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Entanglement of Photon-Subtracted Two-Mode Squeezed Thermal State and Its Decoherence in Thermal Environments

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Abstract

Using a non-Gaussian operation—photon subtraction from two-mode squeezed thermal state (PS-TMSTS), we construct a kind of entangled state. A Jacobi polynomial is found to be related to the normalization factor. The negativity of Wigner function (WF) is used to discuss its nonclassicality. The investigated entanglement properties turn out that the symmetrical PS-TMSTS may be more effective than the non-symmetric for quantum teleportation. Then the time evolution of WF is used to examine the decoherence effect, which indicates that the characteristic time of single PS-TMSTS depends not only on the average photon number of environment, but also on the average photon number of thermal state and the squeezing parameter.

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Acknowledgements

This project was supported by the National Natural Science Foundation of China (Grant no. 11264018), and the Natural Science Foundation of Jiangxi Province of China (No. 20132BAB212006) as well as the Sponsored Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University.

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Authors and Affiliations

  1. Center for Quantum Science and Technology, College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, 330022, China

    Hao-Liang Zhang, Yin-Quan Hu, Fang Jia & Li-Yun Hu

  2. Key Laboratory of Optoelectronic and Telecommunication of Jiangxi, Nanchang, 330022, China

    Hao-Liang Zhang & Li-Yun Hu

Authors
  1. Hao-Liang Zhang
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  2. Yin-Quan Hu
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  3. Fang Jia
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Correspondence to Li-Yun Hu.

Appendices

Appendix A: Derivation of Eq. (16)

Substituting Eq. (15) into Eq. (14) and using Eq. (11), we have

$$\begin{aligned} W ( \alpha,\beta ) =&\tilde{A}_{1}M_{m,n}^{-1}\int \frac {d^{2}z_{1}d^{2}z_{2}}{\pi^{2}}\vert z_{1}\vert ^{2m}\vert z_{2}\vert ^{2n} \\ &{} \times\exp \bigl[ \tilde{A}_{2} \bigl( z_{1}^{\ast}z_{2}^{\ast }+z_{1}z_{2} \bigr) -\tilde{A}_{3} \bigl( \vert z_{1}\vert ^{2}+\vert z_{2}\vert ^{2} \bigr) \bigr] \langle z_{1},z_{2}|\Delta_{a} \bigl( \alpha, \alpha^{\ast} \bigr) \Delta _{b} \bigl( \beta, \beta^{\ast} \bigr) |z_{1},z_{2}\rangle \\ =&\tilde{A}_{1}M_{m,n}^{-1}e^{-2 ( \vert \alpha \vert ^{2}+\vert \beta \vert ^{2} ) } \frac{\partial ^{2m+2n}}{\partial\tau^{m}\partial s^{m}\partial\tau^{\prime n}\partial s^{\prime n}}\int\frac{d^{2}z_{1}d^{2}z_{2}}{\pi^{4}}e^{- ( \tilde {A}_{3}+2 ) \vert z_{1}\vert ^{2}- ( \tilde {A}_{3}+2 ) \vert z_{2}\vert ^{2}} \\ &{} \times e^{ ( s+2\alpha^{\ast}+\tilde{A}_{2}z_{2} ) z_{1}+ ( \tau+2\alpha+\tilde{A}_{2}z_{2}^{\ast} ) z_{1}^{\ast}+ ( s^{\prime}+2\beta^{\ast} ) z_{2}+ ( \tau^{\prime}+2\beta ) z_{2}^{\ast}} \vert _{\tau, s,\tau^{\prime},s^{\prime}=0} \\ =&W_{0} ( \alpha,\beta ) F_{m,n} ( \alpha,\beta ), \end{aligned}$$
(A1)

where W 0(α,β) is defined in Eq. (17), and

$$ F_{m,n} ( \alpha,\beta ) =M_{m,n}^{-1} \frac{\partial ^{2m+2n}}{\partial\tau^{m}\partial s^{m}\partial\tau^{\prime n}\partial s^{\prime n}}e^{K_{1} ( s\tau+s^{\prime}\tau^{\prime} ) +K_{3} ( ss^{\prime}+\tau\tau^{\prime} ) +R_{1}s+R_{2}\tau +R_{3}s^{\prime}+R_{4}\tau^{\prime}} \vert _{\tau,s,\tau ^{\prime },s^{\prime}=0}, $$
(A2)

and

$$ R_{1}=2 \bigl( K_{1}\alpha+K_{3} \beta^{\ast} \bigr) =R_{2}^{\ast },\qquad R_{3}=2 \bigl( K_{1}\beta+K_{3}\alpha^{\ast} \bigr) =R_{4}^{\ast}, $$
(A3)

as well as

$$ K_{1}=\frac{\bar{n}-\sinh^{2}r}{2\bar{n}+1},\qquad K_{3}=\frac{\sinh r\cosh r}{2\bar{n}+1}. $$
(A4)

Expanding the partial exponential items in Eq. (A2), then Eq. (A2) becomes

$$\begin{aligned} F_{m,n} ( \alpha,\beta ) =&M_{m,n}^{-1} \frac{\partial ^{2m+2n}}{\partial\tau^{m}\partial s^{m}\partial\tau^{\prime n}\partial s^{\prime n}}\sum_{l,j=0}^{\infty} \frac{K_{1}^{l+j}}{l!j!} ( s\tau ) ^{l} \bigl( s^{\prime} \tau^{\prime} \bigr) ^{j} \\ &{} \times\exp \bigl[ K_{3} \bigl( ss^{\prime}+\tau \tau^{\prime} \bigr) +R_{1}s+R_{1}^{\ast} \tau+R_{3}s^{\prime}+R_{3}^{\ast} \tau^{\prime } \bigr] _{\tau,s,\tau^{\prime},s^{\prime}=0} \\ =&M_{m,n}^{-1}\sum_{l,j=0}^{\infty} \frac{K_{1}^{l+j}}{l!j!}\frac {\partial ^{2l+2j}}{\partial R_{1}^{l}\partial ( R_{1}^{\ast} ) ^{l}\partial R_{3}^{j}\partial ( R_{3}^{\ast} ) ^{j}} \\ &{} \times\frac{\partial^{2m+2n}}{\partial\tau^{m}\partial s^{m}\partial \tau^{\prime n}\partial s^{\prime n}}e^{K_{3} ( ss^{\prime }+\tau\tau^{\prime} ) +R_{1}s+R_{3}s^{\prime}+R_{1}^{\ast}\tau +R_{3}^{\ast}\tau^{\prime}}\bigg\vert _{\tau,s,\tau^{\prime },s^{\prime}=0}. \end{aligned}$$
(A5)

Further using the generating function of two-variable Hermite polynomials,

$$ \frac{\partial^{m}}{\partial\tau^{m}}\frac{\partial^{n}}{\partial \upsilon^{n}}e^{-A\tau\upsilon+B\tau+C\upsilon}\bigg\vert _{\tau=\upsilon=0}= ( \sqrt{A} ) ^{m+n}H_{m,n} \biggl( \frac {B}{\sqrt{A}},\frac{C}{\sqrt{A}} \biggr), $$
(A6)

Equation (A5) can be put into the following form

$$\begin{aligned} F_{m,n} ( \alpha,\beta ) =&\frac{K_{3}^{m+n}}{M_{m,n}}\sum _{l,j=0}^{\infty}\frac{ ( -K_{1} ) ^{l+j}}{l!j!}\frac{\partial ^{2l+2j}}{\partial R_{1}^{l}\partial ( -R_{1}^{\ast} ) ^{l}\partial R_{3}^{j}\partial ( -R_{3}^{\ast} ) ^{j}} \\ &{} \times H_{m,n} \biggl( \frac{R_{1}}{i\sqrt{K_{3}}},\frac{R_{3}}{i\sqrt {K_{3}}} \biggr) H_{m,n} \biggl( -\frac{R_{1}^{\ast}}{i\sqrt {K_{3}}},-\frac{R_{3}^{\ast}}{i\sqrt{K_{3}}} \biggr). \end{aligned}$$
(A7)

Using the relation

$$ \frac{\partial^{l+k}}{\partial x^{l}\partial y^{k}}H_{m,n} ( x,y ) =\frac{m!n!}{ ( m-l ) ! ( n-k ) !}H_{m-l,n-k} ( x,y ), $$
(A8)

thus we can obtain Eq. (18).

Appendix B: Derivation of Eq. (27)

Using the displacement operator \(D_{a} ( \alpha ) =e^{-\vert \alpha \vert ^{2}/2}e^{\alpha a^{\dagger}}e^{-\alpha^{\ast}a}\) and \(D_{b} ( \beta ) =e^{-\vert \beta \vert ^{2}/2}e^{\beta b^{\dagger}}e^{-\beta^{\ast}b}\), the CF of PS-TMSTS is given by

$$\begin{aligned} & \chi_{E} ( \alpha,\beta ) \\ &\quad =\operatorname{tr} \bigl[ D_{a} ( \alpha ) D_{b} ( \beta ) \rho ^{\mathit{SS}} \bigr] \\ &\quad =A_{1}M_{m,n}^{-1}e^{- ( \vert \alpha \vert ^{2}+\vert \beta \vert ^{2} ) /2}\operatorname{tr} \bigl[ e^{\alpha a^{\dagger }}e^{-\alpha ^{\ast}a}e^{\beta b^{\dagger}}e^{-\beta^{\ast}b}a^{m}b^{n} {\,:\,} e^{A_{2} ( a^{\dagger}b^{\dagger}+ab ) -A_{3} ( a^{\dagger }a+b^{\dagger}b ) }{\,:\,} a^{\dagger m}b^{\dagger n} \bigr] \\ &\quad =A_{1}M_{m,n}^{-1}e^{- ( \vert \alpha \vert ^{2}+\vert \beta \vert ^{2} ) /2} \frac{\partial^{2m+2n}}{\partial ( -\alpha^{\ast} ) ^{m}\partial ( -\beta^{\ast} ) ^{n}\partial\alpha^{m}\partial\beta^{n}} \\ &\qquad{} \times \operatorname{tr} \bigl[ e^{-\alpha^{\ast}a}e^{-\beta^{\ast}b}{ \,:\,} e^{A_{2} ( a^{\dagger}b^{\dagger}+ab ) -A_{3} ( a^{\dagger }a+b^{\dagger}b ) }{\,:\,} e^{\alpha a^{\dagger}}e^{\beta b^{\dagger}} \bigr]. \end{aligned}$$
(B1)

In a similar way to derive Eq. (10), using (10), one can directly obtain

$$\begin{aligned} \chi_{E} ( \alpha,\beta ) =& M_{m,n}^{-1}e^{- ( \vert \alpha \vert ^{2}+\vert \beta \vert ^{2} ) /2} \frac{\partial^{2m+2n}}{\partial ( -\alpha^{\ast} ) ^{m}\partial ( -\beta^{\ast} ) ^{n}\partial\alpha^{m}\partial\beta^{n}} \\ &{} \times\exp \bigl[ \tilde{B}_{1} \bigl( \beta \bigl( - \beta^{\ast } \bigr) +\alpha \bigl( -\alpha^{\ast} \bigr) \bigr) + \tilde{B}_{2} \bigl( \alpha \beta+ \bigl( -\alpha^{\ast} \bigr) \bigl( -\beta^{\ast} \bigr) \bigr) \bigr] _{\tau,s,\tau^{\prime},s^{\prime}=0}. \end{aligned}$$
(B2)

Taking the following transformations

$$\begin{aligned} -\alpha^{\ast} \longrightarrow&\tau-\alpha^{\ast},\qquad\alpha \longrightarrow\alpha+s, \\ \\ -\beta^{\ast} \longrightarrow&\tau^{\prime}-\beta^{\ast},\qquad \beta \longrightarrow\beta+s^{\prime}, \end{aligned}$$
(B3)

which leads to

$$\begin{aligned} & \exp \bigl[ \tilde{B}_{1} \bigl( \beta \bigl( -\beta^{\ast} \bigr) +\alpha \bigl( -\alpha^{\ast} \bigr) \bigr) +\tilde{B}_{2} \bigl( \alpha \beta+ \bigl( -\alpha^{\ast} \bigr) \bigl( - \beta^{\ast} \bigr) \bigr) \bigr] \\ &\quad \longrightarrow\exp \bigl[ -\tilde{B}_{1} \bigl( \vert \alpha \vert ^{2}+\vert \beta \vert ^{2} \bigr) +\tilde {B}_{2} \bigl( \alpha^{\ast}\beta^{\ast}+\alpha\beta \bigr) +\tilde {B}_{1} \bigl( s\tau+s^{\prime}\tau^{\prime} \bigr) +\tilde {B}_{2} \bigl( ss^{\prime}+\tau\tau^{\prime} \bigr) \bigr] \\ &\quad\hphantom{\longrightarrow}{} \times\exp \bigl[ s \bigl( \beta\tilde{B}_{2}-\alpha^{\ast} \tilde {B}_{1} \bigr) +\tau \bigl( \alpha\tilde{B}_{1}- \beta^{\ast}\tilde {B}_{2} \bigr) +\tau^{\prime} \bigl( \beta\tilde{B}_{1}-\alpha^{\ast }\tilde{B}_{2} \bigr) +s^{\prime} \bigl( \alpha\tilde{B}_{2}-\beta ^{\ast} \tilde{B}_{1} \bigr) \bigr], \end{aligned}$$
(B4)

thus Eq. (B2) becomes Eq. (27).

Appendix C: Derivation of Eq. (37)

Substituting Eq. (16) into Eq. (36), we have

$$\begin{aligned} & W_{m,n} ( \gamma,\eta,t ) \\ & \quad =\frac{4 ( 2\bar{n}+1 ) ^{-2}\pi^{-2}}{ ( 2\mathfrak {N}+1 ) ^{2}T^{2}M_{m,n}}e^{-2\frac{\vert \gamma \vert ^{2}+\vert \eta \vert ^{2}}{ ( 2\mathfrak{N}+1 ) T}}\frac{\partial^{2m+2n}}{\partial\tau^{m}\partial s^{m}\partial \tau ^{\prime n}\partial s^{\prime n}}\exp \biggl[ g_{1} \bigl( s\tau +s^{\prime }\tau^{\prime} \bigr) + \frac{g_{2}}{2} \bigl( ss^{\prime}+\tau\tau ^{\prime} \bigr) \biggr] \\ & \qquad{} \times\int\frac{d^{2}\alpha d^{2}\beta}{\pi^{2}}\exp \bigl[ - \bigl( 2g_{0}+g_{3}e^{-\kappa t} \bigr) \vert \alpha \vert ^{2}+ \bigl( 2g_{2} \beta+g_{3}\gamma^{\ast}+2g_{1}s+g_{2} \tau^{\prime} \bigr) \alpha \bigr] \\ & \qquad{} \times\exp \bigl[ + \bigl( 2g_{2}\beta^{\ast}+g_{3} \gamma +2g_{1}\tau +g_{2}s^{\prime} \bigr) \alpha^{\ast} \bigr] \\ & \qquad{} \times\exp \bigl[ - \bigl( 2g_{0}+g_{3}e^{-\kappa t} \bigr) \vert \beta \vert ^{2}+ \bigl( g_{3} \eta^{\ast}+g_{2}\tau +2g_{1}s^{\prime } \bigr) \beta \\ & \qquad{}+ \bigl( g_{3}\eta+g_{2}s+2g_{1} \tau^{\prime} \bigr) \beta ^{\ast} \bigr] _{\tau,s,\tau^{\prime},s^{\prime}=0} \\ & \quad =\frac{ ( 2\bar{n}+1 ) ^{-2}\pi^{-2}G^{-1}}{ ( 2\mathfrak{N}+1 ) ^{2}T^{2}M_{m,n}}\exp \biggl[ -\Delta_{1} \bigl( \vert \gamma \vert ^{2}+\vert \eta \vert ^{2} \bigr) + \frac {g_{2}g_{3}^{2}}{2G} \bigl( \gamma^{\ast}\eta^{\ast}+\gamma\eta \bigr) \biggr] \\ & \qquad{} \times\frac{\partial^{2m+2n}}{\partial\tau^{m}\partial s^{m}\partial \tau^{\prime n}\partial s^{\prime n}}\exp \biggl[ \frac{\mu_{1}\mu _{2}}{4G} \bigl( s \tau+s^{\prime}\tau^{\prime} \bigr) +\frac{g_{2}\mu _{2}^{2}}{8G} \bigl( ss^{\prime}+\tau\tau^{\prime } \bigr) +\frac{2\gamma\mu_{1}+\eta^{\ast }g_{2}\mu_{2}}{4G}g_{3}s \biggr] \\ & \qquad{} \times\exp \biggl[ \frac{2\gamma^{\ast}\mu_{1}+\eta g_{2}\mu _{2}}{4G}g_{3}\tau+ \frac{2\eta\mu_{1}+\gamma^{\ast }g_{2}\mu_{2}}{4G}g_{3}s^{\prime}+ \frac{2\eta ^{\ast}\mu_{1}+\gamma g_{2}\mu_{2}}{4G}g_{3}\tau^{\prime} \biggr] _{\tau,s,\tau ^{\prime},s^{\prime}=0}, \end{aligned}$$
(C1)

where (g 0,g 1,g 2,g 3) and (μ 1,μ 2,G,Δ1) are defined in Eqs. (39) and (40), respectively. In a similar way to deriving Eq. (18), we can further insert Eq. (C1) into Eqs. (37)–(38).

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Zhang, HL., Hu, YQ., Jia, F. et al. Entanglement of Photon-Subtracted Two-Mode Squeezed Thermal State and Its Decoherence in Thermal Environments. Int J Theor Phys 53, 2091–2107 (2014). https://doi.org/10.1007/s10773-014-2015-y

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