Damping Law of Classical Weyl Correspondence of Density Operator in Amplitude Dessipative Channel

Abstract

For developing quantum mechanics theory in phase space, we expolre how does an initial density operator ρ ₀’s classical Weyl correspondence function h ₀ evolve into h _t in a damping channel. We derive the following integration transformation that relating them

$h_{t}\left (\alpha ,\alpha ^{\ast } \right ) =\frac {2}{T}\int \frac {d^{2}\beta } {\pi } h_{0}\left (\beta ,\beta ^{\ast } \right ) \exp \left [ -\frac {2}{T} \left (\alpha ^{\ast } -\beta ^{\ast } e^{-\kappa t}\right ) \left (\alpha -\beta e^{-\kappa t}\right ) \right ]$

this is the damping law of classical Weyl correspondence of density operator in amplitude dessipative channel. The integration method within Weyl ordered product of operators and Wigner operator’s Wel ordering form ${\Delta } \left (\alpha ,\alpha ^{\ast } \right ) =\frac {1}{2}{{\begin {array}{l}:\\:\end {array}}}\delta \left (\alpha ^{\ast } -a^{\dag } \right ) \left (\alpha -a\right ){{\begin {array}{l}:\\:\end {array}}}$ is essential to our derivation.