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The Journal of Chemical Physics

Time correlation functions for mixed quantum-classical systems

J. Liam McWhirter

Published Apr 8, 1998• 8 citations
Publisher

Abstract

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.

DOI
10.1063/1.475978

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Solving quantum dynamics is an exponentially difficult problem. Thus, an exact numerical solution is inaccessible for any condensed matter system.\nA promising approach is to divide the system into a quantum subsystem containing degrees of freedom which are of greater interest or those which have more profound quantum character (e.g., have smaller mass) and a classical bath containing the rest of the system.\nImposing such a partition and treating the bath classically results in quantum-classical dynamics. The quantum-classical Liouville equation is a general equation in the Hilbert space of quantum degrees of freedom while it resides in the phase space of the classical degrees of freedom.\nAny numerical solution to this equation requires representation of the quantum subsystem in some basis. Solutions to this equation have been already proposed in the subsystem, adiabatic and force bases, each with its own cons and pros.\n\nIn this work, the quantum-classical equations of motion are cast in the subsystem basis and subsequently mapped to a number of fictitious harmonic oscillators.\nThe result is quantum-classical dynamics in the mapping basis which treats both quantum and classical degrees of freedom on the same footing, i.e., in phase space. Neglecting a portion of the back reaction of the quantum-subsystem to classical bath results in an expression for the time evolution of an operator (density matrix) equal to its Poisson bracket with the Hamiltonian.\nThis equation can be solved in terms of characteristics to provide a computationally tractable method for calculating quantum-classical dynamical properties. The expressions for expectation values and correlation functions in this formalism are derived.\nCalculations on spin-boson system, barrier crossing models---the so called Tully models---and the Fenna-Mathews-Olson pigments show very good agreement between the results of this method and numerical solutions to the Schrödinger equation.

•Aug 31, 2011•0 citations
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.
We review the Pechukas stationary phase analysis which yields the semiclassical reduced propagator. This propagator describes the dynamics of a mixed quantum-semiclassical system. In addition, we review a variant of this analysis, presented by us in an earlier article [J. Chem. Phys. 108, 5683 (1998)], which yields a semiclassical reduced propagator written in terms of connected and partially connected “classical” paths. These paths are obtained by solving the concatenation of several short time interval Pechukas equations. We argue and then demonstrate numerically that the “energy” along these paths is generally piecewise conserved: conserved across one short time interval, but not across several such intervals. In our review of these analyses, we relax the assumption made by Pechukas that the magnitude of the transition amplitude associated with the quantum subsystem varies much more slowly with changes in the classical subsystem’s trajectory than its phase. As our analyses demonstrate, this assumption serves to simplify the evaluation of the normalization path integral; the stationary phase paths are not affected by the making of this assumption. Solving the Pechukas equation subject to the initial configuration and velocity of the classical subsystem yields a collection of “classical” paths; the solution is nonunique. We provide a short time uniqueness theorem pertaining to the class of functional differential equations to which the Pechukas equation belongs; then, we review the sample problem that Pechukas used to first demonstrate this nonuniqueness. The theorem and the sample problem allow us to identify which parts within the Pechukas equation’s structure are responsible for the nonuniqueness in its solution. This nonuniqueness is verified numerically. Here, we show that the “energy” is conserved along each of the “classical” paths in the collection. However, the “energies” of any two paths in this collection will be the same only if the initial state specified in the Pechukas equation is an instantaneous adiabatic eigenstate of the quantum subsystem Hamiltonian which contains the interaction potential between the quantum and classical subsystems.
In a recent letter [Chem. Phys. Lett. 291, 101 (1998)] we presented a semiclassical methodology for calculating influence functionals arising from many-body anharmonic environments in the path integral formulation of quantum dynamics. Taking advantage of the trace operation associated with the unobservable medium, we express the influence functional in terms of a single propagator along a combined forward–backward system path. This propagator is evaluated according to time-dependent semiclassical theory in a coherent state initial value representation. Because the action associated with propagation in combined forward and backward time is governed by the net force experienced by the environment due to its interaction with the system, the resulting propagator is generally a smooth function of coordinates and thus amenable to Monte Carlo sampling; yet, the interference between forward and reverse propagators is fully accounted for. In the present paper we present a more elaborate version of the semiclassical influence functional formalism, along with algorithms for evaluating the coherent state transform of the Boltzmann operator that enters the influence functional. This factor is evaluated by performing an imaginary time path integral, and various approximations of the resulting expression as well as sampling schemes are discussed. The feasibility of the approach is demonstrated via numerous test calculations involving a two-level system coupled to (a) a dissipative harmonic bath and (b) a ten-dimensional bath of coupled anharmonic oscillators.
A derivation of the equations of motion for a mixed quantum-classical system in a dissipative environment is presented. The classical environment of the quantum subsystem is taken to be composed of two parts: the first part, termed the classical subsystem, comprises those degrees of freedom that are coupled directly to the quantum subsystem, whereas the second part is a large bath that is coupled only to the classical subsystem. Projection operator methods are used to eliminate the bath degrees of freedom yielding a dissipative equation of motion for the coupled quantum and classical subsystems.
The conditions under which an open quantum-mechanical system may be described by mixed quantum-classical dynamics are investigated. Decoherence is studied using influence functional methods in a model composite quantum system comprising two coupled systems, A and C, interacting with a harmonic bath with Ohmic and super-Ohmic spectral densities. Subsystem A is directly coupled to subsystem C, while C is coupled directly to the bath. Calculations are presented for a model where subsystem A is taken to be a two-level system which is bilinearly coupled to a single harmonic oscillator C subsystem. The loss of quantum coherence in each subsystem is discussed in the extreme nonadiabatic regime where the intrinsic dynamics of subsystem A is essentially frozen. Subsystem C is shown to lose its coherence rapidly, while subsystem A maintains coherence for longer time periods since C modulates the influence of the bath on A. Thus, one may identify situations where the coupled AC system evolution effectively obeys mixed quantum-classical dynamics.
We revisit Caldeira-Leggett's quantum master equation representing mixed quantum-classical theory, but with limited applications. Proposed is a Fokker-Planck quantum master equation theory, with a generic bi-exponential correlation function description on semiclassical Brownian oscillators' environments. The new theory has caustic terms that bridge between the quantum description on primary systems and the semiclassical or quasi-classical description on environments. Various parametrization schemes, both analytical and numerical, for the generic bi-exponential environment bath correlation functions are proposed and scrutinized. The Fokker-Planck quantum master equation theory is of the same numerical cost as the original Caldeira-Leggett's approach but acquires a significantly broadened validity and accuracy range, as illustrated against the exact dynamics on model systems in quantum Brownian oscillators' environments, at moderately low temperatures.
Solving quantum dynamics is an exponentially difficult problem. Thus, an exact numerical solution is inaccessible for any condensed matter system.\nA promising approach is to divide the system into a quantum subsystem containing degrees of freedom which are of greater interest or those which have more profound quantum character (e.g., have smaller mass) and a classical bath containing the rest of the system.\nImposing such a partition and treating the bath classically results in quantum-classical dynamics. The quantum-classical Liouville equation is a general equation in the Hilbert space of quantum degrees of freedom while it resides in the phase space of the classical degrees of freedom.\nAny numerical solution to this equation requires representation of the quantum subsystem in some basis. Solutions to this equation have been already proposed in the subsystem, adiabatic and force bases, each with its own cons and pros.\n\nIn this work, the quantum-classical equations of motion are cast in the subsystem basis and subsequently mapped to a number of fictitious harmonic oscillators.\nThe result is quantum-classical dynamics in the mapping basis which treats both quantum and classical degrees of freedom on the same footing, i.e., in phase space. Neglecting a portion of the back reaction of the quantum-subsystem to classical bath results in an expression for the time evolution of an operator (density matrix) equal to its Poisson bracket with the Hamiltonian.\nThis equation can be solved in terms of characteristics to provide a computationally tractable method for calculating quantum-classical dynamical properties. The expressions for expectation values and correlation functions in this formalism are derived.\nCalculations on spin-boson system, barrier crossing models---the so called Tully models---and the Fenna-Mathews-Olson pigments show very good agreement between the results of this method and numerical solutions to the Schrödinger equation.