PHYSICAL REVIEW A VOLUME 44, NUMBER 12 15 DECEMBER 1991 Quantum Brownian particle and memory efFects J. R. Brinati Instituto de Fssica, Universidade Federal do Rio de Janeiro, Caixa Postal 68.528, 21945 Rio de Janeiro, Rio de Janeiro, Brazil S. S. Mizrahi B.M. Pimentel Instituto de FI'sica Teorica, Universidade Estadua/ Paulista, Rua Pamplona 145, 01405 Sao Paulo, SB'o Paulo, Brazil (Received 15 March 1991) The quantum Brownian particle, immersed in a heat bath, is described by a statistical operator whose evolution is ruled by a generalized master equation (GME). The heat bath's degrees of freedom are con- sidered to be either white-noise or colored-noise correlated, while the GME is considered under either the Markov or non-Markov approaches. The comparisons between these considerations are fully developed, and their physical meaning is discussed. PACS number(s): 05.40.+j I. INTRODUCTION In recent years great interest [1—7] arose in considering non-Markovian processes in quantum systems interacting with a heat bath or reservoir, since memory effects hap- pen to be important and cannot be disregarded in the so- called generalized master equation (GME), which de- scribes the time evolution of some system of interest S. A paradigm of a system commonly studied is the quan- tum Brownian particle (4), represented by a harmonic oscillator (HO) of frequency Qo and unit mass. Further- more, the system 4 is considered to interact with a medi- um, the reservoir %, constituted of a very large number of independent HO's of frequencies co„. In principle, the total Hamiltonian of /+A is quite simple, but the deter- mination of the dynamical evolution of 4 is almost im- possible to solve exactly (and is not necessary) due to the complexity involved in manipulating a great number of coupled integro-differential equations. Therefore, ade- quate approximations are introduced in the formalism in order to attain solutions that make transparent the essen- tial physical aspects of the time evolution of S. Among the interesting aspects that emerge in this study, one cites the thermal relaxation process that occurs in three different approaches: the drastic Markov approximation (MA) or white-noise MA (WNMA), corresponding to zero correlation time between the reservoir operators; the weaker MA or colored-noise MA (CNMA), which con- siders a finite correlation time; and finally the more pre- cise non-Markov (NM) approach or colored-noise NM (CNNM). These concepts will be precisely defined later. Another interesting aspect to be treated corresponds to the memory effects in the dynamical evolution of the Brownian particle energy and its position-momentum correlation. Unlike other authors [6], we choose to attri- bute the Markov or non-Markov nomenclature to the system of interest S and not to the reservoir, since this is more similar to the original concepts [8]; however, we distinguish between white-noise and colored-noise ap- proaches in the treatment of the reservoir operator corre- lations. Our treatment difFers from that of Refs. [1] and [7] in the picture we work: specifically, we obtain a GME in c numbers that is used to calculate the mean values, whereas they solve Langevin equations. Among the con- siderations in our treatment we cite the following. (1) The 4+% interaction is in the rotating-wave approxima- tion (RWA) [9], meaning the absence of rapidly oscillat- ing terms. It is worth noting that our treatment is equivalent to the so-called modified RWA of Ref. [1]. (2) The weak-coupling approach, introduced in the GME of Peiers [10], takes into account terms up to second order in the interaction between 4 and %. Since we work with the hypothesis that the reservoir remains in a global thermal equilibrium at any time, then this approach is sufficient to ensure an irreversible evolution [11] of the system S. It is worthwhile to observe that in the above sense the interaction is weak for A, whereas the damping may be strong or weak for 4, as discussed in Ref. [7]. Under these conditions the dynamical behavior of the quantum Brownian particle becomes characterized by four parameters: the natural frequency Qo, the damping constant y, which by its turn depends on the squared coupling constant ~K„~, the memory correlation time w„ and the reservoir temperature T. The aim of the present work is to compare the CNMA, considered by Velasco et al. [2,5,6], with the less- stringent CNNM approach. Moreover, besides the usual— I ~~/~, [1,2,5 —7] memory kernel e ', which corresponds to a Cauchy distribution for the bath frequency spectrum, we also consider the memory kernel sin(tlat, )/t, correspond- ing to a square distribution, which has not been con- sidered in those references. The paper is organized as follows. In Sec. II the gen- eralized master equation is introduced. In Sec. III the Hamiltonian is constructed, and in Sec. IV we define the 7923 1991 The American Physical Society 7924 J. R. BRINATI, S. S. MIZRAHI, AND B. M. PIMENTEL HO frequency distributions of the reservoir to be used. Finally, Sec. V is devoted to the results and discussions. II. THE GENERAI. IZED MASTER EQUATIDN The exact Liouville equation for S+%, in the interac- tion picture, is given by &pq(&) i = [ Vg(t), pg(0)] Bt cy, so that the present behavior of 4 does not depend on the past history, since Eq. (2) becomes a linear differential equation with constant parameters. (b) Colored noise for the thermal bath and p&&(t) local in time or Markovian (CNMA) consists of keeping the actual K(t —r') as a memory kernel while taking off p»(t) from the integral, so that Eq. (2) becomes a linear differential equation in p»(t), but with time-dependent parameters. (c) Colored noise for the thermal bath and p&&(t) non- local in time or non-Markovian (CNNM) keeps Eq. (2) without any approximation involving the time. where we assume a factorized form for the initial state P(0) =p+(0) p&(T) and consider the reservoir statistical operator p&( T) representing an equilibrium canonical en- semble. Now, in order to get a master equation for the reduced statistical operator, we take the trace over the reservoir variables and introduce the basic irreversibility hypothesis [11]p&(t)=p&&(t) p&&(T). Then III. THE MODEI. HAMII. TONIAN P=N, +8~+ V, (5) We consider one specific harmonic oscillator, the quan- tum Brownian particle (4), interacting with a heat bath (% ), represented by a very large number of HO's in equi- librium at some temperature T. The model Hamiltonian 1s (2) where where the memory kernel depends on the Liouville operators L;„, =[V, ], . with Lo =L++L&, H, H&, and V are, respectively, the system, the thermal bath, and the interaction Hamiltoni- an. This approach is equivalent to the Born approximation for the exact kernel of the GME obtained by Peiers [10] according to the Zwanzig projection technique [12], A'~ = g co„b„b„ are the system and reservoir Hamiltonians, respectively, and Q,~ and ~„are the oscillator frequencies. The in- teraction between system and reservoir is considered in the RWA, namely, P = y(X„*b„'A +Sr„b„A"), where E„and K„* are coupling constants. In the above equations we considered that fi= 1 and the operator com- mutation rules are bosonic type. Substituting the Hamiltonian (5) into the GME (2), we arrive at the following equation: where ~E„~ is the modulus of the coupling constant defined in Eq. (8). In treating the integro-differential equation (2) there are cases that permit di6'erent approaches to it. For in- stance, the thermal-bath HQ frequencies can be con- sidered as a density distribution that may be constant for any frequency (white noise) or with some structure (colored noise). On the other hand, the system S can be considered as being local in time [substituting in Eq. (2) p&&(t') by p+&(t)], the Markov approximation, or can keep the nonlocal character already present in Eq. (2), the non-Markov approach. In what follows our aim is to make a quantitative as well as a qualitative comparison between the cases below. (a) White noise for the thermal bath, which implies the Markov approximation, consists of assuming in Eq. (2) that the kernel k(t —t')=5(t —t')k. The 5 function means that the reservoir spectrum is Hat for any frequen- X [ A, A fp»(t') ] X [At, Apg~(t')] j+H. c. ), where oo I gz, (t —r')= g ~X(co )~ [I+n(Pcs )]e are c-number kernel functions and (lob) $,2(t t')= g /K(co )f n(P~ —)e, (10a) QUANTUM BROWNIAN PARTICLE AND MEMORY EFFECTS 7925 stands for the quanta mean value of the mth HO and p=(k&T); kz is the Boltzmann constant. The func- tions g)2 and g2, are correlation functions of the reservoir operators computed at difFerent times. Now we seek expressions for the mean values of quan- tities of interest belonging to 4, namely, ( A ), ( A tA ), ( A ), and their Hermitian conjugated. These quantities are essential to calculate the evolution in time of the ener- gy, position, momentum, and p-q correlation in the cases we mentioned in the preceding section. An adequate way to manipulate Eq. (9) is to work with the coherent-state labels as a mapping from a q-number to a c-number pic- ture, namely, A~a, As(t) ~P~, (a, a*;t); A, (t —t')= f defog(co)IK(co)l e 0 A,(t t—')= f d~g(~)lK(~)l'n(p~)e (16b) u(s)= u (0) s +A, (s) (17a) To characterize a colored noise, the function g(co) is assumed to have some shape or structure distinct from the Oat shape corresponding to the white noise, which characterizes a Markovian process of system S. Now, to calculate the mean value of some function f (a, a*) given by Eq. (1S), one takes its Laplace trans- form, which results in an algebraic equation and thereaf- ter turns back with the Mellin transform. For the above-mentioned cases we get the following Laplace transforms: (a) f (a,a')=a Calling the Laplacian transform Xt,((a)t )=u (s), we obtain this mapping leads us to the c-number GME aPss{t) dt' A, t —t' aI'gg t' Bt 0 Ba +c.C. where the kernel functions are now (13) (b) f (a,a*)=a Calling Xt, ( ( a ), ) = U (s), then v (0)U(s)= s +2A)(s) (c) f(a, a")=a'a Calling Xt, ( ( a*a ), ) = u) (s), then su) (0)+2 ReA2(s ) u) (s) = s [s+2ReA, (s)] (17b) (17c) A, (t —t') = [g»(t —t') —g),(t t')]e— (et@ —no)( t t )— (14a) where A, (s) and A2(s) are the Laplace transforms of A, (t t') and A2(t —t'—), respectively. For the above expressions [(17a)—(17c)], the Mellin transform (f(aa')), =——f ds e"[X t [(f( a a)) ]t] = g ~K(co )~ n(Pro )e (14b) d(f(a,a*)),= f P@s(a,a*,t)f(a, a") . (1S) Then any mean value of a function involving a and o. is calculated as a classical mean value; for instance, is performed once the function g(co) is defined before- hand. Considering that g(it))=go for all values of the fre- quencies, we obtain the %'NMA, and the calculated mean values of the selected quantities mentioned in Sec. III are IV. THE HO FREQUENCY DISTRIBUTIONS OF THE RESERVOIR In Eqs. (14) the discrete sums over the frequencies of the HO's are assumed to be closely spaced with a density distribution g (co) such that one is permitted to substitute the sums by integrals, ~f defog(co) 0 and the kernel functions, Eqs. (14), become (19a) (A'), =(A'),e ' ( A A ),=[(A A )()—n(PQ„)]e r'+n(PQ ) (19b) (19c) where y=2rtgo~K(Qo)~ is the damping constant and ( )0 stands for the initial mean values. In order to calcu- late the mean values of the Hermitian conjugated quanti- ties of 2 and A, it is sufhcient to take the complex con- jugate of ( A ), and ( A ),. In the following we are going to consider two specific 7926 J. R. BRINATI, S. S. MIZRAHI, AND B. M. PIMENTEL shapes for g(co) and compare the two approaches men- tioned in Sec. II. In all the cases considered, the mean values of the selected quantities are calculated up to first order in ~„ the heat-bath correlation time, assumed to be much smaller than the system characteristic times. A. Square distribution The distribution function is considered to have a Aat shape for a range of frequencies only, presenting a cutoff in the limits of its domain, i.e., g(~)=go 1 1+(co—Qo) r, (24) with a peak at +=00. All the mean values are calculat- ed, as in the previous case, up to first order in v.„besides involving also a first order in the ratio y /Qo. For the CNMA we get B. Cauehy distribution The Cauchy distribution function is a smooth continu- ous function for all values of frequencies, i.e., go no —&, '~~~no+ g(~)= ' 0 otherwise, with width 2&, For the CNMA we obtain ( A ),=( A ),[1+k[/, (0) y, (y—t/k)]}e (A'), =& A'&, II+2k[y, (0)—y, (yt/k)]}e ' (20) (21a) (21b) ( A ),= ( A ) 1+—[(r(t)—o.(0)]k zy(—/2 annot 2 (A'), =& A'), I 1+k[~(t)—~(0)] —zrt —2&QO( (A'A &, =[& A'A &, —n(PQ, )] (25a) (25b) (A A), =[(A A)o —n(PQ0)] X [1+2k [y, (0)—y, (yt /k) ] }e ~'+ n (pQO), (2 1c) where k =y~, is a factor that characterizes the memory effects in the time evolution of the mean values. For the CNNM approach the mean values are X [1+kRe[o(t) —o(0)] +k(l —e r' ")}e ~'+n(PQ0), where the integral iXt /7. oo eo(t) = —— dx x (1+x ) is calculated by numerical methods and Q O~c z =1+ ln 2m 1+Q()r, (25c) (26) 1+—(1 y t /2—) e kg, (yt/k) e— (22a) where zy is a pole that occurs in the Mellin transform, Eq. (18). The CNNM approach gives (A'&, =(A'), 1+ (1 y t) e-2k 1+—[1— (0)] 1—k zest 0 2 e —zyt/2 2k', (yt/k) e— (22b) 2 —(1—k/2)7t/k I I i, ' 0 2 '' (27a) ( A tA ),= [( A A )0—n(PQO)] 1+ (1—yt) e r' 2k', (yt/k)—2k 7T ( A ), = ( A )0( [1+k[1—o (0)](1 zyt) }e- —(( k)ytlk+k (—t)) ' o( (27b) +n(pQO) . In Eqs. (20) and (21), y, (y) is a function defined as 1 X,(y)= —[cos(y)+y si(y)] . (23) ( A "A ),=[& A'A &,—n(PQ, )] X ( I 1+k [ 1 Reer (0) ]—( 1 y t ) }e- —ke " "'~' "+k Reo(t))+n(PQ ) . (27c) QUANTUM BROWNIAN PARTICX.E AND MEMMORY EFFECTS 7927 0.06 0.3 0.04 0.1 0.02 0.00 U' + U G4 —0.1 —0.02 0 I 4 —0.3 0 4 FIG. 1IG. 1 ~ Time evolution of the Brownian ap ri ution function, Eq. (20'. Th sponds to the CNN q, The solid line corre- M approach and the da CNMA these curves the %'NMA e independent variabl was subtracted. ' bley =yt. In all FIG. distribution function, E . 20. e p-q correlation fG. . Time evolution of th or the square 1. ion, q. 20). All data are the same as in Fig V. RESULTS AND DISCUSSION To characterize the behavior of the uantum particle we calculate the dynamical e p —q correlation. These quantities ar to th in the preceding section, as follows ained (E ),=Ao( ( A A ),+—') 2 and &pq+qp &, =i(& A~ &, — & A which resent a cc aractenstic transient behavior for~ ~ of the cases pointed out in Sec. II. Figures 1 and 2 exhibit the ener versus uare an Cauchy distribution We consider the G ' utions, respectively. i er e auber coherent states.h , considered as a back r was subtracted from th CNMAe and CNNM e c gl ound, in order to stress th ' d'eir i erent behavior. expressions For the square spectrum case, Fig. l, the CNNM ap- 0.06 0.3 / / 0.04 f I 0.02 0.00 0.1 C4 U" + cT —0 I I I i I'I I[ I [ I I [ I I I Ii —0.02 0 I 4 —0.3 0 I 4 FIG. 2.. 2. Time evolution of the Brownian art' the Cauch distrib t fri u ion unction, E . same as in Fig. 1. q. (24). All data are the FIG.G. . Time evolution of the p-q correlation for 1. 7928 J. R. BRINATI, S. S. MIZRAHI, AND B. M. PIMENTEL proach shows remarkable oscillations (solid line), even asymptotically, in comparison to the CNMA (dashed line), which presents no significant oscillatory structure. Figure 2 compares the energy versus time behavior for the Cauchy distribution. The CNNM curve presents a behavior similar to the square case, but now the oscilla- tions are weaker, while the CNMA has a smooth struc- ture. Asymptotically, all the curves of Figs. 1 and 2 go to zero, which corresponds to the thermal equilibrium ener- gy The p-q correlations are depicted in Figs. 3 and 4 for the square and Cauchy distributions, respectively, and again we subtracted the WNMA from the CNMA and CNNM expressions. In both figures we observe an oscil- latory behavior for the CNNM (solid line) and the CNMA (dashed line). For initial times, the oscillation amplitudes are more pronounced for the CNMA than for the CNNM approach, and this behavior changes as time increases. For the square distribution, Fig. 3, while the CNMA amplitudes decrease monotonically with time, the CNNM exhibits an exotic oscillatory behavior. In all the calculations, to get the curves of these figures we used n(PQ o)= 0. 58, ( 3 3 &o=1, and k=0.2. For the Cauchy distribution we fixed the ratio y/Go=0. 2. We can have a better qualitative insight of the different characteristics of the Markov approximation and non- Markov approach when we consider the zero-time delay of the correlations and J(t)=(A 3&,—&A &, &A&, . Concerning the Markov approximation and an initial coherent state, we can verify, from Sec. IV, that the quantity I(t) is null, regardless of whether it is con- sidered a white-noise or colored-noise bath (either square or Cauchy distributions). On the other hand, the strength of the correlation J (r) depends essentially on the temperature through n (/3Qo); specifically, when T=O, J(t)=0 at any time, meaning that an initial coherent state remains coherent as it evolves. For the non-Markov approach and an initial coherent state we have a qualitatively different situation. The correlations I (t) and J(t) are time dependent, being pro- portional to the constant k =y~„' besides this, the strength of J(t) also depends on n (/30o) in a similar way as in the CNMA. So one verifies that even at T=O, an initial coherent state has its coherence destroyed as it be- gins to evolve. This fact is crucial to discern between a Markov and non-Markov treatment of a system under study. ACKNOWLEDGMENT This work was supported in part by Conselho Nacional do Desenvolvimento Cientifico e Tecnologico, Brazil. [1]K. Lindenberg and B. J. West, Phys. Rev. A 30, 568 (1984). [2] J. Breton, A. Hardisson, F. Mauricio, and S. Velasco, Phys. Rev. A 30, 542 {1984). [3] V. Capek and V. Szocs, Czech. J. Rev. B 36, 1182 (1986). [4] J. R. Brinati, B. M. Pimentel, S. S. Mizrahi, and S. A. Carias de Oliveira, Can. J. Phys. 66, 1044 {1988). [5] S. Velasco, J. A. White, and A. C. 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