Quantum Brownian motion of a heavy particle: An adiabatic expansion

Physica 91A (1978) 485-506 © North-Holland Publishing Co.

QUANTUM BROWNIAN MOTION OF A H E A V Y P A R T I C L E : AN A D I A B A T I C E X P A N S I O N G. ICHE and P. NOZIERES lnstitut Laue Langevin, 156X, 38042 Grenoble Cedex, France Received 11 November 1977

We consider Brownian motion of a heavy particle (mass M), under conditions where the particle itself displays quantum effects, either because T is smaller than characteristic frequencies, or because the thermal wave packet size, h l ~ , is bigger than characteristic lengths. A direct space method is developed which clearly separates wave packet spread from recoil effects (in the coupling to the heat bath): for a heavy particle, the latter are small, while the former are important in a quantum regime (a plain expansion in IIM is thus precluded). Expanding in powers of the recoil corresponds to an adiabatic expansion. The zeroth order yields an effective "potential" which is defined unambiguously for arbitrarily strong particle-bath coupling. The next term describes friction. A highly degenerate bath usually results in a collision integral which may be obtained from suitably renormalized golden rule arguments. An exception is the degenerate Fermi gas, in which one cannot separate scattering processes, there being no distinct time scales. Such an approach clearly emphasizes the various time scales of interest; it also points to the importance of coherence effects in the evolution of the Brownian particle wave packet.

1. Introduction T h e c l a s s i c a l t h e o r y o f B r o w n i a n m o t i o n is a well e s t a b l i s h e d field. F o r a h e a v y B r o w n i a n p a r t i c l e w i t h m a s s M m u c h larger t h a n the t y p i c a l m a s s e s m o f the h e a t b a t h , an a d i a b a t i c e x p a n s i o n in p o w e r s o f m / M is p o s s i b l e , h o w e v e r s t r o n g t h e p a r t i c l e - b a t h i n t e r a c t i o n V. P h y s i c a l l y , s u c h an e x p a n s i o n relies on t h e e x i s t e n c e o f t w o v e r y d i f f e r e n t time s c a l e s : a f a s t scale f o r the h e a t b a t h d e g r e e s o f f r e e d o m (time o f c o r r e l a t i o n o f the r a n d o m f o r c e ) , a n d a s l o w s c a l e f o r t h e B r o w n i a n p a r t i c l e ( r e l a x a t i o n time o f its v e l o c i t y ) , In t h e e n d , o n e o b t a i n s the w e l l - k n o w n F o k k e r - P l a n c k e q u a t i o n , d e r i v e d e x p l i c i t l y in t h e r e m a r k a b l e p a p e r o f R e s i b o i s a n d L e b o w i t z l ) . In t h e i r w o r k the e x p a n s i o n in p o w e r s o f m / M (i.e. in t h e r e c o i l o f t h e B r o w n i a n p a r t i c l e ) p r o c e e d s in t w o s t e p s . (i) In z e r o t h o r d e r r e c o i l is n e g l e c t e d ; a p a r t i c l e at p o i n t R is in local e q u i l i b r i u m w i t h the b a t h . It feels an e f f e c t i v e p o t e n t i a l e q u a l to the free e n e r g y F ( R ) o f the b a t h , c a l c u l a t e d in all o r d e r s o f the c o u p l i n g V. 485

486

G. ICHE AND P. NOZIERES

(ii) Small departures from local equilibrium give rise to dissipating terms in the Liouville equation. For large M, relaxation of these terms is negligible and they reduce to the usual diffusive-type F o k k e r - P l a n c k collision integral. Consider for instance the disorption of a particle chemisorbed on a solid surface. As pointed out by d'Agliano et al.2), the same interaction provides the binding potential well and the thermal agitation that ultimately leads the adparticle out. An expansion in powers of V would clearly be meaningless. Only a mass expansion allows us to separate a strong static binding from a weak fluctuating friction. Such a classical analysis relies entirely on the concept of local equilibrium. When the particle lies at R (whatever its momentum), the heat bath almost has the canonical distribution corresponding to the hamiltonian H ( R ) . The bath may be classical [as in the original work of Resibois and Lebowitz~)] or quantum [it is then described by a density matrix p(R, p)~)]: it does not matter as long as the particle is classical, with a well-defined position R. The real problem arises when the particle is quantum, being described by a wave packet with finite extension. In a statistical system it is described by a delocalized density matrix, p(R, R'). Local equilibrium is then meaningless, as one does not know where the particle lies exactly: at R? at R'? in between? An adiabatic expansion, if possible, must be constructed along entirely new lines. In practice, such a quantum regime prevails in either of two cases. (i) When the particle is bound in a potential well, with oscillation f r e q u e n c y 1), the classical description breaks down if h f ~ > T.

(1)

In typical situations, hI~ ~ Eo(m/M) ~/2, where E0 is some bath characteristic energy (Debye temperature for a phonon bath, Fermi energy for an electron bath). (ii) A quantum description is also required when the size of a thermal wave packet exceeds the range of the interaction forces, a: T ~ h2/Ma 2. (2) This regime is reached even for uniform systems. Typically, h2/a 2~ mEo: condition (2) is less restrictive than (1). In both cases we need a genuine quantum Liouville equation for the particle reduced density matrix

fiR, R') = TrB p(R, R') = (RITrB plR')

(3)

(the trace being taken over the bath coordinates). The purpose of the present paper is to construct such a quantum Liouville equation for f, and to discuss the relevance of an adiabatic expansion. There have been in the past many attempts at a quantum theory of that type - e i t h e r on specific models4), or on more formal grounds~). Our work attempts to complement these approaches by putting the emphasis on the physical aspects of a mass ratio expansion: time scales, characteristic lengths, nature of quantum effects, etc.

QUANTUM BROWNIAN MOTION OF A HEAVY PARTICLE

487

We use the non-equilibrium perturbation theory devised originally by Feynman et al.6), Schwinger 7) and KjeldyshS). Such a formalism is particularly appropriate to our problem because it gives direct access to the density matrix, incorporating from the outset any interference effect in the bathparticle interaction. The calculation is carried out to all orders in the interaction strength V (it is thus applicable to a chemisorption problem). Once again, introducing such a formalism in quantum transport theory is not new. Indeed, Kjeldysh s) developed his perturbation scheme precisely in order to study the non-linear Liouville equations that occur in high field conductivity. Even earlier Koustantinov and Perel 9) devised an essentially similar perturbation scheme leading to transport equations. A somewhat different but closely related approach is that of Baym and Kadanoff~°), who used double time Green's functions. Most of these works are either concerned with formalism, or deal with a "quasi-particle limit", in which a dressed particle m o v e s freely over long mean free paths. In contrast, we shall apply Kjeldysh's formalism to the adiabatic limit of a heavy Brownian particle, thereby gaining more insight in the underlying physics. In dealing with the small parameter m/M, we must carefully separate two types of effects. (i) Those effects arising from the finite extention of the wave packet at a given time: i.e. R ~ R'. This spread is not small in the quantum regime, and we cannot expand as far as it is concerned. Since R - R' - (h/X/M-T) any brute force expansion in powers of 1/M is precluded. (ii) Those effects arising from the particle recoil over the duration time tc of a typical collision (the correlation time mentioned earlier). The corresponding lengths [R(t)-R(t-to)] or [R'(t)-R'(t-to)], are unrelated to the wave packet width (the latter had infinite time to build up). For a heavy particle we hope the recoil will be small, providing the basis for an adiabatic expansion. As we shall see in detail, this hope is not really justified. At least part of the collision kernels must have a long time memory, t¢-h/T; in view of the uncertainty principle, this is the only way the bath can let the particle know what its temperature is. Over such long times, recoil is large and expansions are bad. We shall see, however, that at low temperatures one can usually bypass this difficulty: one recovers a master equation of the Pauli type with suitably renormalized energies and transition probabilities. Such a picture breaks down for metallic electron heat baths, where anomalies similar to the Kondo effect occur at low temperatures. In the end, the conclusions of this adiabatic expansion are somewhat disappointing. The final Liouville equation for [ is either nearly obvious (one could guess it from purely phenomenological arguments) or untractable (for electron thermostats). We nevertheless feel that the present approach is useful, for a number of reasons. (1) It provides a unified language, yielding in appropriate limits the classical

488

G. ICHE AND P. NOZ1ERES

F o k k e r - P l a n c k equation, the quantum master equation, friction of a fast particle, etc. (2) It clearly separates wave packet effects from recoil effects. Being formulated in real space and time coordinates it is physically quite transparent. (3) It stresses a n u m b e r of effects due to the loss of coherence of the heat bath when it is coupled to a wave packet, effects which are s o m e w h a t surprising at first s i g h t - y e t physically enlightening. (4) It rigorously justifies the use of a zeroth order adiabatic expansion in constructing the potential well profile in n o n - h o m o g e n e o u s problems. In this way we c o m p l e m e n t the usual treatments of quantum transport.

2. Calculation of the reduced density matrix in the Kjeldysh formalism We write the hamiltonian of the coupled particle bath system as

H = H B + H o + V(R), 2

Ho = ~ M + U ( R ) , where HB is the bath hamiitonian and U ( R ) any one-body external potential acting on the particle. The coupling to the bath, V(R), is arbitrarily strong. The full density matrix, o(R, R') is represented in coordinate space for the Brownian particle in an arbitrary basis for the heat bath. The reduced density matrix, f(R, R'), is defined by eq. (3). We want to derive a Liouville equation for ~fl~t. We shall be mostly concerned with free relaxation of f in the absence of any external field. This relaxation occurs o v e r a characteristic time tr; it proceeds to an equilibrium distribution f0 = TrB e ~u (which is not trivial because of the non-commutation of H0 and V). The initial departure from equilibrium may be due either to an initial value f:~ f0 at some time t0, or to the previous excitation of the system by an external hamiltonian, Hext, which has been switched off while the relaxation is studied. In either case an intrinsic free relaxation is meaningful only if tr'> to: otherwise one could not separate excitation from the subsequent relaxation (in the initial value problem, we would retain m e m o r y of the launching at time to). If tr ~ to, a global treatment is needed, as was done for instance in the linear response formalism of Kubo~); however, for studying diffusion in the strong coupling regime, this is not much help. Here, we do not consider such a general case, and we assume that

tr ~> tc ~ h/T.

(4)

QUANTUM BROWNIAN MOTION OF A HEAVY PARTICLE

489

f H ext

V

Fig I We shall see that (4) is usually fulfilled for a heavy Brownian particle at low temperature (in a highly degenerate heat bath). Consider, for instance, a system "pre-excited" by a hamiltonian Hext. We assume that V is established adiabatically from t = -oo, at which time Hext--0. The system is launched at time t = - m in thermal equilibrium Po =

e x p [ - B ( H 0 + Ha)] Tr[...]

At time t the density matrix is p(t) =

U(t,-oo)p0U(-~,

t),

(5)

where U is the evolution operator appropriate to the total hamiltonian /4 = H +/-/ext. In standard perturbation methods one would expand U in powers of V and H~xt. In order to obtain p, we must expand the two factors in (5); hence, a double perturbation expansion which is the gist of Kjeldysh's formalism. As an example consider a Brownian particle linearly coupled to a bath of phonons (transposition to other heat baths is straightforward). We may depict the expansion by diagrams, the building blocks of which are the vertices shown in fig. 1. The full line is the Brownian particle propagator, ~0(t)= exp[-iH0t], which always propagates forward in time since there is only one such particle in the problem (in our first quantized representative, the drone fermion line is not necessary). The wavy line is a phonon; the cross is a coupling to /'/ext. A given matrix element (m,RllU(tl; t2)lm2R2) will be represented by the diagrams of fig. 2. Box A contains any number of vertices V and H , xt (including phonon closed loops). The particle line goes from R, at time t~ to R2 at time t2. The dangling phonon lines on the left determine the bath state m l, those on the right correspond to the state m2. stote

RI

mn

state rn 2

t

I

a

~

P

I

tI

t2

I

Fig 2

490

G. ICHE AND P. NOZIERES

~?

R l state rn I I I I

~

R' ) state m' I

Fig 3

We now consider the operator p(R, R', t), given by (5). It will involve two such set 1 of diagrams, one for U(t, -oo), one for U ( - ~ , t). We draw them on top of each other, the term variable running horizontally from left to right for both. The matrix product UpoU means that the dangling lines at time t = -oo are contracted in pairs, one from each half of the picture, with statistical weight n,, equal to the phonon occupancy at temperature T. The matrix elements (mIp(R, R', t)[rn') are thus represented by diagrams such as that of fig. 3. The upper particle line stops at R , the lower line stops at R'. Dangling phonon lines originating from the upper (lower) half of the picture characterize the state m(m'). The reduced density matrix [(R, R', t) is obtained by taking the trace over the bath coordinates - i.e. by contracting the dangling lines of states m and m' in fig. 3. Thus, f(R, R', t) is exactly given by the diagrams of fig. 4, which contain only closed phonon loops: hence the Kjeldysh version of perturbation expansions. Here we only gave a qualitative sketch of the argument, trying to emphasize the main physical steps; we refer to Kjeldysh's work for a more extensive discussion of technical aspects. The essential features of this approach are as follows. (1) We introduce a double trajectory of the Brownian particle; the upper one x(t), ending up at R, describes the " h i s t o r y " of the bra (in the sense of F e y n m a n ' s sum over paths). The lower one x'(t), ending up at R', corresponds to the ket evolution. At a given time, [ x ( t ) - x ' ( t ) ] characterizes the delo-

calization of the wave packet.

R ~

Fig 4

R~

QUANTUM BROWNIAN MOTION OF A HEAVY PARTICLE

491

(2) The Brownian particle propagator is e -in°' on top e ~H°' on bottom. Over short times it is nearly local, the recoil [x(t) - x(t - z)] being negligible. For a free particle, for instance, one verifies easily that

[ M

~d/2

~iM(x2_xl)2]"

G0(t; x,x2) = ~2--i-~-~t) expL2-~

Our diagrammatic representation explicitly displays the recoil effects on either the top and bottom parts (bra and ket expressions). These recoil effects are clearly separated from the wave packet spread (x - x' at a given t). (3) The effect of the heat bath on f is contained in an arbitrary number of closed loops. For our linearly coupled phonon bath these closed loops are single phonon lines. In more complicated cases (e.g. scattering of electrons or phonons by the Brownian particle), they may be of arbitrary order: yet they remain closed. We note the existence of "cross loops" connecting the upper and lower halves of the diagram: these describe the " c o h e r e n c e effects" in the behavior of the bra and ket expansions in f. Such coherence effects are automatically included in the Kjeldysh language. (4) The contribution of phonon progagators D is calculated according to Kjeldysh recipes, the operators being ordered along a hairpin time axis which goes from left to right on the upper half, and then back to t = -oo on the lower half. Propagators connecting two upper vertices are causal De; those connecting two lower points are " a n t i c a u s a l " / ) c ; finally, propagators connecting top to bottom are pure spectral densities, D ÷= (a*(t)a(O)) or D - = (a(O)a*(t)). L o w e r vertices are opposite in sign to the upper ones. We refer to Kjeldysh for a complete description. Here we note only that the 2 x 2 matrix structure of propagators applies to D. This formalism of Kjeldysh is unfamiliar to many, and at first sight it may appear uselessly complicated. It should be realized that, on the contrary, it provides the natural language in our problem, since it yields direct access to the density matrix. Wave packet delocalization, recoil and coherence effects are clearly spelled out, and the corresponding physical picture is quite transparent (as usual when one works with real space and time variables!). We wish to calculate the time derivative

aflat

=

L(f),

where L is the full Liouville operator for the Brownian particle. Part of the time dependence arises from the last propagator e ±iH°t on the right of fig. 4. The corresponding contribution to L is simply Loft) = i[f, H0].

(6)

It describes the free evolution of the particle, uncoupled to the heat bath. The latter affect L only via the boundary on the integration of various vertices, which extends from - oo to t. Deriving with respect to t means that any one of these vertices is fixed at the integration boundary t. Thus, we obtain another contribution, LI, given by all the diagrams of figs. 5a and b, where the

492

G. 1CHE AND P. NOZIERES I

4

o ',:'

b) Fig 5

right-most vertex is at t, on either top or bottom. Note that this formal result is exact, valid to all orders of perturbation theory. In principle we can construct a simple Dyson equation for LI. In fig. 5 we go b a c k w a r d in time, starting from t, and we identify the first time ( t - r) at which there is no phonon line at all, of any type. An example is shown in fig. 6a. At that earlier time the particle is at x on the upper line, x' on the lower one. By summing o v e r all diagrams between ( t - r ) and t, we define an irreducible kernel £(x, x', t - r; R, R ' t ) which describes the effective interaction of the particle density matrix with the heat bath. Note that the interaction £ is retarded ( r # 0), and also that it necessarily involves two space variables at each end: these are essential in order to take care of coherence effects. (In practice the left-most vertex of Y, may also be on top or bottom, so that E contains four terms.) Left of ( t - r ) we recognize the diagrams that would contribute to f ( x , x', t - r). We can thus write the Liouville equation as i -~- = [H0;/] +

/

dx dx'

~ , (x, x', t - r, R, R ' , t ) f ( x , x ' t - r).

(7)

0

The last term of (7) looks like a retarded collision integral, which in principle is rigorously exact. For strong coupling such a result remains formal, because the irreducible kernel ~ must be calculated to all orders in V. (A diagram is irreducible if it cannot be split by cutting the two particle lines at the same time t. As a result, the diagram of fig. 6b is irreducible if t, < t2, while it is reducible when tt > t2.) The calculation of £ is thus a hard task, and (7) is of little use as it stands. We

I

t~

I

I

,~

IR'

I

i

i

t-T

t

t-T

i I

a)

I t2

b) Fig 6

j

t

QUANTUM BROWNIAN MOTION OF A HEAVY PARTICLE

493

need some other argument to proceed further; for instance, the assumption that the Brownian particle is heavy and slow moving. Let tc be the characteristic time range of X describing the duration of a given collision, tc may be an atomic time of the bath, hlOo(or h/Ev for an electron bath); it may also be a thermal time hiT, i.e. the time that a thermal wave packet takes to sweep past a given point. We shall see later how to decide between these two possibilities: for the moment we note only that tc is a bath quantity, independent of M. We assume the external field Hext has been switched off long before ( t - t O : we thus consider only free relaxation, meaningful if tr'> t~. (As a consequence the collision kernel contains only particle-bath vertices V.) Over a time t c a heavy particle should not recoil much. If for a moment we assume this is true, we see the possibility of a new adiabatic expansion, in powers of the recoil, instead of the interaction strength. We shall first construct such an expansion, and then discuss its validity.

3. The Liouville operator in the absence of recoil

In zeroth order approximation we neglect recoil completely in the collision operator X. As a consequence the Brownian particle remains fixed at R at every vertex on the upper half of the diagram, while it is at R' for every vertex of the lower half. (Note that the wave packet structure, R ~ R', has been preserved despite the fact that we ignore recoil.) As viewed from the heat bath viewpoint, the particle is a structureless scattering object with no memory between successive c o l l i s i o n s - a rather unusual scatterer, however, since the potential is not the same on the bra and ket sides of the expansion. In principle we might go on calculating Y.(R, R', t), inserting the final result into (7). Such a procedure is possible, yet definitely awkward, as one immediately faces complicated transient effects due to the finite duration t of X. If we are to neglect recoil, it is much simpler to calculate [(R, R', t) directly. There is no correlation between different vertices and, consequently, disconnected closed loops of bath lines (phonons, electrons, etc.) simply factorize. The usual linked cluster theorem applies, despite the fact that R ~ R': the important fact is that "no recoil = no m e m o r y " . The corresponding density matrix f(R, R', t) is thus t

where C(R,R') is the contribution of all connected closed loops involving both V(R) and V(R') vertices. The last time i n t e g r a t i o n - s a y on the right most v e r t e x - h a s been singled out in (8); as a result of time invariance C(R, R') is a time independent quantity.

494

G. ICHE AND P. NOZIERES

From (8) it follows at once that

af(R, R', t) - C ( R , R')J:(R, R', t). at

(9)

In the absence of recoil, the Liouville operator is thus a mere multiplication for C(R, R'). Eq. (9) is our real zeroth order result. Strictly speaking (8) is not a consistent approximation, since recoil is certainly not negligible when one goes back far enough in time. On the other hand, when calculating af/at we sample only times close to t (within a range - t o ) , and the neglect of recoil makes sense. Eq. (9) is the first correction to the free particle Liouville operator L0. A few general remarks, first. The right-most vertex of C yields a factor [+ V(R)] if on top and [ - V(R')] if on bottom, the propagators being the same in both cases. Thus, the identity

C(R,R)=-O

(10)

holds (as pointed out by Kjeldysh). Eq. (10) guarantees that the trace of f (i.e. the normalization of the density matrix) is conserved. By definition C is given by exp f C(R" R') = Tra(URPoU+R,), J

where UR is the evolution operator of the bath given the Brownian particle is fixed at R. Using the unitarity of UR, it follows from this identity that

C(R,R') = C*(R'; R).

(11)

Eq. (11) guarantees that [ remains hermitian through time. Eq. (9) is thus consistent with the obvious symmetry requirements. A more detailed calculation of C(R, R') usually depends on the nature of the heat bath. Simple conclusions emerge only in two limiting cases: (i) at zero temperatures, and (ii) at very high temperatures. We shall discuss these two limits successively, beginning with the low temperature case. Consider for example a heat bath of electrons, scattering off the impurity (a similar argument would hold for phonons). In calculating C(R, R') we Fourier transform from times to frequencies. A closed loop involves a sequence of electron propagators that meander from top to bottom of the diagram, with constant frequency ~ throughout the loop. E v e r y propagator from top to bottom involves a statistical factor [ l - n ( ~ ) ] , where n is the Fermi distribution; propagators from bottom to top carry the particle density n(~). As a result closed loops that couple the top and bottom are at least of order n(1 - n): they vanish identically at zero temperature. In the limit T = 0, C0(R, R') involves only diagrams that contain either only V(R) vertices (loops in the upper half) or only V(R') vertices (loops in the lower half). Physically the disappearance of "cross loops" should be traced to the absence of zero energy excitations in the

G. ICHE AND P. NOZIERES

495

non-degenerate ground state of the electron heat bath: it is clearly a general feature at T ~ 0 , which is not specific to our model. In F e y n m a n ' s 6) terminology we would say there are no "influence functional" contributions to C at zero temperature. Now, closed loops conducted on V(R) alone are just those that would appear in a standard perturbation expansion at T----0: they yield within a factor i the change in ground state energy of the bath, ,~Eo(R), due to the perturbation V(R). We conclude that, to all orders in V, the recoilless closed loop contribution at zero t e m p e r a t u r e is simply C0(R, R') = - i [ E 0 ( R ) - Eo(R')],

(12)

where Eo(R) is the local ground state energy of the bath when the Brownian particle is fixed at position R. The corresponding contribution to (9) may be viewed as the c o m m u t a t o r of .f(R, R') with an effective local potential Eo(R). Thus, despite the fact that the Brownian wave packet is very delocalized, we demonstrate that, at T = 0 and to leading order in recoil, the concept of an effective local hamiltonian

H ~ = Ho + Eo(R ) is v a l i d - a statement, in fact, of the B o r n - O p p e n h e i m e r In the opposite limit of very high temperature the classical, and the w a v e packet is small: R ~ R'. We can ( R - R ' ) , as is done in detail in section 5. To lowest closed loop reduces to

Co(R, R') = - i ( R - R') grad F,

approximation. Brownian particle is expand in powers of order, the recoilless (13)

where F(R) is the free energy of the heat bath when the particle is at R. Once again, F(R) behaves as an effective potential acting on the Brownian particle. We find the same result as at T = 0, but for a completely different reason. H e r e , eq. (13) follows from the expansion in powers of (R - R'), i.e. f r o m our assumption of a small wave packet. At T = 0, eq. (12) was a consequence of the sharpness of the Fermi distribution: . f ( l - [ ) = 0. We may illustrate this point by considering a very unphysical situation, namely a free electron heat bath coupled to the particle by a central potential V(R). We assume that V has only an 1 = 0 phase shift that depends on R. We replace V(R') by some potential V'(R) located at the same place as the first, but having different magnitude. The phase shifts of these two potentials are respectively 8 and 8'. The closed loop contribution can be calculated explicitly and turns out to be

'I

+~

C(R, R') = ~

d~ Im log{1 + n(~)(e -2i~8-8'~- 1)}.

At high T we expand in powers of (8 - 3'), which yields +ee

C= f n(8-~$')d~ J

71"

496

G. ICHE AND P. NOZIERES

i.e. (12). At T = 0 ( 3 - 6 ' ) may be l a r g e - b u t the same result holds because n is either 0 or I. In the intermediate temperature range the concept of an effective potential is not valid: the adiabatic approximation does not work. The recoilless closed loop, C(R, R'), is not the difference of two quantities, one at R and one at R', and thus the corresponding contribution to the Liouville operator is not a commutator. Incidentally, C(R, R') even has an imaginary part: Im C ~ (R - R') 2. In view of (9), Im C leads to an exponential decay of f when R ~ R'. Put another way, if there were no recoil effect at all the density matrix f(R, R') would spontaneously shrink, the wave packet becoming smaller and smaller. Such a conclusion is at first sight very surprising. However, its physical meaning is fairly clear. Assume that we launch at time t = 0 a spread out wave packet with density matrix f(R, R'). This implies a certain amount of phase coherence between the bra and ket sides of the full density matrix p(R, R'), such that a finite f ( R , R ' ) persists after taking the trace out of the bath coordinates. As time goes on, the bra vector evolves according to hamiltonian H ( R ) , while the ket is governed by H(R'). When R = R', any phase log of the bra is cancelled in the ket, and p retains its original coherence. If, instead, R ~ R', then the bra and the ket evolve differently. They lose their coherence and as a consequence the trace f(R, R') decays. (Of course, such a loss of coherence is irrelevant at T = 0 as there is nothing to lose: there is no statistical ensemble, and the wave function is uniquely defined.) From this vantage point the evolution of the reduced density matrix f(R, R', t) appears as a compromise between two conflicting tendencies: (i) the wave packet tends to shrink because of the loss of coherence of the heat bath response as time goes on; and (ii) this shrinkage is opposed by the kinetic energy of the Brownian particle, which wants instead to make the wave packet as large as possible. Such a picture is rather unusual but, right or wrong, it deserves reflection.

4. Low temperature collision integral We now proceed to take recoil into account. For that purpose we must first extract from the closed loops the large contribution found in the preceding section, which would preclude any perturbation expansion. To do this we devise the following subtraction procedure Any closed loop that sits across the upper and lower halves of the diagram is small at low temperature: we leave it as it is. A closed loop sitting only on the upper branch is large at T = 0. Let x be the position of the Brownian particle at the sight most vertex on the upper time axis. Because of recoil the other vertices will correspond to different positions x2. . . . . x,. To the closed loop contribution we add and subtract the

QUANTUM BROWNIAN MOTION OF A HEAVY PARTICLE

497

same quantity

[C(xl . . . . . x,) - C0(Xl . . . . . xl)] + C0(Xl . . . . . xl),

(14)

where Co is calculated at T = 0. The last term of (14) acts as an effective local potential AE0(R) acting on the Brownian particle: we incorporate it into the bare hamiltonian Ho(R) which is thereby replaced by H ~ = H o + E o ( R ) . Hopefully, the first two terms of (14) will largely cancel each other; that will be the basis of our recoil expansion, a procedure which clearly is meaningful only at very low temperatures. Once the subtraction has been performed, we return to the procedure of section 2, defining an irreducible self energy E(x, x', R, R', t, t') and a corresponding Liouville equation similar to (7): the only difference is that the major part of • has been transf~erred to H0. The importance of recoil depends on the time range of ~ ( t - t'), which in turn is controlled by the long time behavior of the heat bath propagators. Consider for instance an electron heat bath: closed loops are built with one electron lines G(r, r', t). Three ranges of time emerge, as sketched in fig. 7: (i) at short time, t ~ l l / E F , G is large, involving constructive interference of m a n y one electron states; (ii) at large times, t>>h/T, G decays exponentially (a consequence of thermal broadening); and (iii) in between, G ~ 1/t. The qualitative behavior of Y.(t - t') follows at once. (i) For small (t - t') all the G lines are " s h o r t " , yielding large factors. The perturbation calculation must be carried out to all orders. On the other hand, recoil is negligible here. The corresponding large contribution to E is eliminated by subtraction. (ii) For (t - t')>> h/T, Y, cuts off naturally. (iii) In the intermediate range h / E F ~ ( t - t ' ) ~ h / T the self-energy behaves as 1/(t - t') 2. The consequences of the integral in (7) guarantees that short time lags are indeed dominant: the effective potential Eo(R) is much more important than collisions. On the other hand, this small tail of Y~ is essential because it provides the relaxation mechanism toward thermal equilibrium: it will give rise to the collision integral ( r e m e m b e r that we need a time G

t file F

hiT

Fig 7

498

G. ICHE AND P. NOZIERES

>~h/T in order to ascertain the energy with an a c c u r a c y T). Similar conclusions would hold for any type of heat bath. Y,(t - t') will always display the following characteristics. (i) A large short time contribution in which we can neglect recoil and which is absorbed in the effective potential Eo(r). (ii) A small tail, which for t < h i t behaves as l/t", the exponent n being found from the density of low lying excitations of the bath, -~"-~. In an electron bath these excitations are electron-hole pairs, and n = I: in a phonon bath n would be ~>2, assuring even faster convergence. The strategy to be used at low temperatures, T, is thus clear: we are interested in E ( t - t') for long times ( t - t') (for which recoil is noticeable). We just saw that "long" bath lines give a small contribution. Moreover, " c r o s s " lines (connecting top to bottom) are known to give a vanishing contribution at T = 0. In leading order we retain only diagrams with a minimum number of such lines. Consider first a phonon heat bath, linearly coupled to the Brownian particle. Closed loops then reduce to a single phonon line. In calculating 33 (t - t') we retain only one such (long) line, any additional phonon line is necessarily short and therefore disappears in the subtraction procedure. Note that there are still four relevant diagrams, each end of the remaining phonon line in Y, being either on the top or on the bottom of the diagram. The situation is slightly more complicated for an electron heat bath (or for a phonon bath with quadratic coupling, yielding phonon scattering rather than emission and absorption). Then, closed loops can have an arbitrary order in V. According to our general recipe we retain only diagrams with two long lines connecting t and t': all the other lines are short, near one end or the other. N e a r each end all the vertices are either on the top half, or on the bottom half, there being no additional cross line. Finally, we are left with the four diagrams of fig. 8: the only difference with an elementary second order

x

a)

b) R~

x'~

x O R ' d)

c) Fig 8

QUANTUM BROWNIAN MOTION OF A HEAVY PARTICLE

499

calculation is that the two vertices are renormalized. Inside each of these two renormalized vertices (black boxes in fig. 8), recoil is negligible: V(x) acts as a static, structureless potential, and the black box is simply the "on the energy shell" scatterin~g matrix t(x) obtained by iterating V. Once again, extra closed loops are short and disappear in the subtraction. Note that the right-hand black box is " n e a r " time t, the left one " n e a r " t', the exact location being irrelevant as long as (t - t') is large (i.e. - h / T ) . The picture that finally emerges is very simple, and nearly self-obvious. The heat bath exerts an effective potential E o ( R ) ( B o r n - O p p e n h e i m e r approximation) on the Brownian particle. In the next order the Brownian particle can excite the heat bath (creating phonons, electron hole pairs, etc.). At low temperature only one excitation is c r e a t e d - n o t because the coupling is weak but as a result of the small phase space available. The price one pays is that the bath-particle interaction is renormalized: instead of the bare coupling, V(R), it is the full t matrix for a fixed particle, t(R), which yields the scattering amplitude for exciting the bath. The situation is similar to that encountered in the Fermi liquid theory where the bath is described in terms of exact quasi-particle states, distorted by the static Brownian particle, whose lifetime is long because the available phase space for decay is small. Our adiabatic expansion program is thus complete. The large "instantaneous" effects are separated from the long time dissipative p a r t - i n a way very similar to the classical theory'). One question remains: to what extent is our analysis of the collision integral a bona fide expansion? Put another way, are the diagrams of E with more "long" lines really smaller than the ones we keep? Every additional "long" line implies integration over an additional vertex, hence a factor - t . For a phonon heat bath that factor is overrun by the asymptotic behavior, lit n, of the phonon propagator. Our expansion is then valid: we can limit ourselves to emission (or scattering) of one phonon only, the higher configuration involving a negligible phase space at low temperatures. On the contrary, in a metallic heat bath the electron propagator behaves as l/t: in that case, adding more "long" lines contribute the same order of magnitude as that of fig. 8. Our deviation of the collision integral is then meaningless (except of course in the case of weak coupling, which is of no concern to us). Note that all these subtracted terms are smaller than the local equilibrium loop E0(R): the concept of an "effective" local potential is sound! But in the next order all the recoil terms are comparable, and we are at a loss to construct a collision integral. The above lack of consequence is another manifestation of the so-called "infrared catastrophy" familiar in degenerate metals. The Brownian particle distorts the electron heat bath, and that distortion has a very long range, due to undamped Friedel oscillations of the screening cloud. When the Brownian particle scatters an electron the screening cloud must readjust i t s e l f - a n d that would take an infinite time at T = 0. During that time the Brownian particle recoils: our while analysis collapses. We can no longer separate two

500

G. ICHE AND P. NOZIERES

" r e c o i l l e s s " steps: excitation and de-excitation of the bath, with a long free p r o p a g a t i o n period in between. Instead, we must follow in detail the motion of the B r o w n i a n particle b e t w e e n t and t', recoil providing a correlation b e t w e e n s u c c e s s i v e scattering events. T h e s e m a y involve the same electron (fig. 9a) or different electrons (fig. 9b and c): in both cases logarithm f a c t o r s build up, and the series does not c o n v e r g e . The difficulty is akin to the K o n d o effect: in both cases we must cope with the transient r e s p o n s e of an electron gas to an impurity with internal degrees of f r e e d o m - i t s spin in the K o n d o effect, its position here. As a result we c a n n o t c o n s t r u c t an explicit collision integral for an electron heat bath. (In fact, the very notion of such a collision term is doubtful since in such a case the relaxation time t is - h / T " one c a n n o t separate excitation and relaxation.l If we disregard this difficulty we are left with a " r e n o r m a l i z e d weak coupling p r o b l e m " . The c o n s t r u c t i o n of a master equation for f is then straightforward. As an e x a m p l e , c o n s i d e r a p h o n o n heat bath linearly coupled to the B r o w n i a n particle. The coupling hamiltonian is aAR)Ib~

+

b'l,

l,

where b* creates a p h o n o n with f r e q u e n c y co~. The s e c o n d order self-energy Y (R, R", x, x", t - t') is given by the f o u r diagrams of fig. 8, the e l e c t r o n - h o l e pair being just replaced by a single p h o n o n line. Before subtraction we m a y write 2 as (R, R ' ; x, x'; t) = G ( x , R, t ) G * ( x ' , R', t){a~(R) - a ~ ( R ' ) } { a , ( x ) D , ( t ) (15)

- a~(x')D~(-t)}.

In this expression we set t - t' = t. D is the p h o n o n p r o p a g a t o r at t e m p e r a t u r e T D , ( t ) = (1 + n,) e *,or,+ n~ e+~'°C

while G ( x , R, t) is the free particle p r o p a g a t o r of the B r o w n i a n particle in the effective hamiltonian H * G ( x , R, t) = (xle

~m'lR).



Fig

9

QUANTUM BROWNIAN MOTION OF A HEAVY PARTICLE

501

(For a more complicated heat bath D, would be the propagator of the leading elementary excitations of the bath, and a , would be renormalized.) In order to exploit E, we first project the density matrix on the basis of the eigenstates of H * . In that new representation G is diagonal:

Grim(t)= e x p ( - i - ~ )

6,m.

The self-energy operator becomes a four-index matrix ,,m, , . , , as sketched in fig. 10. That matrix should be convoluted with/..,(t'), according to (7). The off-diagonal elements of [ oscillate in time: their contribution to the master equation for the diagonal part [., will thus enter in higher order of perturbation theory. Since the subtracted self-energy is small, we can ignore such corrections; we retain only the diagonal elements of the self-energy, E,".~, that couple the occupation fm to f,. The smallness of the non-diagonal elements of [ in a quantum transport equation is well known: see for instance the "classic" of Kohn and Luttinger~2). This crucial step is made possible by our subtraction procedure which eliminated the largest part of the particlebath coupling, leaving only a small residue which we treat as a perturbation. Finally, we note that the diagonal probability f, will evolve on a scale t much longer than the "thermal width" hit of the self-energy (otherwise a relaxation equation would have no meaning). Thus, we can integrate E (t) from t = 0 to oo: only the "zero f r e q u e n c y " part of E is relevant in the master equation. This would be wrong for the non-diagonal elements: hence the need for subtraction. From then on the calculation is just straightforward algebra. Let us define the matrix element

f

V~. = J dRd~*(R)a~(R)d~v(R).

(16)

The two diagrams of fig. 8a and d yield a contribution to E ~ (after integration over t): =I&""~]IV~'pl2 E , p~. !

(

-

l+n~ nv - c.c.} Ep-oJ~+in ~E,. - Ep+oo~+i-q

p D~

m I

a)

n ~ ~ . ~

m

nl

m I

b) Fig I0

(17a)

502

G.

ICHE

AND

P.

NOZIERES

while those of fig. 8b and c yield a term 1 ,V~,,12{ l+n~ n~ i E, - E m - o~ + in + E, - E m + to~ + i T

c.c.

}

(17b)

(see fig. 10). We note that the principal parts drop out of (17): in the master equation for the diagonal elements f, only irreversible effects arising from the 6-functions survive. This was indeed to be expected: the principal parts, as usual, provide a reversible contribution to the effective potential, which has been incorporated H * . The latter enters in the c o m m u t a t o r [ H * , f], which has no diagonal element, in agreement with (17). Such a conclusion would not hold for the off-diagonal elements f,m" then the principal parts of the energy denominators would yield a finite contribution [to ( E , - Em)f,m], much larger than the dissipative part with which we are concerned. It is precisely in order to eliminate the large reactive off-diagonal term that we p e r f o r m e d subtraction. Before proceeding further, we must subtract from (17a) the corresponding contribution at T = 0 and with no recoil, which is equal to 1 6 " " - p~, -VmP

I

-~o~+in

c.c..

That expression vanished because there is no zero f r e q u e n c y phonon. (Of course, the off-diagonal elements of the subtracted Y. do not vanish: they contribute to [H*,f].) Finally, the diagonal master equation reduces to the trivial weak coupling result

= ~ W,,,~f, + Z W,~,f,,,, rn

rn

(18)

where the transition probability is given by the golden rule of perturbation theory W,,,

27r =~-~,lV~,,,12[(l+n,)6(Em-E,+hto~)+n~6(E,,-E,-hw~)].

(19)

v

In a sense it is s o m e w h a t ridiculous to have gone through such a long analysis in order to obtain a nearly trivial result, Yet, what is not completely obvious is that (18) and (19) do not rely on a weak coupling between particle and bath. They involve renormalized quantities, as in a Landau theory. What we have achieved is a clear separation of a strong effective potential from a w e a k transition probability. The final result is just what c o m m o n sense would dictate.

5. Classical limit

It is instructive to see how our general quantum formalism reduces to the classical limit of Resibois and Lebowitz in the limit of high temperatures. In such a case all the lengths of the problem are short: the wave packet is small

QUANTUM BROWNIAN MOTION OF A HEAVY PARTICLE

503

(R ~ R'), and the duration of a collision, h/T, is short enough that the recoil of the Brownian particle (x - R), is also small. Our subtraction procedure can be carried o u t - b u t it is useless: we may as well directly expand Y. in powers of the various distances~ treating the Stokes and the irreversible terms together. Let us take as an origin the center of the wave packet as time t: R0 = ~(R + R'). The perturbation V ( x ) is then expanded as V ( x ) = V ( R o ) - (x - R o ) F + • • ",

(20)

where F = - V V is the force operator. V ( R o ) is incorporated in the hamiltonian H * of the Brownian particle: the remaining correction ( x - R 0 ) f is small, and can be treated as a perturbation. In first order in F the self-energy is an instantaneous multiplicative factor ~ . " ~ (R, R ' ) = ( F ) ( R - R ' ) ,

(21)

the average being taken in the local equilibrium distribution of the bath, e x p [ - / 3 H ~ ] . Eq. (21) is the first term in an expansion of our closed loops in powers of (R - R'). The classical Liouville equation is obtained via a Wigner transformation in which (R

R')~-ih

0 ap

The first order E "~ thus gives a contribution + ( F ) ~ p/ ,

(22)

which is the usual Stokes term in the transport equation [it is easily verified that - ( F ) is the gradient of the heat bath free energy F(R)]. Eq. (22) is the effective potential of Resibois and Lebowitz. Note that it is defined for arbitrarily large V(R): we expand in powers of (R - R'), not of V! In second order we find a contribution from the term - ( x - R0f in (20), treated within the H a r t r e e - F o c k approximation. That contribution is zero because we took R0 halfway between R and R'. We are left with the linear perturbation ( x - R0) treated to second order. The corresponding contribution to Y~ is (R, R'; x, x', t) = 1 2G(x, R, t ) G * ( x ' , R ' , t ) [ R - R'] × [(x - R0)x(t) - (x' - R0)x(-t)],

(23)

where x ( t ) is the f o r c e - f o r c e correlation function. For simplicity we consider only the case of an h o m o g e n e o u s heat bath, for

504

G. ICHE AND P. NOZIERES

which the propagator G is that of a free particle

G(x,R,t)=

exp ~ ( R -

x) 2 .

(241

In the presence of an effective potential V ( r L the propagator G would acquire an extra phase factor (for small t): e x p [ - ~it { V ( R ) + V(x)}]. That factor is not negligible for t ~ h/T. Indeed, it is that term which is responsible for the replacement of F by the force fluctuation [ F - ( F ) ] in 1". We set x0= ~(x + x'), P = R - R', ~ = x - x'. The product GG* then reduces to

M

GG* = 2~'ht exp ~

j

( R 0 - Xo)(p - ~) •

For a small t, G G * is very peaked. According to (7) it must be convoluted with the density matrix f ( x , x ' ) ~ f ( x o , r,). The latter is a rapidly varying function of ~ (the wave packet is sharp), but a slowly varying function of xo: in leading order in recoil we are thus allowed to do the replacement

f(xo, ~) ~ f(Ro, ~). We can now perform the integration over x0 in (23), which yields

8 ht f dxo ~ = ~ { - ~ 6'(p - ~){x( t ) - x ( - t )} + z 8(0 - sr)[x(t) + x ( - t ) ] }. (25) We express the force correlation function in terms of its spectral density

x ( t ) = jf ~ J(to)

[e i,~,+ei,~, e ~h~] dto.

(26)

0

Moreover, we r e m e m b e r that f varies slowly during a collision: we may carry the t integration in Y~, which yields two factors:

A =

if

dt x(t)+x(-t)2

= 2¢r ~lim=oJ(to)Toh

0

and

(27) B = ( i dt[x(t) -

x ( - t ) ] = 2rr lim J(to) _ hA ,o=o w

J

A plain Fourier transformation in which ~ ' ~ - i h T p ap

and

a +~_ ~-~=

T "

QUANTUM BROWNIAN MOTION OF A HEAVY PARTICLE

505

leads at once to the usual classical F o k k e r - P l a n c k equation:

c~f + p__ ~f + F_~p F = A F 1 aT

m oR

[-M--f

,9

,~ + c~2 e ,"

]

fj.

(28)

The classical limit thus appears naturally. We note that the diffusion term in (28), -O2flOp :, arises from the difference ( x - x ' ) in the bracket of (23): it is thus a consequence of the wave packet delocalization. In contrast, the "drift" term, - cgpf/ap, arises from ( x 0 - R0): it appears as a result of the recoil of the Brownian particle (as confirmed by the factor 1/M). The physical origin of these two terms is thus different (their balance leads to the usual Maxwell equilibrium distribution).

6. Conclusion

The net result of our long discussion is s o m e w h a t disappointing. On the positive side we have shown that for a heavy Brownian particle in a degenerate heat bath one can unambiguously define an effective adiabatic potential, even for strong coupling V, and that despite the fact that the Brownian particle wave packet is widely spread out. The effective potential goes smoothly into the classical driving force when T ~> 0D, EF. The "zeroth order" term of the adiabatic expression is thus well defined. The next t e r m - t h e collision i n t e g r a l - r e d u c e s to an obvious golden rule transition probability for a degenerate phonon bath (with suitably renormalized amplitudes, "h la Landau"). In the intermediate temperature range, T - 01), we can say n o t h i n g - n e i t h e r can we for a metallic electron bath in which difficulties akin to the K o n d o effect appear. One could try different types of expansion in order to cope with the electron case. For instance, consider a situation in which the adiabatic potential well V ( R ) is deep and narrow. The Brownian particle then remains close to the bottom R0 of the well, and we can expand in powers of the displacement r = R - R 0 . The relaxation of the Brownian particle may be viewed as a hybridization of the corresponding oscillator with the heat bath excitation modes. [Rather than working with a Liouville equation, it is simpler to calculate directly the Brownian particle propagator (r(t)r(O)).] Here, it is the dynamic localization of the particle which makes the problem tractable. Quite generally we can treat situations in which the wave packet is large, while the recoil is small. Conversely, one can devise an expansion in powers of the w a v e packet width [ x ( t ) - x ' ( t ) ] , for arbitrarily large recoil. But when the two lengths are large, we can do nothing (save by phase space arguments that do not apply to an electron bath). The main interest of our discussion is to provide a clear physical picture of what happens in real space and time, and also to introduce a unified language which can be adapted to all the relevant limits (classical, quantum, etc.)

506

G. ICHE AND P. NOZIERES

References 1) P. R6sibois and J. Lebowitz, Phys. Rev. 139 (1965) 1101. 2) E.G. d'Agliano, W.L. Schaich, P, Kumar and H. Suhl, Proc. 24th Nobel Syrup. on Collective Properties of Physical Systems (Academic Press, New York, 1973). 3) See, for instance, P. R6sibois and H.T. Davis, Physica 30 (1964) 11)77. 4) G.W. Ford, M. Mac and P. Mazur, J. Math. Phys. 6 (1%5) 504. 5) The leading role in this respect was played by the Brussels school. See, for instance, R. Dagonnier and P. R6sibois, Bull. Acad. Roy. Sci. Belg. 52 (1966) 229, 1475. 6) R.P. Feynman, R.W. Hellwarth, C.K. Iddings and P. Platzmann, Phys. Rev. 127 (1962) 1004. 7) J. Schwinger, J. Math, Phys. 2 (1961) 407. 8) L.V. Kjeldysh, JETP-Sov. Phys. 20 (1965) 1018. 91 O.V. Konstantinov and V.I. Perel. JETP-Sov. Phys. 12 (1961) 142. To our knowledge this work is the first to use the double time perturbation scheme that applies to non-equilibrium situations. 10) G. Baym and L. Kadanoff, Quantum Statistical Mechanics (W.A. Benjamin, New York, 1%2). 111 R. Kubo, J. Phys. Soc. Jap. 12 (1957) 570; 12 (1957) 1203. 12) A.A. Abrikosov, Physics 2 (1965) 5. 13) W. Kohn and J.M. Luttinger, Phys. Rev. 108 (1957) 590; 109 (1958) 1892.