Some Evolution Formulas on the Optical Fields Propagation in Realistic Environments

Abstract

Evolution formulas of the density operator, the photon number distribution, and the Wigner function are derived for the problem on the optical fields propagation in realistic environments. Using the idea “reservoir modeled by beam splitter (BS)” and the Weyl expansion of the density operator, we obtain these formulas cleverly, which are very useful for quantum optics and quantum statistics. As an application, we study the time evolution of the photon number distribution and the Wigner function for single-photon-added coherent state in thermal environment.

Appendices

Appendix A: Some Character of Squeezed Thermal State

In this appendix, we give a good expression of the density operator and then calculate the CF and WF for squeezed thermal state.

1.1 1) Density operator

Recalling the P-function of squeezed thermal state ρ _b

$$ \rho_{b}=\frac{1}{\bar{n}}\int \frac{d^{2}\eta} {\pi} e^{-\frac{1}{\bar{n}} \left\vert \eta \right\vert^{2}}S\left( \lambda \right) \left\vert \eta \right\rangle \left\langle \eta \right\vert S^{\dag} \left( \lambda \right) . $$

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and noticing the squeezing operator $S\left (\lambda \right ) =\int \frac {dx}{ \sqrt {e^{-\lambda } }}\left \vert \frac {x}{e^{-\lambda } }\right \rangle \left \langle x\right \vert$ with $\left \vert x\right \rangle =\pi ^{-\frac {1}{ 4}}e^{-x^{2}/2+\sqrt {2}xb^{\dag } -b^{\dag 2}/2}\left \vert 0\right \rangle$, leading to

$$ S\left( \lambda \right) \left\vert \eta \right\rangle =\frac{1}{\sqrt{\mu} } e^{-\frac{\left\vert \eta \right\vert^{2}}{2}-\frac{\nu} {2}\eta^{2}+\frac{ \eta} {\mu} b^{\dag} +\frac{\nu} {2}b^{\dag 2}}\left\vert 0\right\rangle , $$

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with $\mu =\cosh \lambda$ and $\nu =\tanh \lambda$, we know

$$\begin{array}{@{}rcl@{}} \rho_{b} &=&\frac{1}{\mu \bar{n}}\int \frac{d^{2}\eta} {\pi} e^{-\left( \frac{1}{\bar{n}}+1\right) \left\vert \eta \right\vert^{2}-\frac{\nu} {2} \left( \eta^{2}+\eta^{\ast 2}\right)} \\ &&\times e^{\frac{1}{\mu} \eta b^{\dag} +\frac{\nu} {2}b^{\dag 2}}\left\vert 0\right\rangle \left\langle 0\right\vert e^{\frac{1}{\mu} \eta^{\ast} b+ \frac{\nu} {2}b^{2}}. \end{array} $$

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This expression of the density operator can help us calculate its statistical properties.

1.2 2) Characteristic function

The CF of squeezed thermal state is given by $\chi _{b}\left (\beta \right ) =$Tr$\left (\rho _{b}D_{b}\left (\beta \right ) \right )$ with $D_{b}\left (\beta \right ) =e^{\left \vert \beta \right \vert ^{2}/2}e^{-\beta ^{\ast }b}e^{\beta b^{\dag } }$ [25]. Thus, we have

$$\begin{array}{@{}rcl@{}} \chi_{b}\left( \beta \right) &=&\frac{1}{\mu \bar{n}}e^{\frac{\left\vert \beta \right\vert^{2}}{2}}\int \frac{d^{2}\eta} {\pi} e^{-\left( \frac{1}{ \bar{n}}+1\right) \left\vert \eta \right\vert^{2}-\frac{\nu} {2}\left( \eta^{2}+\eta^{\ast 2}\right)} \\ &&\left\langle 0\right\vert e^{\frac{1}{\mu} \eta^{\ast} b-\beta^{\ast} b+ \frac{\nu} {2}b^{2}}e^{\frac{1}{\mu} \eta b^{\dag} +\beta b^{\dag} +\frac{ \nu} {2}b^{\dag 2}}\left\vert 0\right\rangle \\ &=&\exp \left( -M\left\vert \beta \right\vert^{2}+N\beta^{2}+N\beta^{\ast} {}^{2}\right) . \end{array} $$

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1.3 3) Wigner function

Using (10) and (26), we obtain the WF of the squeezed thermal state as follows

$$ W_{\rho_{b}}(\varepsilon )=\frac{2}{\pi \left( 2\bar{n}+1\right)} e^{-\frac{ 2\cosh 2\lambda} {2\bar{n}+1}\left\vert \varepsilon \right\vert^{2}+\frac{ \sinh 2\lambda} {2\bar{n}+1}\left( \varepsilon^{2}+\varepsilon^{\ast} {}^{2}\right)} . $$

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1.4 1) Normalization factor

Using Tr$\left (\rho _{a}\right ) =1$, we have the normalization factor

$$\begin{array}{@{}rcl@{}} {\Gamma} &=&e^{-\left\vert \kappa \right\vert^{2}}\frac{d^{2}}{dh_{1}ds_{1}}\left\langle 0\right\vert e^{\left( h_{1}+\kappa^{\ast} \right) a}e^{\left( s_{1}+\kappa \right) a^{\dag} }\left\vert 0\right\rangle |_{s_{1}=h_{1}=0} \\ &=&e^{-\left\vert \kappa \right\vert^{2}}\frac{d^{2}}{dh_{1}ds_{1}} e^{\left( h_{1}+\kappa^{\ast} \right) \left( s_{1}+\kappa \right)} |_{s_{1}=h_{1}=0} \\ &=&1+\left\vert \kappa \right\vert^{2}, \end{array} $$

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1.5 2) characteristic function

Substituting $D_{a}\left (\alpha \right ) =e^{\frac {\left \vert \alpha \right \vert ^{2}}{2}}e^{-\alpha ^{\ast } a}e^{\alpha a^{\dag } }$ and (29) into $\chi _{in}\left (\alpha \right ) =$Tr$\left (\rho _{a}D_{a}\left (\alpha \right ) \right )$, we obtain the CF as follows

$$\begin{array}{@{}rcl@{}} \chi_{in}\left( \alpha \right) &=&\frac{e^{\frac{\left\vert \alpha \right\vert^{2}}{2}-\left\vert \kappa \right\vert^{2}}}{\Gamma} \frac{ d^{2}}{dh_{1}ds_{1}}e^{\left( h_{1}+\kappa^{\ast} -\alpha^{\ast} \right) \left( s_{1}+\kappa +\alpha \right)} |_{s_{1}=h_{1}=0} \\ &=&\frac{e^{-\frac{\left\vert \alpha \right\vert^{2}}{2}+\alpha \kappa^{\ast} -\kappa \alpha^{\ast} }}{\Gamma} \left( \alpha \kappa^{\ast} -\kappa \alpha^{\ast} -\left\vert \alpha \right\vert^{2}+\left\vert \kappa \right\vert^{2}+1\right) , \end{array} $$

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1.6 3) Wigner function

Substituting (29) into (10), we obtain the WF

$$\begin{array}{@{}rcl@{}} W_{\left\vert \psi_{a}\right\rangle} (\varepsilon ) &=&\frac{ 2e^{2\left\vert \varepsilon \right\vert^{2}-\left\vert \kappa \right\vert^{2}}}{\pi {\Gamma}} \frac{d^{2}}{dh_{1}ds_{1}}e^{\left( h_{1}+\kappa^{\ast} -2\varepsilon^{\ast} \right) \left( 2\varepsilon -s_{1}-\kappa \right) \allowbreak} |_{s_{1}=h_{1}=0} \\ &=&\frac{2e^{-2\left\vert \kappa -\varepsilon \right\vert^{2}}}{\pi {\Gamma}} \left( \left\vert \kappa \right\vert^{2}-2\kappa \varepsilon^{\ast} -2\varepsilon \kappa^{\ast} +4\left\vert \varepsilon \right\vert^{2}-1\right) . \end{array} $$

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Appendix B: Some Character of Single-Photon-Added Coherent State

In this appendix, we derive the normalization factor, the CF and the WF for the SPACS, whose density operator $\rho _{a}=\left \vert \psi _{a}\right \rangle \left \langle \psi _{a}\right \vert$ can be expressed as

$$ \rho_{a}=\frac{e^{-\left\vert \kappa \right\vert^{2}}}{\Gamma} \frac{ d^{2}}{dh_{1}ds_{1}}e^{\left( s_{1}+\kappa \right) a^{\dag} }\left\vert 0\right\rangle \left\langle 0\right\vert e^{\left( h_{1}+\kappa^{\ast} \right) a}|_{s_{1}=h_{1}=0}. $$

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