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arXiv:1402.6205 · Physical Review B

Time-local master equation connecting the Born and Markov approximations

Christian Karlewski, Michael Marthaler

Published Sep 4, 2014• 30 citations
arXivHTMLPDF

Abstract

We present an exact expansion of the master equation for an open quantum system. The resulting equation is time local and enables us to calculate clearly defined higher-order corrections to the Born-Markov approximation. In particular, we show that non-Markovian terms are of the same order of magnitude as higher-order terms in the system-bath coupling. As a result, we emphasize that analyzing non-Markovian behavior of a system implies going beyond the Born approximation. Additionally, we address with this approach the initial-state problem occurring in non-Markovian master equations.

DOI
10.1103/physrevb.90.104302

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We present an exact expansion of the master equation for an open quantum system. The resulting equation is time local and enables us to calculate clearly defined higher-order corrections to the Born-Markov approximation. In particular, we show that non-Markovian terms are of the same order of magnitude as higher-order terms in the system-bath coupling. As a result, we emphasize that analyzing non-Markovian behavior of a system implies going beyond the Born approximation. Additionally, we address with this approach the initial-state problem occurring in non-Markovian master equations.
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We introduce simple diagrams and rules to calculate the exact time-local master equation for an arbitrary open quantum system interacting with a bath. In it, the time-convolutionless generator is expanded in the system-bath coupling to all orders, and the nthn\mathrm{th}nth-order contribution is expressed by the superoperators generated from the system operator in the system-bath coupling and the associated nnn-point nonequilibrium correlation functions of the environment operators. This theory generalizes previous attempts to obtain the time-convolutionless generator and provides a general framework to study the non-Markovian, nonequilibrium dynamics of open quantum systems.
We introduce the NCA and NCA-Markov dynamical maps for open quantum systems, which go beyond weak-coupling master equations replacing the Born approximation with a non-crossing approximation (NCA). We apply these NCA maps on a prototype model of open quantum systems, the Ohmic spin-boson model at zero temperature, showing that they can qualitatively reproduce its non-perturbative behaviour, including its crossover from a coherent to an incoherent dynamics and its localization quantum phase transition, which are entirely missed by perturbative master equations. Remarkably, the time-local NCA-Markov map qualitatively captures this physics entirely, opening up the possibility to address strong-coupling effects at a numerical cost comparable to standard Markovian master equations.
Abstract Master equations are a useful tool to describe the evolution of open quantum systems. In order to characterize the mathematical features and the physical origin of the dynamics, it is often useful to consider different kinds of master equations for the same system. Here, we derive an exact connection between the time-local and the integro-differential descriptions, focusing on the class of commutative dynamics. The use of the damping-basis formalism allows us to devise a general procedure to go from one master equation to the other and vice versa, by working with functions of time and their Laplace transforms only. We further analyze the Lindbladian form of the time-local and the integro-differential master equations, where we account for the appearance of different sets of Lindbladian operators. In addition, we investigate a Redfield-like approximation, that transforms the exact integro-differential equation into a time-local one by means of a coarse graining in time. Besides relating the structure of the resulting master equation to those associated with the exact dynamics, we study the effects of the approximation on Markovianity. In particular, we show that, against expectation, the coarse graining in time can possibly introduce memory effects, leading to a violation of a divisibility property of the dynamics.
Beyond the conventional quantum regression theorem, a general formula for non-Markovian correlation functions of arbitrary system operators both in the time- and frequency-domain is given. We approach the problem by transforming the conventional time-non-local master equation into dispersed time-local equations-of-motion. The validity of our approximations is discussed and we find that the non-Markovian terms have to be included for short times. While calculations of the density matrix at short times suffer from the initial value problem, a correlation function has a well defined initial state. The resulting formula for the non-Markovian correlation function has a simple structure and is as convenient in its application as the conventional quantum regression theorem for the Markovian case. For illustrations, we apply our method to investigate the spectrum of the current fluctuations of interacting quantum dots contacted with two electrodes. The corresponding non-Markovian characteristics are demonstrated.
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.