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    Journals >Acta Physica Sinica >Volume 68 >Issue 3 >Page 034202-1 > Article
    • Acta Physica Sinica
    • Vol. 68, Issue 3, 034202-1 (2019)
    Deterministic quantum entanglement among multiple quantum nodes
    Yan-Hong Liu1, Liang Wu1, Zhi-Hui Yan1、2, Xiao-Jun Jia1、2、*, and Kun-Chi Peng1、2
    Author Affiliations
  • 1Institute of Opto-Electronics, State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China
  • 2Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
    • show less
    DOI: 10.7498/aps.68.20181614 Cite this Article
    Yan-Hong Liu, Liang Wu, Zhi-Hui Yan, Xiao-Jun Jia, Kun-Chi Peng. Deterministic quantum entanglement among multiple quantum nodes[J]. Acta Physica Sinica, 2019, 68(3): 034202-1 Copy Citation Text show less

    Abstract

    Quantum entanglement is a significant quantum resource, which plays a central role in quantum communication. For realizing quantum information network, it is important to establish deterministic quantum entanglement among multiple spatial-separated quantum memories, and then the stored entanglement is transferred into the quantum channels for distributing and transmitting the quantum information at the user-control time. Firstly, we introduce the scheme of deterministic generation polarization squeezed state at 795 nm. A pair of quadrature amplitude squeezed optical fields are prepared by two degenerate optical parameter amplifiers pumped by a laser at 398 nm, and then the polarization squeezed state of light appears by combining the generated two quadrature amplitude squeezed optical beams on a polarizing beam splitter. Secondly, we present the experimental demonstration of tripartite polarization entanglement described by Stokes operators of optical field. The quadrature tripartite entangled states of light corresponding to the resonance with D1 line of rubidium atoms are transformed into the continuous-variable polarization entanglement via polarization beam splitter with three bright local optical beams. Finally, we propose the generation, storage and transfer of deterministic quantum entanglement among three spatially separated atomic ensembles. By the method of electromagnetically induced transparency light-matter interaction, the optical multiple entangled state is mapped into three distant atomic ensembles to build the entanglement among three atomic spin waves. Then, the quantum noise of entanglement stored in the atomic ensembles is transferred to the three space-seperated quadrature entangled light fields through three quantum channels. The existence of entanglement among the three released beams verifies that the system has the ability to maintain the multipartite entanglement. This protocol realizes the entanglement among three distant quantum nodes, and it can be extended to quantum network with more quantum nodes. All of these lay the foundation for realizing the large-scale quantum network communication in the future.

    Keywords

    $\begin{aligned} & {{\hat S}_0} = \hat a_{\rm H}^†{{\hat a}_{\rm H}} + \hat a_{\rm V}^†{{\hat a}_{\rm V}},{{\hat S}_2} = \hat a_{\rm H}^†{{\hat a}_{\rm V}}{\rm{e}^{\rm{i}\theta }} + \hat a_{\rm V}^†{{\hat a}_{\rm H}}{\rm{e}^{ - \rm{i}\theta }}, \\ & {{\hat S}_1} = \hat a_{\rm H}^†{{\hat a}_{\rm H}} - \hat a_{\rm V}^†{{\hat a}_{\rm V}},{{\hat S}_3} = (\hat a_{\rm H}^†{{\hat a}_{\rm V}}{\rm{e}^{\rm{i}\theta }} - \hat a_{\rm V}^†{{\hat a}_{\rm H}}{\rm{e}^{ - \rm{i}\theta }})/\rm{i}, \end{aligned} $(1)

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    $\begin{split}& {V_0} = {V_1} = {\alpha ^2}({\Delta ^2}{\hat X_{\rm H}} + {\Delta ^2}{X_{\rm V}}), \\ & {V_2} = {\alpha ^2}{\cos ^2}\theta ({\Delta ^2}{\hat X_{\rm H}} + {\Delta ^2}{\hat X_{\rm V}}) \\ & \quad\quad + {\alpha ^2}{\sin ^2}\theta ({\Delta ^2}{\hat Y_{\rm H}} + {\Delta ^2}{\hat Y_{\rm V}}),\\ & {V_3}(\theta ) = {V_2}\left(\frac{{\text{π}}}{2} - \theta \right),\end{split}$ ()

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    $\begin{gathered} {I_1} \!\equiv \! \frac{{{{\text{δ}}^2}({{\hat S}_{{2_{{d_2}}}}} - {{\hat S}_{{2_{d3}}}}) + {{\text{δ}}^2}({g_1}{{\hat S}_{{3_{{d_1}}}}} + {{\hat S}_{{3_{{d_2}}}}} + {{\hat S}_{{3_{d3}}}})}}{{4\left| {\alpha _c^2 - \alpha _a^2} \right|}} \! \geqslant \!1, \\ {I_2} \! \equiv \! \frac{{{{\text{δ}}^2}({{\hat S}_{{2_{{d_1}}}}} - {{\hat S}_{{2_{d3}}}}) + {{\text{δ}}^2}({{\hat S}_{{3_{{d_1}}}}} + {g_2}{{\hat S}_{{3_{{d_2}}}}} + {{\hat S}_{{3_{d3}}}})}}{{4\left| {\alpha _c^2 - \alpha _a^2} \right|}} \! \geqslant \!1, \\ {I_2} \! \equiv \! \frac{{{{\text{δ}}^2}({{\hat S}_{{2_{{d_1}}}}} - {{\hat S}_{{2_{d2}}}}) + {{\text{δ}}^2}({{\hat S}_{{3_{{d_1}}}}} + {{\hat S}_{{3_{{d_2}}}}} + {g_3}{{\hat S}_{{3_{d3}}}})}}{{4\left| {\alpha _c^2 - \alpha _a^2} \right|}} \! \geqslant \!1, \\ \end{gathered} $(2)

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    ${{\text{δ}}^2}({\hat S_2}) + {{\text{δ}}^2}({\hat S_3}) \geqslant \sum\nolimits_k {{P_k}} ({\text{δ}}_k^2({\hat S_2}) + {\text{δ}}_k^2({\hat S_3})),$(3)

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    $ \begin{aligned} {I_1} & \geqslant {P_1}{I_{1,1}} + {P_2}{I_{1,2}} + {P_3}{I_{1,3}} \\ & \geqslant {P_1}{I_{1,1}} + {P_2}{I_{1,2}} \geqslant {P_1} + {P_2}. \end{aligned} $(4)

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    $\hat \psi (z,t) = \cos \theta (t)\hat E(z,t) - \sin \theta (t)\sqrt N {\hat \sigma _{1,3}}(z,t),$(6)

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    $\cos \theta (t) = \frac{{\varOmega (t)}}{{\sqrt {{\varOmega ^2}(t) + {g^2}N} }},$(7)

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    $\sin \theta (t) = \frac{{g\sqrt N }}{{\sqrt {{\varOmega ^2}(t) + {g^2}N} }},$(8)

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    $\begin{split} & \hat X_{\rm{L}}^{{\rm{out}}} = \hat X_{\rm{L}}^{{\rm{in}}} + \kappa \hat P_{\rm{A}}^{{\rm{in}}},\hat P_{\rm{L}}^{{\rm{out}}} = \hat P_{\rm{L}}^{{\rm{in}}},\\ & \hat X_{\rm{A}}^{{\rm{out}}} = \hat X_{\rm{A}}^{{\rm{in}}} + \kappa \hat P_{\rm{L}}^{{\rm{in}}},\hat P_{\rm{A}}^{{\rm{out}}} = \hat P_{\rm{A}}^{{\rm{in}}}. \end{split}$(9)

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    $\hat P_{\rm{A}}^{{\rm{final}}} = \hat P_{\rm{A}}^{{\rm{in}}} - gx = \hat P_{\rm{A}}^{{\rm{in}}}(1 - g\kappa ) - g\hat X_{\rm{L}}^{{\rm{in}}},$(10)

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    ${H_I} = {\rm{i}}\hbar \eta {A_{\rm{W}}}{\hat a^†}{\hat S^†} - {\rm{i}}\hbar {\eta ^ * }A_{\rm{W}}^*\hat S\hat a,$(11)

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    $\begin{split} & \hat a_i^{{\rm{out}}}(t) = \hat a_i^{{\rm{in}}}\cosh (\eta {A_{\rm{W}}}t) + \hat S_i^{{\rm{i}}{{\rm{n}}^†}}\sinh (\eta {A_{\rm{W}}}t),\\ & \hat S_i^{{\rm{out}}}(t) = \hat S_i^{{\rm{in}}}\cosh (\eta {A_{\rm{W}}}t) + \hat a_i^{{\rm{i}}{{\rm{n}}^†}}\sinh (\eta {A_{\rm{W}}}t). \end{split}$(12)

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    $\begin{split}V &={{\text{δ}}^2}(\hat X_{{\rm{a}}i}^{{\rm{out}}} - \hat X_{{\rm{s}}i}^{{\rm{out}}}) + {{\text{δ}}^2}(\hat Y_{{\rm{a}}i}^{{\rm{out}}} + \hat Y_{{\rm{s}}i}^{{\rm{out}}}) \\ &= 4{{\rm{e}}^{ - 2\eta {A_W}\tau }} = 4{{\rm{e}}^{ - 2r}}.\end{split}$(13)

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    $\begin{split} {\text{δ}} {{\hat Y}_3} = \frac{1}{{\sqrt 2 }}({\text{δ}} \hat Y_1^{{\rm{out}}} + {\text{δ}} \hat Y_2^{{\rm{out}}}),\\ {\text{δ}} {{\hat X}_4} = \frac{1}{{\sqrt 2 }}({\text{δ}} \hat X_1^{{\rm{out}}} - {\text{δ}} \hat X_2^{{\rm{out}}}). \end{split}$(14)

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    $\hat S_1^{{\rm{final}}}(t) = \hat S_1^{{\rm{out}}} - \sqrt 2 {g_1}{\text{δ}} {\hat X_4} + \sqrt 2 {\rm{i}}{g_2}{\text{δ}} {\hat Y_3},$(15)

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    $V' = 4\left[ {(1 + {g^2})\cosh (2r) - 2g\sinh (2r)} \right].$(16)

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    ${g^{{\rm{opt}}}} = \frac{{\sinh (2r)}}{{1 + \cosh (2r)}}.$(17)

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    ${H_{{\rm{EIT}}}} = {\rm{i}}\hbar \kappa {A_{\rm{C}}}\hat a{(0)_{\rm{S}}}{\hat J^†} - {\rm{i}}\hbar \kappa {A_{\rm{C}}}\hat J\hat a(0)_{_{\rm{S}}}^†.$(18)

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    $\hat X{(t)_{{\rm{A}}j}} = \sqrt {{\eta _{\rm{M}}}} \hat X{(0)_{{\rm{L}}j}} + \sqrt {1 - {\eta _{\rm{M}}}} \hat X_{{\rm{A}}j}^{{\rm{vac}}},$(19)

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    $\hat P{(t)_{{\rm{A}}j}} = \sqrt {{\eta _{\rm{M}}}} \hat P{(0)_{{\rm{L}}j}} + \sqrt {1 - {\eta _{\rm{M}}}} \hat P_{{\rm{A}}j}^{{\rm{vac}}}.$(20)

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    $\begin{split} I{(t)_{{\rm{A}}1}} \!=\; & \left\langle {{{\text{δ}}^2}(\hat X{{(t)}_{{\rm{A}}2}} - \hat X{{(t)}_{{\rm{A}}3}})} \right\rangle \Big/2 \\ & + \left\langle \!{{{\text{δ}}^2}({g_{{\rm{A}}1}}\hat P{{(0)}_{{\rm{A}}1}} \!+\! \hat P{{(0)}_{{\rm{A}}2}} + \hat P{{(0)}_{{\rm{A}}3}})} \!\right\rangle \!\!\Big/2 \!\geqslant 1, \\ I{(t)_{{\rm{A}}2}} \!=\; & \left\langle {{{\text{δ}}^2}(\hat X{{(t)}_{{\rm{A}}1}} - \hat X{{(t)}_{{\rm{A}}3}})} \right\rangle \Big/2 \\ &+ \left\langle \!{{{\text{δ}}^2}(\hat P{{(0)}_{{\rm{A}}1}} \!+\! {g_{{\rm{A}}2}}\hat P{{(0)}_{{\rm{A}}2}} + \hat P{{(0)}_{{\rm{A}}3}})} \!\right\rangle \!\!\Big/2\! \geqslant 1, \\ I{(t)_{{\rm{A}}3}} \!=\; & \left\langle {{{\text{δ}}^2}(\hat X{{(t)}_{{\rm{A}}1}} - \hat X{{(t)}_{{\rm{A}}2}})} \right\rangle \Big/2 \\ &+ \left\langle \!{{{\text{δ}}^2}(\hat P{{(0)}_{{\rm{A}}1}} \!+ \!\hat P{{(0)}_{{\rm{A}}2}} \!+\! {g_{{\rm{A}}3}}\hat P{{(0)}_{{\rm{A}}3}})} \right\rangle \!\!\Big/2 \geqslant 1. \end{split} $(21)

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    $\begin{split} & \hat X{(t)_{{\rm{L}}j}} = - \sqrt {{{\eta '}_{\rm{M}}}} \hat X{(t)_{{\rm{A}}j}} + \sqrt {1 - {{\eta '}_{\rm{M}}}} \hat X_{{\rm{L}}j}^{{\rm{vac}}},\\ & \hat P{(t)_{{\rm{L}}j}} = - \sqrt {{{\eta '}_{\rm{M}}}} \hat P{(t)_{{\rm{A}}j}} + \sqrt {1 - {{\eta '}_{\rm{M}}}} \hat P_{{\rm{L}}j}^{{\rm{vac}}}, \end{split}$(22)

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    $\begin{split} I{(t)_{{\rm{L}}1}} \!=\; & \left\langle {{{\text{δ}}^2}(\hat X{{(t)}_{{\rm{L}}2}} \!-\! \hat X{{(t)}_{{\rm{L}}3}})} \right\rangle \Big/2 \\ & + \left\langle \!{{{\text{δ}}^2}({{g'}_{{\rm{L}}1}}\hat P{{(0)}_{{\rm{L}}1}} \!+\! \hat P{{(0)}_{{\rm{L}}2}} \!+\! \hat P{{(0)}_{{\rm{L}}3}})} \!\right\rangle \!\Big/2 \geqslant \!1, \\ I{(t)_{{\rm{L}}2}} \!=\; & \left\langle {{{\text{δ}}^2}(\hat X{{(t)}_{{\rm{L}}1}} - \hat X{{(t)}_{{\rm{L}}3}})} \right\rangle \Big/2 \\ & + \left\langle \!{{{\text{δ}}^2}(\hat P{{(0)}_{{\rm{L}}1}} \!+\! {{g'}_{{\rm{L}}2}}\hat P{{(0)}_{{\rm{L}}2}} \!+\! \hat P{{(0)}_{{\rm{L}}3}})} \!\right\rangle \!\Big/2 \geqslant \!1, \\ I{(t)_{{\rm{L}}3}} \!=\;& \left\langle {{{\text{δ}}^2}(\hat X{{(t)}_{{\rm{L}}1}} \!-\! \hat X{{(t)}_{{\rm{L}}2}})} \right\rangle \Big/2 \\ & + \left\langle \!{{{\text{δ}}^2}(\hat P{{(0)}_{{\rm{L}}1}} \!+\! \hat P{{(0)}_{{\rm{L}}2}} + {{g'}_{{\rm{L}}3}}\hat P{{(0)}_{{\rm{L}}3}})} \!\right\rangle \!\Big/2 \geqslant \!1. \end{split} $(23)

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    Yan-Hong Liu, Liang Wu, Zhi-Hui Yan, Xiao-Jun Jia, Kun-Chi Peng. Deterministic quantum entanglement among multiple quantum nodes[J]. Acta Physica Sinica, 2019, 68(3): 034202-1
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