Time-convolutionless master equation: Perturbative expansions to arbitrary order and application to quantum dots

TL;DR

This paper develops a perturbative expansion for the time-convolutionless quantum master equation's generator, simplifying calculations of open quantum system dynamics and applying it to quantum dots with coherences.

Contribution

It introduces a recursive, lower-term perturbative expansion for the TCL master equation generator, applicable to systems with coherences, and relates it to a generalized T-matrix.

Findings

Derived a perturbative expansion with fewer terms than ordered-cumulant expansion.

Showed divergence cancellation in the generator, unlike T-matrix rate equations.

Applied the method to quantum dots, including coherences.

Abstract

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 $T$-matrix. Even though the $T$-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.