Entanglement, Bell violation, and phase decoherence of two atoms inside an optical cavity

The simple system of two atoms couple to single mode optical cavity with the phase decoherence is studied for investigating the entanglement and Bell violation between atoms and cavity or between two atoms. We show that in the resonance case (i) atom-field entanglement rapidly decays with phase decoherence and disappears in the stationary state, (ii) atom-atom entanglement is more robust against phase decoherence and survives in the stationary state. In the nonresonance case, the pairwise atom-atom and atom-field entanglement is sensitive to the detuning parameter and is not completely destroyed during evolution. On the other hand, violation of Bell-CHSH inequality is very fragile against the phase decoherence and finally disappears in the stationary state. The phenomenon that the more Bell violation, the less entanglement, or vice versa in such a realistic physical system, is revealed. This phenomenon maybe is the consequence of the choice of concurrence as the entanglement measure and the observables to build the Bell-CHSH inequality. The genuine three-partite entanglement is also analyzed by making use of the state preparation fidelity. It is shown that the genuine three-partite entanglement can appear in the evolution of the system even in the presence of the phase decoherence.