Quantum Brownian motion in a periodic potential: a pedestrian approach

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Quantum Brownian motion in a periodic potential: a pedestrian approach

C. Aslangul, N. Pottier, D. Saint-James

To cite this version:

C. Aslangul, N. Pottier, D. Saint-James. Quantum Brownian motion in a periodic po- tential: a pedestrian approach. Journal de Physique, 1987, 48 (7), pp.1093-1110.

�10.1051/jphys:019870048070109300�. �jpa-00210531�

Quantum Brownian motion in a periodic potential: ^a pedestrian approach

C. Aslangul (1,*), N. Pottier (1) ^{and D.} Saint-James (2,**)

(1) Groupe ^de Physique ^{des Solides} de l’Ecole Normale Supérieure (+), Université Paris VII, 2, place Jussieu, 75251 Paris Cedex 05, ^France

(2) Laboratoire de Physique Statistique, Collège ^de France, 3,

d’Ulm, ⁷⁵⁰⁰⁵ Paris, ^France

(Requ ^{le 26} janvier 1987, accepté ^{le 6}

1987)

Résumé.

La dynamique ^d’une particule quantique couplée ^à

environnement de manière ohmique ^et

déplaçant ^dans

potentiel périodique ^{est étudiée au} moyen d’un développement ^{direct de} perturbations ^de l’équation ^{du mouvement} par rapport à l’amplitude des ondulations du potentiel. Ainsi,

^faisant uniquement appel ^{à la} mécanique quantique élémentaire, ^nous calculons la mobilité de cette particule Brownienne dans les deux cas d’une force extérieure finie et constante et d’une petite ^force appliquée harmonique. La mobilité

statique ^est reliée à la mobilité d’une particule

^un réseau décrit en liaisons fortes par ^une transformation de dualité établie antérieurement. De plus, ^{un calcul de} réponse ^{linéaire en} présence ^d’une petite ^force appliquée harmonique ^montre que la particule

comporte ^{comme une} particule ^libre équivalente

coefficient de frottement modifié (dépendant ^{de la} fréquence), ^{et, en} conséquence, ^avec

spectre ^{de bruit} modifié, ^ces

deux quantités ^étant reliées par le théorème de fluctuation-dissipation.

Abstract.

The dynamics ^of

quantum particle coupled ^{to an} environment in an ohmic way and moving ⁱⁿ

periodic potential is examined by ^{means of a direct} perturbation expansion ^{of the} equation of motion with respect ^{to the} amplitude ^{of the} potential (corrugation strength). Thus, by using only elementary quantum mechanics, ^we calculate the mobility of this Brownian particle ^{in both cases of}

finite and constant external force and of

small applied ^harmonic force. The static mobility ^{is found to} be related to that of

particle ^{in a} tight-binding ^lattice by

duality transformation previously demonstrated. Besides, it is shown by

^linear

response calculation in the presence of

small applied harmonic force that the particle behaves like

equivalent ^free particle ^{with a modified} (frequency-dependent) ^friction coefficient, and, consequently,

modified noise spectrum, ^both being ^related by ^the fluctuation-dissipation ^theorem.

Classification

z

Physics ^Abstracts 05.30

05.40

1. Introduction.

In the present paper, ^we would like to investigate, using only elementary quantum mechanics, ^{the mo-}

tion in a periodic potential ^{of a} particle ^{in the}

presence of ohmic dissipation, ^{that is of a} particle

described by ^a Langevin equation ^{with a constant}

friction coefficient _or, which amounts to the _same, with white noise in the classical (high-temperature)

limit. For such a Brownian particle, ^the Heisenberg representation ^{of the} coordinate q (t ) obeys ^the equation ^{of motion}

In equation (1), q ^denotes the friction coefficient,

V (q ) ^{is the} periodic potential ; F s (t) ^{is a} systematic (not random) applied ^{external force and} Fr (t ) ^{is a}

random force with known spectral properties.

As is well known, ^this equation ^{of motion can be}

obtained by properly coupling ^the particle ^{to a bath}

of an infinite number of degrees ^of freedom, ^the coupling ^to the environment providing ^{both the}

friction term and the fluctuating ^{force. A} very simple

model which has been extensively ^{studied in the last} past years ^is the Caldeira-Leggett ^model [1], ⁱⁿ

which the coordinate q ^{of the} particle ^{is linearly} coupled ^{to an infinite set} of harmonic oscillators.

The Hamiltonian of the total system (particle plus bath) ^is given by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019870048070109300

For such a linear coupling, ^{one obtains an} equation

of motion for q (t ) ^{which in} general ^{differs from} equation (1) since it contains a retarded friction

term. The microscopic expressions ^{for both the}

friction term and the fluctuating ^{force can be ob-}

tained [2] : ^the fluctuating ^{force is a} linear combi- nation of the creation and annihilation operators ^of the boson bath and the friction coefficient is linked to the spectral density ^{of the} fluctuating ^force by ^the fluctuation-dissipation ^theorem. Following [1], ^one

then goes ^{to a} continuous representation ^{for the bath}

modes and sets the simple prescription ^{about the}

bath density ^{of modes} p (w ) ^{and the} coupling

constant G «(JJ ) ^{of the} particle ^{to the bath :}

where Ie (CJJ / CJJ e) is ^some cut-off function such that

Ie (0) ^{= 1 and} decreasing ^{on a} frequency range ^of order wc (ohmic dissipation model). By rejecting

CJJ e ^to infinity, ^{it is} possible ^{to derive an} equation ^of

motion of type (1), ^{in which the friction term is}

instantaneous.

In the absence of _any potential, ^{or in a} quadratic potential, ^the solution of equation (1) ^is fairly simple, ^{since this} equation ^is linear with respect ^to

the particle coordinate. One can either calculate the time evolution of average quantities, ^{such as the}

average particle position, and thus obtain the

mobility, ^or calculate correlation functions, ^which yield ^{for instance the} expression ^of the diffusion coefficient. The properties ^{of a} damped quantum free particle ^{and of a} damped quantum ^harmonic oscillator have been investigated ^{in detail} along

these lines by several authors [2-6].

However, for other types ^of potentials, ^non-

linearities arise so that a solution of equation (1) ^can only be obtained by approximate ^{methods ; this is}

the case of the periodic potential ^{of the form}

studied first by ^{M. P. A. Fisher and W.} Zwerger [7],

and later on by ^W. Zwerger [8]. Actually, ^the

calculations performed ⁱⁿ references [7] ^and [8]

primarily ^{concern a} tight-binding ^{lattice. Indeed, as}

was shown by ^{A. Schmid} [9] ^and by ^{M. P. A. Fisher}

and W. Zwerger [7], there exists a duality ^transform-

ation between the periodic potential (4) ^{and an} appropriate tight-binding ^{model, the} hopping ^inte- gral being proportional ^{to V. This} duality ^transform-

ation yields ^a relation between the mobilities of the

particle ⁱⁿ both situations.

Besides, ^a direct calculation of the mobility ^{of the} particle ^{in a} periodic potential of the form (4) ^has

been recently proposed by U. Eckem and F. Pelzer

[10]. They ^use field-theoretical methods based on

the Keldysh formalism and they ^{determine the} generating functional for real-time finite-tempera-

ture correlation functions ; ^the mobility of the par- ticle is expressed ^{in terms of a} self-energy, ^which

itself is calculated through ^a perturbation expansion

with respect ^{to the} potential strength. ^An expression

for the frequency-dependent particle mobility ^{in the}

linear response range is thus obtained.

Actually, ^{since the} quantum statistical properties

of a free particle ^{in the} presence of ohmic dissipation

are well known [2, 3], ^{one is led to think that for}

some

weak » potentials, ^the effect of the potential

on the particle may be treated as a perturbation

around its free Brownian motion. The potential (4)

can easily ^{be handled in this} way and the clue ^{to a} solution of this problem ^{lies in the} following ^{fact :}

since the fluctuating ^force Fr (t ) acting ^{on the} quantum ^Brownian particle ^{is a} linear combination of the boson bath creation and annihilation operators, ^all the statistical properties ^{of not} only

the free particle position operator qo (t ) ^but equally

of sin { Qo (t ) } ^{can be known} (where ^{we have}

defined the dimensionless position operator Qo (t )

(2 7T fa) qo (t )). ^{Indeed, in the} following ^we

show that the average value and the correlation functions of sin { Qo (t )} ^{can be} expressed simply ⁱⁿ

terms of the corresponding quantities ^for qo (t ) ^itself.

This property ^{is the} quantum analog ^{of the} following

well-known property ^{of classical random} processes : if Fr (t ) ^{is a Gaussian} fluctuating ^force, qo (t ) ^{is also a}

Gaussian random process, and the statistical proper- ties of _exp {i Qo (t )} ^can be calculated in terms of the first two cumulants [11, 12]. ^{However, in the quan-}

tum case, the calculations are made a little more

difficult since the position operators ^{at different}

times do not commute [13]. ^{In the} present paper, ^we take advantage ^{of the} Gaussian-like property ^{of the}

boson variables to calculate by ^{a direct} simple perturbation expansion ^of equation (1) ^with V (q ) ^as given by equation (4) ^all the statistical properties ^of

the particle position ^and velocity ^{at the lowest non-}

zero order with respect ^{to the} corrugation strength.

If we assume that the particle and the bath have been in thermal equilibrium in the infinite past, by

this extremely straightforward ^and transparent ap-

proach ^{we recover} the results of the field-theoretical method developed in reference [10]. Our calcula- tions can be achieved either for a finite external force (non-linear situation), ^{or in a} linear response limit.

The paper ^is organized ^as follows : in section 2, ^we outline the principle ^{of the} perturbation expansion

of the equation ^{of motion.}

In section 3, ^we restrict the study ^{to the case of a}

finite and constant external force Fs; the static non-

linear mobility ^{of the} particle in the presence of ohmic dissipation ^{in a} periodic potential ^{is calculated}

and is shown to be related, ^as expected, ^{to the}

mobility ^{in a} tight-binding ^lattice [7, 8,14] by ^the duality transformation [7, 9].

In section 4, ^{we consider the case of a small} harmonic applied ^force F s (t) ^{and we derive the} frequency-dependent mobility ^and diffusion coef- ficients. The results obtained can be given ^a very

simple physical interpretation : in the presence of a

periodic potential, ^the particle ^{behaves like an} equivalent ^free particle (with possibly ^a mass renor-

malization) subjected ^{to a modified} (frequency-de- pendent) ^friction coefficient, and, consequently, ^{to a}

modified noise spectrum, ^both being ^related by ^the fluctuation-dissipation ^theorem.

2. The perturbation expansion.

2.1 THE MODEL. - With the potential V (q ) ^as given by ^formula (4), equation (1) ^{reads :}

where a is the period ^{of the} potential ^{and V is an}

energy characterizing ^the amplitude of its variations

(corrugation strength). ^{We shall see in the} sequel

that the motion of the particle obeying equation (5)

can be handled as a perturbation ^{around the Brow-}

nian motion of the same particle subjected ^{to the}

same friction and to the same noise, ^{but in the}

absence of potential, provided ^{that V remains much}

smaller than an energy of ^the order of 1í’Y, ^with

y = 11 / M. ^{Note that, since} y-’ represents ^the relaxation time of the free Brownian particle, ^this

condition on the amplitude ^{V of the} potential ^also

means that the time scale associated with V must remain much larger ^{than the} relaxation time of the free Brownian particle.

2.2 SUMMARY OF THE FREE PARTICLE RESULTS.

For a free particle ^{with ohmic} dissipation, ^the equation ^{of motion reads} simply :

Let us recall that since in equation (6), the friction

term is instantaneous, ^{the free} particle ^{results have}

to be used in the infinitely ^short memory time limit

(WC --+ 00).

To specify ^{the statement of the} problem ^com- pletely, ^{it is} necessary ^to give ^some information about the initial preparation ^{of the} system (particle plus bath). ^{We shall assume, as in our} study ^{of the} quantum ^{Brownian motion of a free} particle [2], ^that

the particle ^{has been} put ^{in contact} with the bath

itself in thermal equilibrium ^at temperature ^{T - at a} time to

^{oo so that at} any finite time the system

(particle plus bath) has reached a thermal equilib-

rium state. This is also the hypothesis ^made by ^U.

Eckem and F. Pelzer in [10]. ^This hypothesis

eliminates transient phenomena and the average

over the particle ^initial velocity ^is automatically

built-in in the calculations. Thus, only ^the average

over the bath variables will have ^to be carried out in the following. Moreover, ^{this choice of} the initial time insures that all the correlation functions will

depend only ^on the time differences. The fluctuating

force Fr (t ) corresponds ^{to a} stationary ^random

process. ^{In terms} of the boson bath creation and annihilation operators, ^its microscopic expression ^is

(see Eq. (2) ^{for the} definition of the coupling

constants Gn). The average value of Fr (t ) ^is equal ^to

zero. In the ohmic dissipation ^model (3) ^with

wc

oo, the spectral density ^of the random force is

[2] : ¹

Note that, owing ^{to the} quantum ^{character of the} bath operators, ^the operators Fr (t ) ^and Fr(O) ^{do not}

commute.

Before proceeding ^{to the} perturbation expansion

of the equation ^{of motion} (5), ^{let us} briefly ^recall

some properties ^{of the} trajectory qo(t) ^{of the free}

Brownian particle. Clearly, qo (t ) ^{is the sum of a} systematic part, ^{which we} shall denote as qOs(t), ^and

of a random part qor (t ). ^{Whereas the} systematic part

depends ^{on the} applied ^{force and} corresponds ^{to the}

average particle trajectory, ^the fluctuating part ^is only ^a function of the bath operators. Namely, ^one gets

and, similarly

QOr(t) ^is equal ^{to zero on} average, ^{and its} quadratic

mean value is given by :

with the function A2(t) ^{as defined} by

Note the appearance of the dimensionless parameter

a

which is a measure of the dissipation strength ⁱⁿ

the periodic potential

The spectral density associated with QOr(t) ^{is :}

At this point ^{it is} important ^to emphasize ^{that the} position operators ^at different times do ^not com-

mute. By using ^{the formula}

(see for instance Ref. [2]), ^one readily ^{shows that}

where the function A1 (t ) ^is given by

Note that, ^since to ^{has been} rejected to - oo,

equation (16) depends only ^on the difference t - t’.

2.3 PERTURBATION EXPANSION.

Let us write the

particle coordinate under the form

with successive terms of increasing ^{order in V.}

Clearly, qo (t ) obeys equation (6) while that the first- order correction q 1 (t) ^satisfies

Equation (19) ^is analogous ^{to the free Brownian}

motion equation (6), ^the quantity

now playing ^{the role of the force.}

Equation (19) ^can formally ^be integrated, yielding

In order to calculate the corresponding first-order correction to the particle mobility, ^{one has to}

evaluate (S (t )) , ^or (exp{:tiQo(t)}). ^{The free}

particle position qo (t ) ^{is the sum of two} commuting quantities, ^the systematic part qos (t ) ^and the fluctuat-

ing part qor(t). Clearly

Since qor (t ) ^{is a} linear combination of the bath

operators, ^{it is} easily ^{seen that}

or, since (Qor(t»

^0,

The r.h.s. of the above equation ^is real, ^{and one} simply gets :

In other words, ^the average first-order force felt by

the particle ^is equal to zero, ^so that there is no first- order correction to the mobility. This is coherent with the fact that the sign ^{of V cannot} play any role in the expression ^of mobility. (Note ^{that the} integral

in the exponent diverges, ^{as a} consequence ^of the ohmic dissipation law).

One must therefore study the second-order correc-

tions. The equation ^{of motion at} second-order is

where the notation sin {Qo (t) + Q 1 (t )} (1) ^means

that only first-order terms must be retained in the

development ^{of this} quantity. ^This development ^has

to be achieved with some caution, owing ^{to the fact}

that the operators qo (t ) ^and q1(t) ^do not commute.

By properly differentiating ^the exponential operators

[15-17], ^{one obtains}

The detailed derivation of this operator identity ^is explained ⁱⁿ appendix ^A.

It is now necessary ^to compute ^{the series of} commutators involved in equation (27). This calculation is carried out in appendix ^{B. As a} result, ^{one finds that the} second-order equation ^{of motion is :}

where the functions 1 (t" - t ) ^and X2 (t " - t ) ^are

power series ^of the function A1 (t" - t ) :

In order to obtain the evolution of the second order correction to the average particle position q2 (t », it ^{remains to} calculate the correlation functions

and

where, ^{as indicated} above, the averages have ^to be carried out over the bath variables.

Two types ^{of terms are a} priori ^{involved in this} calculation, ^{that is either terms of the} type

in which the two exponentials ^have arguments ^{of the}

same sign, ^{or terms of the} type

with different signs ^{of the} arguments. ^{The com-}

mutator of the position operators ^at different times

being ^{a c-number} (see Eq. (16)), ^{one has,} according

to the Glauber formula

Note that only ^{the first factor in} equation (30) ^{has to}

be averaged. Proceeding ⁱⁿ analogy ^{with the calcu-}

lation done to obtain formula (24), ^{one can} easily ^be

convinced that the terms with arguments ^{of the same} sign ^{involve a} divergent integral ^{in the} argument ^of

an exponential, ^{and thus} give ^{a null} contribution to the correlation functions. The only non-zero con-

tributions come from the terms with arguments ^of different signs, ^or, following ^a terminology ^{used in}

references [7-10], ^from the « neutral charge ^con- figurations

^{As a} result, ^one gets :

so that:

Finally, by taking ^{into account the} developments (29) ^the equation ^{of motion} (28) ^{can be} averaged

into

To proceed further in the calculation, ^{one has to} specify ^{the detailed} time-dependence ^{of the} applied

external force FS (t ). ^{In section 3 we} shall first examine the case of a constant external force, ^not necessarily small ; then, ^{in section} 4, ^{we shall} study

the linear response ^{to a small} applied ^harmonic

force.

3. Finite constant external force.

In this section, ^we discuss in detail the behaviour of the mobility ^{when the} applied ^{external force} FS (t )

^{F is} time-independent. ^No hypothesis ^will

be made about its intensity.

3.1 CALCULATION OF THE MOBILITY.

We have

(see Eq. (9)) : ¹

Inserting this result in formula (32), it appears that,

in the presence of ^{a constant} external force, ^the correlation functions

and

depend only ^{on the time} difference t" - t ; ^{this is a} consequence of the choice of the initial time to

^{oo. This means that at} any finite time a

stationary regime ^is obtained ; ^{this is even more} evidently displayed ^{in the} following.

Indeed, by using ^the explicit ^form (17) ^{of the}

function A1 (t ), ^the equation ^{of motion} (33) ^is easily

rewritten as :

The r.h.s. of the above equation being ^{a constant,}

the time-independent particle velocity ^is equal ^to

The corresponding ^static non-linear mobility A ^{in the} periodic potential ^{is thus :}

where

denotes the free particle mobility ^{and E}

^{Fa is the}

energy drop ^{over one} period ^{of the} potential. ^Note

that in formula (37), ^the correction to the free

particle mobility JL 0 involves the ratio V / liy, ^which

indeed plays ^{the role of an} expansion parameter ⁱⁿ the problem. ^{This will} be discussed more thoroughly

later.

3.2 DUALITY RELATION WITH THE MOBILITY IN A TIGHT-BINDING LATTICE.

The result of (37) ^for

the particle mobility ^{in the} periodic potential (4),

calculated to second order with respect ^{to the}

corrugation strength, ^{involves an} integral ^{which is}

identical to the mobility ^{of a} particle ^{on a} tight- binding lattice, ^{at the same} temperature ^{and with the}

same friction coefficient q [7, 14], provided ^{that an} appropriate correspondence between the other par- ameters of both models is taken into account. This is in accordance with a general theorem of A. Schmid [9] ^and of M. P. A. Fisher and W.

Zwerger [7], ^who have established that there exists a

duality transformation between the periodic poten- tial (4) ^{and an} appropriate tight-binding model, ^the hopping integral being proportional ^{to V. In its most} general ^{form the} duality ^relation [7, 9] ^reads (with

the appropriate parameters)

As ’a matter of fact, either in the tight-binding

lattice [14], ^{or, as above, in the} periodic potential (see ^formula (37)), ^we have shown that the particle mobility ^{can be obtained} by ^a very simple ^{and direct} perturbative ^{treatment of the} equations ^{of motion.}

Similar results are obtained for the periodic potential

in reference [10] by using ^the Keldysh method ; however, ^{the formalism} used in this paper ^{is rather}

complicated. Although ^the integral ⁱⁿ (37) ^{can be} directly calculated, ^we shall take advantage ^{of the} correspondence (39) ^{to use} the results for the

mobility ^{in a} tight-binding ^lattice already ^obtained

for instance in references [7] ^and [14].

First, ^{let us} briefly recall what are the correspon- dence rules between the two models : while the temperature ^{and the friction} coefficient ^{are the}

same in both problems, ^{the other} parameters ^are related in a non-trivial manner. VTB

V /2 ^{is the} tight-binding hopping integral ; ^the dimensionless

dissipation strength ^and the lattice spacing ^{in the} tight-binding ^{lattice are} given by « TB = 1 / a ^and

aTB

a/« respectively. Moreover, ^{the term corre-} sponding ^{to the} product p «(ù ) 1 G «(ù ) 12 in ^{the tight-}

binding ^{model is not} simply given by IíTJ (ù / 7T ^{as in}

the periodic potential ^{but is of the form}

where

in other words, ^a Lorentzian cut-off is automatically

built-in in the tight-binding model, ^with the cut-off

frequency

cTB

y

’T1 / M, ^{which is not present in}

the periodic potential.

Thus the form of the functions A1 (t ) ^and A2 (t ),

involved in equation (37), ^{and defined} by equations (17) ^and (12) respectively ^{is a} priori imposed by ^the equation ^{of motion of the} particle ^{and should in} principle ^{be used in order to calculate} J.L TB’ However,

we have shown in reference [14] ^{that the} mobility ⁱⁿ

the tight-binding lattice is in fact largely independent

of the precise ^form of the cut-off function provided

that one assumes that both the hopping integral VTB ^{and the} energy drop I £ I TB = ^{a I F} TB ^{are much}

smaller than hw _cTB’ In terms of the parameters ^{of the}

periodic potential, ^{it means} that, ^{if one assumes that}

both V and

are much smaller than hy, the results will be largely independent ^{of the} precise ^{form of} the cut-off function. We shall take advantage ^{of this} property ^to

use at sufficiently ^low temperatures ^{a different cut-} off function (namely ^an exponential one,

fcTB (- / y )

exp (-

/ y ), for which the analytic

calculation has been performed ⁱⁿ [14]). ^Thus,

instead of using ^the expressions (17) ^and (12) ^{for the}

functions Al (t ) ^and A2 (t ), ^{we shall write :}

and, ^for kB ^{T «} hy,

Let us now examine the physical meaning ^{of the}

two conditions required ^for deriving ^results indepen-

dent of the precise form of the cut-off function.

First, the ratio Vlhy clearly plays the role of the

expansion parameter ^{in the} problem ; ^{indeed we}

have supposed ^{that the} particle trajectory ^{does not}

much move away from the free Brownian particle trajectory. ^This evidently implies that the time scale associated with V must remain much larger ^{than the}

relaxation time M/q

y -1 ^{of the free Brownian}

particle. Similarly, the second condition I E I ^{« a 9y}

can be rewritten as

which means that the characteristic acceleration time of the particle by ^the applied ^{force must} equally

remain much larger ^{than the} relaxation time y -1 ^of

the free Brownian particle. Note that the two conditions V 9y and I e I

^{9y , which insure the}

irrelevance (except for numerical factors) ^{of the} specific ^{form of} the cut-off function fcTB’

^{must be}

satisfied in our perturbative approach. However,

f cTB ^is by construction a Lorentzian and, ^{in the} framework of a more refined calculation, ^the equiva-

lence between two different cut-off functions is not evident [8].

3.3 RESULTS FOR THE MOBILITY. - At this stage, ^it would be sufficient to refer to [7] ^and [14] ^which

treat the tight-binding ^{lattice. Here we} shall however

quote the main results for completeness ^{and we} postpone ^the physical discussion of the results as

well as of their validity ^{to the} subsequent paragraphs.

3.3.1 Low temperatures (kB ^T 9y ).

^{In this case}

we use the exponential ^cut-off function ; ^{the func-}

tions A1 (t ) ^and A2(t) ^are given by equations (40)

and (41). ^The computation ^{is then} straightforward

and yields ^the

^and T-dependent mobility ^tt ( E, T).

For the sake of simplicity, ^{we shall} only give ^{here the}

results corresponding ^{to the two} limits T -+ 0 and

e -+

0, although ^{the full} expression ^{can be written}

down without difficulty (see for instance Eq. (33) ⁱⁿ

Ref. [14]).

(i) Zero-temperature ^limit

One obtains :

In this formula, it appears that, apart ^{from the} negligible exponential ^factor exp {- I e I / a fry} - ^1,

the mobility depends only ^{on the} parameter

a result which has been demonstrated by ^W. Zwerger [8] by using scaling arguments. ^{From now on, we} shall drop ^the exponential ^{factor in} equation (43).

Note that, ^for

1, ^the correction to 110 goes ^to

infinity ^{when s -} 0, ^{so that the linear} mobility ^does

not exist at zero temperature. ^This point ^{is to be}

discussed later on.

(ii) ^Finite temperature : linear limit

At finite temperature, ^{a linear} mobility ^{can be}

defined and reads

3.3.2 High temperatures (kB T > hy ).

^{In this}

case we use the Lorentzian cut-off function. The functions At (t) ^and A2 (t) ^{are now taken as} given by equations (17) ^and (12). ^The s-dependence ^{of the} mobility ^is negligible ^{in this} temperature range [14].

In order ^to obtain a classical result, ^{we follow the} procedure ^of references [7] ^and [8]. ^We replace ^the

factor sin {At (t)} ⁱⁿ equation (37) by ^its argument

and we approximate ^{the function} A2 (t) by

Two different types ^{of behaviour are} observed, according ^{to the value of the} parameter

introduced by ^{M. P. A.} Fisher and W. Zwerger [7]

(w o

(2 w la) J V / M ^denotes the small classical oscillation frequency ^{around one of the} potential minima).

When the parameter ^{K is much smaller than} 1,

A2 (t) - Ky t, ^{and one can} replace ^sin f A, (t)) ^{by its}

argument provided ^that kB T > 9y. ^{One obtains}

When K is much higher ^than 1, A2 (t) - Ky 2 t2/2

and one can replace ^sin fA,(t)l ^{by its} argument

provided ^that kB T > Eo

^{4 7T 2} IP / Ma2. Eo ^{is the}

localization _{energy in} a well of the potential ^and plays ^an important ^{role in the} problem, ^{as we shall}

see in the subsequent discussion. One obtains

Note that (47) ^and (48) ^{are classical results, in}

accordance with references [7] ^and [8]. This is the

reason why ^we used the classical parameter WO instead of (1/h) .JVEo.

3.4 DISCUSSION OF THE VALIDITY OF THE CALCU- LATION.

Let us now discuss the conditions of

validity ^{of this} perturbation expansion. ^{As indicated} above, ^{the ratios} V /9y and 1 8 1 /a hy ^{must be small.}

A necessary condition for the validity ^{of the} pertur- bation treatment is that the mobility ^{in the} periodic potential ^{does not much} differ from the free particle mobility. ^Since the result of the calculation takes the form of (39), it will be valid only ^as long ^as

,"TB (with the convenient parameters) remains much smaller than _go.

3.4.1 Low temperatures (kB ^{T 9y} ).

(i) Zero-temperature ^limit

By looking ^{at the} appropriate expression (43) ^of

the mobility, ^{one obtains the} validity ^condition

For all

1, this condition is automatically ^fulfilled

since in our treatment both factors in equation (49)

have to be much smaller than 1, whereas for

« >

1, ^it implies ^{that the} applied force should be

high enough ^so that the energy drop ^{over one} period

satisfies

1

(As ^{indicated above,} the linear mobility ^{stricto sensu}

cannot be defined in this case).

This can be physically understood as follows : when a

1, ^the mobility ^{in the} tight-binding ^lattice (where aTB 1) ^{increases when the} applied ^{force F}

decreases while the mobility ^{in the} periodic potential

decreases. The system ^{thus exhibits a} tendency

towards localization. Note however that a completely

localized state would be meaningless ^{in our} perturba-

tive approach. ^{In other words, formula} (50) ^states

that the applied force has to be strong enough ^to prevent ^the particle ^from localizing.

(ii) ^Finite temperature : linear limit

At finite temperatures, ^{a linear} mobility ^{can be}

defined (Eq. (44)) ^{and we will restrict the discussion}

to this case. The validity ^{condition reads :}

For all

1, this condition is automatically ^fulfilled since ^both factors in equation (51) ^{are small, whereas,}

for a

1, ^it implies ^{that the} temperature ^{should be} high enough ^{so that}

This can be physically understood as follows : when

a >

1, ^the mobility ^{in the} tight-binding ^lattice

(where a TB 1) increases when the temperature

decreases while the mobility ^{in the} periodic potential

decreases. Formula (52) simply ^{states that, in order}

to obtain a sufficiently high mobility, ^the tempera-

ture has to be high enough ^to prevent ^the particle

from localizing

3.4.2 High temperatures (kB T > 1í’Y).

^{In the} high-temperature range kB T >> hy, ^{two different} expressions ^{for the} mobility ^{are obtained} according

to the value of the parameter K ^{as defined} by equation (46).

When K 1, ^the validity condition is automati-

cally ^{fulfilled as soon as the ratio} V /9y ^{is a small}

parameter.

When K > 1, ^the quantity (VlkB T)312 (wo/y )

must remain much lower than 1. One must then have

which is not a very restrictive condition since

To conclude this discussion, ^{let us make some}

comments about the physical meaning of the condi- tion V 1í’Y. It may ^{seem at} first sight ^{that this}

condition does not involve the period ^{of the} potential

or the localization _energy Eo ^{in a} well of the

potential. However, ^{it is} interesting ^to rewrite the above condition in terms of

Indeed, 1í’Y ==

a Eo/2 ^{7T, so that the} condition V 9y ^transforms

into

For

1, this condition automatically ^{insures that}

V Eo, ^{which means there are no bound states and}

the mobility ^{does not differ} very ^much from the

mobility go of the free particle. ^{On the} contrary, ^for

a >

1, ^condition (53) ^{does not insure} automatically

that there are no bound states. In this _case, condi- tions (50) ^{on the} applied ^{force or} (52) ^{on the} temperature ^{mean that the} applied ^{force or the}

temperature ^{must be} sufficiently high ^{to extract the} particle ^{from the} possible ^{bound state, or, in other} words, ^{that the bound state} becomes irrelevant. The

perturbation ^treatment around the free Brownian motion is then justified.

The inequality V « 9wo (= ,,IVEO) (equivalent

to V Eo) ^{could have been used to define a}

«

weakly corrugated potential » [10]. Note that the

periodic potential ^{has then no bound states. The} foregoing discussion shows that potentials ^of larger amplitudes ^can indeed be studied by ^a perturbational

treatment around the free Brownian motion ; ^{this is}

the case if

1 and if the applied ^{force or the}

temperature ^are sufficiently high.

To conclude this discussion, ^{we can} say that the

validity ^{domain of} the method is in fact _very large,

and only ^{excludes, when a}

1, ^{a small} region ^{in the} vicinity ^{of T or F} equal ^{to zero.}

3.5 DISCUSSION OF THE RESULTS.

The discussion of the behaviour of the mobility ^{as a function of the} temperature ^{and of the} applied ^{external force will}

not be undertaken here in detail, ^{since this has} already ^been achieved in references [7, 8] (periodic potential ^and tight-binding lattice) ^and [14] (tight- binding lattice).

Let us however make some comments about our

results.

3.5.1 Classical regime.

^At high temperatures, that is when kB T > hy, ^a comparison ^{with the}

results of a standard classical Langevin equation ^is possible. ^{Two different} types ^{of behaviour are} observed, according ^to the value of the parameter K (Eq. (46)). ^{We have} by hypothesis V « 9y « kB ^T.

(i) ^{Let us first} study ^{the case where the} parameter K is much smaller than 1 ; ^{this can occur} only ^if wo/y JV /kB T, ^{which means that the classical}

oscillator at the bottom of a potential ^{well is} strongly

overdamped. The classical regime ^{is a diffusive one,} governed by ^a Smoluchowski equation [7, 8], and, ^at high temperatures, ^the mobility > tends towards the free particle mobility tLo ^as T-2 (see ^formula (47)).

(ii) If instead the parameter K ^{is much} larger ^than 1, ^the only requirement ^on w o/ y ^{is that}

which is not very restrictive. The mobility > ^then

tends towards the free particle mobility uo ^as

T-312 (see ^formula (48)).

3.5.2 Quantum regime.

^{Let us first} concentrate

on the case kB ^T« hy. ^{When this condition is} fulfilled, ^the mobility ^is given by equation (44).

However, ^the A-dependence ^{of this} expression ^is

rather intricate, ^since

llh. ^For a > 1, ^formula (44) gives

so that quantum ^{effects are} certainly important ^for

Note the singularity ^{in h In A in formula} (54), ^which

is not uncommon in that kind of problem.

As shown by ^W Zwerger [8], ^for kB T >> hy, ^and

in the overdamped ^limit (K 1), ^which implies

a > 1, ^the quantum corrections are in h2. _Thus,

when one goes from high (kB T > 9y ) ^{to low} (kB ^TT hy ) temperatures, ^the quantum corrections in the overdamped ^limit cross over from a h2

behaviour to a h In h one. This is explained ^{in more}

detail in appendix ^C.

When quantum ^{effects can be} neglected, ^formula (54) ^{reduces to} the classical result (47). ^{It is} interesting ^{to note that formulae} (44) (and (54)

which is deduced from it) ^{on the one hand, and the}

classical result (47) ^{on the other hand were derived} by using ^two different cut-off functions.

Finally, ^{let us} compare ^our result (37) ^{with that of}

U. Eckern and F. Pelzer [10]. They ^{have found a}

correction to the friction coefficient, ^{due to cor-} rugation, and it is easily ^{seen that,} for low values of this correction, their result and our result are

identical. They ^{show that, since} they formally ^obtain

a correction to the friction coefficient

and not to the mobility

^{their result for the} mobility ^is always finite, ^but they ^{do not} assign precise validity ^{limits to}

their calculation. In our approach, ^the validity

domains of the two calculations of the mobility ^{in the}

periodic potential ^{and in the} tight-binding ^{lattice are}

the _same, provided ^one takes into account the one-

to-one correspondence ^between parameters ^indi- cated above. The validity ^{domain of the} present calculation of the mobility ^{in the} periodic potential ^is precisely delineated : ^the calculation is valid either for

1 (and ^{then the} corrugation ^{of the} potential

is necessarily weak), ^{or for a}

1, ^where potentials

of larger amplitudes ^can be studied provided only

that the vicinity ^{of either T or F} equal ^{to zero is}

excluded.

4. Linear response ^{to a} small applied ^harmonic

force.

We shall now assume that a small harmonic force F s (t)

^{F e"’ is} applied ^{to the} system ^{and we shall} study the linear response. Let ^us recall that, ^{for a}

free particle, ^the frequency-dependent mobility, ^or complex admittance J.L 0 (Cl) [18], ^{is defined} by

with

4.1 CALCULATION OF THE MOBILITY.

We use the results of section 2, ^{and in} particular, ^the equation ^of

motion (33). ^{We have} (see Eq. (9))

At the lowest order in F, ^the correlation functions which are involved in equation (33), ^{that is}

and

reduce to (see ^formula (32))

By using ^the explicit ^form (17) ^{of the} function Al(t), ^the equation ^{of motion} (33) ^can be rewritten

as

This formula shows that a stationary regime ^is obtained, ^{in which}

The second-order correction Ag (,w’) ^{to the free} particle mobility ^reads

Note that, ^{in the} static limit w - 0, ^{the same result is}

’

recovered either from equation (60) ^{or from} equation (37) ^{with F --+ 0.}

One can equally ^{write the} mobility tk ( w ) ^{in the}

form :

where Aq (w ) ^is complex ^and frequency-dependent.

Formula (60) ^{shows that, at} second order with respect ^to V, Aq (w ) ^is given by

The real part d 1] , (ClJ) ^of Aq (w ) modifies the friction coefficient q ^{of the} particle ^{whereas the} imaginary part Aq " (w ) may be interpreted ^{as a}

mass renormalization term. Before giving any ^ex-

plicit ^results for Aq’ (w ) ^{and Aq} "(w ), ^{let us remark}

that equations (62) ^and (37) ^involve integrals ^{of the}

same type,

playing the role of (27T/a)(F/1]).

Therefore, ^{if we assume that}

is much smaller than y, the results will be largely independent ^{of the} particular form of the cut-off function ; ^{this will}

allow us to use for computational facilities at zero or low temperatures kB ^{T h-y an} exponential ^cut-off

function.

Formula (61) ^{states in} particular ^{that, in the}

presence of ^the periodic potential (4), ^the particle

behaves like an equivalent free ^Brownian particle

with a modified (frequency-dependent) friction coeffi-

cient and a mass renormalization term (possibly frequency-dependent). ^{On account} of ^the fluctuation- dissipation theorem, ^this implies ^that the random

force (and ^{thus the noise which is its} spectral density)

is modified by ^the presence of ^the potential.

We shall examine this point ^more thoroughly ⁱⁿ

the following paragraph.

4.2 NOISE IN THE PRESENCE OF THE PERIODIC POTENTIAL.

Generally speaking, the second fluc-

tuation-dissipation ^{theorem states} that, ^{in a} dissipa-

tive system, the friction coefficient q is linked to the

spectral density of the random force Fr(t) by

This is the case of the free Brownian particle [2]. ^As

demonstrated above, in the presence of the periodic potential, ^the friction coefficient q is modified by ^the

addition of â17’ (Cù ), ^{which in turn} implies ^{that the}

noise is modified by ^the addition of the quantity

We know [7, 14] ^{that this} quantity ^can equally ^be expressed ^{in the} equivalent ^form

in which one recognizes (by using the Wiener- Khintchine theorem) ^the spectral density ^{of the}

random force

This result however, could have been derived in a

completely different way, by looking ^{at the stochastic} equation ^{of motion}

similar to equation (5), but in which the particle

evolves under the sole action of the random force

F, (t) ^{due to the} coupling with the bath. At the lowest perturbation order, ^{the force term due to the} potential ^{is a} stochastic term equal ^to

where the quantity F,(t) ⁺ F’(t) ^now plays ^{the role}

of the random force. This scheme will be consistent with the preceding ^{one if it can be shown that the} spectral density ^{of the} quantity F,(t) + F’(t) ^is actually equal ^{to the sum of} expressions (63) ^and (64) ; ^{in other words, the two random forces} Fr(t) ^and F’(t) ^{must be} uncorrelated.

The expression ^{to be} analysed ^{is the} correlation function

The random force F,(t) ^{is a linear} combination of the bath creation and annihilation operators (see Eq. (7)) ; by using ^the following property ^{of boson} operators (where ^{0 and qi are} any complex quantities)

which states that these average values ^are both

proportional ^to the average value of

one proves that expression (69) ^is proportional ^to

the average value of exp { i QOr (t )}, ^{that is} actually equal ^{to zero} (see Eq. (24)). ^{Therefore the random}

forces F,(t) ^and F’(t) ^{are, as} expected, ^uncor-

related. At second order with respect ^to the corruga- tion strength, ^{the noise in the} system ^{is thus the sum} of expressions (63) ^and (64) (or (65)).

4.3 RESULTS FOR Aq’(W ) ^{AND Aq} " ( w ). ^{- In this} paragraph ^we only give ^the analytic results for

å’T1 ’ (CJJ) and ^The physical implications ^of

these results on the transport coefficients, ^{as well as}

the validity ^{of the} calculation, ^{will be discussed in}

the following paragraphs. ^{As in the static case, we}

shall successively ^examine the low and high tempera-

ture situations.

4.3.1 Low temperatures (kB ^T hy). ^{- We use in}

this case the exponential ^cut-off function ; ^{the func-}

tions Al(t) ^and A2(t) ^are given by equations (40)

and (41).

(i) Zero-temperature ^limit

The real part Aq’(w ) ^{of Aq} (w ) ^is equal ^to

Exactly ^{as in formula} (43), ^the exponential ^{factor is}

very close ^to 1 and will be dropped ^{out in the} sequel.

The imaginary part Aq " (w ) of Aq (w ) is given by