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Europhysics Letters (EPL)

Stochastic simulation of quantum dissipative dynamics

Y Zhou, YiJing Yan, Jiushu Shao

Published Oct 4, 2005• 71 citations
Publisher

Abstract

As a system can be separated from its heat bath via a decoupling scheme based on the Hubbard-Stratonovich transformation, the quantum dissipative dynamics is exactly described by the average of the random system moving in the stochastic field induced by the bath. As a test of this stochastic formalism, a direct numerical implementation is employed to simulate the spontaneous decay of a two-state atom. In contrast, the validity of a deterministic algorithm based on the stochastic scheme is investigated. Moreover, a mixed random-deterministic approach is established and its efficiency is exemplified by studying the exact dynamics for the spin-boson model at zero temperature.

DOI
10.1209/epl/i2005-10262-4

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As a system can be separated from its heat bath via a decoupling scheme based on the Hubbard-Stratonovich transformation, the quantum dissipative dynamics is exactly described by the average of the random system moving in the stochastic field induced by the bath. As a test of this stochastic formalism, a direct numerical implementation is employed to simulate the spontaneous decay of a two-state atom. In contrast, the validity of a deterministic algorithm based on the stochastic scheme is investigated. Moreover, a mixed random-deterministic approach is established and its efficiency is exemplified by studying the exact dynamics for the spin-boson model at zero temperature.
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.
Based on the Hubbard-Stratonovich transformation, the dissipative interaction between the system of interest and the heat bath is decoupled and the separated system and bath thus evolve in common classical random fields. This manipulation allows us to establish a novel theoretical methodology by which the reduced density matrix is formulated as an ensemble average of its random realizations in the auxiliary white noise fields. Within the stochastic description, the interaction between the system and the bath is reflected in the mutually induced mean fields. The relationship between the bath-induced field and the influence functional in the path integral framework is revealed. As a demonstration of this approach, we derive the exact master equations for two model systems.
The exact dynamics of a system coupled to an environment can be described by an integro-differential stochastic equation for the reduced density. The influence of the environment is incorporated through a mean field which is both stochastic and nonlocal in time and where the standard two-time correlation functions of the environment appear naturally. Since no approximation is made, the presented theory incorporates exactly dissipative and non-Markovian effects. Applications to the spin-boson model coupled to a heat bath with various coupling regimes and temperature show that the presented stochastic theory can be a valuable tool to simulate exactly the dynamics of open quantum systems. Links with the stochastic Schrödinger equation method and possible extensions to "imaginary time" propagation are discussed.
The methodology of stochastic description for dissipation, a generic scheme to decouple the interaction between two subsystems, is applied to the study of dissipative dynamics in quantum optics. It is shown that the influence of the coupled thermal or vacuum field on the quantum mode can be exactly represented by the induced stochastic fields. The quantum mode thereby satisfies a stochastic differential equation and dissipation effect due to the coupling with the environment is obtained through statistical averaging. Within the framework of stochastic description, it is demonstrated how to derive the master equation for a single optical mode interacting with the bosonic bath. A numerical algorithm for solving the master equation in which the coefficients are determined by a set of integral equations is discussed and a comparison with the known results is displayed. The derivation of the master equation for the spontaneous decay of two-state atoms in the vacuum is also presented.
Spin-boson models involving many interacting spins and bosons are ubiquitous in quantum simulation platforms. At the same time, characterizing the dynamics of these quantum systems represents a significant challenge. Here, we consider general spin-boson models where bosons are subject to Markovian dissipation (e.g., due to cavity loss). We present an exact hybrid quantum-classical stochastic approach where the solution of a classical stochastic equation—mimicking the bosonic modes—is input into a quantum stochastic equation for the spins. Furthermore, the spins are effectively decoupled for each stochastic realization, which nevertheless comes at the expense of sampling over unphysical states. In contrast with existing stochastic approaches based on the influence functional formalism, we place no restriction (factorizability or Gaussianity) on the initial state, or the spin-boson coupling (except that it be linear in the bosonic operator). Markovian dissipation, being at the heart of our approach, renders the stochastic equations Markovian even in the strong coupling regime. Furthermore, it ensures hermiticity (though not positivity) of the density matrix for each realization, thus improving the convergence of stochastic sampling. Interestingly, we find a condition on the classical simulability of the system based solely on the single atom cooperativity even in a many-body setting. We benchmark and showcase the utility of our approach in several examples, specifically in cases where a direct numerical computation is unfeasible.
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.