Necessary condition for the thermalization of a quantum system coupled to a quantum bath

A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ${\rho }^{\mathcal{S}}\left(t\right)$ approaches an equilibrium state ${\rho }_{\text{eq}}^{\mathcal{S}}$, the latter depending only on macroscopic characteristics of the bath (e.g., temperature) but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present paper we concentrate on one aspect of thermalization, namely, on the system initial state independence (ISI) of ${\rho }_{\text{eq}}^{\mathcal{S}}$. A necessary condition for the system ISI is derived in the quantum framework. We use the derived condition to prove the absence of the system ISI in a specific class of models. Namely, we consider a single spin coupled to a large bath, the interaction term commuting with the bath self-Hamiltonian (but not with the system self-Hamiltonian). Although the model under consideration is nontrivial enough to exhibit the decoherence and the approach to equilibrium, the derived necessary condition is not fulfilled and thus ${\rho }_{\text{eq}}^{\mathcal{S}}$ depends on the initial state of the spin.