Open quantum systems beyond Fermi's golden rule: Diagrammatic expansion of the steady-state time-convolutionless master equation

TL;DR

This paper introduces a diagrammatic method for computing steady-state properties in open quantum systems using the time-convolutionless master equation, enabling efficient higher-order calculations and physical interpretation.

Contribution

We develop a diagrammatic approach to evaluate the steady-state TCL generator based on operators, facilitating higher-order perturbative calculations and physical insights.

Findings

Recovered exact expansion to next-to-leading order for a non-interacting level

Low diagram complexity enables extension to higher orders

Method provides a simple physical interpretation of diagrams

Abstract

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 non-interacting 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.