Evolution law of Wigner function in laser process

Abstract

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 form\mathrm{\Delta }(\alpha ,{\alpha }^{*},t)={\displaystyle \frac{T}{\pi }}:\exp [-\mathrm{2}T(a^{†}{\mathrm{e}}^{-(\kappa -g)t}-{\alpha }^{*})-(a{\mathrm{e}}^{-(k-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=\mathrm{0},\mathrm{\Delta }(\alpha ,\alpha ,t)\to {\displaystyle \frac{1}{\pi }} : \exp [-\mathrm{2}(a^{†}-{\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.

Keywords

Kraus operator / Wigner operator / laser process

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