Continuous variable entanglement in the system of a mesoscopic Josephson junction with a thermal cavity field

The dynamical behavior of entanglement for the system of a mesoscopic Josephson junction coupled to a single-mode optical cavity is investigated. It is found that if the cavity field is initially prepared in a thermal state and the mesoscopic Josephson junction in its lowest energy state, the coupling system can evolve into a two-mode Gaussian mixed state. The time-dependent characteristic function in the Wigner representation for the Gaussian state is analytically obtained and the linear entropy exchange between the cavity field and the mesoscopic Josephson junction subsystems is found. Furthermore, we investigate the influence of the temperature on the entanglement of the coupling system and find that the entanglement at a certain time exhibits the critical behavior with respect to the temperature of the cavity field.