Evolution law of Wigner function in laser process

Abstract

Based on the density operator’s operator-sum representation recently 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 form $\Delta (\alpha ,\alpha ^* ,t) = \tfrac{T} {\pi }:\exp [ - 2T(a^\dag e^{ - (\kappa - g)t} - \alpha ^* )(ae^{ - (\kappa - g)t} - \alpha )]$:, 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 $t = 0,\Delta (\alpha ,\alpha ^* ,t) \to \tfrac{1} {\pi }:\exp [ - 2(a^\dag - \alpha ^* )(a - \alpha )]$;, which is the initial Wigner operator. Using this formalism the evolution law of Wigner functions in laser process can be directly obtained.