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The Journal of Chemical Physics

Sudden representation and sudden approximation quantal generalized master equation

C. K. Chan, Donald Jack Kouri

Published Aug 15, 1985• 1 citation

Abstract

The application of the sudden approximation in the derivation of a quantal generalized master equation (GME) is examined. Two different types of physical systems are considered. One is a composite system comprised of a fast primary system and slow bath compared to the time the former is coupled to the latter. The other is a composite system comprised of a slow primary system and fast bath. The resulting sudden GME’s for both cases contain non-Markovian memory kernels. In the second case, the memory kernel can be further approximated by a Markovian form. The resulting Markovian-sudden GME is identical to the GME obtained by using the adiabatic elimination method for removing the (fast) stochastic bath coordinates. Using a representation of the Schrödinger propagator for the density operator analogous to the recently developed (energy) sudden representation of the Schrödinger propagator for the wave function, the exact GME is recast into a form such that when the memory kernel and the inhomogeneity term of the equation are expanded in a perturbation series, the zeroth order equation is in the sudden approximation form. Finally, a harmonic oscillator coupled linearly to a bath of harmonic oscillators is used as an illustration. The behavior of the bath correlation functions in the Markovian and the sudden limits is examined. The reduction of the exact GME to the sudden approximation form is also considered.

DOI: 10.1063/1.449363

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The application of the sudden approximation in the derivation of a quantal generalized master equation (GME) is examined. Two different types of physical systems are considered. One is a composite system comprised of a fast primary system and slow bath compared to the time the former is coupled to the latter. The other is a composite system comprised of a slow primary system and fast bath. The resulting sudden GME’s for both cases contain non-Markovian memory kernels. In the second case, the memory kernel can be further approximated by a Markovian form. The resulting Markovian-sudden GME is identical to the GME obtained by using the adiabatic elimination method for removing the (fast) stochastic bath coordinates. Using a representation of the Schrödinger propagator for the density operator analogous to the recently developed (energy) sudden representation of the Schrödinger propagator for the wave function, the exact GME is recast into a form such that when the memory kernel and the inhomogeneity term of the equation are expanded in a perturbation series, the zeroth order equation is in the sudden approximation form. Finally, a harmonic oscillator coupled linearly to a bath of harmonic oscillators is used as an illustration. The behavior of the bath correlation functions in the Markovian and the sudden limits is examined. The reduction of the exact GME to the sudden approximation form is also considered.

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Memory functions in time-convolution Generalized Master Equations (GME) for probabilities of finding a general system (interacting by a general coupling with a true thermodynamic bath) in individual states are considered without resorting to any approximation. After taking the thermodynamic bath limit, time integrals from zero to infinite times of the memories are considered. It is argued that these integrals entering, e.g., the usual naive Markov approximation converting GME the Pauli master (PME) equations are exactly zero. This implies long-time tails of memories (unobtainable by perturbational expansions) and slower-than-exponential long-time asymptotics of relaxation.

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.

Abstract Memory functions in time‐convolution Generalized Master Equations (GME) for probabilities of finding a general system (interacting by a general coupling with a true thermodynamic bath) in individual states are considered without resorting to any approximation. After taking the thermodynamic bath limit, time integrals from zero to infinite times of the memories are considered. It is argued that these integrals entering, e.g., the usual naive Markov approximation converting GME the Pauli master (PME) equations are exactly zero. This implies long‐time tails of memories (unobtainable by perturbational expansions) and slower‐than‐exponential long‐time asymptotics of relaxation.

Based on the stochastic unravelling of the reduced density operator in the Feynman path integral formalism for an open quantum system in touch with harmonic environments, a new non-Markovian stochastic Schrödinger equation (NMSSE) has been established that allows for the systematic perturbation expansion in the system-bath coupling to arbitrary order. This NMSSE can be transformed in a facile manner into the other two NMSSEs, i.e., non-Markovian quantum state diffusion and time-dependent wavepacket diffusion method. Benchmarked by numerically exact results, we have conducted a comparative study of the proposed method in its lowest order approximation, with perturbative quantum master equations in the symmetric spin-boson model and the realistic Fenna-Matthews-Olson complex. It is found that our method outperforms the second-order time-convolutionless quantum master equation in the whole parameter regime and even far better than the fourth-order in the slow bath and high temperature cases. Besides, the method is applicable on an equal footing for any kind of spectral density function and is expected to be a powerful tool to explore the quantum dynamics of large-scale systems, benefiting from the wavefunction framework and the time-local appearance within a single stochastic trajectory.

We report a study of a stochastic Schrödinger equation corresponding to the Redfield master equation with slipped initial conditions, which describes the dynamics of a slow subsystem weakly coupled to a fast thermal bath. Using the projection-operator method of Feshbach, we derive a non-Markovian stochastic Schrödinger equation of the generalized Langevin type, which simulates the time evolution of the quantum wave functions of the subsystem driven by the fluctuating bath. For δ-correlated baths, the non-Markovian stochastic Schrödinger equation reduces to the previously derived Markovian one. Numerical methods are proposed to simulate the time evolution under our non-Markovian stochastic Schrödinger equation. These methods are illustrated with the spin-boson model.