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Quantum-classical limit of quantum correlation functions

Sergi, A., Kapral, R.

Published Oct 22, 2004• 24 citations
OpenAlex

Abstract

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.

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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.
Quantum time correlation functions are often the principal objects of interest in experimental investigations of the dynamics of quantum systems. For instance, transport properties, such as diffusion and reaction rate coefficients, can be obtained by integrating these functions. The evaluation of such correlation functions entails sampling from quantum equilibrium density operators and quantum time evolution of operators. For condensed phase and complex systems, where quantum dynamics is difficult to carry out, approximations must often be made to compute these functions. We present a general scheme for the computation of correlation functions, which preserves the full quantum equilibrium structure of the system and approximates the time evolution with quantum-classical Liouville dynamics. Several aspects of the scheme are discussed, including a practical and general approach to sample the quantum equilibrium density, the properties of the quantum-classical Liouville equation in the context of correlation function computations, simulation schemes for the approximate dynamics and their interpretation and connections to other approximate quantum dynamical methods.
The statistical mechanics of systems whose evolution is governed by mixed quantum-classical dynamics is investigated. The algebraic properties of the quantum-classical time evolution of operators and of the density matrix are examined and compared to those of full quantum mechanics. The equilibrium density matrix that appears in this formulation is stationary under the dynamics and a method for its calculation is presented. The response of a quantum-classical system to an external force which is applied from the distant past when the system is in equilibrium is determined. The structure of the resulting equilibrium time correlation function is examined and the quantum-classical limits of equivalent quantum time correlation functions are derived. The results provide a framework for the computation of equilibrium time correlation functions for mixed quantum-classical systems.
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.
The quantum-classical Liouville equation describes the dynamics of a quantum subsystem coupled to a classical environment. It has been simulated using various methods, notably, surface-hopping schemes. A representation of this equation in the mapping Hamiltonian basis for the quantum subsystem is derived. The resulting equation of motion, in conjunction with expressions for quantum expectation values in the mapping basis, provides another route to the computation of the nonadiabatic dynamics of observables that does not involve surface-hopping dynamics. The quantum-classical Liouville equation is exact for the spin-boson system. This well-known model is simulated using an approximation to the evolution equation in the mapping basis, and close agreement with exact quantum results is found.
We consider the time correlation function of observables pertaining to a (quantum subsystem +bath), where the bath is coupled to a reservoir with many degrees of freedom. Integrating over the coordinates of this reservoir and assuming no initial correlations between the (quantum subsystem+bath) and the reservoir, we obtain an expression for the time correlation function that contains an influence functional. We then take the semiclassical and Fokker–Planck limits while modeling the reservoir with an Ohmic continuum of harmonic oscillators coupled bilinearily to the coordinates of the bath. The semiclassical limit is taken using a variant of Pechukas’ stationary phase analysis of the reduced propagator that yields a time correlation function written in terms of connected “classical” paths. These paths are got by solving the concatenation of several short-time interval Pechukas equations; as a result, the determination of these paths is more feasible than the determination of the “classical” path associated with a single long-time interval Pechukas equation. This concatenation includes the dissipative and stochastic forces associated with a classical Brownian particle. We then use decoherence arguments derived from an inspection of the influence functional to eliminate the phase interference structure of the bath. This elimination yields a mixed quantum-classical time correlation function that can be evaluated using nonadiabatic mixed quantum-classical dynamics schemes similar to those proposed recently by Webster and Tully.
Mixed quantum-classical equations of motion are derived for a quantum subsystem of light (mass m) particles coupled to a classical bath of massive (mass M) particles. The equation of motion follows from a partial Wigner transform over the bath degrees of freedom of the Liouville equation for the full quantum system, followed by an expansion in the small parameter μ=(m/M)1/2 in analogy with the theory of Brownian motion. The resulting mixed quantum-classical Liouville equation accounts for the coupled evolution of the subsystem and bath. The quantum subsystem is represented in an adiabatic (or other) basis and the series solution of the Liouville equation leads to a representation of the dynamics in an ensemble of surface-hopping trajectories. A generalized Pauli master equation for the evolution of the diagonal elements of the density matrix is derived by projection operator methods and its structure is analyzed in terms of surface-hopping trajectories.