Thermalization of small quantum systems: From the zeroth law of thermodynamics

Thermalization of isolated quantum systems has been studied intensively in recent years and significant progresses have been achieved. Here, we study thermalization of small quantum systems that interact with large chaotic environments under the consideration of Schr\"{o}dinger evolution of composite systems, from the perspective of the zeroth law of thermodynamics. Namely, we consider a small quantum system that is brought into contact with a large environmental system; after they have relaxed, they are separated and their temperatures are studied. Our question is under what conditions the small system may have a detectable temperature that is identical with the environmental temperature. This should be a necessary condition for the small quantum system to be thermalized and to have a well-defined temperature. By using a two-level probe quantum system that plays the role of a thermometer, we find that the zeroth law is applicable to quantum chaotic systems, but not to integrable systems.