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

Carsten Timm
Phys. Rev. B 83, 115416 – Published 10 March 2011

Abstract

The master equation describing the nonequilibrium 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.

  • Received 16 November 2010

DOI:https://doi.org/10.1103/PhysRevB.83.115416

©2011 American Physical Society

Authors & Affiliations

Carsten Timm*

  • Institute for Theoretical Physics, Technische Universität Dresden, DE-01062 Dresden, Germany

  • *carsten.timm@tu-dresden.de

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Issue

Vol. 83, Iss. 11 — 15 March 2011

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