Evolution Law of the Optical Field of Degenerate Parametric Amplifier in Dissipative Channel

We explore the time-evolution law of the optical field of degenerate parametric amplifier (DPA) in dissipative channel. It turns out that its density operator at initial time ρ ₀ = A exp(E ^? a ^?2) exp(a ^? alnλ) exp(E a ²) evolves into (rho (t)= frac {A}{lambda ^{prime }}) (exp left (frac {E^{ast }e^{-2kappa t}a^{dag 2}}{ lambda ^{prime 2}}right )exp left {a^{dag }aln frac {[lambda -(lambda ^{2}-4|E|^{2})T]e^{-2kappa t}}{lambda ^{prime 2}}right } exp left (frac { Ee^{-2kappa t}a^{2}}{lambda ^{prime 2}}right ),) where κ is the damping constant of the channel, T = 1 ? e ^?2κt , and (lambda ^{prime }equiv sqrt {(1-lambda T)^{2}-4|E|^{2}T^{2}}.) We employ the method of integration (or summation) within an ordered (normally ordered or antinormally ordered) of operators to overcome the obstacles in the process of calculation.