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Exact Time Evolution and Master Equations for the Damped Harmonic Oscillator Abstract CORE Metadata, citation and similar papers at core.ac.uk Provided by CERN DocumentExact Server Time Evolution and Master Equations for the Damped Harmonic Oscillator Robert Karrlein and Hermann Grabert Fakult¨at f¨ur Physik der Albert–Ludwigs–Universit¨at, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany (August 26, 1996) Abstract Using the exact path integral solution for the damped harmonic oscillator it is shown that in general there does not exist an exact dissipative Liouville operator describing the dynamics of the oscillator for arbitrary initial bath preparations. Exact non-stationary Liouville operators can be found only for particular preparations. Three physically meaningful examples are exam- ined. An exact new master equation is derived for thermal initial conditions. Second, the Liouville operator governing the time-evolution of equilibrium correlations is obtained. Third, factorizing initial conditions are studied. Ad- ditionally, one can show that there are approximate Liouville operators inde- pendent of the initial preparation describing the long time dynamics under appropriate conditions. The general form of these approximate master equa- tions is derived and the coefficients are determined for special cases of the bath spectral density including the Ohmic, Drude and weak coupling cases. The connection with earlier work is discussed. Typeset using REVTEX 1 I. INTRODUCTION Recently the problem of the reduced dynamics of a quantum system in contact with a reservoir has gained renewed interest [1–3]. While the foundations of quantum dissipative processes were layed already in the sixties [4], this early work was mainly concerned with weakly damped systems and has relied on the Born and Markov approximations. In this conventional approach the dynamics of the dissipative quantum system is described in terms of quantum master or Langevin equations. Important progress beyond the limitations of the weak coupling approach was made in the eighties, in particular path integral techniques were shown to be powerful means to describe quantum dissipative systems [5]. This has lead to unexpected new results, as for instance the non-exponential decay of correlation functions in the low temperature range [6,7], not directly available within the conventional master equation approach. The new insight gained also provided a basis for profound critique [1,3,8,9] of concepts de- veloped in the context of the quantum master equation approach such as complete positivity or the quantum regression theorem. However, the results of the path integral approach were rarely used to derive improved master equations valid in the range of strong damping and/or low temperatures. In this paper we shall address this problem for the ubiquitous quantum dissipative system, the damped harmonic oscillator. Based on available exact results [10] we derive new generalized quantum master equations describing the exact relaxation of mean values and the time evolution of equilibrium correlation functions. Whenever appropriate, the new results will be confronted and compared with earlier findings. More specifically, the article is organized as follows. In the following section the micro- scopic model of a harmonic oscillator coupled to a harmonic bath is introduced. The exact time evolution of the reduced density matrix of the oscillator [10] will briefly be summarized. In section III we examine whether the exact time evolution of the density matrix is compat- ible with a generalized quantum master equation with a time dependent Liouville operator. It is shown that in general there is no Liouville operator independent of the initial prepa- 2 ration. In section IV we study specific types of initial preparations. Exact time-dependent Liouville operators describing the time evolution of thermal initial conditions and of cor- relation functions are derived. In the classical limit these Liouville operators are shown to reduce to the Adelman Fokker-Planck operator [11]. We also examine the case of factorizing initial conditions. In section V we investigate particular types of damping leading to a time evolution of the density matrix governed approximately by a time-independent Liouville operator. Earlier results by Talkner [9] and by Haake and Reibold [12] are recovered. We discuss in detail the limit of a weakly damped oscillator and derive a generalization of the Agarwal equation [13]. Additional time coarse-graining is shown to lead to the quantum-optical master equation by Weidlich and Haake [14]. Finally in section VI we present our conclusions. II. MODEL HAMILTONIAN AND EXACT TIME EVOLUTION The standard microscopic model [15–17] for the damped harmonic oscillator starts out from a Hamiltonian H composed of the oscillator part p2 M H = + ω2q2, (1) O 2M 2 0 the bath Hamiltonian N 2 pn mn 2 2 HR = + ωnxn, (2) n=1 2mn 2 X and an interaction part N N 2 2 cn HOR = −q cnxn + q 2 . (3) n=1 n=1 2mnωn X X This model describes an oscillator with the associated classical equation of motion t 2 Mq¨(t)+ dsγ(t − s)˙q(s)+ω0q(t)=0 (4) Z0 where 3 N 2 1 cn γ(t)= 2 cos(ωnt)(5) Mn=1 mnωn X is the damping kernel. The dynamics of the density matrix of the entire system (oscillator and bath) is given by W (t)=exp(−iHt/h¯)W (0) exp(iHt/h¯), (6) from where the reduced density matrix of the oscillator is obtained by tracing out the bath modes ρ(t)=TrRW(t). (7) The path integral technique allows for a complete solution to this problem for a large class of initial states. Since the derivation is expounded in [10], we merely state the result 0 0 0 0 0 0 0 0 ρ(qf ,qf,t)= dqidqidqd¯ q¯J(qf,qf,t,qi,qi,q,¯ q¯)λ(qi,qi,q,¯ q¯)(8) Z 0 0 where λ(qi,qi,q,¯ q¯) is the so-called preparation function defining the initial non-equilibrium state W (0) as a modification of the equilibrium state Wβ =exp(−βH)/Tr exp(−βH)inthe 0 subspace of the oscillator. This modification can be produced by operators Oj, Oj that act upon the particle only leaving the reservoir coordinates unchanged 0 W (0) = OjWβOj. (9) j X We then have 0 0 0 0 0 hq|W (0)|q i = dqd¯ q¯λ(q, q , q,¯ q¯)hq¯|Wβ|q¯i. (10) Z where 0 0 0 0 0 λ(q, q , q,¯ q¯)= hq|Oj|q¯ihq¯|Oj|q i. (11) j X Several relevant examples for λ(q, q0, q,¯ q¯0) are discussed in [10] and special cases will be 0 0 0 considered below. J(qf ,qf,t,qi,qi,q,¯ q¯) is the propagating function describing the time 4 evolution of the reduced density matrix. In terms of the transformed coordinates r = (q + q0)/2,x=q−q0,itreads 1 i J(xf,rf,t,xi,ri,x,¯ r¯)= exp Σ(xf ,rf,t,xi,ri,x,¯ r¯) (12) N(t) h¯ where 2 2 h¯r¯ hpi 2 Σ(xf ,rf,t,xi,ri,x,¯ r¯)=i 2 + x¯ 2hq i 2¯h ! +M[xf rf f1(t)+xirif2(t)−xirff3(t)−xfrif4(t)] i + M x2R++(t)+2x xR+−(t)+x2R−−(t) (13) 2 i f i f h i + − + − +M r¯(xiC1 (t)+xfC1 (t)) − ix¯(xiC2 (t)+xfC2 (t)) h i and 2πh¯ N(t)= (2πhq2i)1/2. (14) M|f3(t)| ±± ± The functions fi(t), R (t), and Ci (t) are given in terms of the symmetrized part S(t)and ¯h the anti-symmetrized part A(t)=−2MG(t) of the equilibrium coordinate autocorrelation function C(t)=hq(t)qi=S(t)+iA(t) in the following way: G˙(t) f (t)=f(t)= , (15) 1 2 G(t) 1 f (t)= , (16) 3 G(t) G˙(t)2 f (t)=−G¨(t)+ , (17) 4 G(t) 2 ++ M hpi f3(t) 2 2 2 2 R (t)= + 2hq iS˙(t)+f3(t) hq i −S (t) , (18) h¯ (M2 hq2i ) h i +− M ¨ ˙ f3(t) ˙ 2 2 2 R (t)= S(t)−f1(t)S(t)− S(t)S(t)+f1(t) hq i −S (t) , (19) h¯ ( hq2i ) h i M p2 1 2 −− h i ˙ 2 2 R (t)= − S(t)−f1(t)S(t) +hq if1(t) , (20) h¯ (M2 hq2i ) h i S(t) C+(t)=f(t) −f(t), (21) 1 3 hq2i 1 S˙(t) S(t) C−(t)= −f(t) +f(t), (22) 1 hq2i 1 hq2i 4 5 2 + M hpi ˙ C2 (t)= +f3(t)S(t) , (23) h¯ "M2 # M C−(t)= S¨(t)−f(t)S˙(t) . (24) 2 h¯ 1 h i In the sequel we shall give results mainly in terms of S(t)andG(t) which both remain finite in the classical limit while A(t) vanishes. The Laplace transforms of G(t)andS(t)readin terms of the Laplace transformγ ˆ(z) of the damping kernel [10,18] ˆ 2 2 −1 G(z)= z +ω0 +zγˆ(z) , h ∞ i ˆ 1 z ˆ ˆ S(z)= 2 2 [G(z) − G(|νn|)]. (25) Mβ n= νn − z X−∞ Finally, ∞ 2 1 2 2 −1 hq i = S(0) = [ω0 + νn + |νn|γˆ(|νn|)] , (26) Mβ n=−∞ X ∞ 2 2 2 ¨ M ω0 + |νn|γˆ(|νn|) hp i = −M S(0) = 2 2 . β n= ω0 + νn + |νn|γˆ(|νn|) X−∞ are the equilibrium variances. III. MASTER EQUATION We now want to investigate the conditions under which the time evolution of the reduced density matrix can be described by a master equation ∂ ∂ ∂ ρ(x, r, t)=L(x, r, , ,t)ρ(x, r, t) (27) ∂t ∂x ∂r with a generally time-dependent Liouville operator L.
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