Quantum Dynamics, Master Equation and Equilibrium for a Qubit Coupled to a Thermal Boson Field

TL;DR

The paper derives an exact, time-local GKLS master equation for a qubit coupled to a thermal bosonic field in the rotating-wave approximation by first obtaining an exact dynamical map for the reduced qubit state. It then analyzes the asymptotics of the dynamical map, showing that equilibration occurs when the determinant $D(t)=\alpha(t)\xi(t)-\gamma(t)\zeta(t)=\alpha(t)+\xi(t)-1$ vanishes as $t\to\infty$, yielding a final state independent of the initial conditions and consistent with thermal equilibrium at the reservoir temperature. A central result is that both the qubit and the reservoir relax to a common thermal state, with the reservoir rapidly returning to its initial thermal state and the qubit dissipating on a macroscopic time scale, highlighting a two-time-scale structure and the pivotal role of on-shell energy-conserving contributions. The approach extends exact solvable models to general thermal baths (beyond vacuum) and clarifies how unitarity and energy conservation govern asymptotic dynamics, while also discussing the vacuum-reservoir exception and connections to Nakajima–Zwanzig-type projections.

Abstract

We analytically derive the exact -- though formal -- master equation for a two-level quantum system (qubit) interacting with a bosonic environment within the rotating-wave approximation, assuming the environment is initially in an arbitrary thermal state. The long-time behavior of the evolution operator governing the dynamics of both the system and the environment is analyzed, and the conditions under which the system approaches thermal equilibrium are examined.