Dynamics of the Density Matrix in Contact with a Thermal Bath and the Quantum Master Equation

TL;DR

This paper analyzes how the density matrix of a quantum system evolves when in contact with a thermal bath, deriving the quantum master equation and clarifying the role of each term in reaching the steady state.

Contribution

It provides an explicit calculation of each term's contribution in the quantum master equation and examines their roles in the steady state of the system.

Findings

Explicitly calculated contributions of quantum master equation terms.

Clarified roles of each term in the steady state.

Analyzed properties of common quantum master equations.

Abstract

We study the structure of the time evolution of the density matrix in contact with a thermal bath in a standard projection operator sheme. The reduced density matrix of the system in the steady state is obtained by tracing out the degree of freedom of the thermal bath from the equilibrium density matrix of the total system. This reduced density matrix is modified by the interaction, and is different from that of the equilibrium of the system alone. We explicitly calculate the contribution of each term in quantum master equation to the realization of the steady state density matrix, and make clear roles of each term. By making use of the role of each term, the properties of the commonly used quantum master equation are examined.