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arXiv:1705.10522 · Physical review. E

Perturbative dynamics of open quantum systems by renormalization group method

Shingo Kukita

Published Oct 9, 2017• 1 citation
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Abstract

We analyze perturbative dynamics of a composite system consisting of a quantum mechanical system and an environment by the renormalization group (RG) method. The solution obtained from the RG method has no secular terms and approximates the exact solution for a long time interval. Moreover, the RG method causes a reduction of the dynamics of the composite system under some assumptions. We show that this reduced dynamics is closely related to a quantum master equation for the quantum mechanical system. We compare this dynamics with the exact dynamics in an exactly solvable spin-boson model.

DOI
10.1103/physreve.96.042113

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We analyze perturbative dynamics of a composite system consisting of a quantum mechanical system and an environment by the renormalization group (RG) method. The solution obtained from the RG method has no secular terms and approximates the exact solution for a long time interval. Moreover, the RG method causes a reduction of the dynamics of the composite system under some assumptions. We show that this reduced dynamics is closely related to a quantum master equation for the quantum mechanical system. We compare this dynamics with the exact dynamics in an exactly solvable spin-boson model.
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We discuss the quantum dynamics of the central spin model in a regime where the central spin and bath are slaved to each other. The exact solution is found when the bath is static and is compared with the effect of an external field, finding that they are inequivalent due to the quantum nature of the environment. When the bath has dynamics, we analyze the differences between the numerical simulation using time-dependent perturbation theory and the equation of motion technique, which shows better accuracy. We demonstrate that the use of dynamical renormalization group (dRG), simultaneously with the equation of motion technique, provides a suitable analytical tool to understand the physics, to capture the main physical processes, and a powerful method to eliminate secular terms. In addition, this approach allows to separate classical nonlinear behavior from corrections due to quantum correlations.
We investigate the exact solution, perturbation theory and master equation of open system dynamics based on our serial studies on quantum mechanics in general quantum systems [An Min Wang, quant-ph/0611216 and quant-ph/0611217]. In a system-environment separated representation, a general and explicit solution of open system dynamics is obtained, and it is an exact solution since it includes all order approximations of perturbation. In terms of the cut-off approximation of perturbation and our improved scheme of perturbation theory, the improved form of the perturbed solution of open systems absorbing the partial contributions from the high order even all order approximations is deduced. Moreover, only under the factorizing initial condition, the exact master equation including all order approximations is proposed. Correspondingly, the perturbed master equation and its improved form different from the existed master equation are given. In special, the Redfield master equation is derived out without using Born-Markov approximation. The solution of open system dynamics in the Milburn model is also gained. As examples, Zurek model of two-state open system and its extension with two transverse fields are studied.
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We outline a non-perturbative approach for simulating the behavior of open quantum systems interacting with a bosonic environment defined by a generalized spectral density function. The method is based on replacing the environment by a set of damped harmonic oscillators - the pseudomodes - thereby forming an enlarged open system whose dynamics is governed by a Markovian master equation. Each pseudomode is connected to one of the poles of the spectral density when analytically continued to the lower-half complex frequency plane. Here, we extend previous results to a completely generic class of open system models, and discuss how our framework can be used as a powerful and versatile tool for analyzing non-Markovian open system dynamics. The effectiveness of the method is demonstrated on the spin-boson model by accurately benchmarking its predictions against numerically exact results.
We present a derivation of the Markovian master equation by the renormalization group method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then by applying the renormalization group method, we show that the dependence of the specific initial time on the perturbative solution can be removed and the Markovian semigroup master equation in the Gorini--Kossakowski--Lindblad--Sudarshan (GKLS) form is obtained in the weak coupling limit.