Intermediate-time dynamics of a particle on a disordered tight-binding lattice: quantum dissipation versus disorder

Physica A 159 (1989) 63-90 North-Holland, Amsterdam

INTERMEDIATE-TIME DYNAMICS OF A PARTICLE ON A DISORDERED TIGHT-BINDING LATTICE: QUANTUM DISSIPATION VERSUS DISORDER

C. ASLANGUL" and N. P O T T I E R Groupe de Physique des Solides de I'Ecole Normale Sup&ieure*, Universit6 Paris" VII, 2 place Jussieu, 75251 Paris Cedex 05, France

D. SAINT-JAMES b

Laboratoire de Physique Statistique, Coll~ge de France, 3 rue d'Ub,,~, 75005 Paris, France

Received 1'/January 1989

We investigate by a generalized Born approximation the quan tum dynamics of a dissipative parucle on a t ight-binding disordered one-dimens.onal lattice. When the friction is ohmic or underohmic (at zero temperature) , the particle evolution is shown to be governed at intermediate t imes by a generalized master equation (i.e. re tarded) for lhe site-diag(,n::! element,, of the density malrix. This ml~dei allo~:s for a discussi~m of lhc interplay ~I the effects ,,f quan tum noise and ~,.f disorder.

In the presencr of ohmic di,,sipali~m, and in disordered ~dllicc ~, x~:;lh a Iinile tip, t imct, ,c tnomcn, t~f the lram, ier ratc~, the mean-square displacement {q ( t ) ) hchaxcs pi~p~qti~midl~ to I :" . as in an ordered lattnce t~ is a ttin~cn~it)ntcs~ tl~ca~tt;~ oi ~tt~¢ t.t~tip[ilig ~t~t:~,~.t~ ~t th !he bath). A localized state exists above the value a = I of the coupling, and a strictly

ditfu~.,e regime takes place for ~r = 112. In disofd-ered lattices where the distr ibution of transfer rates diverges like IV '~ with

{I ~: ~ < l, the t i ~ ¢ dependence of (q '{ t~) depends ~m the tv, o parameters ~ and ~. Above c~ = 1, a localized slate stb'.! exists: the diffusive regime occurs al~mg a certain line in the {~. #~ plane. Depend ing t~n both ~ ae.~ ~u. ~arious subdiffusi~c ~r superdiffusixc regimes arc

~blained.

!. Introduction

d i s o r d e r e d t i g h t - b i n d i n g t a i t i c c in t h e [nc>cn~ .c t~I t~hmi~ di>~l[~.ati~;l~ :~',~-

twofold. First, this study can bc vic~ved as the extension to lhc disordered c,~sc

" Also at Universit6 Paris VI, 5 place Jussieu, 75252 Paris Cedcx qb5, Fiance " Also at Universit6 Paris VIi, 2 place Jussicu, 75251 Paris ( ' edcx ~)5. I r a n ¢ c

* Laboratoire associ6 au C.N.R.S.

0378-4371/89/$03.50 © Elsevier Science Publishers B.\'. (North-Holland Physics Publishing Division)

64 C. Aslangul etal. / Quantum dissipation versus disorder

of that of an ohmic quantum particle on an ordered tight-binding lattice. We show that, when the friction is ohmic or underohmic (at zero temperature), the particle evolution is governed, in an intermediate range of time values which will be made precise below, by a generalized master equation (i.e. retarded) for the site-diagonal elements of the density matrix. On a disordered lattice, these transfer rates are random variables. Therefore, and this is the second motivation, the evolution equation can be considered as a generalization to a quantum noise; situation of the classical master equation (non-retarded) governing the excitation dynamics of a particle on a random one-dimensional system in the presence of classical noise. Let us now develop these two points of view in more detail.

The dynam.~cs of a quantum particle on an ordered one-dimensional tight- binding lattice in the presence of ohmic dissipation has been the object of recent extensive studies [1-8]. Whereas at high temperatures (i.e. for k~T >> htoc), where t% is some characteristic frequency linked to the phonon bath, the dynamics is classical-like and a standard diffusive regime is found, at low temperatures kBT ~ hi%), strikingly different features do e m e r ~ . In particular, at zero temperature, a transition towards localization takes place, when the interaction with the phonon bath is sufficiently strong, in proper notations when a parameter a is equal to one. For a > 1, the particle is strictly localized, which means that the mean-square displacement q2(t) tends towards a finite value. For a < 1, the particle is delocalized, but the different authors who tackled the problem seem at first sight to disagree on the behaviour of q2(t).

According to Guinea [3] and to Weiss and Wollensak [8], q2(t) behaves like In t for c~ = 1/2 (as if the particle were not subjected to an external potential), while Zaikin and Panyukov [7] assert that q2(t).--t. Actually all these authors claim to have an exact t reatment for this special value of a, at least in the limit ~%--~oo. For any a < 1, Zaikin and Panyukov [7] argue that qZ(t) follows a power-law--- t 2(~-"~, a result which correctly matches with the well-known a = 0 limit (q2(t)---t2). This in contradiction with the statement by Weiss and Wollensak [8] claiming that, for a , ~ l , the spreading of the probability distribution is slower than diffusive. A precise discussion of the pros and the cons about this problem is beyond the scope of this article. Let us only remark that both treatments [7] and [8] use the same path integral techniques, so that the apparent discrepancies in the results could be explained by an incorrect evaluation in ref. [8] of the time scale beyond which the claimed time behaviour actually shows up. Indeed, a more precise evaluation by Zaikin and Panyukov [9] of the validiiy condition given in ref. [8] for the logarithmic increase of q2(t) (for a = 1/2) re~eals that, for a N-site lattice, the quantity Ft must be very large as compared to a rapidly increasing function of N (thc notation of ref. [8] is used). For instance, with N = 7 (respectively N = 11), one

C. Aslangul eta/. / Quantum dissipation versus disorder 65

obtains the validity condition l"t ~> exp(2rr-') (respectively l"t ~ exp(5rr:)), but not simply Ft~> 1, as it is claimed in ref. [8]. Thus in an infinite lattice the logarithmic increase of qZ(t) as found in ref. [8] is valid at infinite times only, while in an intermediate range of time values (where t may be large as compared to F -~ but non infinite) the behaviour of q2(t) follows a power- law---t 2tl-a), as found in ref. [7].

In two former papers [6], we applied the second order Born approximation to this problem, and we found the same result as Zaikin and Panyukov [7]. In other words, subdiffusive regimes (i.e. q2(t) growing slower than t) are obtained for 1.2 < a < 1 while superdiffusive regimes (with q2(t) growing faster than t) are obtained for 0 < a < 1/2. A strictly diffusive regime only exists for a = 1/2. We are aware of the fact that the Born approximation certainly fails at infinite times; however, both this approach and the infinite resummation achieved in ref. [7] provide for q2(/) the same behaviour---t 2{l-c'). This fact seems to indicate that this is nevertheless a satisfactory approximation - at least for the calculation of qa( t ) - in the intermediate range of time values, the evaluation of which will be made more precise later. This is the reason why we propose an application of this method to the behaviour of a quantum particle in a random one-dimensional lattice, coupled to a phonon bath by the so-called ohmic friction. All our results are to be considered as describing intermediate- time features of the particle dynamics, in the above sense. Since the striking result of localization is strictly obtained only for T = 0, we shall restrict our calculation to this situation.

The excitation dynamics of a classical particie in a random one-dimensional lattice has been studied in great detail by various authors [10-14]. They were able to determine the long-time properties of various autocorrelation functions, among which the mean-square displacement (q2(t)). As a result, the dynamics depends primarily on the probability distribution of the transfer rates, and, particularly, on the form of this distribution near the origin. In the so-called class (i) lattices [11] where the first inverse moment of this distribution is finite, one obtains behaviours which are akin to those in an ordered lattice. In class (ii) lattices [11], where the probability of small transfer rates is much higher,

When the classical noise is replaced by a quantum one, do these behaviov.rs subsist. In particular, do class (i) lattices present the same bchaviours a~ quoted above for an ordered lattice? What can be the consequence of the interplay between the effects of quantum noise and the effects of disorder on

class (ii) lattices? In the next section, we describe the model and we show that. in the presence

of under-ohmic or ohmic friction, the particle dynamics is indeed governed at intermediate times by an evolution equation invol,,ing only the site-diagonal

66 C. As langul et al. I Quan tum dissipation versus disorder

elements of the density matrix, that is by a master equation. This equation is retarded, as opposed to the classical ease where it is instantaneous. In section 3, we extend to the quantum particle an effective medium approximation introduced in refs. [10,11] for the classical case. Classes (i) and (ii) of disordered lattices are successively considered. In section 4, an exact calculation of the evolution of (q2(t)) in class (i) lattices is presented.

2. The model

For a given configuration of the lattice, the particle plus bath Hamiltonian is given by

= ( A , , C . + I C . + h.c.) + a ~ h G ~ ( b ~ + b,,) + ~ h t % b ~ b v + Cste, " " (1)

where A denotes a random overlap integral and a is the lattice spacing. Only overlaps between nearest-neighbour sites are taken into account in the particle Hamiltonian. In eq. (1),

- = n C , , C , , t."t n

pictures the particle coordinate. The added constant stands for renormalization purposes [15]. The distribution of the overlap integrals will be made precise later. By transforming H into Htr = S H S -1, with

S = exp [-~q- ~ G--z~(b a , t%

- b*~)], (2)

one readily obtains for the transformed Hamiltonian

H,~ Hin t + H . h ( B C + + B C ) + ~ * = = + _ h w , , b ~ b , , , (3)

where

B_+ = exp[_+~ mGv (b, , -b'v)] (4) v O0 v

and

r - , ¢ t c . = c_ = c + . (5) n

C. Aslangui et al. / Quantum dissipation versus disorder 67

For an infinite lattice, q and C. satisfy the following comnmtation relations:

q- c . ] = + c . (6)

Let us now write down the evolution equation of the reduced particle density matrix in the new representation• In the Born approximation, it takes the form [6]:

t

Op i 1 f 0--7 = o l -

O

0

with Hint as defined by

I ~ IP

dl" Tr, {H+.t(-r)IH,.,. p ( t - ~')P.I + h.c.}.

(7)

F

H~.,= H~ . , - (H, . t )b=h[(B_ - ( B _ ) b ) C + +(B+ - (B+) . )C_]

= hI~B_C+ + 8B+C_I. (s)

The average values ( . . . ) , are taken with respect te the badl, assumed to be in thermal equilibrium at temperature T. Note tha'. thi- stage, only a given configuration of the overlap integrals has been co~,idcred: the average with respect to the disorder has not yet been carried out.

As well known [15], the central ingredient in such dissipative models is the product of the density of modes of the bath t i r e s the squared coupling constant which, in the continuum limit, produces a smooth function of the frequency A(to). Since behaviours for time values such that oct,> 1 will be concerned, it is sufficient to know the behaviour of A(¢o) at frequencies w ~ toe. In most cases, A(to) may be assumed to behave as to 5 at low frequencies (and to be cut at high frequencies by a cutoff function f,). The ohmic dissipation model corresponds to 6 = 1; the models where 6 < 1 and where 6 > 1 are referred to as respectively the subohmic and superohmic dissipation models [16,17].

When (B+_) b = 0 , there is no proper evolution term in eq. (7), i.e. the first term vanishes. This is realized when 6 ~ 1 (at T = 0) or when ~ ~ 2 tat finite T) [16, 17]. In the following sections, we shall mostly analyse the ohmic dissipation case, which has been the object of extensive investigations in ordered lattices. Since the most striking features of the ohmic dissipation appear at T = 0, we shall restrict ourselves to the zero-temperature case in the analysis will be carried out in the next sections. Note however that any finite tempera-

ture could be treated by the same methods. For the remainder of this section, we shall write down the evolution equation

of the density matrix when the evolution is entirely due to the relaxation terms

68 C. A s l a n g u l et al. I Q u a n t u m d i s s i p a t i o n ve r sus d i s o r d e r

(i.e. when ( B+ ) , = 0). One gets

t

O__.p = j d , ((sB+ p ( t - , ) C + l

Ot o

- c_p(t-

+ p(t- )C_l

- (~B_O')~B+)b[C_, C+p(t- r)]}. (9)

In an ordered lattice, the operators C+ and C_ commute, and it is then easy to directly derive from eq. (9) the evolution equations of the physical quantities of interest, namely of the mean-square displacement qZ(t); this method was used in ref. [6]. When the lattice is disordered, the same type of treatment is not so obviously tractable and it is easier to resort to another method.

Let us rewrite eq. (9) on the basis of the site-localized states {In)}. By taking the detailed expression of the correlation functions of the B operators, one readily shows that the site-diagonal elements p , ( t )= (n lp ( t ) l n ) of the reduced density matrix obey the generalized (i.e. retarded) master equation

dp. dt

t

= 2 f d,r 4)(r)[A2._t[p._,(t- ~') - p , , ( t - ~')] 0

+ A 2 . [ p , , + , ( t - r ) - p . ( t - r ) ] ] . (10)

with the kernel ~b(t) as defined by

~b(t) = cos{ A l(t) } exp{- A 2(0} (11)

In the ohmic model, the functions A l(t ) and A2(t ) are given by [6]

A i( t ) = 2a tan -l coot,

A,t t~-- . ,~ In(1 4-,,,2t2~ 4- 9 , in [ sinh(t /rr) ] z x ' " . . . . , " " - - c " , . . . . . . L t / r T j '

(12a)

k B T - - <~ tl m ¢19h~

(We have introduced the "temperature time" r r = h / k B T and a cutoff function fc = exp(-oJ/%), the precise form of which has no bearing on the dynamics for %t ~> 1 [6]; the parameter a is , dimensionless measure of the coupling strength with the phonon bath.)

Let us now discuss the validity of the above approximation. Clearly, in eqs. (7), (9) or (10), only the first terms of a Born perturbation expansion are

C. Aslangul et al. I Quantum dissipation versus disorder 69

retained. Some condition of the form A~- c '~ 1 is therefore required for these equations to be valid, where :A is some average overlap integral and r, a correlation time linked to the correlation function of the operators ~iB._.. Whatever r c is finite, the calculation is expected to be valid provided that A is small enough. Therefore a necessary condition of validity of eqs. (7), (9) or (10) is I- c finite, the problem being of course the proper definition of %. No definite criterion does exist for this problem since the integrands in these equations are not simple exponentials. Therefore there is no general agreement among authors about the definition of ~'¢. In particular, when T = 0 , the function exp{-A2( t )} behaves like a power law, proportionally to t -2" for tOct >> 1. Clearly, in order to get a finite correlation time, one has to require that the integrals should decrease sufficiently fast with time, which imposes some lower bound on t~.

Let us first comment on the case c~ > 1. As it will be seen below, the particle is localized in that case. Clearly, this result is not barred by the failure of the Born approximation at infinite times. It is a well accepted fact indeed that, in a double-well or in a periodic lattice, the particle is localized. Therefore one expects that it is also localized in a lattice with a bond disorder, since the localized particle cannot explore long distances and thus cannot experience the

effect of disorder. In the following discussion, we shall concentrate on the case a < 1. Once the

condition Ar c "~ 1 is realized, one has to ask the question of the domain ef validity in time of eqs. (7), (9) or (10). This domain is difficult to delineate precisely. Ordinarily in such problems a renormalized time scale arises beyond which the validity of a calculation using a second-order Born expansion cannot be safely ascertained (see for instance the discussion given in ref. [7a]). Here

this time scale is A~ -~ such that

ot 1

tr(12o, cos(o ,l 2,,-°, (13,

However, for an ordered lattice, Zaikin and Panyukov [7], using an infinite resummation procedure, have obtained results for which the validity domain in

time is actually aiven bv

1

m a x o

Note that eq. (14) leads to a larger domain that Art <~ 1. As far as (q2(t)) iS concerned, it turns out that the Born approximation leads to the same result as ref. [7]. Therefore we hopefully adopt, also for a disordered lattice, the

70 C. Aslangul et al. / Quantum dissipation versus disorder

pragmatic point of view according to which the validity domain in time of the behaviour found for the mean-square displacement is given by eq. (14). All the results which will be derived below in the framework of the Born approximation are assumed to hold in the intermediate range of time values, limited as indicated by eq. (14) on the side of large time values (and as it will be indicated below on the side of small time values). Anyhow our aim in this paper is just to give general trends on the combined effects of disorder and dissipation.

Let us now come back to eq. (10). This retarded equation turns out to be a generalization of the instantaneous master equation, widely used in the study of the dynamics of classical excitations in disordered one-dimensional lattices

[10-14],

dp,, = W,, ,,_ (p . - p,,) + W,, .+t(p,,+~ - p,,) (15) dt . t - 1 . ,

where p,, denotes the occupancy probability of site n and W,,.,,+~ = W,,+t., , denotes the random transfer rate between sites n and n + 1. One can also remark that, when a--->0, ~b(t)---> 1 for all t, so that eq. (10) becomes a second-order equation of the form

d'p,, dr2 = w , , , , _ , ( p , , , - p , , ) + w, , , ,+, (p , ,+,-p, , ) , (16)

with the initial condition

dp,, (t = 0) = 0 (17) dt

Eq. (14) represents the evolution of the displacement in a chain of randomly coupled harmonic oscillators [10]. In the original tight-binding model, the particle is then decoupled from the phonon bath and there is no damping. However, a direct derivation from the Liouville equation shows that additional terms involving off-diagonal couplings arise in the equivalent of eq. (16). This displays the failure of the Born approximation at least near a = 0. We shall v v t J t l t t J t ~ , t t t . V I I t l l l , . " J ]75U11111. l¢~l t~ , . , l . i ~ t L V ~ L , I t l I K , I K . , ~ , U I I K . , L ~ . I t l l l l l l d ~ l l l K . ~ t l l d t "v'f ir i] cl~

amounts to modify continuously the "order" of the master equation (10). Indeed we shall scc in the following sections that for the time values of interest the dominant terms of the solutions of eq. (10) and of the first-order equation (15) behave in the same way for c~ = 1/2. Moreover for a f> 1, the localization found below can be viewed as arising from a zeroth-order equation.

It is important to note that eq. (10) does not come out from the phenomcnological considerations, but has been derived from a microscopic model of a particle, coupled to a bath of phonons in an underohmic or in an

C. Ashmgul et al. I Quantum dissipation ve r sus disorder 71

ohmic way, and moving on a one-dimensional disordered tight-binding lattice. The present underlying microscopic model forces the transfer rates to be symmetric. In the usual classical context, it is in principle possible to study the case of asymmetric transfer rates. This has been done in particular in ref. [14]. However, in the case of eq. (16), the requirement of symmetric transfer rates insures that the site-diagonal elements of the density matrix are positive.

Only the site-diagonal elements of the density matrix are involved in eq. (10); this is a direct consequence of the fact that the average values over the bath of the B operators are equal to zero, and thus, this is only true, at T = 0, for underohmic or ohmic models (6 ~< 1). At T = 0, this property no longer holds in superohmic dissipation models (6 > 1). Note that, at finite temperature, this property remains valid for 6 ~< 2. When the average values ( B. )I, are different from zero, the non-diagonal elements of the density matrix are coupled to the diagonal ones, so that the simple form of a master equation is lost. This is reminiscent of the fact that, for an underohmic or an ohmic particle in a double-weP, potential [17] at T= 0, the expected value of the particle coordinate is decoupled from the other dynamical variables, which is

not true for a superohmic particle. Considering the intermediate-time regime as defined above, we now have to

solve at best the retarded master equation (10) for a given configuration ef the transfer rates, in other words to calculate the mean-square displacement q-'(t) in this situation, or, at least, the configuration averaged quantity (q ' ( t ) ) . For this purpose, we shall first resort to an effective medium approximation t EMA) introduced in refs. [10, 11] for the corresponding classical problem, which yields the time behaviour of (q2(t)) (section 3). Then, in section 4, we present an exact calculation in class (i) lattices, which confirms the result of the EMA.

Let us now derive the basic equations which will be used in the next two sections. As in the classical case, we suppose that the initial condition is

p,,(t = 0) = 6,,.,, • (18)

For each function of time f(t), it will be convenient to in.troducc il.~ Lap!ace , . . . . . ¢ ~ , . , ~ . L-'{ - , ~ I . I ¢ . . l l l a l l , , J l l l l l I,<.~.Jl,,,

/

F(z) = j dt exp ( - z t ) f t t ) .

II

Rc z > ~ . rig)

The Laplace transform of the retarded master equation (,1()) roads

zP, , (z) - p, , ( t = o)

= 2¢,( . r . ) lJ - " ,I,D,, , ( z ) - l ' , , ( . : ) l + J ' ! P . , . , . ( -~ ...... f ) . i - ) ! l , i2~.~i


where we have introduced the Laplace transform q~(z) of the function 4~(t) as defined by eq. (11). The detailed expression of ~(z) has been calculated in ref. [ls]:

1 (z)'o-' z} a'rr + - - F(1 - 2a)

toe

+ m ~.. _ toe n=0

. s i n { Z + n~r} ,oo T

n - 2 a + l (21)

The Laplace transform of the classical master equation (15) is

zP. (z ) - p . ( t=O) = W . . . _ , [ P . _ , ( z ) - P,,(z)] + W..,,+t[P.+,(z ) - P.(z)]. (22)

In the classical problem, one usually assumes that the transfer rates W..,.+~= W.+t,. are independent random variables, distributed according to a given probability density n(W). By comparing eqs. (20) and (22), one sees that in the quantum case the role of W...+~(=W.+~..) is played by the quantity 2q~(z)('4.) 2. In order to retain time-independent random variables, it will be convenient in the quantum problem to put

W. - 2"42" ~o¢ (23)

3. The effective medium approximation

The study which follows will be restricted to the intermediate-time regime as defined above; in particular, the time values will be bounded by the inequality (14).

3.1. The classical EMA

Before dealing with the quantum problem, let us briefly recall the principle of the effective medium approximation (EMA) as developed for the classical situation in refs. [10, 11]. If one introduces the quantities G+(n >10) and G~ (n ~< 0) as defined by

. . ,t.-

- = w , ( p - p )/p G n z,n_+l n-+-I n (24)

C. Aslangui et al. / Quantum dissipation versus disorder 73

they obey thc rccursion relations

[ , 1 ] - G2 = W,, .± ! z + G,± !

1

• ( 25 )

In the EMA, the random lattice is approximated by an effective ordered lattice, in which all the W,.,+t's are replaced by a z-dependent effective transfer rate W~ff(z). This effective transfer rate is determined by a convenient- ly chosen self-consistency condition• As discussed in refs. [10, 11], this self- consistency condition is not unique; a possible choice is to construct it from the recursion relation (25). If all the G', 's are to be replaced by g~,(z), this last quantity is determined by

f 1 ] gctt = d W n( W ) -~ + . (26) g e f f q- Z

0

Note that this amounts to average over all the possible configurations of the lattice. From now on we will denote by ( . . . ) the average over the distribution n ( W ) of the transfer rates.

3.2. The E M A in the quantum case

In the quantum case, the self-consistency condition reads

gef f = dW n(W) Wq~(z)¢o c q g¢ff + z o

(27)

with q~(z) as given by expression (21). Following refs. [10, 11], we shall successively consider two different classes

of lattices, according to the probabilty distribution of the Wn's.

(i) Class (i) lattices" in these lattices, the first inverse moment of the probabili-

ty density is finite, that is

m_, = f dW - - - ~ < (28)

0

(ii) Class (ii) lattices" in these lattices, the probability density n(W) diverges at the origin like a negative power of W,


n(W)={(l -0, Ix) W-/.t 0 ~< W < W,n,

w l - p "

m

otherwise,

(29)

the exponent Ix being such that 0 < Ix < 1. The sharp cutoff at W= W m is chosen for convenience, but clearly any sufficiently fast decreasing cutoff function of W could be used for the analysis at times t >> W~ ~. (The values IX > 1 are excluded since n ( W ) would not be integrable; for /z <0 , the probability density (29) would describe a particular case of a class (i) lattice.)

In class (i) lattices, we shall denote by m k and kth-moment of the distribution when it exists (k is a relative integer):

m =f t . (30) \ \ ¢D c

0

In class (ii) lattices, all these moments are simply proportional to powers of

Win. We shall proceed to an asymptotic analysis of the EMA condition, as given

by eq. (26) (classical case) or by eq. (27) (quantum case). By asymptotic analysis, we mean an analysis for times t >> m_ 1 (in class (i) lattices) or t >> Wm ~ (in class (ii) lattices), yet times bounded as indicated by the inequality (14). It is possible to give in each class of lattices a precise condition of validity of the expansion, involving all the parameters of the problem, in particular a (in class (i) lattices), or both c~ and Ix (in class (ii) lattices). For the sake of simplicity, we shall give in each class of lattices an approximate condition of validity, involving only the moments of the probability density.

3.3. Diffusion coefficient and mean square displacement in class ( i) lattices

(a) Classical case At small z (that is for jzj <~(m_~)-z), the asymptotic analysis of the EMA

,:,tuauOt~ 1,~oj reveals that its solution behaves as [i0, 11]

] I/2 z g~rf "" ~ , (31)

provided that the first inverse moment m_ l of the probability distribution of transfer rates be finite, which is by definition realized in class (i) lattices. One can define the average complex diffusion coefficient (D(z)} as the Laplace

C. Aslangu! et al. I Quantum dissipation versus disorder 75

transform of the velocity autocorrelation function. It is related tt, ,g,.t,(z) by [10, III

( D ( Z ) ) -" g e f f ( g e f f "t" Z ) / Z . (32)

For small z, it is given by

1 ( O ( z ) ) . - - ~ , (33)

m - 1

so that for times t>> m_, the mean-square displacement ( [ q ( t ) - q(0)] 2) -~

(q2(t) } behaves as

2 a 2 ( q 2 ( t ) ) -'- ~ t . (34)

m _ 1

Thus, in these lattices, the behaviour is strictly diffusive, as in an ordered lattice. Note that the appearance of m_ 1 is sensible since when m_ ~ is large one expects a bottleneck effect due to the relatively high weight of the small transfer rates in the distribution n(W). Indeed m_ t measures the time scale for exploring distances much larger than the unit cell.

( b ) Q u a n t u m case

Two different situations have to be distinguished, depending on the coupling strength with the bath.

c. < 1 : ~(z) behaves like z 2~-t for [ z [ ~ % (see eq. (21)). The asymptotic ~malysis for [z[ ,.~ (re_t) -~ of the E M A condition (27) reveals that its solution behaves as

"rr z (35) g e f f " " _ 2 s i n ( a w ) F ( 2 a )

The diffusion coefficient (D(z ) ) is then, for small z , given by

2~ -- I

" " i .

( D ( z ) ) m , 2 s i n ( a ~ ) F ( 2 a ) %,

so that for times t ~ m _ ~ the mean-square displacement ( q : ( t ) ) behaves as

( q2(t ) ) ... a 2 cos(ce rr) , w ~ ( 1 / A 2 ) ( 1 / 2 - c~)(1 - c~) ( % t ) - " - " ' (37}

76 C. Aslangul et al. / Quantum dissipati6,n versus disorder

As in an ordered lattice [6], ~he quantum dt~mped particle does not undergo a strictly diffusive motion, the latter being only obtained for a = 1/2, in which ease

4a a "tr ( q2(t)) ,,- to~(lla2) -~ ¢o¢t. (38)

Thus asymptotically for times t >> m_t , the particle in class (i) lattices exhibits the same behaviour as in an ordered lattice. The particle is delocalized: (q2(t)) follows a power-law--- t 2(l-a). Subdiffusive regimes (i.e. (q2(t) ) growing slower than t) are obtained for 1/2 < a < 1 while superdiffusive regimes (with (q2(t)) growing faster than t) are obtained for 0 < t~ < 1/2. A strictly diffusive regime only exists for a = 1/2, with a diffusion coefficient equal to

1 ';,r (D(z)) -- m_~ 2 ' (39)

which can be related to the fact that ~(z)---> const. (=~r/2toc) in that case (see eq. (21)). Note that, when t~ = 0 +, a quasi-coherent motion with (q2(t)) --- t 2 is recovered; this result would indeed be correct in an ordered lattice but seems dubious at first sight in a one-dimensional disordered lattice, for which a localization due ;o disorder is expected for a = 0. This result is probably related to the failure of the Born approximation mentioned above.

At this point a comment on the small time scales of this problem is in order. In our dissipative model of a particle on a disordered tight-binding lattice, two frequency scales are involved. The first one, denoted as toe, is a cutoff frequency characteristic of the phonon bath; the other one, denoted as (m_~) -~, involves both the overlap integrals and toe (see eq. (23)). In deriving the evolution equation of the particle density matrix at the Born approximation order, we have assumed that the overlap integrals A n remain much smaller than %; from eq. (23), one sees that this in turn implies that (m_~) -t .~. %. Since the asymptotic analysis of the EMA condition is carried out for Izl (m_,)- ' , it is well justified in this analysis to replace q~(z) by an approximate expression valid for I zl ,o=.

> 1: ~ (z ) behaves like q~,z for ]zl'~ ~o¢, where ~ is a constant (see eq. (21)). The foregoing asymptotic analysis is no more valid, since the ratio (g~ff + z ) /W~(z )% behaves like a constant. The EMA condition (27) now reads

? 1

g~. = (g~ff + z) J dW n(W) (40) o 1 + g,m + z

Wq~ o z ~o~

C. Aslangul et al. / Quantum dissipation versus disorder 77

and has the solution g~n = go z. Eq. (40) yields

go = q,o,Oo f dW n(W) o

W

W~ow¢ 1 +

l + g 0

(41)

One easily verifies that for t~ - 1 >> 1 the solution of eq. (41) is approximately given by

go-.- m l OoWc (42)

and the diffusion coefficient by

( D ( z ) ) .-. m , ~o t%Z . (43)

As for the mean-square displacement, it is equal to

( q 2 ( t ) ) . . ~ a 2 (A 2) 1 (44) 2 to c (c~ - 1/2)(c~ - 1)

In the opposite limit a - 1 ,~ 1, the solution of eq. (41) is -vv,'"'~"~";'""~l',~.,,,,,,.,.-,?

given by

( -.- ~ ( 4 5 ) go m_ t

and the diffusion coefficient by

1 ( D ( z ) ) --- ~ OotOcz . (46)

m -1

The mean-square displacement is equal to

9

a" 1 (qZ(t)) --- ,o~_(!/A2,3 ( ce - 1/2)(ce -- 1) " (47)

As in an ordered lattice [6], the particle is localized for a > 1. due to the ohmic friction: (qZ(t)) tends towards a constant. This constant diverges for a = 1; by comparing eqs. (37) and (47), it is seen that on b( '~h sides of a = 1, (q2( t ) ) has the same dependence with respect to the moments of the probability density

and to a. Note that the solution (41) of eq. (40) does not rely on an asymptotic

expansion of the E M A condition. Since we used the form ~)z for @(z); it is

78 C. Aslangul et al. I Quantum dissipation versus disorder

required only that Izl Therefore the expressions (44) or (46) of { qZ(t)) are valid as soon as t ~> co ~~, whieh is less restrictive than t-> m_ ~. The particle is then locked in a localized state. Loosely speaking, one can say that the effects of the ohmic friction (localization) are felt by the particle before those of disorder. In other words, the particle, being localized, cannot explore long distances and thus cannot experience the effect of disorder as would do a classical particle (or a quantum particle with a < 1).

Finally, let us underline that formulas (46) and (47) correspond to a rather large extent of the particle on the lattice, (hence the presence of re_l), while formulas (43) and (44) represent a very localized state in which the particle explores only the close vicinity of its starting point (hence the presence of rn t). This is reminiscent of the fact that, for short times, the dynamics of the particle can be obtained as a power-series expansion involving the moments of positive order [10, 11].

3.4. Diffusion coefficient and mean-square displacement in class ( ii) lattices

(a) Classical case In class (ii) lattices, the probability density of the transfer rates diverges at

the origin like W -~' with 0 < / z < 1 (eq. (29)). At small z (for Izl Win), the asymptotic analysis of the EMA equation (26) shows that its solution behaves as [10, 11]

I I

I g~r, "" I_ av(1 - tx) Wm • (48)

The diffusion coefficient { D(z)) is then, for small z, given by

2 /.t

(D(z ) ) "" [ at(1 - / x ) Wmm

and the behaviour for times t ~> W~ t of the mean-square displacement (q2(t)) by

, [ s in ( r r t z ) ] - - - (q2(t) ) --- 2a" 7r(1 - / z )

2 ( l - N ) 2-/a (],Vm g ) 2 - . ( 5 0 )

Due to the disorder, the particle undergoes a subdiffusive motion. As compared to class (i) lattices, the probability of small transfer rates is much higher in class (ii) lattices, which slows down the motion, all the more so as ~ is close to 1 [10, 11]. Note that in ref. [10] the particle is said to be localized in this case. We prefer to reserve the tram of localization for the situations in which


(q:( t ) ) tends towacds a constant. As stated above, [or/~ < 0, the piobabdity density (29) actually describes a class (i) lattice. In that case, a careful analysis shows that the asymptotic behaviour of gen is no more given by eq. (48); one instead recovers eq. (31). Therefore, the validity of eqs. (48)-(50) is limited to 0 < / z < l .

(b) Quantum case Again, two different situations have to be distinguished, depending on the

coupling strength with the bath.

c~ <1: ~(z) behaves like z 2~-1 for Izl ,oo. The asymptotic analysis for Izl Wm of the EMA condition (27) shows that its solution behaves as

[2 g e f f " " sin(a'rr)F(2c~)

( 2 a - 1 ) ( 1 - ~ )

1 !

(51)

The diffusion coefficient (D(z)) is then, for small z, given by

2 ( 1 - ~ )

"IT

2 ( 2 a - 1 ) ( 1 - ~ )

. /a

sin(wtt) ]Y-~ ( z 2-u

x (52)

and the behaviour for times t >> Wm ~ of the mean-square displacement (q2(t)) by

2 ( l - i t )

( q2(t)) " 2a2 2sin(~)F(2c~)

2 sin(rr~t) ]2-~,

X

\ \ q - - |

2 - g 2-~u

2 ( l - ~ t ) 2 ( l - u )

As could be expected, the time behaviour of (q2(t)) depends on the two parameters c~ and Ix. Various subdiffusive or superdiffusive regimes are obtained according to their values. A purely diffusive regime is obtai~,ed u hen

80 C. Aslangul et al. I Quantum dissipation versus disorder

(D(z)) tends towards a constant for small z, or (q2(t)) behaves like t for large t, that is when

1 - / z = 1. (54) 4 ( 1 - a) 2 - / z

Eq. (54) delineates a curve in the (a,/~) plane, which can be referred to as the "diffusive line" (see fig. 1). By construction, in class (ii) lattices, the values of /x are restricted to the interval (0, 1) (see the discussion at the end of the preceding paragraph). For the present time only the values of a between 0 and 1 are considered. The diffusive line (54) intersects the/~ = 0 axis at the point of abscissa a = 1/2, where the diffusive regime is found in the absence of disorder [6]. Above the diffusive line, that is, when the parameter/~ is higher (larger probability of small transfer rates), and/or when the parameter a is higher (stronger coupling with the phonon bath), the behaviour of the particle is subdiffusive. Conversely, below this line, the behaviour of the particle is superdiffusive. Clearly, as a general trend, the presence of disorder slows down the particle motion, as could be expected in class (ii) lattices in which the most important W's are near W=0 . When a - ~ 0 , one gets (q2(t))---t ~ with/3 <2 , but, as in class (i) lattices, this result is wrong, due probably to the failure of the Born approximation.

a > 1: ~(z) behaves like ~oz for [z I ,~ to c, where '/~0 is constant. The EMA condition (27) takes the form of eq. (40), with the probability distribution of transfer rates n(W) as given by eq. (29). Here also, as in class (i) lattices, there exists a solution geff "-go Z" Eq. (40) yields

~.~

1.0

Isupercl!l't'usive\ I. regimes ~

O 0..5

Localization

Fig. 1. Diffusive line in class (ii) lattices, as given by formula (54). Remind that the vicinity of cr = 0 is excluded.


i%, go = @oo¢(l- -' I dW w

0

W

W(/)0 co ¢ 1+

l + g 0

(55)

The solution of eq. (55) can be found in the same way as the solution of eq. (41) in class (i) lattices. When a - 1>> 1, the mean-square displacement (qZ(t)) is approximately given by formula (44). When a - 1 .~ 1 one finds

(q2(t))-.-2a2[Wm (oc

2( I -~ . )

(1 - 2 : ) ( 2 - 2c~)

2 sin(artx) ]2- ,

~(1 - p,) (56)

Thus, in class (ii) lattices, the situation for a > 1 is exactly the same as in class (i) lattices (see above), or as in an ordered lattice [6]: due to the ohmic friction, the particle is localized for (~ > 1, which means that (q2(t)) tends towards a constant. This constant diverges for (~ = 1; by comparing eqs. (53) and (56), it is seen that on both sides of a = 1, (q2(t)) has the same dependence with respect to the parameters of the problem. This divergence is less severe than in class (i) lattices (see eqs. (37) and (47)), which is linked to the fact that class (ii) lattices are in some sense more disordered than class (i) lattices.

The localization threshold due to ohmic friction is thus insensitive to disorder in the transfer rates (at least in the types of disordered lattices studied above) in the sense that it always takes place at the value c~ = 1 of the coupling with the phonon bath.

4. Exact interme. ~e-time dynamics in class (i) lattices

As seen above, in the framework of the EMA calculation, the time behaviour of the mean-square displacement (q2(t)) in class (i) lattices does not differ very much from that in an ordered lattice. This is valid for both the classical and the quantum cases.

In the classical case in which the particle evolves according to the classical master equation (15). exact calculations have been achieved for class (i) lattices [10, 11, 13, 14], which essentially yield the same result as the EMA calculation. Along similar lines, we shall in the present section propose an exact derivation of the time behaviour of the mean-square displacement, appropriate to both :he classical and the quantum cases. In the quantum case, the particle obeys the generalized master equation (10), established for an infinite lattice. The study which follows will be restricted to the intermediate-


time regime; in particular the time values will be bounded by the inequality (14).

Following Derrida [14], we first replace the infinite disordered chain by a periodic one consisting of elementary units containing N sites, in which we compute the mean-square displacement. At the end of the calculation, N will be allowed to go to infinity, which restores the original disordered chain. As usual, is assumed that these two limiting procedures may be commuted. Note that, at this stage, only a given configuration of the overlap integrals is considered. No average with respect to the disorder is carried out.

For the periodic chain one has

W,+Nk = W,, (k integer, - ~ < k <~ + ~ ) . (57)

(The definition of W,, in the quantum case is given in eq. (23).) As in ref. [14], we introduce the quantities

r , ( t ) = ~ p,,+Nk(t) (58) k-'~ - ~

and

s,,(t)= ~ (n+ Nk)p,,+Nk(t ) . (59)

Actually, the knowledge of the quantities r,,(t) and s , ( t ) is sufficient to compute the mean-square displacement.

For the purpose of formal simplicity, let us introduce an N-dimensional Hilbert space spanned by N site-localized states {In)} with n = 0 , . . . N - 1 . One can define the vectors

N - 1

IR(z)) = R,,(z)ln) (60) n = l)

and

N - I

IS(z)) = ~ S , , ( z ) ln ) . (61) n = 0

As demonstrated above, the Laplace transform P,,(z) of p , ( t ) obeys eq. (20). Due to this equation, one gets

IR(z)) = [z + W 4 ~ ( z ) ~ ! - ~ l r ( t = O ) ) . (62)

C. Aslangul et ai. / Quantum dissipation versus disorder 83

The operator W can easily be expressed in terms of two auxiliary operators WD and C, defined as follows:

N - I

wD= Z W.In)<,,I. (63) n=O

N - I

C = ~ In + 1 ) { nl. (64) n=0

On the basis of the localized states {In )}, W D is represented by a diagonal matrix, the elements of which are the transfer rates I4,',. In a class (i) lattice where the first inverse moment m_ ! of the probability density of the W,,'s is finite, none of the W,,'s vanishes since the probability to have a vanishing W,, is zero, and therefore W D is a non-singular operator. As for the operator C, it is a unitary operator, associated with a circular permutation of the basis unit vectors. One can write

W = ( 1 - C)WD(1 - C * ) . (65)

As previously indicated, we assume that, at time t = 0. the particle is deposited at one lattice site. for instance n = 0; the initial density matrix is then given by formula (18). For a given configuration of the transfer rates, the mean-square displacement q2(t) can be calculated as a function of the ,rite-

diagonal elements of the reduced density matrix as

-I- ~¢:

qZ(t ) = a 2 ~'~ nZp.(t). (66)

One easily shows that, in Laplace variables.

[ Nlz~_-{l N - 1

O"(z) = z-'~(z)w c 2(W n - W,_,)S,,(z)+ ~ (W,, - - ) tZ =: (b

+ w,,_,)n,,(z) 1 (67)

or, in operator notations.

N - 1

Q2(z)= Z <nlr(~)) (6s~ n =0

with

IT(z)) : a2z-'~'(z)o~¢ [2(CWD - WD C: ) lS ( z ) ) + (cw,~ + w . c *)1 R(:)) 1.

84 c. Aslangul et al. / Quantum dissipation versus disorder

It is straightforward to show that

Is(z)) = ~(z),o¢ 1 z + W~(z)¢o¢ ( c w D - WDC*)IR(z) ) , (70)

so that

IT ( z ) ) = aZz - l~ ( z ) t% [CW D + WDC*

+ 2q~(z)to¢(CW D - WDC* ) z + wq,(z)toc (CWD- WDC*)]IR(z)).

(71)

Eqs. (68) and (71) allow for an easy derivation of the exact time behaviour of q2(t).

It is important to note that the operator W possesses a non-degenerate zero eigenvalue, which corresponds to the conservation of Tr[p(t)]. The associated normalized eigenvector I fo) is given by

N - 1

Ifo} = N - ' / 2 Z I n } . (72) n = 0

By introducting the projectors P= If,)(f~] and Q = 1 - P , one gets

N - I N - 1

Z R.(z) = Z ( n l R ( z ) ) - N'J2(fol(Z + Wq~(z)~oc)-'lr(t = 0)) n = O n = O

= N"2z -' (fo]r(t=O)) = z - ' , (73)

which means that Tr[p( t ) ]= 1 for all times. Note that If,) is equally an eigenveetor of C and C*, namely

ClJ;> = c*l~,> = IJi,> (74)

and that, in the subspace generated by Q, the operator W does have an inverse. In terms of [f~), one derives the following exact expression of the Laplace transform of the mean-square displacement"

Q2(z ) --a2z-,~(z)tocN ,/2 (folCW D + WDC t

z + Wq)(z)o~ ( c w , , - w . c t ) l R ( z ) ) .

(75)


For clarity, we shall now successively discuss this exact expressimJ in the two cases a < 1 and a > 1.

a < l : q~(z) behaves like Z 2 a - I for ]zl,~o¢ (see eq. (21)). A very simple reasoning on the density matrix shows that the particle is delocalized. Let us rewrite eq. (62) as

(P 1 ) IR(z)) = 7 + Q z + W~(z)¢o< Q Ir(t=0)). (76)

For small z one gets

IR(z)) "- -P Ir(t - O) . Z

(77)

Otherwise stated, R.(t)--> 1 IN at large times. Thus, although the transfer rates are random variables, the equilibrium state is symmetric. The particle is delocalized . By using eqs. (60), (73) and (74), one finds, provided that Izl is much smaller than the smallest non-zero eigenvalue of W:

Q2(z)---2a2z-2@(Z)<o<(f, IWD + W D ( 1 - C+)(QWQ)- ' (C - 1)WDIf,,).

This matrix element is easily computed by taking into account the relation between the inverse of a projected operator and the projected of the inverse

(QMQ) -~ = QM-1Q _ Q M - ' P ( P M - ' P ) - ' P M - ' Q . (79)

In the present case we have

1 QW;'PW~)'Q (80) _ I q (QWDQ) -I =QWDIQ TrWD

so that we obtain

• • • r ~1

l Q-(z).-.- 2a-z -@(z)~< N N - I q 1

t l - (I

and

q-'(t) ---

1

a-" c o s ( ~ ) ~( x-, ( 1 / 2 - c~)(1 - a) (wct)-

E w,7'] ~") (82)

86 C. Aslangul etal. I Quantum dissipation versus disorder

Remember that no configuration average has been carried out. The above results correspond to a given configuration of the transfer rates inside the elementary unit of N sites, which is periodically repeated.

Taking now, as in ref. [14], the limit N---~ ~ in expression (82), one gets for an infinite lattice

a cos(a r) q2(t)" 2¢o¢m_ (1/2"= a) (eoct)" , (83)

which is identical to the expression (37) of (qZ(t)) obtained within the EMA (see also eqs. (23) and (28)). Thus, as far as the dominant term is concerned, the EMA gives the exact result for the mean-square displacement. Note that the above asymptotic analysis is rigorously valid in a periodic lattice with an elementary cell of N sites provided that Izl remains much smaller than the smallest non zero eigenvalue of W. In the N---~ ~ limit, the validity of the analysis requires that the probability density n ( W ) must go to zero rather quickly (m_~ has to remain finite). This requirement excludes the lattices where some of the W,,'s can vanish, among which class (ii) lattices. In class (i) lattices, the occurrence of m_~ in eq. (83) allows one to think that the calculation is valid when t>> m_~, but we have no definite proof of this physically reasonable assertion. (Moreover, this was also the condition of validity derived in the framework of the effective medium approximation (see section 3).)

When a < 1, the particle is delocalized and therefore loosely speaking, it "sees" the totality of the infinite lattice. This is the reason why, when N is allowed to go to infinity, the configuration average is automatically achieved; in other words qZ(t) for a given (infinite) configuration of the transfer rates is identical to the configuration averaged quantity (q2(t)). This is linked to the fact that, for a < 1, the initial conditior, is completely forgotten.

¢r > 1" ~(z ) behaves like ~ z for Iz{ '~ ¢oc, where q~0 is a constant depending on ¢r (see eq. (21)). A very simple reasoning on the density matrix shows that the particle i~ IocnliTe d Eq. rt~-~ ¢ . . . . . iI . . . . . . . . . . ~,.,..) yields ,,.,, ~,,,,,,, z

l IR(z) ) .~ z(1 Ir(t = 0 ) ) . (84) + W~o¢o¢)

The initial condition is not forgotten. This is most clearly seen by examining the limit c~ - 1 >> 1. in which eq. (84) is easily solved. One gets

r,,(t = + ~ ) = r,( t = 0 ) = 6,,,. (85)

C. Aslangul et al / Quantum dissipation versus disorder 87

The particle remains completely localized on the initial site. This is evidently an extreme situation; when a - 1 is not much larger than 1, there is a spreading of the particle on the sites not too far from the initial site. Eq. (75) yields

QZ(z) -.- aZz-'Cl, omcN"Z(fol [WD(I + C*)

1 1 + 2@oWD(1-C*) 1 + W~ot % (CWD-WDCT) 1 1

+ 10). (86)

On this expression, one sees that q Z(t) tends towards a finite constant. The particle is localized. In the two limits a - 1 <~ 1 and c~ - 1 >> 1 this constant can be easily made explicit. When a - 1 <~ 1, it is given by

a 2 1 qZ(t) -.. . (87)

[ N- , ] ( a - - 1 / 2 ) ( a - - a ) 2 1 n = O

In this limit (just above the localization transition), the N sites of the unit cell are actually " seen" by the particle and the configuration average is automatically achieved by allowing N to go to infinity. In the opposite limit a - 1 >> l,

one finds

A ~ + A 2 , -1 1 (88)

q2(t) " a- 2002 (a - 1 / 2 ) ( a - 1) "

The particle is strongly localized and does not explore the disorder on a large scale: even when N is allowed to go to infinity, the above result remains configuration-dependent. Otherwise stated, the initial condition is not forgotten. If now one carries out the configuration average over the transfer rates, one recovers an expression identical to the EMA reseat (eq. (44)).

Finally, let us note that in any case the ordered lattice can be recovered by setting W D proport ional to the unit matrix. When this done. all our previous

results [6] are retrieved.

5. Conclusion

In the present pap. ~,', we i,ave stt, di,-d *he intermediate-t ime dynamics of a quantum particle on a tight-binding disordered one-dimensional lattice in the

presence of ohmic dissipation.


The particle is described by a disordered tight-binding Hamiltonian; the dissipation is accounted for by a linear coupling between the particle and a bath of phonons. From the particle plus bath Hamiltonian, the time evolution of the reduced particle density matrix is derived in the Born approximation. The validity of this calculation has been discussed in detail; as a result, it may be considered as valid in an intermediate range of time values, the extent of which increasing with the coupling constant.

The central ingredient in such dissipative models is the product of the density of modes of the bath times the squared coupling constant, a function of frequency which may be assumed in most cases to behave as co ~. It has been shown that the behaviours observed crucially depend on the value of the exponent 8 [16, 17]. The most studied ohmic dissipation model corresponds to 8 - - ) .

We have demonstrated that in the presence of ohmic dissipation, the particle density matrix obeys a generalized master equation (i.e. retarded), involving only the diagonal elements of the density matrix. This master equation does not come out from phenomenological considerations, but derives from a microscopic model; it contains by construction symmetric transfer rates. Such a master equation equally holds at T = 0 for the so-called underohmic model (8 < 1). At finite temperature, it remains valid for 8 ~< 2.

In the remainder of the paper, we have chosen to study only the ohmic dissipation model at zero temperature, since, in an ordered lattice, it is in the ohmic case and at zero temperature that the most striking effects of these dissipative models occur.

In a first step, we have extended to the quantum problem an effective medium approximation (EMA) developed for the classical situation in refs. [10, 11]. We have considered two classes of lattices, according to the probability distribution of the transfer rates. In class (i) lattices, the first inverse moment of the probability density is finite. In class (ii) lattices, the probability density n (W) of the transfer rates diverges at the origin like a negative power of W, i.e. n(W) --- W --~' with 0 < Ix < 1. As a iesult, in both classes of lattices, a localization due to ohmic friction takes place above the value a = 1 of the coupling with the phonon bath. One therefore can conclude that the Iocaliza- tLUlt LLIIK;blIUILI due to/ U I I I I I I U I I ILSLIL}ll iS~ ill LI1US(2 OTIL3-Ll l lTICFIS1OIIal l a [ [ l C @ S ,

insensitive to disorder in the transfer rates (at least in the types of disordered lattices studied above).

The dyn~,mics depends primarily on the form of the probability distribution of the transfer rates near the origin, i.e. on the class of lattices. We have determined the time-behaviour of the mean-square displacement (qZ(t)). In class (i) lattices, the particle is found to behave as in an ordered lattice. For c~ <1 (but not too small), it is delocalized: { q (t)) follows a power-law---

C. Aslangul et al. / Quantum dissipation versus disorder S9

t 2(1-"~. For a > 1, it is localized: (q2( t ) ) tends towards a constant. In any case,

the amplitude of { qZ(t)) depends on some moinents of the distribution of the transfer rates. In class (ii) lattices, the behaviour of the mean-square displacement {q2(t)) depends on the two parameters a and t~, so that various subdiffusive or superdiffusive regimes are obtained. As a general trend, in class (ii) lattices, the presence of disorder slows down the particle motion, due to the fact that the most important W's are near W = 0

In a second step, we have presented an exact derivation of the time behaviour of (qZ(t)) in class (i) lattices; this derivation extends to the quantum problem a method proposed in ref. [14] for the classical situation. In the range of time values of interest, the expression of the mean-square displacement ! xq2(t)) obtained by this exact derivation coincides, for all values of c~, with

that of the E M A calculation. A comment on the sma!l t ime scales of this problem is in order. As

announced in the introduction, this study can be viewed, either as the extension to the disordered case of the study of an ohmic quantum particle on an ordered tight-binding lattice, or as a generalization to a quantum noise situation of the classical master equat ion governing the excitation dynamics of a particle on a random one-dimensional system in the presence of classical noise. Two characteristic small time scales emerge in this problem: the smaller o,,e, denoted as to~-I, characterizes the noise; the lar~er,_ one. denoted as m :

-~ (in class (ii) lattices) characterizes the disorder. (in class (i) lattices), or as W m In the delocalized regime (a < 1), the particle must have enough time to explore the disorder before the asymptotic regime is fully established: this is the reason why the corresponding expression of the mean-square displacement (q2(t)) is valid only for times t-> m_~ (in class (i) lattices) or Wm 1 (in class (ii) lattices). In the localized regime (a > 1), the particle is locked in a localized state, as a consequence of the ohmic friction, before having enough time to

- 1 explore disorder. This regime is established as soon as t > w c .

Finally, let us comment on the interplay between noise and disorder as pictured by this model. The localization is produced by the ohmic dissipation, and the localized threshold of the particle only depends on the pa r ame~r s characterizing the interaction with the bath. This has also been observed in other situations (see for instance ref. [16]), so that we think that ~ = 1 may be considered as a universal criterion for quantum localization in a dissipative ohmic model. In the localized regime, the disorder modifies the extent of the particle on the lattice. In the delocalized regime, in class (i) lattices, the exponent characterizing the t ime-dependence of the mean-square displacement (q2(t)) depends only on c~ and the disorder modifies the amplitude of ( q z ( t ) ) in class (ii) lattices the exponent depends on both a' and/x , in such a way that

the disorder sire" v-s down the motion.


Acknowledgements

We want to thank Dr. H. Grabert, Drs. U. Weiss and M. Wollensak for having drawn our attention on the limitations of the Born approximation for large times and small a values and for valuable remarks. We are specially grateful to Drs. Zaikin and Panyukov for an illuminating remark on the relevant time scales and detailed comments on their calculation, and for having allowed us to quote these arguments in this paper. We are also indebted to Dr. P. Philips for interesting discussions and preprints on this and related subjects.

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Intermediate-time dynamics of a particle on a disordered tight-binding lattice: quantum dissipation versus disorder