Quantum thermalization and equilibrium state with multiple temperatures

A large class of isolated quantum systems in a pure state can equilibrate and serve as a heat bath. We show that, once equilibrium is reached, any of its subsystems, which are much smaller than the isolated system, can be thermalized such that the subsystem is governed by the Gibbs distribution. Within this theoretical framework, the celebrated superposition principle of quantum mechanics leads to a prediction of a thermalized subsystem with multiple temperatures when the isolated system is in a superposition state of energy eigenstates of multiple distinct energy scales. This multiple-temperature state is at equilibrium, completely different from a non-equilibrium state that has multiple temperatures at different parts. Feasible experimental schemes, in particular with ultra-cold atoms, to verify this prediction are discussed.