Field-theoretical approach to open quantum systems and the Lindblad equation

TL;DR

This paper introduces a systematic field-theoretical framework for open quantum systems, deriving a non-Markovian master equation and its Markovian Lindblad form using diagrammatic perturbation theory and approximations.

Contribution

It develops a novel field-theoretical approach to derive non-Markovian and Markovian master equations for open quantum systems, including the Lindblad form, based on condensed-matter methods.

Findings

Derivation of a Dyson equation for the transmission matrix of open quantum systems.

Systematic diagrammatic expansion for the kernel of the master equation.

Recovery of the Lindblad master equation under the rotating-wave approximation.

Abstract

We develop a systematic field-theoretical approach to open quantum systems based on condensed-matter many-body methods. The time evolution of the reduced density matrix for the open quantum system is determined by a transmission matrix. Developing diagrammatic perturbation theory, invoking Wick's theorem in connection with a Caldeira-Leggett quantum oscillator environment in thermal equilibrium, the transmission matrix satisfies a Dyson equation characterized by an irreducible kernel. Unlike the Nakajima-Zwanzig and standard approaches, the Dyson equation is equivalent to a general non-Markovian master equation for the reduced density matrix, incorporating secular effects and independent of the initial preparation. The kernel is determined by a systematic diagrammatic expansion in powers of the interaction. We consider the Born approximation for the kernel. Applying a condensed-matter pole or, equivalently, a quasiparticle-type approximation, equivalent to the usual assumption of a timescale separation, we derive a master equation of the Markov type. Furthermore, imposing the rotating-wave approximation,we obtain a Markov master equation of the Lindblad form. To illustrate the method, we consider the standard example of a single qubit coupled to a thermal heat bath.