On the Redfield and Lindblad master equations

TL;DR

This paper resolves a discrepancy in the Redfield equation, demonstrating its formal equivalence to the Lindblad equation without the rotating wave approximation by enforcing energy conservation.

Contribution

It identifies and corrects a discrepancy in the Redfield equation, establishing its equivalence to the Lindblad equation without the rotating wave approximation.

Findings

Redfield equation does not represent a proper quantum map

Discrepancy linked to diagrammatic expansion and quasi-particle approximation

Energy conservation resolves the discrepancy and links Redfield to Lindblad equations

Abstract

In a previous work we developed a field theoretical approach to open quantum systems using condensed matter methods. In the Born approximation we derived the Redfield equation on the basis of a multi-oscillator bath, a Dyson equation, a diagrammatic expansion and a quasi-particle approximation. In addition applying a rotating wave approximation we obtained the Lindblad equation describing a proper quantum map. The issue regarding the additional rotating wave approximation was left as an open problem. The present work addresses the open problem and presents new results. We identify a discrepancy in the popular and standard Redfield equation. The discrepancy is associated with the well-known fact that the Redfield equation does not represent a proper quantum map. The discrepancy is related to the diagrammatic expansion and a consistency requirement in the quasi-particle approximation. The explicit resolution of this discrepancy is obtained by imposing energy conservation on the Born level. As a result we obtain formal equivalence between the energy-conserving Redfield equation and the Lindblad equation without invoking the rotating wave approximation. We provide a detailed mapping of the field theoretical approach to the standard microscopic derivation in the theory of open quantum systems.