Complete Positivity and Thermal Relaxation in Quadratic Quantum Master Equations

TL;DR

This paper develops a systematic method to derive quantum master equations that are completely positive and describe thermal relaxation, focusing on quadratic Hamiltonians and their application to quantum heat conduction models.

Contribution

It introduces a CPTP criterion for quantum master equations derived from generalized Brownian motion, enabling their use in interacting many-body systems.

Findings

Established a CPTP criterion for quadratic quantum master equations

Demonstrated applicability to network models of heat conduction

Ensured thermal relaxation description regardless of system Hamiltonian

Abstract

The ultimate goal of this paper is to develop a systematic method for deriving quantum master equations that satisfy the requirements of a completely positive and trace-preserving (CPTP) map, further describing thermal relaxation processes. In this paper, we assume that the quantum master equation is obtained through the canonical quantization of the generalized Brownian motion proposed in our recent paper [T. Koide and F. Nicacio, Phys. Lett. A 494, 129277 (2024)]. At least classically, this dynamics describes the thermal relaxation process regardless of the choice of the system Hamiltonian. The remaining task is to identify the parameters ensuring that the quantum master equation meets complete positivity. We limit our discussion to many-body quadratic Hamiltonians and establish a CPTP criterion for our quantum master equation. This criterion is useful for applying our quantum master equation to models with interaction such as a network model, which has been used to investigate how quantum effects modify heat conduction.