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Journal of Physics A Mathematical and General

Memory effects upon the relaxation dynamics of the Wigner distribution function

E. Susana Hernández, Horacio M. Cataldo

Published Sep 7, 1991• 4 citations
Publisher

Abstract

The solutions of the generalized Fokker-Planck equation that arises in the semiclassical description of a quantal harmonic coordinate immersed in an arbitrary heat bath are investigated in the non-Markovian regime. The treatment consists of retaining a finite number of terms of the analytic expansion of the Laplace-transformed memory kernel around the origin which leads to approximate relaxation frequencies involving different degrees of memory. For a regime of highly inelastic collisions the authors extend previous findings concerning the relation between Markovian and non-Markovian relaxation frequencies and, for the usual elastic coupling, they consider fermion and phonon environments showing that the former involves a much more complex dynamics with a non-exponential time decay. Focusing upon the exponential part of the decay, they find that for both classes of reservoirs the first two non-Markovian corrections have well defined signs that increase the relaxation frequencies.

DOI
10.1088/0305-4470/24/17/027

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The solutions of the generalized Fokker-Planck equation that arises in the semiclassical description of a quantal harmonic coordinate immersed in an arbitrary heat bath are investigated in the non-Markovian regime. The treatment consists of retaining a finite number of terms of the analytic expansion of the Laplace-transformed memory kernel around the origin which leads to approximate relaxation frequencies involving different degrees of memory. For a regime of highly inelastic collisions the authors extend previous findings concerning the relation between Markovian and non-Markovian relaxation frequencies and, for the usual elastic coupling, they consider fermion and phonon environments showing that the former involves a much more complex dynamics with a non-exponential time decay. Focusing upon the exponential part of the decay, they find that for both classes of reservoirs the first two non-Markovian corrections have well defined signs that increase the relaxation frequencies.
We develop a generalized quantum Fokker-Planck theory in a non-Gaussian-Markovian model bath. The semiclassical bath adopted in this work is charactered by three parameters. One denotes the strength of system-bath coupling and the other two are chosen to interpolate smoothly the solvation dynamics between the long- and short-time regimes. The fluctuation-dissipation relation in this model bath is analyzed in detail. Based on this model bath, we derive two sets of coupled Fokker-Planck equations. These two equation sets are equivalent in the second order of system-bath coupling but different in the higher orders. The corresponding reduced Liouville equation in one set of the Fokker-Planck formulation is characterized by a memory relaxation kernel, while that in the other is by a local-time relaxation tensor. Each resulting set of Fokker-Planck equations involves only the reduced density operator and a series of well-characterized Hilbert-space relaxation operators. The present theory is valid for arbitrary time-dependent Hamiltonians and is applicable to the study of quantum coherence and relaxation in various dynamic systems.
We analyze the irreversible behavior of a quantum harmonic oscillator immersed in an arbitrary thermal reservoir. For the fully coupled (FC) model, i.e., the coupling between subsystems linear in the coordinate of the oscillator, we construct a generalized master equation for the reduced density operator of the oscillator. We apply the rotating-wave approximation (RWA) and investigate its connection with the FC master equation. By means of the Wigner distribution, we obtain the corresponding Fokker-Planck equations in the semiclassical representation. We find that, when the RWA is not imposed, the correct classical limit is obtained (Kramer's equation). Then, we establish the conditions for the equivalence between the FC Fokker-Planck equation and that obtained in the RWA. Quantum maps for a kicked oscillator immersed in a heat bath are obtained in both cases. These maps are studied in the classical and semiclassical limits, and it is shown that they coincide for low kicking frequency compared to the damping rate.
The heat-bath frequency distribution, necessary to maintain the canonical commutation relations for all time for a quantum mechanical oscillator with non-Markovian features like memory-dependent damping, is shown to satisfy certain constraint relations. An algorithm is given to explicitly find the heat-bath frequency distribution in terms of a series expansion for all processes where the timescale for the non-Markovian memory kernel is much smaller than the inverse of the strength of damping term. In the non-Markovian case, the heat-bath distribution function exhibits dependence on the system characteristics. The KS periodicity conditions on the system Green functions as the system approaches equilibrium are established for the present case.
We study the dynamics of relaxation and thermalization in an exactly solvable model of a particle interacting with a harmonic oscillator bath. Our goal is to understand the effects of non-Markovian processes on the relaxational dynamics and to compare the exact evolution of the distribution function with approximate Markovian and non-Markovian quantum kinetics. There are two different cases that are studied in detail: (i) a quasiparticle (resonance) when the renormalized frequency of the particle is above the frequency threshold of the bath and (ii) a stable renormalized "particle" state below this threshold. The time evolution of the occupation number for the particle is evaluated exactly using different approaches that yield to complementary insights. The exact solution allows us to investigate the concept of the formation time of a quasiparticle and to study the difference between the relaxation of the distribution of bare particles and that of quasiparticles. For the case of quasiparticles, the exact occupation number asymptotically tends to a statistical equilibrium distribution that differs from a simple Bose-Einstein form as a result of off-shell processes whereas in the stable particle case, the distribution of particles does not thermalize with the bath. We derive a non-Markovian quantum kinetic equation which resums the perturbative series and includes off-shell effects. A Markovian approximation that includes off-shell contributions and the usual Boltzmann equation (energy conserving) are obtained from the quantum kinetic equation in the limit of wide separation of time scales upon different coarse-graining assumptions. The relaxational dynamics predicted by the non-Markovian, Markovian, and Boltzmann approximations are compared to the exact result. The Boltzmann approach is seen to fail in the case of wide resonances and when threshold and renormalization effects are important.
In this paper we investigate the non-Markovian dynamics of a qubit by comparing two generalized master equations with memory. In the case of a thermal bath, we derive the solution of the recently proposed post-Markovian master equation [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101(R) (2005)] and we study the dynamics for an exponentially decaying memory kernel. We compare the solution of the post-Markovian master equation with the solution of the typical memory kernel master equation. Our results lead to a new physical interpretation of the reservoir correlation function and bring to light the limits of usability of master equations with memory for the system under consideration.