Partial decoherence and thermalization through time-domain ergodicity

An approach, differing from two commonly used methods (the stochastic Schrödinger equation and the master equation) but entrenched in the traditional density matrix formalism, is developed in a semiclassical setting in order to go from the solutions of the time-dependent Schrödinger equation to decohering and thermalized states. This is achieved by utilizing the time ergodicity, rather than the sampling (or ensemble) ergodicity, of physical systems. We introduce the formalism through a study of the Rabi model (a two-level system coupled to an oscillator) and show that our semiclassical version exhibits, both qualitatively and quantitatively, many features of state truncation and equilibration. We then study the time evolution of two qubits in interaction with a bosonic environment, such that the energy scale of one qubit is much larger and that of the other is much smaller than the environment's energy scale. The low-energy qubit decoheres to a mixture, while the high-energy qubit is protected through the adiabatic theorem. However, an interqubit coupling generates an overall decoherence and leads, for some values of the coupling, to long-term revivals in the state occupations.