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The Journal of Chemical Physics

Quantum-classical approximation beyond Redfield theory

ANATOLE A. NEUFELD

Published Jul 17, 2003• 17 citations
Publisher

Abstract

A quantum-classical approximation, capable of describing the evolution of open quantum systems well beyond the applicability limits of Redfield theory is suggested. The theory is based on the short lifetime of the quantum correlations between the quantum and the classical subsystem, caused by energy dispersion (per degree of freedom) of the canonical bath. The resulting quantum-classical approximation has the form of two auxiliary differential equations and fully accounts both for the arbitrary long memory of the heat reservoir and detailed balance. These equations allow direct solution in the time domain without constructing/diagonalizing Liouville space operators, and, in combination with molecular dynamics techniques to simulate bath dynamics, may be applied to quantum subsystems with a fairly large number of levels. A simple example of a two-level system, coupled to a single correlation time canonical bath, was considered to demonstrate different regimes of approaching the canonical equilibrium state.

DOI
10.1063/1.1587124

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A quantum-classical approximation, capable of describing the evolution of open quantum systems well beyond the applicability limits of Redfield theory is suggested. The theory is based on the short lifetime of the quantum correlations between the quantum and the classical subsystem, caused by energy dispersion (per degree of freedom) of the canonical bath. The resulting quantum-classical approximation has the form of two auxiliary differential equations and fully accounts both for the arbitrary long memory of the heat reservoir and detailed balance. These equations allow direct solution in the time domain without constructing/diagonalizing Liouville space operators, and, in combination with molecular dynamics techniques to simulate bath dynamics, may be applied to quantum subsystems with a fairly large number of levels. A simple example of a two-level system, coupled to a single correlation time canonical bath, was considered to demonstrate different regimes of approaching the canonical equilibrium state.
A fully classical limit of the recently published quantum-classical approximation [A. A. Neufeld, J. Chem. Phys. 119, 2488 (2003)] is obtained and analyzed. The resulting kinetic equations are capable of describing the evolution of an open system on the entire time axis, including the short-time non-Markovian stage, and are valid beyond linear response regime. We have shown, that proceeding to the classical mechanics limit we restrict the class of allowed correlations between an open system and a canonical bath, so that the initial conditions and the relaxation operator has to be appropriately modified (projected). Disregard of the projection may lead to unphysical behavior, since mechanism of the decay of some correlations is essentially of quantum-mechanical nature, and is not correctly described by classical mechanics. The projection (quantum correction to the kinetics) is particularly important for the non-Markovian regime of relaxation towards canonical equilibrium. The conformity of the developed method to the conventional approaches is demonstrated using a model of Brownian motion (heavy particle in the bath of light ones), for which the obtained non-Markovian equations are reduced to the standard Fokker-Planck equation in phase space.
A phase-space representation of Bloch–Redfield theory is used to describe the dynamical evolution of quantum dissipative systems. The resulting Liouville operator equations are capable of incorporating both the master equation in eigenstate space and the stochastic equation in classical phase space, and thus provide a useful framework for mixing classical, semiclassical, and quantum dynamics for simulating complicated dissipative systems. In addition, the proper limit of quantum dissipation, the approximate nature of the second-order cumulant truncation, the detailed balance of quantum correlation functions, and the reduction of dissipation by a transformation of the bath Hamiltonian are investigated within the framework of phase-space Bloch–Redfield theory.
The conditions under which quantum-classical Liouville dynamics may be reduced to a master equation are investigated. Systems that can be partitioned into a quantum-classical subsystem interacting with a classical bath are considered. Starting with an exact non-Markovian equation for the diagonal elements of the density matrix, an evolution equation for the subsystem density matrix is derived. One contribution to this equation contains the bath average of a memory kernel that accounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating a Markovian approximation on this term in the evolution equation. The resulting subsystem density matrix equation is still non-Markovian due to the fact that bath degrees of freedom have been projected out of the dynamics. Provided the computation of nonequilibrium average values or correlation functions is considered, the non-Markovian character of this equation can be removed by lifting the equation into the full phase space of the system. This leads to a trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results are illustrated by computations of the rate constant of a model nonadiabatic chemical reaction.
A quantum dissipation theory is constructed with the system–bath interaction being treated rigorously at the second-order cumulant level for both reduced dynamics and initial canonical boundary condition. The theory is valid for arbitrary bath correlation functions and time-dependent external driving fields, and satisfies correlated detailed-balance relation at any temperatures. The general formulation assumes a particularly simple form in driven Brownian oscillator systems in which the correlated driving-dissipation effects can be accounted for exactly in terms of local-field correction. Remarks on a class of widely used phenomenological quantum master equations that neglects the bath dispersion-induced dissipation are also made in contact with the present theory.
A quantum-classical limit of the canonical equilibrium time correlation function for a quantum system is derived. The quantum-classical limit for the dynamics is obtained for quantum systems comprising a subsystem of light particles in a bath of heavy quantum particles. In this limit the time evolution of operators is determined by a quantum-classical Liouville operator, but the full equilibrium canonical statistical description of the initial condition is retained. The quantum-classical correlation function expressions derived here provide a way to simulate the transport properties of quantum systems using quantum-classical surface-hopping dynamics combined with sampling schemes for the quantum equilibrium structure of both the subsystem of interest and its environment.
We propose a method which combines the quantum-classical mapping approach to surface hopping (MASH) with the dissipative quantum dynamics of the Lindblad master equation. Like conventional surface-hopping methods, our approach is based on classical trajectories coupled to the dynamics of a quantum subsystem. However, instead of evolving the subsystem wavefunction according to the time-dependent Schrödinger equation, we use stochastic quantum trajectories derived from secular Redfield theory. This enables the simulation of open quantum systems coupled simultaneously to Markovian quantum baths and anharmonic non-Markovian classical degrees of freedom. Applications to the spin--boson model and to the cavity-enhanced fluorescence of an electronically nonadiabatic molecule show excellent agreement with fully quantum-mechanical benchmarks.