Solution of quantum Langevin equation

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May 15, 2004 - Langevin equation has been recently developed Phys. Rev. E 65 ... for quantum analogue of Kramers' equation for a nonlinear potential which ...
JOURNAL OF CHEMICAL PHYSICS

VOLUME 120, NUMBER 19

15 MAY 2004

Solution of quantum Langevin equation: Approximations, theoretical and numerical aspects Dhruba Banerjee Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India

Bidhan Chandra Bag Department of Chemistry, Visva Bharati, Shantiniketan 731 235, India

Suman Kumar Banika) and Deb Shankar Rayb) Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700 032, India

共Received 30 September 2003; accepted 27 February 2004兲 Based on a coherent state representation of noise operator and an ensemble averaging procedure using Wigner canonical thermal distribution for harmonic oscillators, a generalized quantum Langevin equation has been recently developed 关Phys. Rev. E 65, 021109 共2002兲; 66, 051106 共2002兲兴 to derive the equations of motion for probability distribution functions in c-number phase-space. We extend the treatment to explore several systematic approximation schemes for the solutions of the Langevin equation for nonlinear potentials for a wide range of noise correlation, strength and temperature down to the vacuum limit. The method is exemplified by an analytic application to harmonic oscillator for arbitrary memory kernel and with the help of a numerical calculation of barrier crossing, in a cubic potential to demonstrate the quantum Kramers’ turnover and the quantum Arrhenius plot. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1711593兴

I. INTRODUCTION

negative in the full quantum domain. Second, although long correlation times and large coupling constants are treated nonperturbatively formally in an exact manner by path integrals, their analytic evaluation beyond harmonic oscillator is generally completed within semiclassical schemes.1,2,5,6,12 This results in situations where the theory fails to retain its validity in the vacuum limit. Third, although the numerical techniques based on the path integral Monte Carlo are very successful in the treatment of equilibrium properties it is often difficult to implement in a dynamical scheme because of the well-known sign problem associated with varying phases in a quantum evolution over paths due to the oscillating nature of the real time propagator.7,8,11 Fourth, the treatment of non-Markovian dynamics within a quantum formulation as retardation effect in a memory functional is very difficult, if not impossible, to handle even numerically.11 Keeping in view of these difficulties it is worthwhile to ask how to extend classical theory of Brownian motion to quantum domain for arbitrary friction and temperature. Based on a coherent state representation of the noise operator and an ensemble averaging procedure using Wigner canonical thermal distribution for harmonic oscillator we have recently15–19 developed a scheme for quantum Brownian motion in terms of a c-number generalized Langevin equation. The approach allows us to use classical methods of nonMarkovian dynamics to derive the generalized quantum Kramers’ equation and the other relevant quantum analogs of classical diffusion, Fokker–Planck and Smoluchowski equations and of the classical expression for reactive flux.20 The object of the present analysis is to explore several approximation schemes for solving c-number generalized quantum Langevin equation coupled to quantum dispersion equations

A system coupled to its environment is the standard paradigm for quantum theory of Brownian motion.1–13 Its overwhelming success in the treatment of various phenomena in quantum optics, transport processes in Josephson junction, coherence effects and macroscopic quantum tunneling in condensed matter physics, electron transfer in large molecules, thermal activation processes in chemical reactions is now well known and forms a large body of current literature. While the early development of quantum optics initiated in the sixties and seventies was based on density operator, semigroup, noise operator or master equation methods3,4 primarily within weak-coupling and Markov approximations, path integral approach to quantum Brownian motion attracted wide attention in the early eighties. Although this development had widened the scope of condensed matter and chemical physics significantly, so far as the large coupling between the system and the heat bath and large correlation times of the noise processes are concerned several problems still need to be addressed. First, a search for quantum analogue of Kramers’ equation for a nonlinear potential which describes quantum Brownian motion in phase space had remained elusive and at best resulted in equation of motion which contains higher 共than second兲 derivatives of probability distribution functions3,14 whose positive definiteness is never guaranteed. As a result these quasiprobability distribution functions often become singular or a兲

Present address: Max-Planck-Institut fu¨r Physik Komplexer Systeme, No´thnitzer Straße 38, 01187 Dresden, Germany. b兲 Electronic mail: [email protected] 0021-9606/2004/120(19)/8960/13/$22.00

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J. Chem. Phys., Vol. 120, No. 19, 15 May 2004

Solution of quantum Langevin equation

for nonlinear potentials depending on the nature of memory kernel and noise strength at arbitrary temperature down to absolute zero. In order to provide a consistency check of the formulation we have analytically solved the problem of damped harmonic oscillator for arbitrary memory kernel, noise strength and temperature and compared with the standard results. The numerical method is exemplified by an application to calculation of quantum rate of barrier crossing in a cubic potential and the results are compared with others.21,35 The approach is classical-looking in form, and is independent of quantum Monte Carlo techniques and takes care of quantum effects due to nonlinearity of the potential order by order with simplicity and accuracy to a high degree.

II. QUANTUM BROWNIAN MOTION A. Coherent state representation of quantum noise and quantum Langevin equation in c-numbers

We consider a particle of unit mass coupled to a medium comprised of a set of harmonic oscillators with frequency ␻ i . This is described by the following Hamiltonian: ˆ⫽ H

pˆ 2 ⫹V 共 xˆ 兲 ⫹ 2

兺j

ˆp 2j

1 ⫹ ␬ j 共 qˆ j ⫺xˆ 兲 2 . 2 2

共1兲

Here xˆ and pˆ are coordinate and momentum operators of the particle and the set 兵 qˆ j , pˆ j 其 is the set of coordinate and momentum operators for the reservoir oscillators coupled linearly to the system through their coupling coefficients ␬ j . The potential V(xˆ ) is due to the external force field for the Brownian particle. The coordinate and momentum operators follow the usual commutation relations 关 xˆ , pˆ 兴 ⫽iប and 关 qˆ k ,pˆ l 兴 ⫽iប ␦ kl . Note that in writing down the Hamiltonian no rotating wave approximation has been used. Eliminating the reservoir degrees of freedom in the usual way3 we obtain the operator Langevin equation for the particle, x¨ˆ 共 t 兲 ⫹



t

0

dt ⬘ ␥ 共 t⫺t ⬘ 兲 xˆ˙ 共 t ⬘ 兲 ⫹V ⬘ 共 xˆ 兲 ⫽Fˆ 共 t 兲 ,

共2兲

where the noise operator Fˆ (t) and the memory kernel ␥ (t) are given by Fˆ 共 t 兲 ⫽

兺j 关 兵 qˆ j 共 0 兲 ⫺xˆ 共 0 兲 其 ␬ j cos ␻ j t⫹pˆ j 共 0 兲 ␬ 1/2 j sin ␻ j t 兴 共3兲

and

␥ 共 t 兲 ⫽ 兺 ␬ j cos ␻ j t, j

共4兲

with ␬ j ⫽ ␻ 2j 共masses have been assumed to be unity兲. Equation 共2兲 is the well known exact quantized operator Langevin equation3 for which the noise properties of Fˆ (t) can be derived using a suitable initial canonical distribution of the bath coordinate and momentum operators at t⫽0 as follows:

具 Fˆ 共 t 兲 典 QS⫽0,

共5兲

1 2

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兵 具 Fˆ 共 t 兲 Fˆ 共 t ⬘ 兲 典 QS⫹ 具 Fˆ 共 t ⬘ 兲 Fˆ 共 t 兲 典 QS其 ⫽

1 2

兺j ␬ j ប ␻ j



coth



ប␻ j cos ␻ j 共 t⫺t ⬘ 兲 , 2k B T

共6兲

where 具 ¯ 典 QS refers to quantum statistical average on bath degrees of freedom and is defined as

具 Oˆ 典 QS⫽

ˆ bath/k B T 兲 ˆ exp共 ⫺H Tr O

共7兲

ˆ bath/k B T 兲 Tr exp共 ⫺H

ˆ ( 兵 qˆ j ⫺xˆ 其 ), 兵 pˆ j 其 , where H bath for any operator O 2 (⫽ 兺 j 兵 (pˆ j /2)⫹1/2␬ j (qˆ j ⫺xˆ ) 2 其 ) at t⫽0. By Trace we mean the usual quantum statistical average. Equation 共6兲 is the fluctuation-dissipation relation with the noise operators ordered appropriately in the quantum mechanical sense. To construct a c-number Langevin equation we proceed from Eq. 共2兲. We first carry out the quantum mechanical average of Eq. 共2兲,

具 xˆ¨ 共 t 兲 典 ⫹



t

0

dt ⬘ ␥ 共 t⫺t ⬘ 兲 具 xˆ˙ 共 t ⬘ 兲 典 ⫹ 具 V ⬘ 共 xˆ 兲 典 ⫽ 具 Fˆ 共 t 兲 典 ,

共8兲

where the quantum mechanical average 具¯典 is taken over the initial product separable quantum states of the particle and the bath oscillators at t⫽0, 兩 ␾ 典 兵 兩 ␣ 1 典 兩 ␣ 2 典 ¯ 兩 ␣ N 典 其 . Here 兩␾典 denotes any arbitrary initial state of the particle and 兩 ␣ i 典 corresponds to the initial coherent state of the ith bath oscillator. 兩 ␣ i 典 is given by 兩 ␣ i 典 n ⫽exp(⫺兩␣i兩2/2) 兺 n⬁ ⫽0 ( ␣ i i / 冑n i !) 兩 n i 典 , ␣ i being expressed in i terms of the mean values of the shifted coordinate and momentum of the ith oscillator, 具 qˆ i (0) 典 ⫺ 具 xˆ (0) 典 ⫽( 冑ប/2␻ i ) ⫻( ␣ i ⫹ ␣ 쐓i ) and 具 pˆ i (0) 典 ⫽i 冑ប ␻ i /2( ␣ 쐓i ⫺ ␣ i ), respectively. It is important to note that 具 Fˆ (t) 典 of Eq. 共5兲 is a classical-like noise term which, in general, is a nonzero number because of the quantum mechanical averaging and is given by ( 具 Fˆ (t) 典 ⬅ f (t)), f 共 t 兲⫽

兺j 关 兵 具 qˆ j 共 0 兲 典 ⫺ 具 xˆ 共 0 兲 典 其 ␬ j cos ␻ j t ⫹ 具 pˆ j 共 0 兲 典 ␬ 1/2 j sin ␻ j t 兴 .

共9兲

It is convenient to rewrite the c-number equation 共5兲 as follows:

具 x¨ˆ 共 t 兲 典 ⫹



t

0

dt ⬘ ␥ 共 t⫺t ⬘ 兲 具 x˙ˆ 共 t ⬘ 兲 典 ⫹ 具 V ⬘ 共 xˆ 兲 典 ⫽ f 共 t 兲 .

共10兲

To realize f (t) as an effective c-number noise we now assume the ansatz that the momenta 具 pˆ j (0) 典 and the shifted coordinates 兵 具 qˆ j (0) 典 ⫺ 具 xˆ (0) 典 其 of the bath oscillators are distributed according to a canonical distribution of Gaussian form as P j ⫽N exp



⫺ 关 具 pˆ j 共 0 兲 典 2 ⫹ ␬ j 兵 具 qˆ j 共 0 兲 典 ⫺ 具 xˆ 共 0 兲 典 其 2 兴 1 2ប ␻ j ¯n j ⫹ 2

冉 冊



共11兲

so that for any quantum mechanical mean value O j ( 具 pˆ j (0) 典 , ( 具 qˆ j (0) 典 ⫺ 具 xˆ (0) 典 ) the statistical average 具 ¯ 典 S is

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具 O j典 S⫽



O j 共 具 pˆ j 共 0 兲 典 , 兵 具 qˆ j 共 0 兲 典 ⫺ 具 xˆ 共 0 兲 典 其 兲

Q 共 t 兲 ⫽⫺

⫻P j 共 具 pˆ j 共 0 兲 典 , 兵 具 qˆ j 共 0 兲 典 ⫺ 具 xˆ 共 0 兲 典 其 兲 ⫻d 具 pˆ j 共 0 兲 典 d 兵 具 qˆ j 共 0 兲 典 ⫺ 具 xˆ 共 0 兲 典 其 .

共12兲

Here ¯n j indicates the average thermal photon number of the jth oscillator at temperature T and ¯n j ⫽1/关 exp(ប␻ j /kBT)⫺1兴 and N is the normalization constant. The distribution 共11兲 and the definition of statistical average 共12兲 imply that f (t) must satisfy

具 f 共 t 兲 典 S ⫽0

共13兲

and

具 f 共 t 兲 f 共 t ⬘ 兲典 S⫽

1 2

兺j ␬ j ប ␻ j



coth



ប␻ j cos ␻ j 共 t⫺t ⬘ 兲 . 2k B T 共14兲

That is, c-number noise f (t) is such that it is zero-centered and satisfies the standard fluctuation-dissipation relation 共FDR兲 as expressed in Eq. 共6兲. It is important to emphasize that the ansatz 共11兲 is a canonical Wigner distribution for a shifted harmonic oscillator34 which remains always a positive definite function. A special advantage of using this distribution34 is that it remains valid as pure state nonsingular distribution function at T⫽0. Furthermore, this procedure allows us to bypass the operator ordering prescription of Eq. 共6兲 for deriving the noise properties of the bath in terms of fluctuation-dissipation relation and to identify f (t) as a classical looking noise with quantum mechanical content. We now return to Eq. 共10兲 to add the force term V ⬘ ( 具 xˆ 典 ) on both sides of Eq. 共10兲 and rearrange it to obtain X˙ ⫽ P, P˙ ⫽⫺

共15兲



t

0

dt ⬘ ␥ 共 t⫺t ⬘ 兲 P 共 t ⬘ 兲 ⫺V ⬘ 共 X 兲 ⫹ f 共 t 兲 ⫹Q 共 t 兲 ,

共16兲

where we put 具 xˆ (t) 典 ⫽X(t) for notational convenience and Q 共 t 兲 ⫽V ⬘ 共 具 xˆ 典 兲 ⫺ 具 V ⬘ 共 xˆ 兲 典

共17兲

represents the quantum correction due to the system degrees of freedom. Equation 共16兲 offers a simple interpretation. This implies that the QGLE is governed by a c-number quantum noise f (t) originating from the heat bath characterized by the properties 共13兲 and 共14兲 and a quantum fluctuation term Q(t) characteristic of the nonlinearity of the potential. To proceed further we need a recipe for the calculation of Q(t). Referring to the quantum nature of the system in the Heisenberg picture, one may write xˆ 共 t 兲 ⫽X⫹ ␦ xˆ ,

共18兲

pˆ 共 t 兲 ⫽ P⫹ ␦ pˆ ,

共19兲

where 具 xˆ 典 (⫽X) and 具 pˆ 典 (⫽ P) are the quantum-mechanical averages and ␦ xˆ , ␦ pˆ are the operators. By construction 具 ␦ xˆ 典 and 具 ␦ pˆ 典 are zero and 关 ␦ xˆ , ␦ pˆ 兴 ⫽iប. Using Eqs. 共18兲 and 共19兲 in 具 V ⬘ (xˆ ) 典 and a Taylor series expansion around 具 xˆ 典 it is possible to express Q(t) as

1

兺 V 共 n⫹1 兲共 X 兲 具 ␦ xˆ n共 t 兲 典 . n⭓2 n!

共20兲

Here V n (X) is the nth derivative of the potential V(X). For example, the second order Q(t) is given by Q(t)⫽ ⫺ 12 V ⵮ (X) 具 ␦ xˆ 2 典 . The calculation of Q(t) therefore rests on quantum correction terms, 具 ␦ xˆ n (t) 典 which are determined by solving the quantum correction equations as discussed in the following section. A digression on initial overall density matrix as well as on c-number Langevin equation may be now in order. We note that in the standard derivation of operator Langevin equation one assumes that the initial (system⫹reservoir) density matrix is separable into that of the system and that of the bath with the latter being an equilibrium density matrix at temperature T. In contrast we assume that initial bath density matrix is diagonal in the coherent state basis. It appears as an ad hoc assumption although it gives correct classical limit, fluctuation-dissipation relation and results in the deep tunneling region. The choice of such initial conditions and how the present Langevin equation differs from the standard approaches can be clarified in the following way. Since the additive noise operators 共3兲 are one photon excitation-de-excitation bath operators, the quantum mechanical average with Fock states 兵 兩 n i 典 其 of the harmonic oscillators of the bath yields zero ( 具 n i 兩 Fˆ i (t) 兩 n i 典 ⫽0) making a further statistical average redundant. The statistical property of the thermal bath only makes its presence felt through two-time correlation functions of the noise operators appropriately ordered. It is therefore difficult to make a one-to-one correspondence between a numerically computable c-number noise 关as f (t) in our case兴 and the corresponding noise operator Fˆ (t) in a nondiagonal basis of the bath states. A diagonal basis serves this purpose in the present context. Second, one may note that the choice of an initial nondiagonal basis generally leads to the time evolution of off-diagonal matrix elements. In the present approach we have avoided the appearance of these elements by expressing the system operator at any time as a quantum mechanical average plus a fluctuation operator 关Eqs. 共18兲 and 共19兲兴 in a nonlinear function of operators. The procedure, however, generates a set of quantum correction equations which are to be solved order by order. Thus the present approach differs from the standard one in the following way. The initial bath density in the diagonal basis allows us to construct the equation of motion for quantum mechanical averages and dispersions for the system driven by a classical looking noise whose correlation is quantum mechanical in its content. In the usual approach 共e.g., Ref. 3兲, on the other hand, one sets up a correspondence between c-numbers and operators via a quasiclassical phase space distribution function whose evolution equation 关which takes care of nonlinearity of the product of operators in terms of higher 共than second兲 derivatives of the distribution function兴 can be expressed equivalently in terms of Langevin equation for c-numbers within the usual approximation schemes. Thus the quantum correction equations do not appear in this case. It is however important to note that the system equilibrates after a while so that the difference in initial conditions becomes immaterial.

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J. Chem. Phys., Vol. 120, No. 19, 15 May 2004

Solution of quantum Langevin equation

Finally a note on noise correlation is relevant. As the theory of Brownian motion at low temperatures is largely stimulated by investigation on resistively shunted Josephson junctions where it has been confirmed by experiments2 that the voltage noise observed at low temperatures has a power spectrum containing a product of frequency dependent damping ␥共␻兲 and an energy factor, ប ␻ coth(ប␻/2k B T) 共which goes over to classical limit of white noise 2⌫k B T for small frequencies where ប ␻ Ⰶk B T and to ⌫ប␻ for high frequency limit where ប ␻ Ⰷk B T, where ⌫ is the dissipation constant in the classical Markovian limit兲. It is thus easy to guess that such an energy factor must have to appear as an integral part of c-number noise correlation of the thermal bath. From a theoretical point of view on the other hand, this energy factor can be recovered as a width of Wigner canonical thermal distribution function of c-numbers obtained as an exact solution of Wigner equation for the harmonic oscillator. The choice of initial conditions and ensemble averaging with a positive definite Wigner distribution can therefore be rationalized both from experimental and theoretical standpoints.

再 冕

␦ xˆ 共 t 兲 ⫽ ␦ xˆ 共 0 兲 1⫺ ␻ 2

t

0



G 共 t ⬘ 兲 dt ⬘ ⫹ ␦ pˆ 共 0 兲 G 共 t 兲 ,

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共25兲

where G(t) is identified as the inverse Laplace transform of ˜ (s) given by G ˜ 共 s 兲⫽ G

1 s 2 ⫹s ˜␥ 共 s 兲 ⫹ ␻ 2

共26兲

with ˜␥ (s) as ˜␥ 共 s 兲 ⫽

冕␥ ⬁

0

共 t 兲 e ⫺st dt

共27兲

the Laplace transform of the friction kernel ␥ (t). Equation 共25兲 yields the relevant quantum correction 具 ␦ xˆ 2 (t) 典 after squaring and quantum mechanical averaging as,

具 ␦˜x 2 共 t 兲 典 ⫽ 具 ␦ xˆ 2 共 0 兲 典 G˙ 2 共 t 兲 ⫹ 具 ␦ pˆ 2 共 0 兲 典 G 2 共 t 兲 ⫹G 共 t 兲 G˙ 共 t 兲 ⫻具 ␦ xˆ 共 0 兲 ␦ pˆ 共 0 兲 ⫹ ␦ pˆ 共 0 兲 ␦ xˆ 共 0 兲 典 .

共28兲

We shall return to this issue in more detail when we discuss the quantum Brownian motion of a damped harmonic oscillator, in Sec. III.

B. Quantum correction equations

We now return to operator equation 共2兲 and put 共18兲 and 共19兲 to obtain

␦ x˙ˆ ⫽ ␦ pˆ , ␦ pˆ˙ ⫹





t

0

共21兲



n⭓2

共22兲

Equations 共21兲–共22兲 form the key element for calculation of the quantum mechanical correction Eq. 共20兲 due to nonlinearity of the system potential V(xˆ ). Note that 具 ␦ xˆ n (t) 典 in Q(t) does not include any statistical averaging and is a dispersion term obtained by pure quantum mechanical averaging which is not to be confused with 具 ␦ xˆ n (t) 典 S . Thus, to obtain 具 ␦ xˆ n (t) 典 we need to perform only a quantum mechanical averaging over the initial product separable bath ⬁ states ⌸ i⫽1 兵 兩 ␣ i (0) 典 其 to get rid of the term Fˆ (t)⫺ f (t). Depending on the nonlinearity of the potential and the memory kernel we consider the following cases separately. 1. Arbitrary memory kernel and harmonic potential

We consider the friction kernel to be arbitrary 共but decaying兲 and the potential as harmonic for which the derivatives higher than second in V(X) in Eq. 共22兲 vanish. The above set of equations may be put in the form 共after quantum mechanical averaging over bath states has been performed兲,

␦ xˆ˙ ⫽ ␦ pˆ , ␦ p˙ˆ ⫽⫺

共23兲

冕␥ t

0

共 t⫺t ⬘ 兲 ␦ pˆ 共 t ⬘ 兲 dt ⬘ ⫺ ␻ ␦ xˆ , 2

共24兲

where ␻ ⫽V ⬙ (X), is a constant. The operator equations 共23兲 and 共24兲 can be solved exactly by Laplace transform technique to obtain 2

For a Lorentzian distribution of bath modes characterized by a density function ␳共␻兲,

␬共 ␻ 兲␳共 ␻ 兲⫽

␥ 共 t⫺t ⬘ 兲 ␦ pˆ 共 t ⬘ 兲 dt ⬘ ⫹V ⬙ 共 X 兲 ␦ xˆ 1 共 n⫹1 兲 V 共 X 兲 ␦ xˆ n ⫽Fˆ 共 t 兲 ⫺ f 共 t 兲 . n!

2. Exponential memory kernel and arbitrary potential

2 ⌫ , ␲ 1⫹ ␻ 2 ␶ 2c

共29兲

where ⌫ is the dissipation constant in the Markovian limit and ␶ c refers to the correlation time, the memory kernel is exponential and ␥ (t) is given by

␥共 t 兲⫽

⌫ ⫺兩t兩/␶ c. e ␶c

共30兲

In this case the quantum correction Eqs. 共21兲–共22兲, after quantum mechanical averaging over bath states, may be rewritten as

␦ x˙ˆ ⫽ ␦ pˆ ,

共31兲

␦ pˆ˙ ⫽⫺V ⬙ 共 X 兲 ␦ xˆ ⫺ 兺

n⭓2

␦ zˆ˙ ⫽⫺

1 共 n⫹1 兲 V 共 X 兲 ␦ xˆ n ⫹ ␦ zˆ , n!

⌫ 1 ␦ pˆ ⫺ ␦ zˆ , ␶c ␶c

共32兲 共33兲

where we have introduced an auxiliary operator ␦ zˆ to bypass the convolution integral in Eq. 共22兲. Making use of Eqs. 共31兲–共33兲 one can derive the quantum correction equations upto second order to obtain

具 ␦ x˙ˆ 2 典 ⫽ 具 ␦ xˆ ␦ pˆ ⫹ ␦ pˆ ␦ xˆ 典 ,

共34兲

˙ ␦ pˆ ␦ xˆ 典 ⫽2 具 ␦ pˆ 2 典 ⫺2V ⬙ 共 X 兲 具 ␦ xˆ 2 典 ⫹ 具 ␦ xˆ ␦ zˆ 具 ␦ xˆ ␦ pˆ ⫹ ⫹ ␦ zˆ ␦ xˆ 典 ⫺V ⵮ 共 X 兲 具 ␦ xˆ 3 典 ,

共35兲

具 ␦ pˆ˙ 2 典 ⫽⫺V ⬙ 共 X 兲 具 ␦ xˆ ␦ pˆ ⫹ ␦ pˆ ␦ xˆ 典 ⫹ 具 ␦ pˆ ␦ zˆ ⫹ ␦ zˆ ␦ pˆ 典 ⫺ 12 V ⵮ 共 X 兲 具 ␦ pˆ ␦ xˆ 2 ⫹ ␦ xˆ 2 ␦ pˆ 典 ,

共36兲

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1 2

˙ ␦ zˆ ␦ pˆ 典 ⫽⫺V ⬙ 共 X 兲 具 ␦ xˆ ␦ zˆ ⫹ ␦ zˆ ␦ xˆ 典 ⫺ V ⵮ 共 X 兲 具 ␦ pˆ ␦ zˆ ⫹ ⫻具 ␦ xˆ 2 ␦ zˆ ⫹ ␦ zˆ ␦ xˆ 2 典 ⫺ ⫺

1 具 ␦ pˆ ␦ zˆ ⫹ ␦ zˆ ␦ pˆ 典 ⫹2 具 ␦ zˆ 2 典 , ␶c

˙ ␦ zˆ ␦ xˆ 典 ⫽ 具 ␦ pˆ ␦ zˆ ⫹ ␦ zˆ ␦ pˆ 典 ⫺ 具 ␦ xˆ ␦ zˆ ⫹ ⫺

具 ␦ z˙ˆ 2 典 ⫽⫺

2⌫ 具 ␦ pˆ 2 典 ␶c 共37兲

⌫ 具 ␦ xˆ ␦ pˆ ⫹ ␦ pˆ ␦ xˆ 典 ␶c

1 具 ␦ xˆ ␦ zˆ ⫹ ␦ zˆ ␦ xˆ 典 , ␶c

⌫ 2 具 ␦ pˆ ␦ zˆ ⫹ ␦ zˆ ␦ pˆ 典 ⫺ 具 ␦ zˆ 2 典 . ␶c ␶c

共38兲 共39兲

Discarding the third order terms in Eqs. 共34兲–共39兲 we obtain a set of closed equations which can be solved numerically along with Eqs. 共15兲 and 共16兲, using suitable initial conditions for calculation of the leading 共second兲 order contribution to quantum correction terms in Q(t) of Eq. 共20兲, i.e., ⫺(1/2)V ⵮ (X) 具 ␦ xˆ 2 (t) 典 . A standard choice20 of initial conditions corresponding to minimum uncertainty states is 具 ␦ xˆ 2 典 t⫽0 ⫽1/2, 具 ␦ pˆ 2 典 t⫽0 ⫽1/2 and 具 ␦ xˆ ␦ pˆ ⫹ ␦ pˆ ␦ xˆ 典 t⫽0 ⫽1.0 with the other moments 关Eqs. 共37兲–共39兲兴 being set at zero. The procedure may be extended to include higher order quantum effects without difficulty and may be easily adopted for numerical simulation of quantum Brownian motion as shown in Sec. IV.

In the quantum overdamped limit we discard the inertial term ␦ x¨ˆ compared to the damping term. The operator equations 共21兲 and 共22兲 then assume a simple form, after quantum mechanical averaging over bath states, as

␥ 0 ␦ xˆ˙ ⫹V ⬙ 共 X 兲 ␦ xˆ ⫹ 兺

n⭓2

1 共 n⫹1 兲 V 共 X 兲 ␦ xˆ n ⫽0, n!

共40兲

where ␥ 0 is the dissipation constant. Equation 共40兲 yields the quantum correction equations order by order to a very high degree. The equations up to third order are shown below in which terms of the order 具 ␦ xˆ 4 典 are to be neglected for closing the system of equations:

具 ␦ x˙ˆ 2 典 ⫽⫺ ⫺

Third order

In this limit ␶ c →0 so that ␥ (t) in Eq. 共22兲 reduces to a delta function ␥ (t)⫽2⌫ ␦ (t), where ⌫ is the Markovian limit of dissipation. The operator equations 共21兲 and 共22兲, after quantum mechanical averaging over bath states, then reduce to

␦ x˙ˆ ⫽ ␦ pˆ ,

共43兲

␦ p˙ˆ ⫽⫺⌫ ␦ pˆ ⫺V ⬙ 共 X 兲 ␦ xˆ ⫺ 兺

n⭓2

1 共 n⫹1 兲 V 共 X 兲 ␦ xˆ n ⫽0. n!

具 ␦ x˙ˆ 3 典 ⫽⫺ ⫺

2V ⬙ 共 X 兲 具 ␦ xˆ 2 典 ␥0 2V ⵮ 共 X 兲 具 ␦ xˆ 3 典 ; 2! ␥ 0

具 ␦ xˆ˙ 2 典 ⫽ 具 ␦ xˆ ␦ pˆ ⫹ ␦ pˆ ␦ xˆ 典 ,

共45兲

˙ ␦ pˆ ␦ xˆ 典 ⫽⫺2V ⬙ 共 X 兲 具 ␦ xˆ 2 典 ⫺V ⵮ 共 X 兲 具 ␦ xˆ 3 典 , 具 ␦ xˆ ␦ pˆ ⫹

共46兲

具 ␦ pˆ˙ 2 典 ⫽⫺2⌫ 具 ␦ pˆ 2 典 ⫺V ⬙ 共 X 兲 具 ␦ xˆ ␦ pˆ ⫹ ␦ pˆ ␦ xˆ 典 ⫺V ⵮ 共 X 兲 具 ␦ xˆ ␦ pˆ ␦ xˆ 典 ,

共47兲

具 ␦ xˆ˙ 3 典 ⫽3 具 ␦ xˆ ␦ pˆ ␦ xˆ 典 ,

共48兲

具 ␦ p˙ˆ 3 典 ⫽⫺3⌫ 具 ␦ pˆ 3 典 ⫺3V ⬙ 共 X 兲 具 ␦ pˆ ␦ xˆ ␦ pˆ 典 ⫹V ⵮ 共 X 兲 ប 2 ,

共49兲

具 ␦ xˆ ␦˙ pˆ ␦ xˆ 典 ⫽⫺⌫ 具 ␦ xˆ ␦ pˆ ␦ xˆ 典 ⫹2 具 ␦ pˆ ␦ xˆ ␦ pˆ 典 ⫺V ⬙ 共 X 兲 具 ␦ xˆ 3 典 ,

共50兲

具 ␦ pˆ ␦˙ xˆ ␦ pˆ 典 ⫽⫺2⌫ 具 ␦ pˆ ␦ xˆ ␦ pˆ 典 ⫹ 具 ␦ pˆ 3 典 共51兲

The equations of higher order can be similarly derived. Summarizing the discussions in the last two sections we now see that the quantum Langevin dynamics in c-numbers can be calculated by solving the Langevin equations 共15兲– 共16兲 for quantum mechanical mean values simultaneously with the quantum correction equations 关e.g., Eqs. 共34兲–共39兲 or Eqs. 共45兲–共51兲兴 which describe quantum fluctuations 共due to nonlinearity of the system兲 around the quantum mechanical mean values. The choice of quantum correction equations rests on the requirement demanded by the nature of the memory kernel and nonlinearity of the potential.

C. The canonical Wigner thermal distribution function for harmonic oscillator and fluctuation-dissipation relation

共41兲

3V ⬙ 共 X 兲 具 ␦ xˆ 3 典 ␥0 3V ⵮ 共 X 兲 具 ␦ xˆ 4 典 . 2! ␥ 0

共44兲

Making use of the above equations we derive the quantum correction equations up to the third order as

⫺2V ⬙ 共 X 兲 具 ␦ xˆ ␦ pˆ ␦ xˆ 典 .

3. Overdamped limit

Second order

4. Markovian limit

共42兲

We have made use of the above scheme in our recent treatment of quantum Smoluchowskii equation as discussed in Ref. 18.

The realization of the stochastic process in a c-number formulation in terms of Eqs. 共15兲–共16兲 essentially depends on the ensemble averaging over the quantum mean values of the coordinates and momenta of the bath oscillators. This is based on a key point of the present analysis—the ansatz 共11兲, the canonical thermal Wigner distribution for shifted harmonic oscillators, which reproduces the exact first and second moments of the c-number noise f (t). To make the perspective more clear in the context of noise properties of the bath we now briefly outline the derivation of the fluctuationdissipation relation in three different ways.

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J. Chem. Phys., Vol. 120, No. 19, 15 May 2004

Solution of quantum Langevin equation

1. Quantum statistical averaging

and

To make the discussion self-contained we first establish that the quantum-statistical average value of a two-time correlation function of the random force operator Fˆ (t) using standard usage of quantum-statistical averaging 具 ¯ 典 QS can be written in the form 共6兲 1 具 Fˆ 共 t 兲 Fˆ 共 t ⬘ 兲 ⫹Fˆ 共 t ⬘ 兲 Fˆ 共 t 兲 典 QS 2 ⫽

兺j





1 ប␻ j ␬ j ប ␻ j coth cos ␻ j 共 t⫺t ⬘ 兲 , 2 2k B T

共52兲

where Fˆ (t) is defined by Eq. 共3兲. Starting from a canonical distribution of bath oscillators at t⫽0 with respect to the free ˆ 0 , with shifted oscillator coorbath oscillator Hamiltonian H dinates qˆ ⬘j (0)⫽qˆ j (0)⫺xˆ (0) as stated earlier in Sec. II, we construct the expectation value of an arbitrary operator according to Eq. 共7兲. As usual Trace implies both quantum mechanical averaging with respect to number states of the oscillators ⌸ n⬁ ⫽0 兵 兩 n 1 典 其 ⌸ n⬁ ⫽0 兵 兩 n 2 典 其 ¯⌸ n⬁ ⫽0 兵 兩 n N 典 其 and the 1 2 N system state 兩␾典 followed by Boltzmann statistical averaging. Making use of the definition of the quantum statistical average Eq. 共7兲 we first derive

具 qˆ ⬘j 共 0 兲 qˆ k⬘ 共 0 兲 典 QS⫽

具 pˆ j 共 0 兲 pˆ k 共 0 兲 典 QS ␻ 2j

⫽ ␦ jk

8965

ប coth共 ␻ j ប/2k B T 兲 2␻ j

共53兲

具 qˆ ⬘j 共 0 兲 pˆ k 共 0 兲 典 QS⫽⫺ 具 pˆ k 共 0 兲 qˆ ⬘j 共 0 兲 典 QS⫽ 21 iប ␦ jk .

共54兲

Since the expectation value of an odd number of factors of qˆ ⬘j (0) and pˆ j (0) vanishes and the expectation value of an even number of factors is the sum of products of the pairwise expectation values with the order of the factors preserved we note that Gaussian property of the distribution of qˆ ⬘j (0) and pˆ j (0) is implied. Relations 共53兲 and 共54兲 directly yield 共52兲. On the other hand quantum mechanical averaging with respect to the number states results in

具 Fˆ 共 t 兲 典 QS⫽0.

共55兲

2. The thermal Glauber – Sudarshan P-function

In this method we first calculate the quantum mechanical average of 1/2关 Fˆ (t)Fˆ (t ⬘ )⫹Fˆ (t ⬘ )Fˆ (t) 兴 with respect to the coherent states of the bath oscillators 兩 ␣ 1 典 兩 ␣ 2 典 ¯ 兩 ␣ N 典 and system state 兩␾典 at t⫽0, where 兩 ␣ i 典 has been defined in the text. 兩 ␣ i 典 is the eigenstate of the annihilation operator aˆ i such that aˆ i 兩 ␣ i 典 ⫽ ␣ i 兩 ␣ i 典 , where qˆ ⬘i (0) and pˆ i (0) are expressed as usual as qˆ ⬘i (0)⫽ 冑ប/2␻ i 关 aˆ †i (0)⫹aˆ i (0) 兴 and pˆ i (0)⫽i 冑ប ␻ i /2关 aˆ †i (0)⫺aˆ i (0) 兴 . We thus note that the coherent state averaging 具¯典 leads to 共we have discarded the cross terms like ␣ j ␣ k* since they will vanish after the statistical averaging that follows in the next step兲

1 具 Fˆ 共 t 兲 Fˆ 共 t ⬘ 兲 ⫹Fˆ 共 t ⬘ 兲 Fˆ 共 t 兲 典 2 ⫽

兺j ⫹



␬j

ប ␬ jប␻ j 共 ␣ * 2 ⫹2 ␣ *j ␣ j ⫹1⫹ ␣ 2j 兲 cos ␻ j t cos ␻ j t ⬘ ⫺ 共 ␣ *j 2 ⫺2 ␣ *j ␣ j ⫺1⫹ ␣ 2j 兲 sin ␻ j t sin ␻ j t ⬘ 2␻ j j 2



iប ␬ ␻ 共 2 ␣ *j 2 ⫺2 ␣ 2j 兲 sin ␻ j 共 t⫺t ⬘ 兲 . 4 j j

共56兲

We now introduce thermal Glauber–Sudarshan distribution3 for the statistical averaging of the quantum mechanical expectation values,

With relations 共58兲, the statistical average 共with respect to the P-function兲 of the quantum mechanical mean value 共56兲 leads us to

P GS共 ␣ j 共 0 兲 , ␣ *j 共 0 兲兲

冉 冊 冋冉

⫽N exp ⫺ ⫽N exp ⫺

1 具具Fˆ 共 t 兲 Fˆ 共 t ⬘ 兲 ⫹Fˆ 共 t ⬘ 兲 Fˆ 共 t 兲 典典GS 2

兩 ␣ j兩 2 ¯n j

具 pˆ j 共 0 兲 典 2 ⫹ ␻ 2j 具 qˆ ⬘j 共 0 兲 典 2 2ប ␻ j¯n j

冊册

,

共57兲

where N is the normalization constant. Noting that with the distribution 共57兲 the normalized statistical averages 具 ¯ 典 GS are given by ¯ j ␦ jk . 具 ␣ 2j 典 GS⫽ 具 ␣ *j 2 典 GS⫽0 and 具 ␣ j ␣ *k 典 GS⫽n



共58兲

1 2

兺j ␬ j ប ␻ j



coth



ប␻ j cos ␻ j 共 t⫺t ⬘ 兲 2k B T

共59兲

and

具具Fˆ 共 t 兲 典典GS⫽0.

共60兲

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8966

Banerjee et al.

J. Chem. Phys., Vol. 120, No. 19, 15 May 2004

3. The canonical thermal Wigner distribution function

In this method we bypass the operator ordering of Fˆ (t)s. Instead, we start from coherent state average of the noise operator Fˆ (t), as given by Eq. 共9兲, i.e., the c-number noise f 共 t 兲⫽

兺j 关 兵 具 qˆ j 共 0 兲 典 ⫺ 具 xˆ 共 0 兲 典 其 ␬ j cos ␻ j t ⫹ 具 pˆ j 共 0 兲 典 ␬ 1/2 j sin ␻ j t 兴

共61兲

and then calculate the product of two-time c-numbers f (t) f (t ⬘ ). Expressing 具 qˆ ⬘j (0) 典 and 具 pˆ j (0) 典 in terms of ␣ j and ␣ *j as stated earlier, we may write

再兺 冉 冑

f 共 t 兲 f 共 t⬘兲⫽

␬j

j

⫹i ⫻



ប 共 ␣ * ⫹ ␣ j 兲 cos ␻ j t 2␻ j j

ប␻ j ␻ j 共 ␣ *j ⫺ ␣ j 兲 sin ␻ j t 2

再兺 冉 冑

⫹i

␬k

k





P 共 q j ,p j 兲 ⫽N 1 exp ⫺

冊冎

冊冎

.

冉 冋冉

⫽N exp ⫺ ⫽N exp ⫺

兩 ␣ j兩 2 ¯n j ⫹1/2



具 pˆ j 共 0 兲 典 2 ⫹ ␻ 2j 具 qˆ ⬘j 共 0 兲 典 2 2ប ␻ j 共 ¯n j ⫹1/2兲

冊册

.

共63兲

Noting that

具 ␣ 2j 典 S ⫽ 具 ␣ *j 2 典 S ⫽0 and 具 ␣ j ␣ k* 典 S ⫽ 共 ¯n j ⫹ 21 兲 ␦ jk ,

共64兲

we obtain on statistical averaging over f (t) f (t ⬘ ),

具 f 共 t 兲 f 共 t ⬘ 兲典 S⫽

1 2

兺j ␬ j ប ␻ j



coth

2ប ␻ j



.

共66兲

共62兲

We now introduce the canonical thermal Wigner distribution function 共11兲 for ensemble averaging 具 ¯ 典 S over the product f (t) f (t ⬘ ) P j 共 ␣ j 共 0 兲 , ␣ *j 共 0 兲兲

具 pˆ j 共 0 兲 典 2 ⫹ ␻ 2j 具 qˆ ⬘j 共 0 兲 典 2

N 1 is the normalization constant which is always a positive definite function in contrast to its excited state counterparts. This is because, according to Wigner et al.,34 ‘‘the incoherence induced by a finite temperature leads to a much smoother distribution function.’’

ប 共 ␣ * ⫹ ␣ k 兲 cos ␻ k t ⬘ 2␻k k

ប␻k ␻ k共 ␣ * k ⫺ ␣ k 兲 sin ␻ k t ⬘ 2

operator ordering of noise as in the former case since we are essentially dealing with a correlation of numbers rather than operators. The uncertainty principle is taken care of through the vacuum term 1/2 in the width of the distribution 共11兲 rather than through operator ordering as in methods I and II. Fourth, the Wigner distribution 共11兲 as well as the thermal Glauber–Sudarshan distribution 共57兲 reduces to classical Maxwell–Boltzmann distribution in the limit ប ␻ j Ⰶk B T. Equation 共57兲 becomes singular in the vacuum limit ¯n →0. The distribution 共11兲 on the other hand remains a valid nonsingular distribution of quantum mean values whose width is simply the natural width due to nonthermal vacuum fluctuations.34 The thermal Wigner distribution then goes over to pure state Wigner distribution for the ground state in the form,



ប␻ j cos ␻ j 共 t⫺t ⬘ 兲 . 2k B T 共65兲

Several points are now in order. First, we note that the fluctuation-dissipation relation has been expressed in several different but equivalent forms in Eqs. 共52兲, 共59兲, and 共65兲. Because of the prescription of operator ordering, the lefthand sides of Eqs. 共52兲 and 共59兲 do not permit us to identify a c-number noise term. Clearly Eq. 共65兲 suits this purpose and quantum noise as a number f (t) can be identified in Eq. 共65兲. The ensemble averaging procedure with Wigner thermal distribution function is therefore followed in this work. Second, the Wigner distribution 共11兲 is distinctly different from the thermal Glauber–Sudarshan distribution 共57兲 because of a vacuum term 1/2 in the width of the distribution 共11兲. Third, the origin of distinction of the methods II and III lies in the fact that in the latter case we need not carry out an

III. APPLICATION TO HARMONIC OSCILLATOR

In order to make a consistency check of the above formulation of quantum Brownian motion in c-numbers, it is pertinent to consider the case of damped harmonic oscillator. The problem has been treated over many years by various group of workers5,13 using several methods, notably by path integrals. For a review we refer to Grabert et al.5 Although exact, we emphasize that because of using different kinds of initial conditions and frequency dependent friction and cutoff frequencies the results obtained in several cases tend to vary. Our aim here is to obtain the result for arbitrary 共decaying兲 memory kernel given that the initial states of the system and the bath are product separable and to show that it corresponds with the standard results known in the literature.5,13 To proceed further we return to our basic equations 共15兲 and 共16兲, which for the case of harmonic oscillator reduce to the following form: X˙ ⫽ P, P˙ ⫽⫺

共67兲



t

0

dt ⬘ ␥ 共 t⫺t ⬘ 兲 P 共 t ⬘ 兲 ⫺ ␻ 2 X⫹ f 共 t 兲 .

共68兲

Here ␻ is the frequency of the harmonic oscillator. Evidently we have Q⫽0. The c-number noise f (t) obeys the usual properties given by Eqs. 共13兲 and 共14兲. The solution X(t) of Eqs. 共67兲 and 共68兲 using the Laplace transform technique is given by ˙ 共 t 兲⫹ X 共 t 兲 ⫽ P 共 0 兲 G 共 t 兲 ⫹X 共 0 兲 G



t

0

dt ⬘ G 共 t⫺t ⬘ 兲 f 共 t ⬘ 兲 , 共69兲

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J. Chem. Phys., Vol. 120, No. 19, 15 May 2004

Solution of quantum Langevin equation

˙ (t)⫽1⫺ ␻ 2 兰 t0 G(t ⬘ )dt ⬘ ; G(t) is defined as the where G Laplace transform of G(s) as given earlier in Eq. 共26兲. On squaring and statistical averaging 具 ¯ 典 S we obtain,

usual quantum statistical averaging defined in Eq. 共7兲兴 by the present scheme. Making use of Eq. 共18兲 具 xˆ 2 (t) 典 QS can be expressed as

具 xˆ 2 共 t 兲 典 QS⫽ 具具共 X 共 t 兲 ⫹ ␦ xˆ 共 t 兲兲 2 典典 S

具 X 2 共 t 兲 典 S ⫽ P 2 共 0 兲 G 2 共 t 兲 ⫹X 2 共 0 兲 G˙ 2 共 t 兲

⫽ 具 X 2 共 t 兲 ⫹ 具 ␦ xˆ 2 共 t 兲 典典 S ⫽ 具 X 2 共 t 兲 典 S ⫹ 具 ␦ xˆ 2 共 t 兲 典 .

˙ 共t兲 ⫹2 P 共 0 兲 X 共 0 兲 G 共 t 兲 G ⫹2

冕 ⬘冕 t

dt

t⬘

0

0

8967

共71兲

dt ⬙ G 共 t⫺t ⬘ 兲

⫻G 共 t ⬘ ⫺t ⬙ 兲 c 共 t ⬘ ⫺t ⬙ 兲 .

共70兲

Here c(t⫺t ⬘ )(⫽ 具 f (t) f (t ⬘ ) 典 S ) refers to the correlation of c-number noise 关Eq. 共14兲兴 due to the heat bath. We now calculate the 具 xˆ 2 (t) 典 QS 关recall that 具 ¯ 典 QS is the

In writing the above expression we have made a clear distinction between a quantum mechanical average 具¯典 and a thermal statistical average 具 ¯ 典 S and their connection to quantum statistical average 具 ¯ 典 QS . The quantity 具 xˆ 2 (t) 典 QS is therefore given by the sum of the right-hand sides of Eqs. 共70兲 and 共28兲. Thus we have,

具 xˆ 2 共 t 兲 典 QS⫽ 兵 P 2 共 0 兲 ⫹ 具 ␦ pˆ 2 共 0 兲 典 其 G 2 共 t 兲 ⫹ 兵 X 2 共 0 兲 ⫹ 具 ␦ xˆ 2 共 0 兲 典 其 G˙ 2 共 t 兲 ⫹G 共 t 兲 G˙ 共 t 兲 具 ␦ pˆ 共 0 兲 ␦ xˆ 共 0 兲 ⫹ ␦ xˆ 共 0 兲 ␦ pˆ 共 0 兲 典 ˙ 共 t 兲 ⫹2 ⫹2 P 共 0 兲 X 共 0 兲 G 共 t 兲 G

冕 ⬘冕 t

dt

0

t⬘

0

dt ⬙ G 共 t⫺t ⬘ 兲 G 共 t ⬘ ⫺t ⬙ 兲 c 共 t ⬘ ⫺t ⬙ 兲 .

Equation 共72兲 is the general expression for quantum statistical average 具 xˆ 2 (t) 典 QS . In the standard notation the above expression can be written as

Putting Eq. 共75兲 in Eq. 共76兲 we obtain after some algebraic manipulations I 0⫽

具 xˆ 2 共 t 兲 典 QS⫽ 具 pˆ 2 共 0 兲 典 G 2 共 t 兲 ⫹ 具 xˆ 2 共 0 兲 典 G˙ 2 共 t 兲 ˙ 共 t 兲 具 pˆ 共 0 兲 xˆ 共 0 兲 ⫹G 共 t 兲 G ⫹xˆ 共 0 兲 pˆ 共 0 兲 典 ⫹2

冕 ⬘冕 t

dt

0

t⬘

0

dt ⬙ G 共 t⫺t ⬘ 兲

⫻G 共 t ⬘ ⫺t ⬙ 兲 c 共 t ⬘ ⫺t ⬙ 兲 .

共73兲

Equation 共72兲 or Eq. 共73兲 is the central result of this section. This expresses the quantum statistical average of xˆ 2 (t) for a damped harmonic oscillator in the non-Markovian regime. That the above expression is exactly equivalent to the standard result, for example,5 can be shown by an analysis of Markovian limit of Eq. 共73兲. Since in this limit the noise is ␦-correlated and we have, ប␻ ␦ 共 t⫺t ⬘ 兲 具 f 共 t 兲 f 共 t ⬘ 兲 典 S ⫽c 共 t⫺t ⬘ 兲 ⫽⌫ប ␻ coth 2k B T

共74兲

冉 冊冋

ប␻ ប␻ coth 2 2k B T





再 冎册

1 1 sin2 ␰ t ⫺⌫t ⫺e ⫹ 1 ␻2 ␻2 ␻2



⌫ 3⌫ 2 2 ⫹ 2 sin ␰ t cos ␰ t 4␰ ␻



ប␻ ប␻ 1 coth • e ⫺⌫t sin2 ␰ t. 2 2k B T ␰ 2

冉 冊

共77兲

The above expression may be put further in the following form after a little bit of algebra, I 0⫽

冉 冊

冉 冊

ប␻ ប␻ ប ប ˙ ⫹⌫G 兲 2 兴 ⫹ coth coth 关 1⫺ 共 G 2␻ 2k B T 2␻ 2k B T ¨ ⫹⌫G ˙ 兲兴 ⫹ ⫻关 2G 共 G

冉 冊

ប␻ ប␻ G 2共 t 兲 . coth 2 2k B T

共78兲

With Eqs. 共76兲 and 共78兲, Eq. 共73兲 results in

具 xˆ 2 共 t 兲 典 QS⫽ 具 pˆ 2 共 0 兲 典 G 2 共 t 兲 ⫹ 具 xˆ 2 共 0 兲 典 G˙ 2 共 t 兲 ⫹G 共 t 兲 G˙ 共 t 兲 ⫻具 pˆ 共 0 兲 xˆ 共 0 兲 ⫹xˆ 共 0 兲 pˆ 共 0 兲 典

and 1 G 共 t 兲 ⫽ sin共 ␰ t 兲 e ⫺⌫t/2 ␰

冕 ⬘冕 t

dt

0

t⬘

0

dt ⬙ G 共 t⫺t ⬘ 兲 G 共 t ⬘ ⫺t ⬙ 兲 c 共 t ⬘ ⫺t ⬙ 兲

ប␻0 ⫽⌫ប ␻ coth 2k B T ⫽I 0 共 say兲 .



t

0



⫹ 具 xˆ 2 典 QS 1⫺

共75兲

with ␰ ⫽ 冑␻ 2 ⫺⌫ 2 /4. With these simplifications the last term in Eq. 共73兲 results in 2

共72兲

G 2 共 t ⬘ 兲 dt ⬘ 共76兲

S 2共 t 兲 2 具 xˆ 2 典 QS



⫹2GS˙ 共 t 兲 ⫹ 具 pˆ 2 典 QSG 2 共 t 兲 .

共79兲

Here we have defined ˙ ⫹⌫G 兲 , S 共 t 兲 ⫽ 具 xˆ 2 典 QS共 G

共80兲

具 xˆ 2 典 QS⫽

ប ប␻ , coth 2␻ 2k B T

共81兲

具 pˆ 2 典 QS⫽

ប␻ ប␻ coth . 2 2k B T

共82兲

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We now take care of the following limits for a consistency check. As t→0 we have ˙ ⫽1, G

¨ ⫽⫺⌫, G

S 共 0 兲 ⫽ 具 xˆ 2 典 QS ,

G⫽0,

共83兲

S˙ 共 0 兲 ⫽0,

共84兲

and as t→⬁ we have ˙ ⫽G ¨ ⫽0. G⫽G

共85兲

From Eqs. 共79兲 and 共85兲 it follows immediately that the thermal equilibrium averages of xˆ 2 and pˆ 2 are given by Eqs. 共81兲 and 共82兲 derived earlier by others, e.g., Caldiera and Leggett.13 Equation 共79兲 is exactly identical to Eq. 共8.29兲 of Grabert et al.5 in the Markovian limit for product separable initial states of the system and the heat bath. Furthermore, Eq. 共84兲 also satisfies their Eqs. 共6.65兲 and 共6.66兲.5 Similarly one can calculate 具 pˆ 2 (t) 典 QS as

具 pˆ 2 共 t 兲 典 QS⫽ 具 P 2 典 S ⫹ 具 ␦ pˆ 2 典 .

共86兲

Repeating the procedure in a similar way we can show that the above expression may be written as a sum of contributions from thermal average value and quantum dispersion. We quote the final results as below: 2

具 pˆ 共 t 兲 典 QS⫽ 2



i, j⫽1



具 x˙ˆ i 共 0 兲 x˙ˆ j 共 0 兲 典 ⫹ 具 pˆ 2 典 QS关 G˙ 2 共 t 兲 ⫹1 兴

S˙ 2 共 t 兲 ˙ S¨ , ⫹2G 具 xˆ 2 典 QS

共87兲

where ˙ xˆ , xˆ 1 ⫽G xˆ 2 ⫽Gpˆ . Equation 共87兲 corresponds to Eq. 共8.26兲 of Grabert et al.5 for factorized initial condition in the Markovian limit. IV. A NUMERICAL SCHEME FOR SOLUTION OF c-NUMBER LANGEVIN EQUATION

Equation 共14兲, the fluctuation-dissipation relation, is the key element for generation of quantum noise. The left-hand side of this equation suggests that f (t) is a ‘‘classical’’ random number and 具 f (t) f (t ⬘ ) 典 S is a correlation function which is classical in form but quantum-mechanical in its content. Since one of the most commonly occurring noise processes is exponentially correlated color noise, we develop the scheme first for this process. This is based on a simple idea that c-number quantum noise may be viewed as a superposition of several Ornstein–Ulhenbeck noise processes. It may be noted that Imamog¨lu,33 a few years ago, has produced a scheme for approximating a class of reservoir spectral functions by a finite superposition of Lorentzian functions with positive coefficients corresponding to a set of fictitious harmonic oscillator modes to incorporate non-Markovian effects into the dynamics. The method is an extension of the stochastic wave function method in quantum optics. The present superposition method is reminiscent of this approach in the time domain. We however emphasize two important points. Because of rotating wave and Born approximations Imamo¨-

glu’s scheme based on master equation is valid for weak system-reservoir coupling, i.e., weak damping regime. The present method on the other hand uses a superposition of noises in the time domain and the quantum Langevin equation is valid for arbitrary coupling since no weak coupling approximation or rotating wave approximation has been used in the treatment. Second, while the stochastic wave function approach as well as the density operator equation have no classical analog, the present scheme makes use of c-numbers to implement the classical numerical method quite freely to solve quantum Langevin equations 共15兲–共16兲. A. Superposition method

We proceed in several steps as follows: We first denote the correlation function c(t⫺t ⬘ )(⫽ 具 f (t) f (t ⬘ ) 典 S ) in the continuum limit as 1 2

c 共 t⫺t ⬘ 兲 ⫽





0

d ␻ ␬ 共 ␻ 兲 ␳ 共 ␻ 兲 ប ␻ coth

⫻cos ␻ 共 t⫺t ⬘ 兲 ,

冉 冊 ប␻ 2k B T

共88兲

where we have introduced the density function ␳共␻兲 for the bath modes. In principle an a priori knowledge of ␬共␻兲␳共␻兲 is essential and a primary factor for determining the evolution of stochastic dynamics governed by Eqs. 共15兲 and 共16兲. We assume a Lorentzian distribution of modes given by Eq. 共29兲 where ⌫ and ␶ c are the dissipation in the Markovian limit and correlation time, respectively. These two quantities and the temperature T are the three input parameters for our problem. Equation 共29兲 when used in Eq. 共4兲 in the continuum limit yields an exponential memory kernel 共30兲. For a given set of these parameters we first numerically evaluate the integral 共88兲 as a function of the time. A typical profile of such a correlation is shown in Fig. 1 for three different temperatures 共solid lines兲 and for ⌫⫽1 and ␶ c ⫽1. In the next step we numerically fit the correlation function with a superposition of several exponentials as c 共 t⫺t ⬘ 兲 ⫽

兺i



Di ⫺ 兩 t⫺t ⬘ 兩 exp ␶i ␶i



i⫽1,2,3,... . 共89兲

The set D i and ␶ i , the constant parameters are thus known. In Fig. 1 we exhibit these fitted curves 共dotted lines兲 against the corresponding solid curves calculated by numerical integration of Eq. 共88兲. It may be checked that at high temperature the required fitting curve is, in general, a single exponential, whereas at very low temperatures down to absolute zero biexponential fitting is an excellent description. At the intermediate temperature one has to use even triexponential fitting, in general. The numerical agreement between the two sets of curves based on Eqs. 共88兲 and 共89兲 in Fig. 1 suggests that one may generate a set of exponentially correlated color noise variables ␩ i according to

␩˙ i ⫽⫺

␩i 1 ⫹ ␰ 共 t 兲, ␶i ␶i i

共90兲

where

具 ␰ i 共 t 兲 典 ⫽0,

共91兲

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J. Chem. Phys., Vol. 120, No. 19, 15 May 2004

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8969

We then set E i ⫽exp(⫺⌬t/␶i), where ⌬t is the integration step length for Eq. 共90兲. Following the integral algorithm of Fox et al.,23 we now generate the next step ␩ i 兩 t⫹⌬t from ␩ i according to the following lines: a⫽random number,

共99兲

b⫽random number,

共100兲



h i⫽ ⫺

2D i 共 1⫺E 2i 兲 ln共 a 兲 ␶i



1/2

cos共 2 ␲ b 兲

共101兲

and

␩ i 兩 t⫹⌬t ⫽ ␩ i E i ⫹h i . FIG. 1. Plot of correlation function c(t) vs t as given by Eq. 共86兲 共solid line兲 and Eq. 共87兲 共dotted line兲 for the set of parameter values mentioned in the text at three different temperatures k B T⫽3.0, 1.0, 0.0 共units arbitrary兲.

具 ␰ i 共 0 兲 ␰ j 共 ␶ 兲 典 ⫽2D i ␦ i j ␦ 共 ␶ 兲

共 i⫽1,2,3,...兲 ,

共92兲

in which ␰ i (t) is a Gaussian white noise obeying Eqs. 共91兲 and 共92兲; ␶ i and D i are being determined from numerical fit. The quantum noise ␩ i is thus an Ornstein–Ulhenbeck process with properties

具 ␩ i 共 t 兲 典 ⫽0,

共93兲





Di ⫺ 兩 t⫺t ⬘ 兩 . 具 ␩ i 共 t 兲 ␩ j 共 t ⬘ 兲 典 ⫽ ␦ i j exp ␶i ␶i

共94兲

Clearly ␶ i and D i are the correlation time and the strength of color noise variables ␩ i , respectively. It is also important to point out that they have nothing to do with true correlation time ␶ c and strength ⌫ of the noise f (t). The c-number quantum noise f (t) due to the heat bath is therefore given by f 共 t 兲⫽



i⫽1,2,...

␩i .

共95兲

Equation 共95兲 implies that f (t) can be realized as a sum of several Ornstein–Ulhenbeck noises ␩ i satisfying 具 f (t) f (t ⬘ ) 典 S ⫽⌺ i 具 ␩ i (t) ␩ i (t ⬘ ) 典 . 22 Having obtained the scheme for generation of c-number quantum noise f (t) we now proceed to solve the quantum Langevin equations 共15兲 and 共16兲 along with the quantum correction equations 共34兲–共39兲. To this end the first step is the integration23,24 of the stochastic differential equation 共90兲. We note that to generate Gaussian white noise ␰ i (t) of Eq. 共90兲 from two random numbers which are uniformly distributed on the unit interval, Box–Mueller algorithm may be used. To be specific we start the simulation with integration of Eq. 共90兲 to generate an initial value of ␩ i as follows: m⫽random number,

共96兲

n⫽random number,

共97兲



␩ i⫽ ⫺

2D i ln共 m 兲 ␶i



1/2

cos共 2 ␲ n 兲 .

共98兲

共102兲

After Eq. 共102兲 the algorithm is repeated back to Eq. 共99兲 and continued for a sufficient time evolution. In the next step we couple the above-mentioned integral algorithm for solving Eq. 共90兲 to standard fourth order Runge–Kutta 共or other兲 algorithm for the solution of Eqs. 共15兲, 共16兲, and 共34兲–共39兲. For an accurate and fast algorithm ␩-integration 共102兲 may be carried out at every ⌬t/2 step, while for Runge–Kutta integration of Eqs. 共15兲–共16兲 and 共34兲–共39兲 step size ⌬t is a good choice. With ⌬t in the range ⬃0.001–0.01 the overall method gives excellent results for the solution of quantum Langevin equation.

B. Spectral method

The method of superposition of Ornstein–Ulhenbeck noises for generation of quantum noise as discussed above primarily pertains to situations where the correlation function can be described by a superposition of several exponentials. Although in overwhelming cases of applications in condensed matter and chemical physics at low-temperature where in general non-Ohmic conditions prevail we encounter such situations, there are cases where the noise correlation exhibits power law behavior or is Gaussian in nature. To extend the present scheme we now show how the spectral method can be implemented for generation of quantum noise. The method had been used earlier by several groups25–28 in the context of classical long and short range temporally correlated noises. We give here a brief outline for its extension to quantum noise. The starting point of our analysis is the knowledge of quantum correlation function c(t⫺t ⬘ ) as given in Eq. 共88兲. Again this depends on ␬共␻兲␳共␻兲, which 关for example, for short range Gaussian correlated noise may be taken as 2⌫ exp(⫺␻2␶2c /4), ⌫ and ␶ c being the strength and the correlation time, respectively兴 must be known a priori. Having known c(t⫺t ⬘ ) as a function of time we first carry out its Fourier transform

具 f 共 t 兲 f 共 t ⬘ 兲 典 S ⫽c 共 t⫺t ⬘ 兲 , C共 ␻ 兲⫽



共103兲

exp共 ⫺i ␻ t 兲 c 共 t 兲 dt

by discretizing the time in N⫽2 n intervals of size ⌬t 共such that ⌬t is shorter than any other characteristic time of the system兲. The result is a string of discrete numbers C( ␻ n ). In discrete Fourier space the noise correlation is given by

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具 f 共 ␻ n 兲 f 共 ␻ m⬘ 兲 典 S ⫽N⌬tC 共 ␻ n 兲 ␦ m⫹n,0 .

共104兲

Therefore the noise in the Fourier domain can be constructed simply as f 共 ␻ n 兲 ⫽ 冑N⌬tC 共 ␻ n 兲 ␣ 共 ␻ n 兲 ,

共105兲

where i⫽0,1,2,...,N;

f 共 ␻0兲⫽ f 共 ␻n兲;

␻ n⫽

2␲n . N⌬t

Here ␣ n are the Gaussian complex random numbers with zero mean and anti-delta correlated noise, such that ␣ n ⫽a n ⫹ib n ; a n and b n 共the real and imaginary parts of ␣ n ) are Gaussian random numbers with zero mean value and variance 1/2,

具 a n 典 ⫽ 具 b n 典 ⫽0; 具 a 2n 典 ⫽ 具 b 2n 典 ⫽ 21 ,

共106兲 where n⫽0;

共107兲

FIG. 2. Plot of quantum turnover at two different temperatures. The rates are normalized by the classical TST value. The solid lines represent the quantum results of Rips et al. 共Fig. 4 of Ref. 21兲 while the dotted lines correspond to the present simulation for ប ␻ b /k B T⫽2.5 共upper pair of curves兲 and ប ␻ b /k B T⫽1.0 共lower pair of curves兲 with V 0 ⫽10k B T.

To begin with we consider a potential of the form V 共 X 兲 ⫽⫺ 13 AX 3 ⫹BX 2 ,

and a 20 ⫽1.

共108兲

The Box–Mueller algorithm may again be used to generate the set of random numbers a n , b n 共and consequently ␣ n ) according to the lines, p n ⫽random number,

共109兲

q n ⫽random number,

共110兲

a n ⫽ 共 ⫺ln p n 兲 1/2 cos 2 ␲ q n ,

共111兲

and similarly for b n . The discrete Fourier transform of the string f ( ␻ n ) is then numerically calculated by the fast Fourier transform algorithm to generate the quantum random numbers f (t n ). The recipe can be implemented for solving Eqs. 共15兲–共16兲 and 共34兲–共39兲 as discussed in the earlier subsection.

V. NUMERICAL APPLICATION TO NONLINEAR OSCILLATOR: BARRIER CROSSING DYNAMICS

We now present the numerical results on c-number quantum Langevin equation for a cubic potential. Our concern here is to solve Eqs. 共15兲–共16兲 for quantum mean values X(⫽ 具 xˆ 典 ) and P(⫽ 具 pˆ 典 ). The quantum dispersion around the mean path are taken into account by following the equations of quantum corrections, e.g., Eqs. 共34兲–共39兲 or Eqs. 共45兲– 共51兲 etc., depending on the situation. Since the system is coupled to heat bath it is also necessary to take statistical mean over several thousands of realizations of trajectories of the quantum mechanical mean values. One of the major aims of this numerical simulation is to show that the method described is Sec. IV A can be implemented very easily to describe both thermally activated processes1,2,11,14 –19,29–32 and tunneling within a unified scheme.

共112兲

where A and B are the constants to be chosen for a fixed set of barrier height V 0 and the frequencies in the well ␻ 0 and at the barrier top ␻ b . The other input parameters for our calculations are temperature T, strength of noise correlation ⌫ and correlation time ␶ c . For the given potential, Q(t) is given by a single quantum correction term Q⫽A 具 ␦ xˆ 2 典 according to Eq. 共20兲. Theoretical consideration demands that the activation barrier V 0 must be much larger than the thermal noise, i.e., V 0 Ⰷk B T. In order to ensure the stability of the algorithm we have kept ⌬t/ ␶ c Ⰶ1, where ⌬t is the integration step size. For a given parameter set we have computed the time for the quantum Brownian particle in the cubic potential starting from the bottom of the metastable well at x⫽0 to reach the maximum at x 0 ⫽2B/A for a single realization of a stochastic path. The statistical average time to reach the barrier top is calculated by averaging over 5000 trajectories. We have used ប⫽k B ⫽1 for the entire treatment. V 0 is expressible as V 0 ⫽8B 3 /3A 2 , or writing ␻ 2b ⫽2B, we have V 0 ⫽ ␻ 2b /3A 2 . To test the efficacy of the method, we now present and compare with others the results obtained by the simulation of the quantum damped cubic potential 共112兲 driven by quantum noise using Eqs. 共15兲, 共16兲, 共45兲–共51兲, and 共90兲. Our results for quantum Kramers’ turnover for the cubic potential is shown in Fig. 2 for Ohmic dissipation on a logarithmic scale. We choose V 0 ⫽10k B T and plot 共dotted lines兲 the barrier crossing rate k 共reciprocal of the time for crossing兲 divided by the transition state theory result k TST at the two different temperatures for the values ប ␻ b /k B T⫽2.5 共upper dotted line兲 and ប ␻ b /k B T⫽1.0 共lower dotted line兲 against the scaled dissipation parameter log(⌫/␻b). The results 共normalized with respect to Ref. 21兲 have been compared with the analytical results of Rips et al. 共Fig. 4 of Ref. 21兲 obtained by a completely different scheme.33,34 The agreement is found to be quite satisfactory. An important advantage of the present method is that it is equipped to deal with vacuum field assisted tunneling as well as the activated tunneling

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J. Chem. Phys., Vol. 120, No. 19, 15 May 2004

FIG. 3. Arrhenius plot for the system with cubic potential (V 0 ⫽5ប ␻ 0 ; ␻ 0 ⫽2) for two different damping constants. The solid lines represent the quantum results for Grabert et al. 共Fig. 10 of Ref. 35兲 while the dotted lines correspond to the present simulation for ⌫⫽2 ␻ 0 ␣ with ␣ ⫽0.5 共upper pair of curves兲 and ␣ ⫽1.0 共lower pair of curves兲.

within an unified description since the Wigner distribution remains a valid nonsingular distribution even at absolute zero and quantum correction to nonlinear potential can be calculated significantly order by order. In Fig. 3 we exhibit 共dotted lines兲 quantum Arrhenius rate in terms of log(k/␻0) as a function of ប ␻ 0 /k B T for the cubic potential with barrier height V 0 ⫽5ប ␻ with ␻ 0 ⫽2, and frequency independent damping ⌫⫽2 ␻ 0 ␣ for two ␣-values, ␣ ⫽0.5 and ␣ ⫽1.0. The quantum corrections have been calculated by employing Eqs. 共45兲–共51兲. The normalized results have been compared with those of Grabert et al. 共Fig. 10 of Ref. 35兲. It is apparent that while at high temperature, the behavior is linear corresponding to the century-old Arrhenius classical result, significant quantum enhancement occurs at low temperature which has the usual form of 1/T 2 . We thus observe that quantum Kramers’ turnover and quantum Arrhenius rate are in close conformity with the earlier results. The numerical method for solving quantum Langevin equation as worked out here is thus capable with dealing with the problem of dynamical theory of reaction rate in terms of a simple algorithm which mimics its classical counterpart and is independent of path integral approaches. VI. CONCLUSION

In this paper we have proposed a simple numerical method within several approximation schemes for solution of the generalized quantum c-number Langevin equation. The method is based primarily on two aspects. First, it has been shown that it is possible to realize quantum noise 共in a quantum Langevin dynamics兲 which satisfies quantum fluctuation-dissipation relation as c-numbers rather than as operators. Second, the quantum correction terms which appear due to the nonlinearity of the potential of the system, can be generated systematically order by order, so that the coupled set of equations for the stochastic dynamics of the system and the correction terms can be solved easily with sufficient degree of accuracy. The method of simulation is equipped to deal with arbitrary correlation time, temperature and damping strength. The mapping of quantum stochastic

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dynamics as a problem of classical simulation as presented here allows us to implement classical methods in quantum simulations. The method is much simpler to handle compared to path integral quantum Monte Carlo as employed earlier in the problem of barrier crossing dynamics.11 We have shown that the exact agreement of our results on damped harmonic oscillators for arbitrary memory kernel and damping strength with the standard results has served as consistency check of the present theoretical framework. Second, our simulation results on quantum barrier crossing dynamics for a particle in a cubic metastable potential agree well with the results obtained earlier. We hope that with the prescription given, many problems with nonlinear potentials where the time correlation may not follow a typical prescribed dynamics can be simplified within the framework of this simulation. ACKNOWLEDGMENTS

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