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The Journal of Chemical Physics

Stochastic simulation of anharmonic dissipation. I. Linear response regime

Yun‐An Yan

Published Nov 28, 2016• 3 citations
Publisher

Abstract

Over decades, the theoretical study of the quantum dissipative dynamics was mainly based on the linear dissipation model. The study of the nonlinear dissipative dynamics in condensed phases, where there exist an infinite number of bath modes, is extremely difficult even if not impossible. This work put forward a stochastic scheme for the simulation of the nonlinear dissipative dynamics. In the linear response regime, the second-order cumulant expansion becomes exact to reproduce the effect of the bath on the evolution of the reduced system. Consequently, a Hermitian stochastic Liouville equation is derived without explicit treatment of the bath. Stochastic simulations for an anharmonic model illustrate that the dynamics dissipated by anharmonic bath exhibits substantial difference on temperature dependence compared to that with the Caldeira-Leggett model.

DOI
10.1063/1.4967812

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Over decades, the theoretical study of the quantum dissipative dynamics was mainly based on the linear dissipation model. The study of the nonlinear dissipative dynamics in condensed phases, where there exist an infinite number of bath modes, is extremely difficult even if not impossible. This work put forward a stochastic scheme for the simulation of the nonlinear dissipative dynamics. In the linear response regime, the second-order cumulant expansion becomes exact to reproduce the effect of the bath on the evolution of the reduced system. Consequently, a Hermitian stochastic Liouville equation is derived without explicit treatment of the bath. Stochastic simulations for an anharmonic model illustrate that the dynamics dissipated by anharmonic bath exhibits substantial difference on temperature dependence compared to that with the Caldeira-Leggett model.
Classical Brownian motion has well been investigated since the pioneering work of Einstein, which inspired mathematicians to lay the theoretical foundation of stochastic processes. A stochastic formulation for quantum dynamics of dissipative systems described by the system-plus-bath model has been developed and found many applications in chemical dynamics, spectroscopy, quantum transport, and other fields. This article provides a tutorial review of the stochastic formulation for quantum dissipative dynamics. The key idea is to decouple the interaction between the system and the bath by virtue of the Hubbard-Stratonovich transformation or Itô calculus so that the system and the bath are not directly entangled during evolution, rather they are correlated due to the complex white noises introduced. The influence of the bath on the system is thereby defined by an induced stochastic field, which leads to the stochastic Liouville equation for the system. The exact reduced density matrix can be calculated as the stochastic average in the presence of bath-induced fields. In general, the plain implementation of the stochastic formulation is only useful for short-time dynamics, but not efficient for long-time dynamics as the statistical errors go very fast. For linear and other specific systems, the stochastic Liouville equation is a good starting point to derive the master equation. For general systems with decomposable bath-induced processes, the hierarchical approach in the form of a set of deterministic equations of motion is derived based on the stochastic formulation and provides an effective means for simulating the dissipative dynamics. A combination of the stochastic simulation and the hierarchical approach is suggested to solve the zero-temperature dynamics of the spin-boson model. This scheme correctly describes the coherent-incoherent transition (Toulouse limit) at moderate dissipation and predicts a rate dynamics in the overdamped regime. Challenging problems such as the dynamical description of quantum phase transition (local- ization) and the numerical stability of the trace-conserving, nonlinear stochastic Liouville equation are outlined.
We use the "generalized hierarchical equation of motion" proposed in Paper I [C.-Y. Hsieh and J. Cao, J. Chem. Phys. 148, 014103 (2018)] to study decoherence in a system coupled to a spin bath. The present methodology allows a systematic incorporation of higher-order anharmonic effects of the bath in dynamical calculations. We investigate the leading order corrections to the linear response approximations for spin bath models. Two kinds of spin-based environments are considered: (1) a bath of spins discretized from a continuous spectral density and (2) a bath of localized nuclear or electron spins. The main difference resides with how the bath frequency and the system-bath coupling parameters are distributed in an environment. When discretized from a continuous spectral density, the system-bath coupling typically scales as ∼1/NB where N B is the number of bath spins. This scaling suppresses the non-Gaussian characteristics of the spin bath and justifies the linear response approximations in the thermodynamic limit. For the nuclear/electron spin bath models, system-bath couplings are directly deduced from spin-spin interactions and do not necessarily obey the 1/NB scaling. It is not always possible to justify the linear response approximations in this case. Furthermore, if the spin-spin Hamiltonian is highly symmetrical, there exist additional constraints that generate highly non-Markovian and persistent dynamics that is beyond the linear response treatments.
We use the “generalized hierarchical equation of motion” proposed in Paper I [C.-Y. Hsieh and J. Cao, J. Chem. Phys. 148, 014103 (2018)] to study decoherence in a system coupled to a spin bath. The present methodology allows a systematic incorporation of higher-order anharmonic effects of the bath in dynamical calculations. We investigate the leading order corrections to the linear response approximations for spin bath models. Two kinds of spin-based environments are considered: (1) a bath of spins discretized from a continuous spectral density and (2) a bath of localized nuclear or electron spins. The main difference resides with how the bath frequency and the system-bath coupling parameters are distributed in an environment. When discretized from a continuous spectral density, the system-bath coupling typically scales as ∼1/NB where NB is the number of bath spins. This scaling suppresses the non-Gaussian characteristics of the spin bath and justifies the linear response approximations in the thermodynamic limit. For the nuclear/electron spin bath models, system-bath couplings are directly deduced from spin-spin interactions and do not necessarily obey the 1/NB scaling. It is not always possible to justify the linear response approximations in this case. Furthermore, if the spin-spin Hamiltonian is highly symmetrical, there exist additional constraints that generate highly non-Markovian and persistent dynamics that is beyond the linear response treatments.
We extend a standard stochastic theory to study open quantum systems coupled to a generic quantum environment. We exemplify the general framework by studying a two-level quantum system coupled bilinearly to the three fundamental classes of non-interacting particles: bosons, fermions, and spins. In this unified stochastic approach, the generalized stochastic Liouville equation (SLE) formally captures the exact quantum dissipations when noise variables with appropriate statistics for different bath models are applied. Anharmonic effects of a non-Gaussian bath are precisely encoded in the bath multi-time correlation functions that noise variables have to satisfy. Starting from the SLE, we devise a family of generalized hierarchical equations by averaging out the noise variables and expand bath multi-time correlation functions in a complete basis of orthonormal functions. The general hierarchical equations constitute systems of linear equations that provide numerically exact simulations of quantum dynamics. For bosonic bath models, our general hierarchical equation of motion reduces exactly to an extended version of hierarchical equation of motion which allows efficient simulation for arbitrary spectral densities and temperature regimes. Similar efficiency and flexibility can be achieved for the fermionic bath models within our formalism. The spin bath models can be simulated with two complementary approaches in the present formalism. (I) They can be viewed as an example of non-Gaussian bath models and be directly handled with the general hierarchical equation approach given their multi-time correlation functions. (II) Alternatively, each bath spin can be first mapped onto a pair of fermions and be treated as fermionic environments within the present formalism.
We investigate the difference between a quantum and classical bath within the Caldeira-Leggett model for dissipative quantum dynamics.It is well known that a Markovian equation of motion for the reduced dynamics can be derived by taking the classical approximation of the harmonic bath correlation function.However, such approximation is only valid at high temperatures, and it is necessary to include the non-Markovian effect of the quantum bath in more general cases.We show that the equation of motion derived for the classical bath can be extended to the exact quantum one, by simply adding a real stochastic process to take account of the difference between the quantum and classical bath correlation functions.Numerical examples in calculating electron and excitation energy transfer dynamics, as well as absorption spectra of molecular aggregates indicate that the proposed method is a valid approach to extend the existing theories to include the quantum effect of the harmonic bath.The possibility of applying a similar idea to account for the difference between zero and high temperature quantum dissipative dynamics is discussed.
Based on a recently developed quantum dissipation formulation [R. X. Xu and Y. J. Yan, J. Chem. Phys. 116, 9196 (2002)], we present a reduced Liouville-space approach to evaluate the response and correlation functions of dissipative systems. The weak system-bath interaction is treated properly for its effects on the initial state, the evolution, and the correlation between coherent driving and non-Markovian dissipation. Numerical demonstration shows this correlated effect cannot be neglected even in the calculation of linear response quantities that do not explicitly depend on external fields. Highlighted in this paper is also the proper choice of theory among various formulations in the weak system-bath interaction regime.