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The Journal of Chemical Physics

Semiclassical description of quantum coherence effects and their quenching: A forward–backward initial value representation study

Haobin Wang, Michael Thoss, Kathy L. Sorge, Ricard Gelabert, Xavier Giménez, William Hughes Miller

Published Feb 8, 2001• 126 citations
Publisher

Abstract

The forward–backward (FB) version of the semiclassical (SC) initial value representation (IVR) is used to study quantum coherence effects in the time-dependent probability distribution of an anharmonic vibrational coordinate and its quenching when coupled to a thermal bath. It is shown that the FB-IVR accurately reproduces the detailed quantum coherent structure in the weak coupling regime, and also describes how this coherence is quenched with an increase of the system–bath coupling and/or the bath temperature. Comparisons are made with other approximations and the physical implications are discussed.

DOI
10.1063/1.1337802

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The forward–backward (FB) version of the semiclassical (SC) initial value representation (IVR) is used to study quantum coherence effects in the time-dependent probability distribution of an anharmonic vibrational coordinate and its quenching when coupled to a thermal bath. It is shown that the FB-IVR accurately reproduces the detailed quantum coherent structure in the weak coupling regime, and also describes how this coherence is quenched with an increase of the system–bath coupling and/or the bath temperature. Comparisons are made with other approximations and the physical implications are discussed.
It is shown that the forward–backward (FB) version of the semiclassical (SC) initial value representation (IVR) is able to describe quantum interference/coherence (i.e., diffraction) of particles transmitted by a two-slit potential. (In contrast, the linearized approximation to the SC-IVR, which leads to the classical Wigner model, is unable to do so.) FB-IVR calculations are also used to describe the (partial) quenching of this interference structure (i.e., “de-coherence”) when the two-slit potential is coupled to a bath of harmonic oscillators.
The decoherence of an anharmonic oscillator in a thermal harmonic bath is examined via a semiclassical approach. A computational strategy is presented and exploited to calculate the time dependence of the purity and the decay of individual matrix elements in the energy representation for a variety of initial states. The time dependence of the decoherence is found to depend on the temperature of the bath, the coupling strength, the initial state of the oscillator, and the choice of quantity measuring the decoherence. Recurrences in the purity and in the off-diagonal matrix elements are observed, as well as the collapse of these matrix elements to the diagonal, providing evidence for the retention of quantum coherence for time scales longer than that indicated by the purity. The results are used to analyze the utility of the Caldeira-Leggett and Redfield models of decoherence and to assess the dependence of dephasing rates on the degree of structure in phase space. In several cases we find that the dephasing dynamics can be described as an initial Zeno-effect regime, followed by a Caldeira-Leggett region, followed by recurrences.
The forward-backward (FB) approximation as applied to semiclassical initial value representations (SCIVR's) has enabled the practical application of the SCIVR methodology to systems with many degrees of freedom. However, to date a systematic representation of the exact quantum dynamics in terms of the FB-SCIVR has proven elusive. In this paper, we provide a new derivation of a forward-backward phase space SCIVR expression (FBPS-SCIVR) derived previously by Thompson and Makri [Phys. Rev. E 59, R4729 (1999)]. This enables us to represent quantum correlation functions exactly in terms of a series whose leading order term is the FBPS-SCIVR expression. Numerical examples for systems with over 50 degrees of freedom are presented for the spin boson problem. Comparison of the FBPS-SCIVR with the numerically exact results of Wang [J. Chem. Phys. 113, 9948 (2000)] obtained using a multiconfigurational time dependent method shows that the leading order FBPS-SCIVR term already provides an excellent approximation.
The forward–backward semiclassical dynamics methodology [J. Phys. Chem. 103, 7753, 9479 (1999)] is reformulated in the interaction representation. The new version of the method allows for a fully quantum mechanical description of a low-dimensional subsystem of interest, along with a semiclassical forward–backward treatment of the solvent coordinates and their coupling to the reference subsystem. Application to the long-time tunneling dynamics in a symmetric double-well system coupled to a harmonic bath shows that the interaction FBSD is capable of capturing quantitatively the tunneling and decoherence effects induced by weakly dissipative environments.
The Herman−Kluk (HK) or coherent state initial value representation (IVR) of semiclassical (SC) theory expresses a time-evolving wave function as a sum (integral) over the initial conditions of classically evolved coherent states (aka minimum uncertainty wave packets). An exact expansion of the wave function can also be made in terms of these same coherent states, thinking of them simply as a (time-dependent) basis set. This paper shows the precise set of approximations that are necessary to degenerate the exact expansion into the SC-IVR result. As a byproduct, we also obtain a generalization of the HK “prefactor”.
Semiclassical time-dependent solutions to the Schroedinger equation can be obtained by use of integral formulas known as initial value representations (IVRs). Over the last few decades IVR formulas have established a sucessful reputation, most notably in the field of physical chemistry, for it is often found that the quantum behavior of molecular systems is very well approximated by semiclassical wavefunctions. The IVR input consists in a swarm of independent classical trajectories, each of which obeys the dynamics dictated by the system’s unquantized Hamiltonian. In a recent work [Aguiar et al, Chem. Phys. 370 (2010)] an IVR expression based on complex classical trajectories was formulated. The complex nature of the classical trajectories in this particular formula is a consequence of employing minimum uncertainty wavepackets (coherent states) to represent the time evolution operator upon which the formula is constructed. Here, an improved and simpler version of this complex IVR formula is presented. In order to test its accuracy, we apply it to a simple model system.