Time-convolutionless master equation for quantum dots: Perturbative expansion to arbitrary order

TL;DR

This paper develops an exact time-convolutionless master equation for quantum dots coupled to leads, providing a divergence-free perturbative expansion that clarifies relations to other master equations and stationary solutions.

Contribution

It derives an exact, divergence-free time-convolutionless master equation for quantum dots, explicitly relating it to T-matrix and Nakajima-Zwanzig formalisms, and analyzes stationary and dynamical solutions.

Findings

Divergences in T-matrix rates cancel in the time-convolutionless generator.

Stationary solutions of the two approaches are identical.

The time-convolutionless generator includes the Nakajima-Zwanzig generator plus explicit corrections.

Abstract

The master equation describing the non-equilibrium dynamics of a quantum dot coupled to metallic leads is considered. Employing a superoperator approach, we derive an exact time-convolutionless master equation for the probabilities of dot states, i.e., a time-convolutionless Pauli master equation. The generator of this master equation is derived order by order in the hybridization between dot and leads. Although the generator turns out to be closely related to the T-matrix expressions for the transition rates, which are plagued by divergences, in the time-convolutionless generator all divergences cancel order by order. The time-convolutionless and T-matrix master equations are contrasted to the Nakajima-Zwanzig version. The absence of divergences in the Nakajima-Zwanzig master equation due to the nonexistence of secular reducible contributions becomes rather transparent in our approach, which explicitly projects out these contributions. We also show that the time-convolutionless generator contains the generator of the Nakajima-Zwanzig master equation in the Markov approximation plus corrections, which we make explicit. Furthermore, it is shown that the stationary solutions of the time-convolutionless and the Nakajima-Zwanzig master equations are identical. However, this identity neither extends to perturbative expansions truncated at finite order nor to dynamical solutions. We discuss the conditions under which the Nakajima-Zwanzig-Markov master equation nevertheless yields good results.