Thermalization of a qubit strongly interacting with a bosonic environment

When a quantum system is placed in a thermal environment, we assume that the system relaxes to the Gibbs state in which decoherence takes place in the system energy eigenbasis. However, when the coupling between system and environment is strong, the thermal state is not necessarily the Gibbs state, and the system density matrix does not have to be diagonal in the energy eigenbasis. The theory of einselection by Zurek suggests that decoherence takes place in the pointer basis rather than in the energy eigenbasis, which can be interpreted as continuous measurement by the environment. However, the actual matrix elements are not known. Based on the theory of environment-induced decoherence, we introduce a couple of propositions: (1) in the strong coupling limit, the Gibbs state is projected to the convex hull spanned by the pointer basis, which necessarily increases the system entropy, and (2) the transition from the Gibbs state to the pointer limit takes place along the projection line perpendicular to the convex hull. We justify these propositions by exact numerical simulation of a qubit interacting with an infinitely large bosonic environment through various coupling Hamiltonians.