Nonclassicality of Coherent Photon-Subtracted Two Single-Modes Squeezed Vacuum State

Abstract

We introduce a two-mode non-Gaussian state, generated by nonlocal coherent photon-subtraction (CPS) from two single-mode squeezed vacuum states (TSSV). Its normalized factor is turned out to be related with a Legendre polynomial. We further investigate the nonclassical properties of the CPS-TSSV through cross-correlation function, antibunching effect, photon number distribution, wave function and Wigner function. It is shown that the CPS operation can produce the vortex state and can exhibit the highly nonclassical properties.

Appendices

Appendix A: Derivation of $\mathcal{P} ( m_{a},n_{b} )$

The photon number distribution (PND) of the CPS-TSSV is defined by $\mathcal{P} ( m_{a},n_{b} ) = \langle m_{a},n_{b}\vert \rho \vert m_{a},n_{b} \rangle$, which means the probability of finding (m _a ,n _b ) photons in the two single-modes field.

$$ \mathcal{P} ( m_{a},n_{b} ) = \bigl\langle m_{a},n_{b}\vert \rho \vert m_{a},n_{b} \bigr\rangle =N_{m}^{-1}\bigl \vert \langle m_{a},n_{b}\vert \vert \psi _{m} \rangle \bigr \vert ^{2}. $$

(43)

First let’s calculate 〈m _a ,n _b ||ψ _m 〉. Considering Eq. (4) and using the completeness relation of the coherent state, we obtain the following results:

(44)

Through expressing $\alpha ^{m_{a}}$ and $\beta ^{n_{b}}$ as$\frac{\partial ^{m_{a}}}{\partial \mu ^{m_{a}}}e^{\mu \alpha }|_{\mu =0}$ and $\frac{\partial ^{n_{b}}}{\partial \nu ^{n_{b}}}e^{\nu \beta }|_{\nu =0}$, and using the integration formula [7], we calculate Eq. (44) as

(45)

Further using the generating function of the Hermite polynomial H _m (q)

$$ H_{m} ( q ) =\frac{\partial ^{m} ( e^{-t^{2}+2qt} ) }{\partial t^{m}}\bigg|_{t=0}, $$

(46)

we can get rid of three partial derivations and introduce one partial derivation on variable t (t is not τ). Then we obtain the final form of 〈m _a ,n _b ||ψ _m 〉 as

(47)

where we define

$$ \begin{aligned} H_{m_{a}}(x,y,t)& \equiv H_{m_{a}} \biggl( -\sqrt{ \frac{\tanh \lambda _{1}}{\tanh \lambda _{1}x^{2}+\tanh \lambda _{2}y^{2}}}xt \biggr) , \\ H_{n_{b}}(x,y,t)& \equiv H_{n_{b}} \biggl( -\sqrt{ \frac{\tanh \lambda _{2}}{\tanh \lambda _{1}x^{2}+\tanh \lambda _{2}y^{2}}}yt \biggr) . \end{aligned} $$

(48)

After substituting Eq. (47) into Eq. (43), we obtain the formula to calculate the photon number distribution (PND) of the CPSSVS.

$$ \begin{aligned}[b] \mathcal{P} ( m_{a},n_{b} ) &= N_{m}^{-1} \bigl \vert \langle m_{a},n_{b}\vert \vert \psi _{m} \rangle \bigr \vert ^{2} \\ & =N_{m}^{-1}P\biggl \vert \frac{\partial ^{m}}{\partial t^{m}} \bigl( H_{m_{a}}(x,y,t)H_{n_{b}}(x,y,t)e^{-t^{2}} \bigr) \big|_{t=0}\biggr \vert ^{2}, \end{aligned} $$

(49)

where

$$ P\equiv \biggl \vert \frac{ ( \tanh \lambda _{1}x^{2}+\tanh \lambda _{2}y^{2} ) ^{m}\tanh ^{m_{a}}\lambda _{1}\tanh ^{n_{b}}\lambda _{2}}{m_{a}!n_{b}!2^{m+m_{a}+n_{b}}\cosh \lambda _{1}\cosh \lambda _{2}}\biggr \vert . $$

(50)

Appendix B: Derivation of WF (36) for CPSSVS

According to Eqs. (34) and $\rho =N_{m}^{-1}\vert \psi _{m} \rangle \langle \psi _{m}\vert$, we have

(51)

where we define

(52)

Using the following formula,

$$ \begin{aligned} e^{xa}f\bigl(a,a^{\dagger }\bigr)e^{-xa}& =f \bigl(a,a^{\dagger }+x\bigr), \\ e^{-xa^{\dagger }}f\bigl(a,a^{\dagger }\bigr)e^{xa^{\dagger }}& =f \bigl(a+x,a^{\dagger }\bigr), \end{aligned} $$

(53)

and the integration formula (7), we obtain

(54)

where

(55)

Further using the formula about Hermite polynomials,

(56)

we can see

(57)

where C is defined in Eq. (39). Substituting Eqs. (54) and (57) into Eq. (51), we can finally obtain Eq. (36).