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The Journal of Chemical Physics

Nonperturbative time-convolutionless quantum master equation from the path integral approach

Guangjun Nan, Qiang Shi, Zhigang Shuai

Published Apr 6, 2009• 18 citations
Publisher

Abstract

The time-convolutionless quantum master equation is widely used to simulate reduced dynamics of a quantum system coupled to a bath. However, except for several special cases, applications of this equation are based on perturbative calculation of the dissipative tensor, and are limited to the weak system-bath coupling regime. In this paper, we derive an exact time-convolutionless quantum master equation from the path integral approach, which provides a new way to calculate the dissipative tensor nonperturbatively. Application of the new method is demonstrated in the case of an asymmetrical two-level system linearly coupled to a harmonic bath.

DOI
10.1063/1.3108521

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The time-convolutionless quantum master equation is widely used to simulate reduced dynamics of a quantum system coupled to a bath. However, except for several special cases, applications of this equation are based on perturbative calculation of the dissipative tensor, and are limited to the weak system-bath coupling regime. In this paper, we derive an exact time-convolutionless quantum master equation from the path integral approach, which provides a new way to calculate the dissipative tensor nonperturbatively. Application of the new method is demonstrated in the case of an asymmetrical two-level system linearly coupled to a harmonic bath.
The Nakajima–Zwanzig generalized quantum master equation provides a general, and formally exact, prescription for simulating the reduced dynamics of a quantum system coupled to a quantum bath. In this equation, the memory kernel accounts for the influence of the bath on the system’s dynamics. The standard approach is based on using a perturbative treatment of the system–bath coupling for calculating this kernel, and is therefore restricted to systems weakly coupled to the bath. In this paper, we propose a new approach for calculating the memory kernel for an arbitrary system–bath coupling. The memory kernel is obtained by solving a set of two coupled integral equations that relate it to a new type of two-time system-dependent bath correlation functions. The feasibility of the method is demonstrated in the case of an asymetrical two-level system linearly coupled to a harmonic bath.
An exact quantum master equation formalism is constructed for the efficient evaluation of quantum non-Markovian dissipation beyond the weak system-bath interaction regime in the presence of time-dependent external field. A novel truncation scheme is further proposed and compared with other approaches to close the resulting hierarchically coupled equations of motion. The interplay between system-bath interaction strength, non-Markovian property, and required level of hierarchy is also demonstrated with the aid of simple spin-boson systems.
A generalized master equation for the spin-boson model is proposed that does not require a perturbative treatment of the interstate or system-bath couplings. The nonperturbative formulation is based on the assumption of a decoupling of system and bath density operators, and results in a generalized memory function that reduces in the limit of weak system-bath coupling to the standard result of the perturbative non-interacting-blip approximation. Numerical studies in comparison to exact path-integral calculations demonstrate that the nonperturbative master equation represents a clear improvement to perturbation theory as long as the overall coupling is still small enough to justify the underlying decoupling assumption. Finally, possible generalizations of the method to the description of a dissipative N-level system and the limitations of the decoupling ansatz are discussed in some detail.
The time-convolutionless master equation provides a general framework to model non-Markovian dynamics of an open quantum system with a time-local generator. A diagrammatic representation is developed and proven for the perturbative expansion of the exact time-local generator for an open quantum system interacting with arbitrary environments. A truncation of the perturbation expansion leads to the perturbative time-convolutionless quantum master equations. We further introduce a nonperturbative approach that approximates the time-convolutionless generator as a nested time-ordered exponential function.
We study the quantum dissipative dynamics of a particle coupled linearly to a set of two-level systems (the heat bath) via the master equation method, which we extract from the path integral formalism independently from the form of the bath spectral density. We compare our results with the standard models based on bosonic heat baths, showing their main differences and similarities. In particular, we study special forms for the spectral density of the bath which give results quite different from the standard models.
The time-convolutionless master equation provides a general framework to model the non-Markovian dynamics of an open quantum system with a time-local generator. A diagrammatic representation is developed and proven for the perturbative expansion of the exact time-local generator for an open quantum system interacting with arbitrary environments. A truncation of the perturbation expansion leads to perturbative time-convolutionless quantum master equations. We further introduce a general iterative approach to construct nonperturbative approximations for the time-local generator as nested time-ordered exponential operators.