Damping law of photocount distribution in a dissipative channel

For a dissipative channel governed by the master equation of density operator}$d\rho /dt=\kappa \left(2a\rho a^{\dagger}-a^{\dagger}a\rho -\rho a^{\dagger}a\right) ,${\small \ we find that photocount distribution formula at time}$t,${\small \}$p\left(n,t\right) =Tr\left\{\rho \left(t\right) \mathbf{\colon}\left(\xi a^{\dagger}a\right) ^{n}e^{-\xi a^{\dagger}a}/n!\colon \right\} ,${\small \ becomes}$p\left(n,t\right) =Tr% \left[ \rho \left(0\right) \mathbf{\colon}\left(\xi e^{-2\kappa t}a^{\dagger}a\right) ^{n}e^{-\xi e^{-2\kappa t}a^{\dagger}a}/n!\colon % \right] ,${\small \ as if the quantum efficiency}$\xi ${\small \ of the detector becomes}$\xi e^{-2\kappa t}${\ This law greatly simplifies the theoretical study of photocount distribution for quantum optical field.