Time Evolution and Characteristic Quantities of Squeezed Chaotic Field in Diffusion Channel

Abstract

In exploring the time evolution law of squeezed chaotic state, described by the density operator,$\rho _{0}=\left (1-e^{k}\right ) S^{\dagger }\left (r\right ) e^{ka^{\dagger }a}S\left (r\right )$, in a diffusion channel, we find two physical quantities characteristic of this physical process, they are

$$\tau=\frac{1}{\left( 2\bar{n}+1\right) e^{-2r}+1}, \theta=\frac {1}{\left( 2\bar{n}+1\right) e^{2r}+1}, $$

where $\bar {n}$ is average photon number of the chaotic field, r is the squeezing parameter and ρ ₀ in normal ordering is

$$\rho_{0}=2\sqrt{\tau \theta}\colon \exp \left[ \frac{1}{2}\left( \tau -\theta \right) \left( a^{\dagger2}+a^{2}\right) -\left( \tau +\theta \right) a^{\dag}a\right] \colon. $$

We find in the diffusion process, τ and 𝜃 evolves into

$$\tau \rightarrow \tau^{\prime}=\frac{\tau}{1+2\kappa t\tau}, \theta \rightarrow \theta^{\prime}=\frac{\theta}{1+2\kappa t\theta}, $$

where κ represent diffusion coefficient, thus

$$\rho \left( t\right) =2\sqrt{\tau^{\prime}\theta^{\prime}}\colon \exp \left[ \frac{1}{2}\left( \tau^{\prime}-\theta^{\prime}\right) \left( a^{\dagger 2}+a^{2}\right) -\left( \tau^{\prime}+\theta^{\prime}\right) a^{\dagger }a\right] \colon, $$

this is the evolution law of squeezed chaotic state in diffusion channel. The photon number of the final state slightly increases by an amount κ t. This diffusion process can be considered a quantum controlling scheme in the way of photon addition by adjusting κ.