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The Journal of Chemical Physics

A semiclassical self-consistent-field approach to dissipative dynamics: The spin–boson problem

Gerhard Stock

Published Jul 22, 1995• 113 citations
Publisher

Abstract

A semiclassical time-dependent self-consistent-field approach for the description of dissipative quantum phenomena is proposed. The total density operator is approximated by a semiclassical ansatz, which couples the system degrees of freedom to the bath degrees of freedom in a self-consistent manner, and is thus in the spirit of a classical-path description. The capability of the approach is demonstrated by comparing semiclassical calculations for a spin–boson model with an Ohmic bath to exact path-integral calculations. It is shown that the semiclassical model nicely reproduces the complex dissipative behavior of the spin–boson model for a large range of model parameters. The validity and accuracy of the semiclassical approach is discussed in some detail. It is shown that the method is essentially based on the assumption of complete randomization of nuclear phases. In particular, the assumption of phase randomization allows one to perform the trace over the bath variables through quasiclassical sampling of the nuclear initial conditions without invoking any further approximation.

DOI
10.1063/1.469778

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A semiclassical time-dependent self-consistent-field approach for the description of dissipative quantum phenomena is proposed. The total density operator is approximated by a semiclassical ansatz, which couples the system degrees of freedom to the bath degrees of freedom in a self-consistent manner, and is thus in the spirit of a classical-path description. The capability of the approach is demonstrated by comparing semiclassical calculations for a spin–boson model with an Ohmic bath to exact path-integral calculations. It is shown that the semiclassical model nicely reproduces the complex dissipative behavior of the spin–boson model for a large range of model parameters. The validity and accuracy of the semiclassical approach is discussed in some detail. It is shown that the method is essentially based on the assumption of complete randomization of nuclear phases. In particular, the assumption of phase randomization allows one to perform the trace over the bath variables through quasiclassical sampling of the nuclear initial conditions without invoking any further approximation.
An efficient method, the self-consistent hybrid method, is proposed for accurately simulating time-dependent quantum dynamics in complex systems. The method is based on an iterative convergence procedure for a dynamical hybrid approach. In this approach, the overall system is first partitioned into a “core” and a “reservoir” (an initial guess). The former is treated via an accurate quantum mechanical method, namely, the time-dependent multiconfiguration self-consistent field or multiconfiguration time-dependent Hartree approach, and the latter is treated via a more approximate method, e.g., classical mechanics, semiclassical initial value representations, quantum perturbation theories, etc. Next, the number of “core” degrees of freedom, as well as other variational parameters, is systematically increased to achieve numerical convergence for the overall quantum dynamics. The method is applied to two examples of quantum dissipative dynamics in the condensed phase: the spin-boson problem and the electronic resonance decay in the presence of a vibrational bath. It is demonstrated that the method provides a practical way of obtaining accurate quantum dynamical results for complex systems.
We study a model for treating dissipative systems, a one dimensional quantum system coupled to a harmonic bath. The dynamics of such a system can be described by Feynman’s path integral expression for the reduced density matrix. In this formulation the interaction of the system with the environment is stored in the influence functional. Recently we showed that fast environmental modes that give rise to correlations in the influence functional which are short range in time can be treated efficiently by a memory equation algorithm, which is a discretized version of a master equation. In this work we extend this approach to treat slow environmental modes as well, thereby efficiently linking adiabatic and nonadiabatic regimes. In this extended method the long range correlations in the influence functional arising from slow bath modes are taken into account through Stock’s semiclassical self-consistent-field approach.
We study a new model for treating quantum dissipative systems, in which the bath is modeled as a collection of spins coupled to the system of interest. We develop a quasiclassical method to study this model, approximating the quantum Heisenberg equations by the classical ones, supplemented with stochastic initial conditions carefully chosen so that the results obtained from the classical equations are as close as possible to the quantum results. Using this method we compare the dynamics of such a spin–spin system with that of a spin–boson system, in which the bath is modeled as a collection of harmonic oscillators. We verify numerically that when the system-bath coupling is spread over many bath spins (the Brownian motion limit), the spin–spin model can be mapped on the spin–boson model (although with a temperature dependent spectral density). We also demonstrate that the two dissipative models are qualitatively very different in a non-Brownian motion regime.
The feasibility of using a basis set approach to the study of quantum dissipative dynamics is investigated for the spin-boson model, a system of two discrete states linearly coupled to a harmonic bath. The infinite Hamiltonian is discretized to a finite number of degrees of freedom. Traditional basis set approach, in a multiconfiguration time-dependent Hartree context, is used to solve the time-dependent Schrödinger equations by explicitly including all the degrees of freedom (“system”+“bath”). Quantities such as the reduced density matrix are then evaluated via a quadrature summation/Monte Carlo procedure over a certain number of time-dependent wave functions. Numerically exact results are obtained by systematically increasing the number of bath modes used to represent the condensed phase environment, as well as other variational parameters (number of basis functions, configurations, etc.). The potential of the current method is briefly discussed.
Phase-space approaches such as the truncated Wigner approximation (TWA) provide an efficient semiclassical framework for performing approximate simulations of the dynamics of open quantum many-body systems outside the reach of exact numerical methods but beyond the mean-field level. For bosonic systems, TWA is known to be equivalent to a Keldysh path-integral formulation truncated at second order in the so-called quantum fluctuations. This semiclassical approach provides an alternative transparent route towards approximate stochastic equations of motion, which can be efficiently solved. Here we establish the corresponding path-integral formulation for interacting open spin-1/21/21/2 systems using the continuous SU(2)\mathrm{SU}(2)SU(2) phase space. We show, in particular, that a consistent treatment of dissipation requires correctly mapping operator products onto the curved spin phase space, leading to stochastic equations that coincide with those obtained from the continuous TWA formulation and thus reproduce the exact dynamics of a single dissipative spin. Our results provide a unified field-theoretic foundation for the TWA to dissipative spin dynamics and offer a systematic starting point for extensions beyond the semiclassical approximation.
A temperature-dependent Schrödinger equation is proposed for the study of quantum dynamics in systems which interact with an external bath. The derivation is based on a time-dependent self consistent field (TDSCF) approximation for a system-bath wave function. As previously shown by Miller, this approximation leads to a quantum mechanical analog to the classical generalized Langevin equation. By replacing the time-evolution of the quantum mechanical bath observables with the corresponding classical trajectories, the fluctuating force and the nonlinear friction kernel in the Langevin-Schrödinger equation become temperature-dependent where the fluctuations intensity is proportional to T. Application of the new equation to the spin-boson model shows agreement with numerically exact path integral calculations for T>0. We relate the success of the TDSCF approximation in this case to the bath-induced noise which diminishes the importance of quantum mechanical system-bath correlations.