Non-classical Effects of the Photon-Added Hadamard Transformed Vacuum State

Abstract

Theoretical analysis are given for the non-classical effects of the photon-added Hadamard transformed vacuum state (PAHTVS) generated by repeatedly acting photon addition operation on a vacuum state which passing through a Hadamard gate. It is shown that the normalization constant of the PAHTVS is a Legendre polynomial. Furthermore, we study the analytical expressions of several quasi-probability distributions for the PAHTVS. We discuss the negative values of the Wigner function for the PAHTVS, which implies the non-classical properties of the PAHTVS.

Appendices

Appendix A: Derivation the explicit form of the wave function of the PAHTVS in Eq. (22)

The coherent state in the coordinate representation can be expressed as

$$ \langle q \vert \alpha \rangle =\pi^{-\frac{1}{4}} \exp \biggl( -\frac{1}{2}\vert \alpha \vert ^{2}- \frac{1}{2} \alpha^{2}-\frac{q^{2}}{2}+\sqrt{2}\alpha q \biggr) . $$

(A.1)

Using Eq. (A.1) and substituting the normal ordering form of Hadamard operator in Eq. (7) into Eq. (22), we have

$$\begin{aligned} & \langle q\vert N_{m}^{-1/2}a^{\dagger m}H\vert 0 \rangle \\ &\quad{} =N_{m}^{-1/2}\frac{2\sigma}{\sqrt{\sigma^{2}+4}}\int\frac{d^{2}\alpha}{\pi} \langle q| \alpha \rangle \langle \alpha \vert \colon a^{\dagger m} \exp \bigl( Aa^{\dagger2} \bigr) \colon|0 \rangle \\ &\quad{} =N_{m}^{-1/2}\pi^{-\frac{1}{4}}\frac{2\sigma}{\sqrt{\sigma^{2}+4}} \frac{\partial^{m}}{\partial k^{m}}\int\frac{d^{2}\alpha}{\pi}\exp \biggl( -\vert \alpha \vert ^{2}+\sqrt{2}\alpha q+k\alpha^{\ast} -\frac{\alpha^{2}}{2}+A \alpha^{\ast2} \biggr) \bigg \vert _{k=0} \\ &\quad{} =W ( \sigma,m ) F ( \alpha,q ), \end{aligned}$$

(A.2)

where we have set

(A.3)

(A.4)

Using the integral formula

$$\begin{aligned} &\int\frac{d^{2}z}{\pi}\exp \bigl( \zeta \vert z\vert ^{2}+\xi z+ \eta z^{\ast}+fz^{2}+gz^{\ast2} \bigr) \\ &\quad{}=\frac{1}{\sqrt{\zeta^{2}-4fg}}\exp \biggl[ \frac{-\zeta\xi\eta+\xi^{2}g+\eta^{2}f}{\zeta^{2}-4fg} \biggr] , \end{aligned}$$

(A.5)

whose convergent condition is $\operatorname{Re} ( \xi+f+g ) <0$, $\operatorname{Re} ( \frac{\zeta^{2}-4fg}{\xi+f+g} ) <0$, or $\operatorname{Re} ( \xi-f-g ) <0$, $\operatorname{Re} ( \frac{\zeta^{2}-4fg}{\xi-f-g} ) <0$, we see

$$ F ( \alpha,q ) =\frac{1}{\sqrt{1+2A}}\exp \bigl( \tau_{2} q^{2} \bigr) \frac{\partial^{m}}{\partial k^{m}}\exp \biggl[ \frac{1}{ 1+2A} \biggl( \sqrt{2}qk-\frac{1}{2}k^{2} \biggr) \biggr] , $$

(A.6)

where $\tau_{2}=\frac{2A-1}{2 ( 1+2A )}$.

Using the generating function of sing variable Hermite polynomials, we get

$$ \frac{\partial^{k}}{\partial t^{k}}\exp \bigl( At+Bt^{2} \bigr) |_{t=0}= ( -i \sqrt{B} ) ^{k}H_{k} \biggl( \frac{iA}{2\sqrt{B} } \biggr), $$

(A.7)

Eq. (A.4) becomes

$$ F ( \alpha,q ) =\frac{1}{\sqrt{1+2A}}\tau_{1}\exp \bigl( \tau _{2}q^{2} \bigr) H_{m} \biggl( \frac{q}{\sqrt{1+2A}} \biggr), $$

(A.8)

where $\tau_{1}= [ \sqrt{1+2A} ( \sqrt{2 ( 1+2A ) } ) ^{m} ] ^{-1}$.

Combing Eqs. (A.3) and (A.8) we get Eq. (22).

Appendix B: Derivation of the Eqs. (32) and (33)

Substituting Eq. (7) into Eq. (32) and using the similar approach to the calculation of Eq. (15), we get

$$\begin{aligned} \bigl\langle a^{\dagger} \bigr\rangle _{H} =&-Nm^{-1} \biggl( \sqrt{\frac{CD}{2} } \biggr) ^{2m+1} \frac{\partial^{2m+1}}{\partial k^{m}\partial l^{m+1}} \\ &{}\times\int\frac{d^{2}\beta}{\pi}\exp \biggl( -\vert \beta \vert ^{2}-k^{2}-l^{2}+\sqrt{\frac{2C}{D}}\beta k-\sqrt{\frac{2C}{D}}\beta^{\ast }l \biggr) \bigg \vert _{k=l=0} \\ =&-Nm^{-1} \biggl( \sqrt{\frac{CD}{2}} \biggr) ^{2m+1} \frac {\partial^{2m+1}}{\partial k^{m}\partial l^{m+1}}\exp \biggl( -k^{2} -l^{2}- \frac{2C}{D}kl \biggr) \bigg \vert _{k=l=0} \\ =&-Nm^{-1} \biggl( \sqrt{\frac{CD}{2}} \biggr) ^{2m+1} \\ &{}\times \sum_{g=0}^{\infty} \biggl( - \frac{2C}{D} \biggr)^{g}\frac{1}{g!}\frac {\partial^{2g}}{\partial\gamma^{g}\partial\gamma^{\ast g}} \frac{\partial ^{2m+1}}{\partial k^{m}\partial l^{m+1}}\exp \bigl[ -k^{2}-l^{2}+\gamma k+ \gamma^{\ast}l \bigr] \bigg \vert _{k=l=\gamma=\gamma^{\ast}=0} \\ =&-Nm^{-1} \biggl( \sqrt{\frac{CD}{2}} \biggr) ^{2m+1} \\ &{}\times\sum_{g=0}^{m} \biggl( -\frac{2C}{D} \biggr) ^{g}\frac{m! ( m+1 ) !}{g! ( m+1-g ) ! ( m-g ) !}H_{m+1-g} ( 0 ) H_{m-g} ( 0 ). \end{aligned}$$

(B.1)

Noticing the orthogonality of Hermite polynomial, we get 〈a ^†〉 _H =0.

Similarly, we can obtain Eq. (33).

Appendix C

Substituting Eq. (7) into Eq. (34) and inserting the completeness relation of coherent state, we have

$$\begin{aligned} & W \bigl( \alpha,\alpha^{\ast} \bigr) \\ &\quad{}= N_{m}^{-1}\frac{4\sigma^{2}}{\sigma^{2}+4} ( -1 ) ^{m}e^{2\vert \alpha \vert ^{2}} \\ &\qquad{}\times \int\frac{d^{2}\beta}{\pi }\vert \beta \vert ^{2m}\exp \bigl[ - \vert \beta \vert ^{2}+A\beta^{\ast2}+A\beta^{2}+2 \bigl( \alpha\beta^{\ast}-\alpha^{\ast} \beta \bigr) +k\beta+l \beta^{\ast} \bigr] \bigg \vert _{k=l=0} \\ &\quad{}= N_{m}^{-1}\frac{4\sigma^{2}}{\sigma^{2}+4} ( -1 ) ^{m}e^{2\vert \alpha \vert ^{2}} \frac{\partial^{2m} }{\partial k^{m}\partial l^{m}} \\ &\qquad{}\times\int\frac{d^{2}\beta}{\pi}\exp \bigl[ -\vert \beta \vert ^{2}+A\beta^{\ast2}+A\beta^{2}+ ( 2\alpha+l ) \beta^{\ast}+ \bigl( k-2\alpha^{\ast} \bigr) \beta \bigr] \bigg \vert _{k=l=0} \\ &\quad{}= N_{m}^{-1}\frac{4\sigma^{2}}{\sigma^{2}+4} ( -1 ) ^{m} \frac {1}{\sqrt{1-4A^{2}}}e^{2\vert \alpha \vert ^{2}} \\ &\qquad{}\times \frac{\partial^{2m}}{\partial k^{m}\partial l^{m}}\exp \biggl[ \frac{1}{1-4A^{2}} \bigl( ( 2\alpha+l ) \bigl( k-2\alpha^{\ast } \bigr) + ( 2\alpha+l ) ^{2}A+ \bigl( k-2\alpha^{\ast} \bigr) ^{2}A \bigr) \biggr] \bigg \vert _{k=l=0} \\ &\quad{}= N_{m}^{-1}\frac{4\sigma^{2}}{\sigma^{2}+4} ( -1 ) ^{m} \frac {1}{\sqrt{1-4A^{2}}}e^{2\vert \alpha \vert ^{2}}\exp K_{1} \\ &\qquad{}\times \sum_{\gamma=0}^{\infty}\frac{1}{\gamma!} \biggl( \frac{1}{4A^{2} -1} \biggr) ^{\gamma}\frac{\partial^{2\gamma}}{\partial K_{2}^{\gamma}\partial K_{2}^{\ast\gamma}} \frac{\partial^{2m}}{\partial k^{m}\partial l^{m}} \exp \bigl( K_{2}k-K_{2}^{\ast}l+K_{3}l^{2}+K_{3}k^{2} \bigr) \bigg \vert _{k=l=0}, \end{aligned}$$

(C.1)

where

$$\begin{aligned} \begin{aligned} K_{1} & =\frac{4}{1-4A^{2}} \bigl( A\alpha^{2}+ \alpha^{\ast2}A-\vert \alpha \vert ^{2} \bigr) , \\ K_{2} & =\frac{2}{1-4A^{2}} \bigl( \alpha^{\ast}-2A\alpha \bigr) , \\ K_{3} & =\frac{A}{1-4A^{2}}. \end{aligned} \end{aligned}$$

(C.2)

Further, using the formula in Eq. (A.7) and the recurrence relation of H _n (x),

$$ \frac{\partial^{l}}{\partial x^{l}}H_{n} ( x ) =\frac{2^{l} n!}{ ( n-l ) !}H_{n-l} ( x ), $$

(C.3)

as well as

$$ H_{m} ( -x ) = ( -1 ) ^{m}H_{m} ( x ), $$

(C.4)

we have

$$\begin{aligned} &W \bigl( \alpha,\alpha^{\ast} \bigr) \\ &\quad{}=N_{m}^{-1} ( -1 ) ^{m}\frac{4\sigma^{2}}{\sigma^{2}+4}\frac{1}{\sqrt{1-4A^{2}}}e^{2\vert \alpha \vert ^{2}}\exp K_{1} \\ &\qquad{}\times \sum_{\gamma=0}^{\infty}\frac{1}{\gamma!} \biggl( \frac{1}{4A^{2}-1} \biggr) ^{\gamma} \frac{\partial^{2\gamma}}{\partial K_{2}^{\gamma}\partial K_{2}^{\ast\gamma}} \frac{\partial^{2m}}{\partial k^{m}\partial l^{m}} \exp \bigl( K_{2}k-K_{2}^{\ast}l+K_{3}l^{2}+K_{3}k^{2} \bigr) \bigg \vert _{k=l=0} \\ &\quad{}= N_{m}^{-1}K_{4}K_{3}^{m}e^{2\vert \alpha \vert ^{2}+K_{1} } \\ &\qquad{}\times \sum_{\gamma=0}^{m}\frac{1}{\gamma!} \bigl[ 4K_{3} \bigl( 1-4A^{2} \bigr) \bigr] ^{-\gamma} \frac{2^{2\gamma} ( m! ) ^{2}}{ ( ( m-\gamma ) ! ) ^{2}}\biggl \vert H_{m-\gamma} \biggl( \frac{iK_{2} }{2\sqrt{K_{3}}} \biggr) \biggr \vert ^{2}, \end{aligned}$$

(C.5)

where $K_{4}=\frac{4\sigma^{2}}{ ( \sigma^{2}+4 ) \sqrt{1-4A^{2}}}$.