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.
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