Time Evolution and Temperature Variation of the Squeezing-Chaotic Mixed Two-Mode Optical Field in One-Mode Diffusion Channel

Abstract

In this paper we theoretically investigate how the squeezing-chaotic mixed two-mode optical field described by the density operator

$$\rho_{0}=\sec h^{2}\lambda \sec h^{2}\tau e^{a^{\dagger }b^{\dagger }\tanh \lambda }\left( \sec h^{2}\lambda \tanh^{2}\tau \right)^{a^{\dagger }a}\left\vert 0\right\rangle_{bb}\left\langle 0\right\vert e^{ab\tanh \lambda } $$

undergoes in a b-mode diffusion channel with the diffusion corfficient κ. We find that in the output state the initial b-mode vacuum |0〉_bb 〈0| has evolved into the chaotic state $\frac {1}{\kappa t + 1}e^{b^{\dagger }b\ln \frac {\kappa t}{ \kappa t + 1}},$ while the squeezing term, $e^{a^{\dagger }b^{\dagger }\tanh \lambda }$$\rightarrow e^{a^{\dagger }b^{\dagger }\frac {\tanh \lambda }{ 1+\kappa t}}$, has been weakened. Measuring b-mode of this new output state leads to a chaotic field with an ascending temperature during the diffusion process, this coincides with the b-mode photon number increasing. We also show that measuring observable in a-mode is not affected by the diffusion in b-mode.

Appendix: Derivation T r ρ 0 = 1 of (1)

We first perform partial tracing over the a-mode, by using the coherent state

$$ \left\vert \alpha \right\rangle_{a}=\exp \left[ -\frac{|\alpha |^{2}}{2} +\alpha a^{\dagger }\right] \left\vert 0\right\rangle_{a},\text{\ } a\left\vert \alpha \right\rangle_{a}=\alpha \left\vert \alpha \right\rangle_{a}\text{ } $$

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whose completeness relation is

$$ \int \frac{d^{2}\alpha }{\pi }\left\vert \alpha \right\rangle_{aa}\left\langle \alpha \right\vert = 1\text{ } $$

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and the normally ordered expansion

$$ f^{^{a^{\dagger }a}}=e^{a^{\dagger }a\ln f}=:\exp \left[ \left( f-1\right) a^{\dagger }a\right] :\text{ } $$

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$$ \left\vert 0\right\rangle_{bb}\left\langle 0\right\vert =:e^{-b^{\dagger }b}: $$

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as well as the method of integration within ordered product (IWOP) of operators we have

$$\begin{array}{@{}rcl@{}} tr_{a}\rho_{_{0}} &=&\sec h^{2}\lambda \sec h^{2}\tau \int \frac{ d^{2}\alpha }{\pi }_{a}\left\langle \alpha \right\vert e^{a^{\dagger }b^{\dagger }\tanh \lambda }:\exp \left[ \left( \sec h^{2}\lambda \tanh^{2}\tau -1\right) a^{\dagger }a\right] : \\ &&\times \left\vert 0\right\rangle_{bb}\left\langle 0\right\vert e^{ab\tanh \lambda }\left\vert \alpha \right\rangle_{a} \\ &=&\sec h^{2}\lambda \sec h^{2}\tau \!\!\int\! \frac{d^{2}\alpha }{\pi }\colon\!\! \exp\! \left[ \!\left( \! \sec h^{2}\lambda \tanh^{2}\tau - 1\!\right) \!|\alpha |^{2} + \left( b\alpha + b^{\dagger }\alpha^{^{\ast }}\right) \tanh \lambda - b^{\dagger }b\!\right] \!\colon \!\! \\ &=&\frac{\sec h^{2}\lambda \sec h^{2}\tau }{1-\sec h^{2}\lambda \tanh^{2}\tau }\exp \left[ b^{\dagger }b\ln \frac{\tanh^{2}\lambda }{1-\sec h^{2}\lambda \tanh^{2}\tau }\right] \end{array} $$

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Moreover, $tr_{a}tr_{b}\rho _{_{0}}$ is normalized

$$ tr_{a}tr_{b}\rho_{_{0}}=\frac{\sec h^{2}\lambda \sec h^{2}\tau }{1-\sec h^{2}\lambda \tanh^{2}\tau }\int \frac{d^{2}\alpha_{b}}{\pi }\left\langle \alpha_{b}\right\vert \colon\!\! \exp\! \left[ \!-\frac{\left( \sec h^{2}\lambda \sec h^{2}\tau \right) b^{\dagger }b}{1-\sec h^{2}\lambda \tanh^{2}\tau } \right] \!\colon\! \left\vert \alpha_{b}\right\rangle = 1=Tr\rho_{0} $$

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thus (1) is qualified to describe a new optical field.