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

Abstract

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 $$\frac{1}{T + 1}:\exp \frac{2}{T + 1}[-(\alpha^{\ast} e^{-\kappa t}-a^{\dag} )(\alpha e^{-\kappa t}-a)]: $$ where T ≡ 1 − e− 2κt. This in turn helps to directly obtain the final state ρ(t) out of the dessipative channel from the initial classical function corresponding to initial ρ(0). Throught the work, the method of integration within ordered product (IWOP) of operators is employed.