Evolution law of Wigner function in laser process

Based on the density operator’s o perator-sum representation r ecently obtained by Fan and Hu for a laser process (Opt. Commun., 2008, 281: 5571; Opt. Commun., 2009, 282: 932; Phys. Lett. B, 2008, 22: 2435), we derive the evolution law of Wigner operator, the law is concisely expressed in the normally ordered formΔ(α,α*,t)=Tπ:exp?[-2T(a?e-(κ-g)t-α*)-(ae-(k-g)t-α)] :, where g and κ are the cavity gain and the loss, respectively, and T≡ (κ-g )(κ+g-2ge^{-2(κ-g) t})^-1. When t=0,Δ(α,α,t)→1π : exp?[-2(a?-α*)-(a-α)] :, which is the initial Wigner operator. Using this formalism the evolution law of Wigner functions in laser process can be directly obtained.