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arXiv:math-ph/0307004 · Journal of Physics A Mathematical and General

A random matrix model of relaxation

Joel L. Lebowitz, Leonid Andreevich Pastur

Published Jan 19, 2004• 19 citations
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Abstract

We consider a two-level system, , coupled to a general n level system, , via a random matrix. We derive an integral representation for the mean reduced density matrix ρ( t ) of in the limit n → ∞, and we identify a model of which possesses some of the properties expected for macroscopic thermal reservoirs. In particular, it yields the Gibbs form for ρ(∞). We also consider an analog of the van Hove limit and obtain a master equation (Markov dynamics) for the evolution of ρ( t ) on an appropriate time scale.

DOI
10.1088/0305-4470/37/5/004

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We consider a two-level system, , coupled to a general n level system, , via a random matrix. We derive an integral representation for the mean reduced density matrix ρ( t ) of in the limit n → ∞, and we identify a model of which possesses some of the properties expected for macroscopic thermal reservoirs. In particular, it yields the Gibbs form for ρ(∞). We also consider an analog of the van Hove limit and obtain a master equation (Markov dynamics) for the evolution of ρ( t ) on an appropriate time scale.
Earlier two of us (J.L. and L.P.) considered a matrix model for a two-level system interacting with a n×nn\times nn×n reservoir and assuming that the interaction is modelled by a random matrix. We presented there a formula for the reduced density matrix in the limit $n\to \infty $ as well as several its properties and asymptotic forms in various regimes. In this paper we give the proofs of the assertions, and present also a new fact about the model.
Abstract This chapter introduces the density operator and the Liouville equation and outlines how reduced dynamical description of open quantum systems naturally leads to the appearance of relaxation processes. (1) The density operator and the quantum Liouville equation (the density matrix for a pure system, statistical mixtures, representations, coherences, thermodynamic equilibrium). (2) An example: The time evolution of a two-level system in the density matrix formalism. (3) Reduced descriptions. (4) Time evolution equations for reduced density operators: The quantum master equation (projection operators, The Nakajima–Zwanzig equation, Derivation of the quantum master equation using the thermal projector, The quantum master equation in the interaction representation, The Markovian limit—the Redfield equation, The secular approximation and the Lindblad equation). (5) The Lindblad equation in the ‘Heisenberg representation’. (6) Relaxation of a two-level system and a harmonic oscillator revisited. (7) The optically driven two-level system in a thermal environment—the Bloch equations. (8) The oscillating dipole model of a two-level molecule. (9) Analogy of a coupled two-level system to a spin ½ system in a magnetic field.
Most traditional discussions of the Markoffian Born approximation of relaxation phenomena (e.g., Bloch equations) are based on the assumption of widely separated time scales in the non-Markoffian weak coupling generalized master equation. By exploiting the physically reasonable Gaussian factorization property of multitime correlation functions which enter through the system reservoir interaction, recent work has shed significant new light on the relationship between the Markoffian-Born approximation and the exact generalized master equation (GME) describing the evolution of the relaxing subsystem. Two limits for which the GME becomes the Born approximation relaxation equation arc discussed: (1) the Van Hove (λ2t) limit rigorously discussed for open systems by Davies, (2) the zero memory (white noise) suggested by Middleton—Schieve and recently discussed for open systems by Gorini—Kossakawsky and Palmer. We have outlined the proof of this limit. We then discuss the physical meaning of these two exact limits in the context of the general theory of kinetic equations of Resibois. The dominant pole contributions are isolated to all orders by means of an expansion in which the terms are in norm proportional to powers of the ratio of the time scales τc/τB where τc is the two point correlation memory time and τB the relaxation time in the Born approximation. Each of the above exact limits corresponds to the vanishing of this ratio. Comments are made on nonexponential decay.
We consider a random matrix model of interaction between a small nnn-level system, SSS, and its environment, a NNN-level heat reservoir, RRR. The interaction between SSS and RRR is modeled by a tensor product of a fixed % n\times n matrix and a N×NN\times NN×N hermitian Gaussian random matrix. We show that under certain "macroscopicity" conditions on RRR, the reduced density matrix of the system $ρ_{S}=\mathrm{Tr}_{R}ρ_{S\cup R}^{(eq)} $, is given by ρS(c)∼exp⁡{−βHS}ρ_{S}^{(c)}\sim \exp {\{-βH_{S}\}}ρS(c)​∼exp{−βHS​}, where HSH_{S}HS​ is the Hamiltonian of the isolated system. This holds for all strengths of the interaction and thus gives some justification for using % ρ_{S}^{(c)} to describe some nano-systems, like biopolymers, in equilibrium with their environment \cite{Se:12}. Our results extend those obtained previously in \cite{Le-Pa:03,Le-Co:07} for a special two-level system.
A non-Markovian master equation is derived for the reduced probability density matrix of a subsystem interacting with a general reservoir of coupled anharmonic modes. Relaxation of the subsystem is mediated through resonant interactions with the reservoir. These interactions correspond to local vibrational motions with finite lifetimes due to diffusion/dephasing. The derivation assumes that the density of these interaction modes is very large so that they can be treated using statistical methods. The resulting master equation is shown to preserve probability, Hermiticity, and translational invariance, and conditions are stated which guarantee the positivity of the reduced density. At long time the master equation reduces to a Markovian equation of dynamical semigroup type. In the high temperature limit the theory is parameter free, and shows good agreement with the exact master equation of a spin-boson system.
Existing Markov models of the relaxation of quantum-mechanical systems are applicable in the approximations where the difference between eigenfrequencies is either much larger than the relaxation rates (global approach) or much lower than these rates (local approach). In this work, a model has been proposed to describe relaxation in systems where the difference between some eigenfrequencies is close to the relaxation rates. It has been shown that the master equation for the density matrix contains coefficients explicitly depending on the time in the interaction representation. Equations for the occupation numbers of the modes of the system have been derived by example of two interacting oscillators. It has been shown that occupation numbers in the proposed approach in the limits of small and large differences between eigenfrequencies compared to the relaxation rates asymptotically match the occupation numbers in the local and global approaches, respectively.