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Physical Review A

Diagrammatic time-local master equation for open quantum systems

Bing Gu

Published Jan 24, 2020• 4 citations
Publisher

Abstract

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 $n\mathrm{th}$-order contribution is expressed by the superoperators generated from the system operator in the system-bath coupling and the associated $n$-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.

DOI
10.1103/physreva.101.012121

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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.
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.
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.
The time-convolutionless quantum master equation is an exact description of the nonequilibrium dynamics of open quantum systems, with the advantage of being local in time. We derive a perturbative expansion to arbitrary order in the system-reservoir coupling for its generator, which contains significantly fewer terms than the ordered-cumulant expansion. We show that the derived expansion also admits a simple recursive formulation. The derived series is then used to describe the nonequilibrium dynamics of a quantum dot, including coherences. We find a relation between the generator and a generalization of the TTT-matrix. Even though the TTT-matrix rate equations are plagued by divergences, we show that these cancel order by order in the generator of the time-convolutionless master equation. This generalizes previous work on the time-convolutionless Pauli master equation, which does not include coherences, and lays the ground for possible numerical evaluations and analytical resummations.
Steady-state observables, such as occupation numbers and currents, are crucial experimental signatures in open quantum systems.The time-convolutionless (TCL) master equation, which is both exact and time-local, is an ideal candidate for the perturbative computation of such observables.We develop a diagrammatic approach to evaluate the steady-state TCL generator based on operators rather than superoperators.We obtain the steady-state occupation numbers, extend our formulation to the calculation of currents, and provide a simple physical interpretation of the diagrams.We further benchmark our method on a single noninteracting level coupled to Fermi reservoirs, where we recover the exact expansion to next-to-leading order.The low number of diagrams appearing in our formulation makes the extension to higher orders accessible.Combined, these properties make the steady-state time-convolutionless master equation an effective tool for the calculation of steady-state properties in open quantum systems.
We develop a recursive perturbative expansion for the time-convolutionless generator of an open quantum system in a generalized Lindblad form. This formulation provides a systematic approach to derive the generator at arbitrary order while preserving a Lindblad-like structure, without imposing assumptions on the system or environment beyond an initially uncorrelated state. The generator is written, at all orders, in a canonical form, which also corresponds to the minimal dissipation condition, which uniquely specifies the decomposition of the generator into Hamiltonian and dissipative contributions. To validate the method and show its effectiveness in addressing non-Markovian dynamics and strong-coupling effects, we compute the generator explicitly up to fourth order.
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.