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The Journal of Chemical Physics

A static ensemble approximation for stochastically modulated quantum systems

Robert Bittl, Klaus J. Schulten

Published Feb 1, 1989• 30 citations
Publisher

Abstract

A short-time approximation for the evolution of quantum systems governed by a Hamiltonian with stochastic time dependence is derived. The evolution operator of the system is replaced by an ensemble of evolution operators with time-independent Hamiltonians, weighted by a distribution function related to the ‘‘line shape function’’ of a randomly modulated harmonic oscillator. The approximation conserves the trace of the density matrix and converges to the exact solution in the case of very slow and in the case of very rapid stochastic modulation. The approximation is applied to hyperfine-induced singlet–triplet transitions of a biradical-like system with fluctuating exchange interaction.

DOI
10.1063/1.456073

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A short-time approximation for the evolution of quantum systems governed by a Hamiltonian with stochastic time dependence is derived. The evolution operator of the system is replaced by an ensemble of evolution operators with time-independent Hamiltonians, weighted by a distribution function related to the ‘‘line shape function’’ of a randomly modulated harmonic oscillator. The approximation conserves the trace of the density matrix and converges to the exact solution in the case of very slow and in the case of very rapid stochastic modulation. The approximation is applied to hyperfine-induced singlet–triplet transitions of a biradical-like system with fluctuating exchange interaction.
Line-shape analysis of transitions in a quantum system is a method widely applied to study relaxation processes on a microscopic scale. In many practical cases, this relaxation arises from stochastic modulation of the Hamiltonian due to interactions with a surrounding heat bath. Using a separable class of stochastic operators, the line-shape problem for a Hamiltonian with a single transition has been solved in closed form. This method is not appropriate for more complicated Hamiltonians with closely spaced multiplets, as the modulation may lead to significant coupling between the individual transitions in the multiplet. In this work, therefore, an investigation of the response function of a modulated multiplet is undertaken. The dynamical equation is derived from the stochastic Liouville equation for the density matrix. It is solved rigorously for two practical examples drawn from magnetic-resonance spectroscopy. These exact results are used to investigate the accuracy of approximations found in the literature.
A stochastic approach to the quantum dynamics randomly modulated in time by a discrete state non-Markovian noise, which possesses an arbitrary nonexponential distribution of the residence times, is developed. The formally exact expression for the Laplace-transformed quantum propagator averaged over the stationary realizations of such N -state non-Markovian noise is obtained. The theory possesses a wide range of applications. It includes some previous Markovian and non-Markovian theories as particular cases. In the context of the stochastic theory of spectral line shape and relaxation, the developed approach presents a non-Markovian generalization of the Kubo-Anderson theory of sudden modulation. In particular, the exact analytical expression is derived for the spectral line shape of optical transitions described by a Kubo oscillator with randomly modulated frequency which undergoes jumplike non-Markovian fluctuations in time.
The time evolution of a randomly modulated quantum harmonic oscillator is studied by introducing the master equation for the reduced density operator s(t)s(t)s(t). The antinormally ordered representation is adopted for s(t)s(t)s(t), and one is thus confronted with the problem of solving the partial differential equation obeyed by the reduced matrix element s‾\overline{s}s of s(t)s(t)s(t). The two limiting cases of short- and long-range correlations of the frequency fluctuations are treated in detail.
The behavior of a three-dimensional, nonrelativistic, quantum mechanical harmonic oscillator is investigated under the influence of three distinct types of randomly fluctuating potential fields. Specifically, kinetic (or transport) equations are derived for the corresponding stochastic Wigner equation (the exact equation of evolution of the phase-space Wigner distribution density function) and the stochastic Liouville equation (correspondence limit approximation) using two closely related statistical techniques, the first-order smoothing and the long-time Markovian approximations. Several physically important averaged observables are calculated in special cases. In the absence of a deterministic inhomogeneous potential field (randomly perturbed, freely propagating particle), the results reduce to those reported previously by Besieris and Tappert.
ABSTRACT Stochastic evolution underpins several approaches to the dynamics of open quantum systems, such as random modulation of Hamiltonian parameters, the stochastic Schrödinger equation (SSE), and the stochastic Liouville equation (SLE). These approaches replace the explicit system–environment coupling with an effective system‐only dynamics, where dissipative behavior emerges from ensemble averaging. Stochastic Hamiltonians, in particular, have long served as phenomenological tools in physical chemistry to include environmental effects without recourse to an explicit microscopic derivation. In this work, we aim at a self‐contained and accessible presentation of these approaches to further elaborate on their common roots in essential concepts of stochastic calculus and to delineate the conditions under which they are equivalent. We also discuss how different formulations naturally lead to different numerical time‐integration schemes, better suited for either classical simulation platforms, based on finite‐difference approximations, or quantum algorithms, that employ random unitary maps. Our analysis aims at providing a unified perspective and actionable recipes for classical and quantum implementations of stochastic evolution in the simulation of open quantum systems.
Our main goal in this paper is to extend to any system of coupled quadratic Hamiltonians some properties known for systems of quantum harmonic oscillators related with the Brownian Quantum Motion model. In a first part we get a rather general formula for the purity (or the linear entropy) in a short time approximation. In a second part we establish a master equation (or a Fokker-Planck type equation) for the time evolution of the reduced matrix density for bilinearly coupled quadratic Hamiltonians. The Hamiltonians and the bilinear coupling can be time dependent. Moreover we give an explicit formula for the solution of this master equation so that the time evolution of the reduced density at time ttt is connected with the reduced density at initial time t0t_0t0​ for $t_0 \leq t