Single-Mode Photon-Addition for the Two-Mode Squeezed State and its Statistical Properties

Abstract

We construct a new optical field, denoted as $\rho _{0}\equiv \left \vert \psi \right \rangle _{ll}\left \langle \psi \right \vert$, $\left \vert \psi \right \rangle _{l}=\frac {\text {sech}^{l}\lambda }{\sqrt {l!} }b^{\dagger l}S_{2}\left \vert 00\right \rangle$, by adding single-mode l-photon to a two-mode squeezed vacuum state, where $S_{2}=\exp \left [ \lambda \left (a^{\dagger }b^{\dagger }-ab\right ) \right ]$ is the two-mode squeezing operator. We find that its partial trace over b-mode will lead to a negative-binomial optical field in a-mode characteristic of l, which exhibits quantum entanglement. The photon number fluctuation of the new optical field both in a-mode and in b-mode is investigated. We employ the summation method within ordered product of operators to proceed our discussion.

Appendix

The second-order degree of coherence is defined as

$$ g^{\left( 2\right) }\equiv\frac{{~}_{{~}_{l}}\left\langle \psi\right\vert \left( a^{\dagger}a\right)^{2}\left\vert \psi\right\rangle_{l}-{~}_{{~}_{l}}\left\langle \psi\right\vert a^{\dagger}a\left\vert \psi\right\rangle_{l}}{\left( {~}_{{~}_{l} }\left\langle \psi\left\vert a^{\dagger}a\right\vert \psi\right\rangle_{l}\right)^{2}} $$

(45)

Since

$$\begin{array}{@{}rcl@{}} \left( a^{\dagger}a\right)^{2} & =&a^{\dagger}a~a^{\dagger}a~\\ & =&a^{\dagger}\left( a^{\dagger}a+1~\right) a~\\ & =&a^{\dagger2}a^{2}+a^{\dagger}a \end{array} $$

(46)

then

$$ {~}_{l}\left\langle \psi\right\vert \left( a^{\dagger}a\right)^{2}\left\vert \psi\right\rangle_{l}-{~}_{{~}_{l}}\left\langle \psi\right\vert a^{\dagger }a\left\vert \psi\right\rangle_{l}={~}_{{~}_{l}}\left\langle \psi\right\vert a^{\dagger2}a^{2}\left\vert \psi\right\rangle_{l} $$

(47)

Using (47), we can convert (45) into

$$\begin{array}{@{}rcl@{}} g^{\left( 2\right) } & =&\frac{{~}_{l}\left\langle \psi\right\vert a^{\dagger2}a^{2}\left\vert \psi\right\rangle_{l}}{\left( {~}_{l}\left\langle \psi\left\vert a^{\dagger}a\right\vert \psi\right\rangle_{l}\right)^{2} }\\ & =&\frac{\left( \frac{1-\gamma}{\gamma}\right)^{2}\left( l+1\right) \left( l+2\right) }{\left[ \frac{l+1}{\gamma}\left( 1-\gamma\right) \right]^{2}}\\ & =&\frac{\left( l+1\right) \left( l+2\right) }{\left( l+1\right)^{2} }\\ & =&\frac{l+2}{l+1}>1 \end{array} $$

(48)