Time Evolution of the Wigner Operator as a Quasi-density Operator in Amplitude Dessipative Channel

For developing quantum mechanics theory in phase space, we explore how the Wigner operator \({\Delta } (\alpha ,\alpha ^{\ast } )\equiv \frac {1}{\pi } :e^{-2(\alpha ^{\ast } -\alpha ^{\dag })(\alpha -\alpha )}\):, when viewed as a quasi-density operator correponding to the Wigner quasiprobability distribution, evolves in a damping channel. with the damping constant κ. We derive that it evolves into $$\frac{1}{T + 1}:\exp \frac{2}{T + 1}[-(\alpha^{\ast} e^{-\kappa t}-a^{\dag} )(\alpha e^{-\kappa t}-a)]: $$