  The quantum thermalization of the Jaynes-Cummings (JC) model in both
equilibrium and non-equilibrium open-system cases is sdudied, in which the two
subsystems, a two-level system and a single-mode bosonic field, are in contact
with either two individual heat baths or a common heat bath. It is found that
in the individual heat-bath case, the JC model can only be thermalized when
either the two heat baths have the same temperature or the coupling of the JC
system to one of the two baths is turned off. In the common heat-bath case, the
JC system can be thermalized irrespective of the bath temperature and the
system-bath coupling strengths. The thermal entanglement in this system is also
studied. A \emph{counterintuitive} phenomenon of \emph{vanishing} thermal
entanglement in the JC system is found and proved.
