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

Abstract

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 $\rho (t)= \frac {A}{\lambda ^{\prime }}$ $\exp \left (\frac {E^{\ast }e^{-2\kappa t}a^{\dag 2}}{ \lambda ^{\prime 2}}\right )\exp \left \{a^{\dag }a\ln \frac {[\lambda -(\lambda ^{2}-4|E|^{2})T]e^{-2\kappa t}}{\lambda ^{\prime 2}}\right \} \exp \left (\frac { Ee^{-2\kappa t}a^{2}}{\lambda ^{\prime 2}}\right ),$ 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.

7 Appendix A

We can prove (10) as follows

$$\begin{array}{@{}rcl@{}} &&\sqrt{(1-\lambda )^{2}-4|E|^{2}} \\ &=&\sqrt{(1-\lambda )^{2}-\frac{\lambda^{2}}{D^{2}}(\omega^{2}-D^{2})\sinh ^{2}(\beta D)} \notag \\ &=&\sqrt{\lambda^{2}[\frac{D^{2}\cosh^{2}(\beta D)-\omega^{2}\sinh ^{2}(\beta D)}{D^{2}}]-2\lambda +1} \notag \\ &=&\sqrt{\frac{D\cosh (\beta D)-\omega \sinh (\beta D)}{\omega \sinh (\beta D)+D\cosh (\beta D)}-\frac{2D}{\omega \sinh (\beta D)+D\cosh (\beta D)}+1} \notag \\ &=&\sqrt{\frac{2D[\cosh (\beta D)-1]}{\omega \sinh (\beta D)+D\cosh (\beta D) }}=2\sqrt{\lambda }\sinh (\beta D/2) \notag \end{array} $$

(43)

so we have

$$ Z(\beta )=\sqrt{\frac{\lambda e^{\beta \omega }}{(1-\lambda )^{2}-4|E|^{2}}}= \frac{e^{\beta \omega /2}}{2\sinh (\beta D/2)} $$

(44)