Quantum dynamics, master equation and equilibrium for a qubit coupled to a thermal boson field

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 that 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.