation is the key mechanismfor state dependent diffusion, it is apparent that the calculation of the transition ratein the presence of state dependent diffusion differs from the calculation of the escaperate for particles subject to thermally uniform noise, in that we may conveniently replacethe potential V (q) by a generalized potential Ψ(q) corresponding to the former case. Asimple way to determine the transition rate is to consider a steady state condition. TheFokker–Planck equation (equation (3.1)) in the overdamped limit can be rewritten in amore compact form as

∂P (q, t)

∂t=

∂

∂q

1

γh(q)

[V ′(q) − QV +

DR

γ

∂

∂q

G(q)2

h(q)

]P (q, t). (4.1)

In the overdamped limit the stationary current from equation (4.1) is given by

J = − 1

γh(q)

[V ′(q) − QV +

DR

γ

d

dq

(G(q)2

h(q)

)]Pst(q) (4.2)

which can be rearranged into the form

d

dq

{Pst(q)L(q) exp

[Ψ(q)

Dq

]}= −Jγh(q)

Dqexp

[Ψ(q)

Dq

]. (4.3)

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Figure 1. A schematic illustration of the energy profile diagram for the decayfrom a metastable state.

In the above expression

Ψ(q) =

∫ q V ′(q′) − QV

L(q′)dq′; L(q) =

G(q)2

h(q). (4.4)

To find a steady state solution of equation (4.3) we assume an absorbing boundary ata coordinate qs past the intervening potential maximum at q = qb (as shown in figure 1).So integrating equation (4.3) between q = qa and q = qs, we have

J =Pst(qa)L(qa) exp [Ψ(qa)/Dq]∫ qs

qa(γ/Dq)h(q) exp [Ψ(q)/Dq] dq

(4.5)

where Dq = DR/γ.The steady escape rate over the top of the potential is given by flux over population:

k =J

na(4.6)

where na is the population around the potential minimum (qa). The zero-current conditiondefines an equilibrium condition in the neighborhood of qa. Thus with J = 0, theprobability distribution function near the potential minimum is given by

Pst(q) = Pst(qa)L(qa)

L(q)exp

[−Ψ(q)

Dq+

Ψ(qa)

Dq

]. (4.7)

The population around the bottom of the potential is given by

na =

∫ q2

q1

Pst(q) dq = Pst(qa)L(qa) exp

[Ψ(qa)

Dq

] ∫ q2

q1

1

L(q)exp

[−Ψ(q)

Dq

](4.8)

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where q1 and q2 are two points around the potential minimum. So the expression for thetransition rate is given by

k =Dq

γI1I2(4.9)

where the integrals I1 and I2 are given by

I1 =

∫ qa

qs

dq h(q) exp

[Ψ(q)

Dq

]I2 =

∫ q2

q1

dq1

L(q)exp

[−Ψ(q)

Dq

]. (4.10)

By linearizing the generalized potential around the maximum and minimum andevaluating the integrals in the usual way, we obtain

k =|Ψ′′(qa)|1/2|Ψ′′(qb)|1/2L(qa)

2πγh(qb)exp

[−

{Ψ(qb) − Ψ(qa)

Dq

}]. (4.11)

Ψ(q) can be simplified further to yield

Ψ(q) =

∫ q V ′(q′) − QV

L(q′)dq′ =

∫ q

dq′ (V ′(q′) − QV )

(g2(q) +

(Q3 − Q2

f

)

g2(q) (1 − γg2(q)τR)

). (4.12)

Here it is important to note that the term L(q) in the denominator of the generalizedpotential contains three items of information about (i) the signature of the non-linearsystem–bath coupling, (ii) the specific quantum correction due to non-linear system–bathcoupling and (iii) the correlation function of the externally driven noise.

Using the above expression for the generalized potential we calculate Ψ′′(qa) andΨ′′(qb) and put them into the expression for the rate constant. The simplified rate constantis given by

k =

(L(qa)

h(qb)

) √(ω2

aΔ1 − Δv1)(ω2bΔ2 − Δv2)

2πγexp

[−

{Ψ(qb) − Ψ(qa)

D0

}](4.13)

where ωa = V ′′(qa) and ωb = V ′′(qb) are the frequencies at the bottom and barrier top,respectively. This is the key result of our paper. Δ1, Δ2, Δv1 and Δv2 are the contributionsdue to quantum correction and are given by

Δ1 =

[g2(q) + Q3 − Q2

f

g2(q)(1 − γg2(q)τR)

]

q=qa

; Δ2 =

[g2(q) + Q3 − Q2

f

g2(q)(1 − γg2(q)τR)

]

q=qb

(4.14)

Δv1 =

[d

dq

{QV

[g2(q) + Q3 − Q2

f

g2(q)(1 − γg2(q)τR)

]}]

q=qa

Δv2 =

[d

dq

{QV

[g2(q) + Q3 − Q2

f

g2(q)(1 − γg2(q)τR)

]}]

q=qb

.

(4.15)

The quantum nature of the transition rate (4.13) is manifested through the quantumcorrection due to non-linearity of the potential and coupling function, and theinhomogeneous quantum diffusion. This rate also contains the signature of non-linearcoupling in the pre-exponential factor and in the generalized potential Ψ(q). Moregenerally, the effect of state dependent diffusion makes its presence felt in the dynamicsof decay.

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In order to check the inherent consistency of the rate expression we now examine thefollowing situations.

(a) First consider that there is no interaction between the bath and the external noise(i.e., De = 0 and the corresponding τe = 0) ε(t). Then the expressions for the quantumcorrection are given by

Δ1 = 1 + Δ′1; Δ2 = 1 + Δ′

2;

where Δ′1 =

[Q3 − Q2

f

(f ′(q) + Qf )2

]

q=qa

; Δ′2 =

[Q3 − Q2

f

(f ′(q) + Qf )2

]

q=qb

;

Δv1 =

[d

dq

{QV

(1 +

Q3 − Q2f

(f ′(q) + Qf)2

)}]

q=qa

;

Δv2 =

[d

dq

{QV

(1 +

Q3 − Q2f

(f ′(q) + Qf)2

)}]

q=qb

;

L(qa) =g2(qa)

h(qa).

So the rate expression can be rewritten as (4.13)

k =

√[ω2

a(1 + Δ′1) − Δv1 ][ω

2b (1 + Δ′

2) − Δv2 ]

2πγh(qb)(1 + Δ′1)

exp

[−

{Ψ(qb) − Ψ(qa)

Dq

}]. (4.16)

The above rate expression is exactly the same as that derived in an earlierpaper [14].

(b) Let us consider that the coupling function is linear, i.e., b2 = 0 and f(q) = q (b1

is assumed to be unity). Then the generalized potential is given by

Ψ(q) =

∫ q

dq(V ′(q′) − QV )

1 − γτR

. (4.17)

In this case, as Qf and Q3 vanish, g(q) = h(q) = 1 and the quantum correction termsdue to non-linear coupling also vanish. In the absence of external noise the transition ratereduces to the following form:

k =

√[ω2

a − Δv1 ][ω2b − Δv2 ]

2πγexp

[−

{Δφ

Dq

}]exp

[−

{V (qb) − V (qa)

Dq

}](4.18)

where

Δφ =

∫ qb

qa

QV (q)

Dqdq. (4.19)

This quantifies the characteristic quantum rate of decay from a metastable state forparticles subject to thermally uniform noise. On the other hand, as the classical limit isapproached all the quantum correction terms (Δv1 , Δv2 and Δφ) due to the non-linearityof the potential tend to vanish and the quantum coefficient (Dq) gets replaced by theclassical one (Dq → kbT ). The transition rate (4.18) reduces exactly to Kramers’ rate inthe overdamped limit:

k =ωaωb

2πγexp

[−

{V (qb) − V (qa)

kbT

}]. (4.20)

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(c) Let us consider the system potential to be a symmetric function, V (q) = V (q+L),with periodicity L and the derivative of the coupling function as also periodic withthe same periodicity as the potential, i.e., f ′(q) = f ′(q + 2π). Such a potential hadbeen a subject of interest in the problem of classical [15] and quantum transport [13] ininhomogeneous media. Furthermore we consider that there is no interacting fluctuationor noise source acting from outside. In the absence of external stochastic modulation(ε(t) = 0) it is easy to check that the rate of transition from valley n to valley n + 1 isexactly equal to the rate of transition from valley n+1 to valley n. As a consequence, theaverage particle velocity in a preferential direction is zero. We thus conclude that thereis no occurrence of current for a periodic potential and periodic derivative of couplingwith the same periodicity. At the macroscopic level this confirms that in the absence ofinteraction with external noise there is no generation of perpetual motion of the secondkind, i.e., no violation of the second law of thermodynamics. Therefore the thermodynamicconsistency based on symmetry considerations ensures the validity of the present scheme.

Before we proceed to numerical results it is important to clarify the issues regardingthe calculation of quantum correction terms. The details of the calculations of quantumcorrection terms are shown in the appendix. One can calculate the value of quantumdispersion terms 〈δqn〉 by direct numerical simulation of the coupled equation (A.10)subject to appropriate boundary conditions. It is also instructive to deal with thequantum correction terms in the analytical way to find the approximate values of quantumdispersion terms. For the overdamped limit we neglect the δ ˙p term from equation (A.9)to obtain

d

dtδq =

1

γ[f ′(q)]2[−V ′′(q)δq − 2γpf ′(q)f ′′(q)δq + (η(t) + π(t)) f ′′(q)δq] + O(δq2).

(4.21)

With the help of equation (4.21) we then obtain the equations for 〈δqn〉:

d

dt〈δq2〉 =

2

γ[f ′(q)]2[−V ′′(q)〈δq2〉 − 2γpf ′(q)f ′′(q)〈δq2〉 + (η(t) + π(t)) f ′′(q)〈δq2〉

]

+ O(〈δq3〉) (4.22)

d

dt〈δq3〉 =

3

γ[f ′(q)]3[−V ′′(q)〈δq3〉 − 2γpf ′(q)f ′′(q)〈δq3〉 + (η(t) + π(t)) f ′′(q)〈δq3〉

]

+ O(〈δq4〉) (4.23)

and so on. (It is apparent from equations (4.22) and (4.23) that in the overdamped limitthe higher order quantum contributions are small since each successive order of correctionis lower than the preceding one by a factor of 1/γ.)

A simplified expression for the leading order quantum correction term 〈δq2〉 canbe estimated by neglecting the higher order coupling terms in the square bracket inequation (4.22) (since the non-linearity of the potential is small, the terms of the order off ′′(x) are small) and taking the average over noise realizations (both internal and external)we can rewrite equation (4.22) as follows:

d

dt〈δq2〉 =

2

γ[f ′(q)]2[−V ′′(q)〈δq2〉 − 2γpf ′(q)f ′′(q)〈δq2〉

]. (4.24)

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The overdamped deterministic mean motion on the other hand gives

q =1

γ[f ′(q)]2

[−V ′(q) − γkBT

f ′′(q)

f ′(q)

](4.25)

(the deterministic part of the Stratonovich prescription (3.2) in the classical limit), sothat we have

dt =γ[f ′(q)]2 dq

[−V ′(q) − γkBT (f ′′(q)/f ′(q))](4.26)

and putting the values of q and dt into the equation (4.24) and then solving the differentialequation we obtain a simplified expression for the leading order quantum dispersion term:

〈δq2〉 =〈δq2〉q=qc[f

′(qc)]4

[f ′(q)]4exp

[2

∫ q

qc

V ′′(q′)

V ′(q′) + γkBT f ′′(q′)f ′(q′)

]. (4.27)

qc is the reference point at which the value of the quantum dispersion term is known. Letus consider the point qc as that at which the quantum correction is minimum; that means〈δq2〉q=qc = �/ω0 and 〈δq2〉/dq = 0. Using this condition one can easily find the value ofqc. The relevant quantum correction terms (A.1), (A.4) and (A.5) can be rewritten forthe metastable potential and square–linear coupling:

QV = −12V ′′′(q)〈δq2〉, (4.28)

Qf = 0, (4.29)

Q3 = [f ′′(q)]2〈δq2〉. (4.30)

(V ′′′′(q) and f ′′′(q) vanish for the chosen metastable potential and square–linear coupling).So by calculating the quantum dispersion term 〈δq2〉 we can find the values ofΔ1, Δ2, Δv1 , Δv2 and Δφ as given by equations (4.15), (4.16) and (4.19).

We now illustrate numerically the behavior of the quantum transition rate given byequation (4.13). The expression reveals that in addition to the barrier height the pre-factoris also affected by the quantum correction terms. The calculated quantum transition rate,equation (4.13), implies that both the barrier height and the frequency factor contain theeffects of quantum corrections due to non-linearity of the system potential and couplingfunction. In figure 2 we present a comparison of the quantum decay rate with theclassical one by plotting k as function of 1/T for two different values of the strengthof the external noise. In the inset of the same figure we also present an Arrhenius plotfor the same parameter set. The effect of quantization of the reservoir and quantumcorrection due to the system non-linearity is apparent in figure 2 in the variation ofln k with 1/T (Arrhenius plot) for varied contributions of the interaction with externalnoise. One observes that in the low temperature regime the classical transition rate issignificantly lower in magnitude than the quantum rate and at higher temperature theeffect of quantization becomes insignificant. From the inset plot it is clear that at thehigh temperature regime the plot exhibits linearity in both cases, which is the standardArrhenius classical result. In the low temperature limit, however, one observes a muchslower variation in the quantum case than the classical one. The quantum behavior canbe interpreted in terms of an interplay between the quantum diffusion coefficient Dq and

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Figure 2. Plot of the transition rate comparing quantum (the profiles with thedotted line and dashed line) and classical (the profiles with the solid line anddashed–dotted line) limits at two different external noise strengths for a fixed setof other parameters a = 0.5, b = 0.15, b1 = 1.0, τe = 0.1, k0 = 1.0 and γ = 1. Theinset profiles present the variation of ln k with 1/T (Arrhenius plot) for the sameparameter set as for the main figure.

the quantum correction due to non-linearity of the system potential appearing in Ψ(q).When the temperature of the system is very low, i.e., in the vacuum limit or in the deeptunneling region, the anharmonic terms in the potential do not contribute significantly.On the other hand as the temperature of the system increases significantly, Dq increases,resulting in decrease of the effective potential, and hence Dq and Q (quantum correctionterms) compete to cancel the effects of each other at higher temperature.

We have examined the effect of non-linear coupling on the quantum decay rate byplotting k as a function b2 (measuring the strength of the non-linear coupling) in figure 3and compared with the corresponding classical result. We observe that with increase ofthe strength of the non-linear coupling, the quantum and the classical rates decrease. Thistype of behavior can be physically explained as follows: with increasing non-linearity inthe coupling function the potential well of the system becomes steeper keeping the barrierheight same. As a consequence the transition rate decreases with the increase in strengthof the non-linearity. Figure 3 also reveals that the effect of non-linearity on the quantumtransition rate is more pronounced than that at high temperature. At the lower value ofthe non-linearity the quantum transition rate is higher than the classical rate, but for largevalues of non-linearity in the coupling function, quantum and classical transition ratestend to equalize. This happens due to the fact that for higher non-linearity the quantumrate decreases more rapidly with the additional contribution of quantum corrections fromnon-linear coupling. In figure 4 we compare the quantum transition rate with classicalone by plotting the transition rate as a function of the strength of the external noise.As expected, the transition rates (both classical and quantum) monotonically increasewith increase of the strength of the external noise. With increase of De, the transitionrate first increases rapidly, then slowly, due to randomization of the system and hence an

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Figure 3. Variation of k as a function of b2, comparing quantum (the profileswith the dotted line and dashed line) and classical (the profiles with the solidline and dashed–dotted line) limits at two different temperatures for a fixed setof other parameters a = 0.5, b = 0.15, b1 = 1.0, τe = 0.1, k0 = 1.0, De = 0.25, andγ = 1.

Figure 4. k versus De plot comparing quantum (the profiles with the dottedline and dashed line) and classical (the profiles with the solid line and dashed–dotted line) limits at two different temperature for a fixed set of other parametersa = 0.5, b = 0.15, b1 = 1.0, b2 = 0.01, τe = 0.1, k0 = 1.0 and γ = 1.

increased contribution from back-reaction. Another interesting observation is that thereexists a critical value of the correlation time (τe) of the external noise at which one findsa minimum in the transition rate–external noise strength plot. Above the critical valueof τe, the transition rare monotonically decreases with De, while below it, the transition

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0.1 0.2 0. .4

0.0050

0.0055

0.0060

3 0

k

De

τe=0.825

τe=0.850

τe=0.875

τe=0.900

Figure 5. k versus De for different values of the correlation time of the externalnoise for a fixed set of other parameters a = 0.5, b = 0.15, b1 = 1.0, b2 = 0.01,T = 0.1, k0 = 1.0 and γ = 1.

rate monotonically increases with De. This is depicted in figure 5. For a closer look wepresent a data set in table 1 for the variation of the transition rate with the strength of theexternal noise at four different values of τe. Therefore the results presented in figure 5 andtable 1 emphasize that at the critical value of the correlation time the plot displaying thequantum decay rate versus external noise strength exhibits a resonance like behavior. Itshould be noted that this is not strictly a resonance in the sense of an increased responsewhen a driving frequency is tuned to the natural frequency of the system. There is,however, a useful analogy to resonance in that sense. The transition rate is maximizedwhen some parameter is turned on near a certain value. We emphasize it as a parametricresonance. The possible reason for such critical variation of the transition rate with De isas follows. The equations (4.11) and (4.12) reveal that the exponential factor of the ratecan be expressed in a simplified form as

k ∼ exp

[− ΔVf

Dqcn(1 − γcnτR)

](4.31)

where ΔVf and cn refer to the effective barrier height and contribution due to non-linearcoupling respectively. So it is expected that with the increase of Dq the transition rateincreases, while it decreases with increasing value of τR. Now from expressions (2.20) forDq and τR it follows that with increasing De both Dq and τR increase. At the very low

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Figure 6. k versus τe plot comparing quantum (the profiles with the dotted lineand dashed line) and classical (the profiles with the solid line and dashed–dottedline) limits at two different temperatures for a fixed set of other parametersa = 0.5, b = 0.15, b1 = 1.0, b2 = 0.01,De = 0.1, k0 = 1.0 and γ = 1.

Table 1. This table presents values of the rate of transition (k) with variationof the noise strength, at four distinct values (around the critical point) of thecorrelation time of the external noise. The other parameters for this data tableare a = 0.5, b = 0.15, b1 = 1.0, b2 = 0.01, T = 0.1, k0 = 1.0 and γ = 1.

τc De k τc De k τc De k τc De k

0.825 0.01 0.005 37 0.850 0.01 0.005 38 0.875 0.01 0.005 36 0.90 0.01 0.005 330.05 0.005 41 0.05 0.005 33 0.05 0.005 25 0.05 0.005 150.1 0.005 47 0.1 0.005 31 0.1 0.005 16 0.1 0.005 010.15 0.005 55 0.15 0.005 34 0.15 0.005 12 0.15 0.004 900.2 0.005 67 0.2 0.005 38 0.2 0.005 09 0.2 0.004 810.25 0.005 82 0.25 0.005 44 0.25 0.005 10 0.25 0.004 750.3 0.005 95 0.3 0.005 52 0.3 0.005 11 0.3 0.004 720.35 0.006 11 0.35 0.005 60 0.35 0.005 15 0.35 0.004 700.38 0.006 19 0.38 0.005 68 0.38 0.005 17 0.38 0.004 690.4 0.006 28 0.4 0.005 72 0.4 0.005 20 0.4 0.004 68

values of τe, the value of τR slowly increases and the increase is much slower than that ofDq. As a result, for a low value of τe the transition rate exponentially increases with thestrength of the external noise. But after some value of τe, τR increases more rapidly thanDq; as a result, in this parameter regime, the transition rate decreases with De and thenincreases after passing through a minimum.

Finally in figure 6 we present the variation of the transition rate with the correlationtime of the external noise, comparing classical and quantum limits. In both quantum and

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classical cases the transition rate decays monotonically with increase of the correlationtime. This result is also consistent with equation (4.31). It can be physically explainedas follows; the rate expression (4.13) can be considered as giving a rate of transitionfrom the metastable state when the system particle is in contact with an effective non-Markovian bath. The strength and correlation of the non-Markovian noise are givenby the equations (2.19) and (2.20). The effectiveness of the random force depends onits non-Markovian nature. With increasing value of the noise correlation time (τR) thenon-Markovian character of the system increases, and the effectiveness of the randomnature of the dynamics is lowered. As a result, the rate of noise-induced hopping fromthe metastable state decreases with increasing value of the noise correlation τe (τR isproportional to τe).

5. Conclusion

In this paper we have studied the quantum stochastic dynamics of noise driven bath in thepresence of non-linear system–bath coupling. On the basis of a standard system–reservoirmodel we have derived the appropriate quantum Langevin equation and associatedFokker–Planck equation, both in weak and strong friction limits, in the presence of finitecorrelation of the external driving force. We have also calculated the rate of decayof a metastable state of the system under the influence of quantum state dependentdiffusion, non-linear dissipation and external driving. Our conclusions are summarizedas follows.

(i) The non-linear interaction between the system and the bath has its imprint inthe expression for the decay rate. The presence of non-linearity in the system–bathcoupling reduces the quantum decay rate. This is due to the fact that non-linear couplingcauses a fluctuation of the potential well and makes it steeper. Furthermore the effectof non-linearity on the quantum rate is more pronounced in comparison to that on theclassical one, and hence non-linearity of the effective potential can be identified as a typicalconsequence of a quantum effect.

(ii) Although, in general, the transition rate monotonically increases with increase ofthe external noise strength, the presence of finite correlation can cause a drastic changein this behavior in certain parameter regimes. We have identified a critical value of thecorrelation time of the external noise at which one finds a minimum in the transitionrate–external noise strength plot. In the upper regime for this critical value of the noisecorrelation, the transition rate monotonically decreases with external noise strength, andbelow the critical value, the nature of the transition rate–external noise strength plot isreversed.

We hope that our observations will be useful for experimental studies on tailoredquantum systems like quantum dots at very low temperature and in spectroscopic studies.

Acknowledgment

The authors wish to thank Professor Deb Shankar Ray for critical comments andsuggestions. JRC is indebted to UGC for financial support. PKG acknowledges supportfrom CSIR, Government of India.

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Appendix. Quantum correction terms

Referring to the quantum nature of the system in the Heisenberg picture we now writethe system operators q and p as q = q + δq and p = p + δp respectively. δq and δprepresent quantum fluctuations around their respective mean values. By construction,〈δq〉 = 〈δp〉 = 0, and they also follow the usual commutation relation [δq, δp] = i�. UsingTaylor series expansions in δq around q and in δp around p, we express QV , Q1, Q2 andQ3, Q4, Q5, and Qf as functions of q, p, 〈δqn〉 and 〈δpn〉 as follows;

QV = V ′(q) − 〈V ′(q)〉

= −∑

n≥2

1

n!V n+1(q)〈δqn〉 (A.1)

Q1 = γ[[f ′(q)]2p − 〈[f ′(q)]2p〉

]

= −γ[2pf ′(q)Qf + pQ3 + 2f ′(q)Q4 + Q5] (A.2)

Q2 = η(t) [〈f ′(q)〉 − f ′(q)]

= η(t)Qf (A.3)

where

Qf =∑

n≥2

1

n!fn+1(q)〈δqn〉 (A.4)

Q3 =∑

m≥1

∑

n≥1

1

m!

1

n!fm+1(q)fn+1(q)〈δqmδqn〉 (A.5)

Q4 =∑

n≥1

1

n!fn+1(q)〈δqnδp〉 (A.6)

Q5 =∑

m≥1

∑

n≥1

1

m!

1

n!fm+1(q)fn+1(q)〈δqmδqnδp〉. (A.7)

The dynamics of these correction terms can be calculated [13] with the help of the followingequations, which can be derived using the operator Langevin equations (2.9a) and (2.9b)and by carrying out quantum mechanical averaging over the initial product separablecoherent bath states:

δq = δp (A.8)

δp = −V ′′(q)δq −∑

n≥2

1

n!V n+1(q) [δqn − 〈δqn〉]

− γ

[2f ′(q)f ′′(q)δq + 2f ′(q)

∑

n≥2

1

n!fn+1(q) [δqn − 〈δqn〉]

+∑

m≥1

∑

n≥1

1

m!

1

n!fm+1(q)fn+1(q) [δqmδqn − 〈δqmδqn〉]

]p

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− γ

[[f ′(q)]2δp + 2f ′(q)

∑

n≥1

1

n!fn+1(q) [δqnδp − 〈δqnδp〉]

+∑

m≥1

∑

n≥1

1

m!

1

n!fm+1(q)fn+1(q) [δqmδqnδp − 〈δqmδqnδp〉]

]

+ η(t)

[f ′′(q)δq +

∑

n≥2

1

n!fn+1(q)[δqn − 〈δqn〉]

]. (A.9)

The operator correction equations can be used to yield an infinite hierarchy of equations.Up to third order we construct, for example, the following set of equations which arecoupled to quantum Langevin equations from (2.14) and (2.15):

d

dt〈δq2〉 = 〈δqδp + δpδq〉 (A.10a)

d

dt〈δqδp + δpδq〉 = −2χ(q, p)〈δq2〉 + 2〈δq2〉 − γ[f ′(q)]2〈δqδp + δpδq〉

− ζ(q, p)〈δq3〉 − 2γf ′(q)f ′′(q)〈δq2δp + δpδq2〉 (A.10b)

d

dt〈δp2〉 = −2γ[f ′(q)]2〈δp2〉 − χ(q, p)〈δqδp + δpδq〉

− 12ζ(q, p)〈δq2δp + δpδq2〉 − 2γf ′(q)f ′′(q)〈δqδp2 + δp2δq〉 (A.10c)

d

dt〈δq3〉 =

3

2〈δq2δp + δpδq2〉 (A.10d)

d

dt〈δp3〉 = −3γ[f ′(q)]2〈δp3〉 − 3

2χ(q, p)〈δqδp2 + δp2δq〉 (A.10e)

d

dt〈δq2δp + δpδq2〉 = −2χ(q, p)〈δq3〉 + 2〈δqδp2 + δp2δq〉 − γ[f ′(q)]2〈δq2δp + δpδq2〉

(A.10f )

d

dt〈δqδp2 + δp2δq〉 = 2〈δp3〉 − 4χ(q, p)〈δq2δp + δpδq2〉 − 2γ[f ′(q)]2〈δqδp2 + δp2δq〉

(A.10g)

where

χ(q, p) = V ′′(q) + 2γpf ′(q)f ′′(q) − η(t)f ′′(q) (A.10h)

ζ(q, p) = V ′′′(q) + 2γpf ′(q)f ′′′(q) + 2γp[f ′′(q)]2 − η(t)f ′′′(q). (A.10i)

For other details we refer the reader to [13].

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