Decoherence analysis of multiple photon annihilation-then-creation coherent state: Amplitude decay and phase damping

We mainly study on the effect of decoherence of the multiple photon annihilation-then-creation coherent state (MPACCS) in two different channels, i.e., amplitude decay and phase damping, by virtue of the time-evolution of the Wigner functions (WFs) of such state. Based on deriving the analytical expression of MPACCS's normalization factor related to the Bell-polynomial, the time-evolution of its WFs in each channel is derived analytically and discussed numerically. After undergoing the amplitude decay channel, the partial negativity of WFs diminishes gradually and the WFs have no negative region when the decay time exceeds a threshold value, while the negativity of WFs in phase damping channel varies more slowly than that of the amplitude decay. Especially, at long times WF evolves into the case of vacuum state in amplitude decay channel and a sum of weight WFs of all the Fock states in phase damping channel.