Time-local master equations: influence functional and cumulant expansion

TL;DR

This paper develops a systematic perturbation expansion method for non-Markovian open quantum systems using influence functional and cumulant expansion, providing explicit formulas for higher-order generators.

Contribution

It introduces an equivalent cumulant expansion approach to derive time-local master equations for non-Markovian dynamics, including explicit second and fourth order formulas.

Findings

Equivalent expressions for the generator in all orders of coupling

Explicit second and fourth order generator formulas

Demonstration of the methods' consistency and applicability

Abstract

The non-Markovian behaviour of open quantum systems interacting with a reservoir can often be described in terms of a time-local master equation involving a time-dependent generator which is not in Lindblad form. A systematic perturbation expansion of the generator is obtained either by means of van Kampen's method of ordered cumulants or else by use of the Feynman-Vernon influence functional technique. Both expansions are demonstrated to yield equivalent expressions for the generator in all orders of the system-reservor coupling. Explicit formulae are derived for the second and the fourth order generator in terms of the influence functional.