Non-classical Properties of Photon-Added Compass State

Abstract

We investigate the observable non-classical features of the photon-added compass state (PACS) by its sub-Poissonian statistics, such as the Mandel’s parameter, second-order correlation function, photon-number distribution and the quasi-probability distribution functions, peculiarly the negativity in the Wigner distribution of the PACS as the specific non-classical features. We study the squeezing properties of the PACS and find the PACS does not show squeezing properties of the quadrature. Finally, we give the non-Gaussianity of the PACS by the fidelity between the PACS and the squeezed coherent state (SCS).

Appendices

Appendix A: Derivation of Eq. (28)

In order to get Eq. (28), we first derive an integral formula

$$\begin{aligned} & \int\frac{d^{2}\beta}{\pi}\beta^{m}\exp \biggl( -\vert \beta \vert ^{2}+\sqrt{2}\beta a+\alpha\beta^{\ast}-\frac{\beta^{2}}{2} \biggr) \\ &\quad{} =\frac{\partial^{m}}{\partial k^{m}}\int\frac{d^{2}\beta}{\pi}\exp \biggl[ - \vert \beta \vert ^{2}+ ( \sqrt{2}q+k ) \beta+\alpha \beta^{\ast}-\frac{\beta^{2}}{2} \biggr] \bigg\vert _{k=0} \\ &\quad{} =\frac{\partial^{m}}{\partial k^{m}}\exp \biggl[ -\frac{1}{2}\alpha^{2}+ ( k+\sqrt{2}q ) \alpha \biggr] \bigg\vert _{k=0} \\ &\quad{} =\alpha^{m}\exp \biggl( -\frac{1}{2}\alpha^{2}+ \sqrt{2}q\alpha \biggr), \end{aligned}$$

(A.1)

where we have used

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

(A.2)

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

Using Eqs. (25) and (A.1), we obtain

$$\begin{aligned} \langle q\vert \alpha,m \rangle ={}&N_{m}^{-1/2}\pi^{-1/4}\exp \biggl[ -\frac{1}{2} \bigl( q^{2}+\vert \alpha \vert ^{2} \bigr) \biggr]\int\frac{d^{2}\beta}{\pi}\beta^{m}\exp \biggl( -\vert \beta \vert ^{2}+\sqrt{2}\beta q-\frac{1}{2}\beta^{2} \biggr) \\ &\quad{}\times \bigl[ \exp \bigl( \alpha\beta^{\ast} \bigr) +\exp \bigl( -\alpha \beta^{\ast} \bigr) +\exp \bigl( i\alpha\beta^{\ast} \bigr) +\exp \bigl( -i\alpha\beta^{\ast } \bigr) \bigr] \\ ={}&N_{m}^{-1/2}\pi^{-1/4}\exp \biggl[ - \frac{1}{2} \bigl( q^{2}+\vert \alpha \vert ^{2} \bigr) \biggr] ( F_{1}+F_{2}+F_{3}+F_{4} ), \end{aligned}$$

(A.3)

where

$$\begin{gathered} F_{1}=\alpha^{m}\exp \biggl( -\frac{1}{2} \alpha^{2}+\sqrt{2}q\alpha \biggr), \end{gathered}$$

(A.4)

$$\begin{gathered} F_{2}= ( -\alpha ) ^{m}\exp \biggl( -\frac{1}{2} \alpha^{2}-\sqrt {2}q\alpha \biggr), \end{gathered}$$

(A.5)

$$\begin{gathered} F_{3}= ( i\alpha ) ^{m}\exp \biggl( \frac{1}{2} \alpha^{2}+\sqrt {2}iq\alpha \biggr), \end{gathered}$$

(A.6)

and

$$ F_{3}= ( -i\alpha ) ^{m}\exp \biggl( \frac{1}{2} \alpha^{2}-\sqrt {2}iq\alpha \biggr) . $$

(A.7)

Substituting Eqs. (A.4)–(A.7) into Eq. (A.3) and after some steps to simplify the equation, we derive the result in Eq. (28) as expected.

Appendix B: Derivation of Eq. (40)

Using Eq. (A.2), we have

$$\begin{aligned} & \int\frac{d^{2}\beta}{\pi}\beta^{m}\exp \biggl( -\vert \beta \vert ^{2}-\frac{\beta^{\ast2}}{2}\tanh\lambda+\beta^{\ast}\alpha\sec h \lambda +\beta\gamma \biggr) \\ &\quad{} =\frac{\partial^{m}}{\partial k^{m}}\int\frac{d^{2}\beta}{\pi}\exp \biggl[ - \vert \beta \vert ^{2}+\beta^{\ast}\alpha\sec h\lambda+\beta ( \gamma+k ) -\frac{\beta^{\ast2}}{2}\tanh \lambda \biggr] \bigg\vert _{k=0} \\ &\quad{} =\exp \biggl( \alpha\gamma\sec h\lambda-\frac{1}{2} \gamma^{2}\tanh\lambda \biggr) \frac{\partial^{m}}{\partial k^{m}}\exp \biggl[ -\frac {1}{2}k^{2}\tanh\lambda+k ( \alpha\sec h\lambda- \gamma\tanh \lambda ) \biggr] \bigg\vert _{k=0} \\ &\quad{} = \biggl( \frac{1}{2}\tanh\lambda \biggr) ^{m/2}\exp \biggl( \alpha\gamma\sec h\lambda-\frac{1}{2}\gamma^{2}\tanh\lambda \biggr) H_{m} \biggl( \frac {\alpha\sec h\lambda-\gamma\tanh\lambda}{\sqrt{2\tanh\lambda}} \biggr), \end{aligned}$$

(B.1)

where in the last step we have used the general generating function of single variable Hermite polynomials

$$ \frac{\partial^{n}}{\partial t^{n}}\exp \bigl( At+Bt^{2} \bigr) \big\vert _{t=0}= ( i\sqrt{B} ) ^{n}H_{n} \biggl( \frac{A}{2i\sqrt{B}} \biggr) . $$

(B.2)

According to Eqs. (39) and (B.2), we have

$$\begin{aligned} \langle \alpha,m\vert S\vert \alpha \rangle ={}&N_{m}^{-1/2}\sec h^{1/2}\lambda\exp \biggl( - \vert \alpha \vert ^{2}+\frac{\alpha^{2}}{2}\tanh\lambda \biggr) \\ &{}\times \int \frac{d^{2}\beta}{\pi}\beta^{m}\exp \biggl( -\vert \beta \vert ^{2}-\frac{\beta^{\ast2}}{2}\tanh\lambda+ \beta^{\ast}\alpha\sec h\lambda \biggr) \\ &{}\times \bigl[ \exp \bigl( \beta\alpha^{\ast} \bigr) +\exp \bigl( -\beta \alpha^{\ast} \bigr) +\exp \bigl( \beta i\alpha^{\ast} \bigr) +\exp \bigl( -i\beta\alpha^{\ast} \bigr) \bigr] \\ ={}&N_{m}^{-1/2} \biggl( \frac{1}{2}\tanh\lambda \biggr) ^{m/2}\sec h^{1/2}\lambda\exp \biggl( -\vert \alpha \vert ^{2}+\frac{1}{2}\alpha^{2}\tanh\lambda \biggr) \\ &{}\times ( K_{1}+K_{2}+K_{3}+K_{4} ), \end{aligned}$$

(B.3)

where

$$ K_{1}=\exp \biggl( \vert \alpha \vert ^{2}\sec h\lambda- \frac{1}{2}\alpha^{\ast2}\tanh\lambda \biggr) H_{m} \biggl( \frac{\alpha\sec h\lambda-\alpha^{\ast}\tanh\lambda}{\sqrt{2\tanh\lambda}} \biggr), $$

(B.4)

$$ K_{2}=\exp \biggl( -\vert \alpha \vert ^{2}\sec h\lambda- \frac{1}{2}\alpha^{\ast2}\tanh\lambda \biggr) H_{m} \biggl( \frac{\alpha\sec h\lambda+\alpha^{\ast}\tanh\lambda}{\sqrt{2\tanh\lambda}} \biggr), $$

(B.5)

$$ K_{3}=\exp \biggl( i\vert \alpha \vert ^{2}\sec h \lambda+\frac{1}{2}\alpha^{\ast2}\tanh\lambda \biggr) H_{m} \biggl( \frac{\alpha\sec h\lambda-i\alpha^{\ast}\tanh\lambda}{\sqrt{2\tanh\lambda}} \biggr), $$

(B.6)

and

$$ K_{4}=\exp \biggl( -i\vert \alpha \vert ^{2}\sec h \lambda+\frac{1}{2}\alpha^{\ast2}\tanh\lambda \biggr) H_{m} \biggl( \frac{\alpha\sec h\lambda+i\alpha^{\ast}\tanh\lambda}{\sqrt{2\tanh\lambda}} \biggr) . $$

(B.7)

Using Eqs. (B.1) and (B.4)–(B.7), we have

$$ F=N_{m}^{-1} \biggl( \frac{1}{2}\tanh\lambda \biggr) ^{m}\sec h\lambda \biggl\vert \exp \biggl( -\vert \alpha \vert ^{2}+\frac{\alpha^{2}}{2}\tanh\lambda \biggr) ( K_{1}+K_{2}+K_{3}+K_{4} ) \biggr\vert ^{2}. $$

(B.8)