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基于级联四波混频过程的量子导引

翟淑琴 康晓兰 刘奎

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基于级联四波混频过程的量子导引

翟淑琴, 康晓兰, 刘奎

Quantum steering based on cascaded four-wave mixing processes

Zhai Shu-Qin, Kang Xiao-Lan, Liu Kui
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  • 多组份量子导引是一种重要的量子资源, 是安全量子通信网络的基础. 本文设计了串联级联四波混频和混合级联四波混频两种不同的方案, 并基于这两种方案分别产生了三组份量子导引和五组份量子导引, 通过构建系统量子态的协方差矩阵, 理论研究了两种方案产生不同模式组合间的导引参数随四波混频过程振幅增益的变化. 结果表明, 利用这两种方案可以实现多种类型的量子导引, 这一结果不仅有助于理解量子导引在多组份系统的分布而且在实际的安全量子通信网络中具有重要的意义.
    Multipartite quantum steering is an important quantum resource and the basis of secure quantum communication network. Multipartite quantum steering can be generated by beam splitter networks, optical frequency comb systems and nonlinear processes. Different types of quantum steering will be produced by different projects. In this paper, we design two different schemes, i.e. series cascaded four-wave mixing and hybrid cascaded four-wave mixing, and based on these two schemes tripartite quantum steering and quinquepartite quantum steering are generated respectively. The steering characters among different users are quantified based on the covariance matrix. In theory, we investigate steering parameters among different modes created by two schemes versus the amplitude gain of four-wave mixing process. We find that one mode can steer the other two modes separately, but the other two modes cannot steer the one mode simultaneously. By comparing the steering characters of joint multimodes to a certain single mode with the individual mode to the single mode respectively, it can be seen that the steerability of the former is stronger than the latter in the whole gain region, and there exists only the steering of joint multimodes to a single mode in the partial gain region. More importantly, the steerability of joint multimodes to a single mode can be enhanced with the quantity of joint multimodes increasing. The results show that multiple types of quantum steering can be realized by using these two schemes, which are helpful in understanding the distribution of quantum steering in multipartite system and have important significance in practical secure quantum communication and quantum secret sharing.
      通信作者: 翟淑琴, xiaozhai@sxu.edu.cn
    • 基金项目: 山西省自然科学基金(批准号: 201801D121121)、国家自然科学基金(批准号: 12074233, 91536222, 11674205)、国家重点基础研究发展计划(批准号: 2016YFA0301404)、2017年山西省高等学校教学改革创新项目(批准号: J2017006)和2020年山西省研究生教育改革研究课题(批准号: 2020YJJG023)资助的课题
      Corresponding author: Zhai Shu-Qin, xiaozhai@sxu.edu.cn
    • Funds: Project supported by the Natural Science Foundation of Shanxi Province, China (Grant No. 201801D121121), the National Natural Science Foundation of China (Grant Nos. 12074233, 91536222, 11674205), the National Basic Research Program of China (Grant No. 2016YFA0301404), the Higher Education Reform and Innovation Project of Shanxi Province, China (Grant No. J2017006), and the Postgraduate Education Reform Research Project of Shanxi Province, China (Grant No. 2020YJJG023)
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    Jones S J, Wiseman H M, Doherty A C 2007 Phys. Rev. A 76 052116Google Scholar

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    Cavalcanti E G, Jones S J, Wiseman H M, Reid M D 2009 Phys. Rev. A 80 032112Google Scholar

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    Cavalcanti D, Skrzypczyk P 2017 Rep. Prog. Phys. 80 024001Google Scholar

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    Bowles J, Vértesi T, Quintino M T, Brunner N 2014 Phys. Rev. Lett. 112 200402Google Scholar

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    He Q Y, Gong Q H, Reid M D 2015 Phys. Rev. Lett. 114 060402Google Scholar

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    Händchen V, Eberle T, Steinlechner S, Samblowski A, Franz T, Werner R F, Schnabel R 2012 Nat. Photonics 6 596Google Scholar

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    Wollmann S, Walk N, Bennet A J, Wiseman H M, Pryde G J 2016 Phys. Rev. Lett. 116 160403Google Scholar

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    Sun K, Ye X J, Xu J S, Xu X Y, Tang J S, Wu Y C, Chen J L, Li C F, Guo G C 2016 Phys. Rev. Lett. 116 160404Google Scholar

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    Branciard C, Cavalcanti E G, Walborn S P, Scarani V, Wiseman H M 2012 Phys. Rev. A 85 010301(RGoogle Scholar

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    Walk N, Hosseini S, Geng J, Thearle O, Haw J Y, Armstrong S, Assad S M, Janousek J, Ralph T C, Symulet T, Wiseman H M, Lam P K 2016 Optica 3 634Google Scholar

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    Kogias I, Xiang Y, He Q Y, Adesso G 2017 Phys. Rev. A 95 012315Google Scholar

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    He Q Y, Rosales-Zárate L, Adesso G, Reid M D 2015 Phys. Rev. Lett. 115 180502Google Scholar

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    Chiu C Y, Lambert N, Liao T L, Nori F, Li C M 2016 npj Quantum Inf. 2 16020Google Scholar

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    Uola R, Costa A C S, Nguyen H C, Gühne O 2020 Rev. Mod. Phys. 92 015001Google Scholar

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    Xiang Y, Kogias I, Adesso G, He Q Y 2017 Phys. Rev. A 95 010101(RGoogle Scholar

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    Deng X W, Xiang Y, Tian C X, Adesso G, He Q Y, Gong Q H, Su X L, Xie C D, Peng K C 2017 Phys. Rev. Lett. 118 230501Google Scholar

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    Armstrong S, Meng W, Teh R Y, Gong Q H, He Q Y, Janousek J, Bachor H A, Reid M D, Ping K L 2015 Nat. Phys. 11 167Google Scholar

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    Cai Y, Xiang Y, Liu Y, He Q Y, Treps N 2020 Phys. Rev. Res. 2 032046(R)

    [23]

    Qin Z Z, Deng X W, Tian C X, Wang M H, Su X L, Xie C D, Peng K C 2017 Phys. Rev. A 95 052114Google Scholar

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    李思瑾 2020 博士学位论文(上海: 华东师范大学)

    Li S J 2020 Ph. D. Dissertation (Shanghai: East China Normal University) (in Chinese)

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    McCormick C F, Boyer V, Arimondo E, Lett P D 2007 Opt. Lett. 32 178Google Scholar

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    Boyer V, Marino A M, Lett P D 2008 Phys. Rev. Lett. 100 143601Google Scholar

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    Ding D S, Zhang W, Zhou Z Y, Shi S, Shi B S, Guo G C 2015 Nat. Photonics 9 332Google Scholar

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    Shu C, Guo X X, Chen P, Loy M M T, Du S W 2015 Phys. Rev. A 91 043820Google Scholar

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    Wang Y F, Li J F, Zhang S C, Su K Y, Zhou Y R, Liao K Y, Du S W, Yan H, Zhu S L 2019 Nat. Photonics 13 346Google Scholar

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    Kogias I, Lee A R, Ragy S, Adesso G 2015 Phys. Rev. Lett. 114 060403Google Scholar

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    Ji S W, Kim M S, Nha H 2015 J. Phys. A 48 135301Google Scholar

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    Reid M D 2013 Phys. Rev. A 88 062108Google Scholar

  • 图 1  (a)串联级联四波混频产生三组份导引示意图. ${\hat a_{{\rm{s0}}}}$是信号光注入; ${\hat a_{{\rm{v1}}}}$, ${\hat a_{{\rm{v2}}}}$是真空模; ${\rm{Pump}}$是泵浦光注入.(b)混合级联四波混频产生五组份导引示意图. ${\hat a_{{\rm{s0}}}}$是信号光注入; ${\hat a_{{\rm{v1}}}}$, ${\hat a_{{\rm{v2}}}}$, ${\hat a_{{\rm{v3}}}}$${\hat a_{{\rm{v4}}}}$是真空模; ${\rm{Pump}}$是泵浦光注入; ${G_i}$为相应的四波混频过程的振幅增益

    Fig. 1.  (a) Schematic of generating tripartite steering using series four-wave mixing (FWM)processes.${\hat a_{{\rm{s0}}}}$ is the seed input;${\hat a_{{\rm{v1}}}}$ and ${\hat a_{{\rm{v2}}}}$ are the vacuum modes;${\rm{Pump}}$ is the pump input.(b)Schematic of generating quinquepartite steering using hybrid cascaded FWM processes.${\hat a_{{\rm{s0}}}}$ is the seed input;${\hat a_{{\rm{v1}}}}$, ${\hat a_{{\rm{v2}}}}$, ${\hat a_{{\rm{v3}}}}$ and ${\hat a_{{\rm{v4}}}}$ are the vacuum modes;${\rm{Pump}}$ is the pump input;${G_i}$ is the amplitude gain of the corresponding FWM processes.

    图 2  (a) $ {G_2} = 1.5$时(1 + 1)型导引参数随$ {G_1}$的变化; (b) $ {G_1} = 1.5$时(1 + 1)型导引参数随$ {G_2}$的变化

    Fig. 2.  (a) The (1 + 1)-type steering parameters versus with $ {G_1}$ for fixed $ {G_2} = 1.5$; (b) the (1 + 1)-type steering parameters versus with $ {G_2}$ for fixed $ {G_1} = 1.5$.

    图 3  (a) ${G_2} = 1.5$时, (2 + 1)型导引参数随${G_1}$的变化; (b) ${G_2} = 1.5$时, (1 + 2)型导引参数随${G_1}$的变化; (c) ${G_1} = 1.5$时, (2 + 1)型导引参数随${G_2}$的变化; (d) ${G_1} = 1.5$时, (1 + 2)型导引参数随${G_2}$的变化

    Fig. 3.  (a) The (2 + 1)-type steering parameter versus with ${G_1}$ for fixed ${G_2} = 1.5$; (b) the (1 + 2)-type steering parameter versus with ${G_1}$ for fixed ${G_2} = 1.5$; (c) the (2 + 1)-type steering parameter versus with ${G_2}$ for fixed ${G_1} = 1.5$; (d) the (1 + 2)-type steering parameter versus with ${G_2}$ for fixed ${G_1} = 1.5$.

    图 4  ${G_1} = {G_2} = {G_3} = 1.5$时, (2 + 1)型导引参数随${G_4}$的变化 (a) $\left( {{C_2}{D_2}} \right)$联合导引${E_2}$以及${C_2}$, ${D_2}$单独对${E_2}$的导引; (b)多种类型两模联合$ \left({A}_{2}{C}_{2}, {A}_{2}{D}_{2}, {B}_{2}{C}_{2}, {B}_{2}{D}_{2}\right)$导引${E_2}$

    Fig. 4.  The (2 + 1)-type steering parameters versus with ${G_4}$ for fixed ${G_1} = 1.5$, ${G_2} = 1.5$, ${G_3} = 1.5$: (a) ${E_2}$ can be steered by $\left( {{C_2}{D_2}} \right)$ jointly and ${E_2}$ can be steered by ${C_2}$, ${D_2}$ individually; (b) ${E_2}$ can be steered by different modes combination $ \left({A}_{2}{C}_{2}, {A}_{2}{D}_{2}, {B}_{2}{C}_{2}, {B}_{2}{D}_{2}\right)$ jointly.

    图 5  ${G_1} = {G_2} = {G_3} = 1.5$时, (1 + 2)型导引参数随${G_4}$的变化 (a) ${E_2}$${A_2}$, ${D_2}$的导引以及${E_2}$$\left( {{A_2}{D_2}} \right)$联合的导引; (b) ${E_2}$${A_2}$, ${B_2}$的导引以及${E_2}$$\left( {{A_2}{B_2}} \right)$联合的导引

    Fig. 5.  The (1 + 2)-type steering parameters versus with ${G_4}$ for fixed ${G_1} = 1.5$, ${G_2} = 1.5$, ${G_3} = 1.5$: (a) The steering from mode ${E_2}$ to individual ${A_2}$, ${D_2}$ and the group of them; (b) the steering from mode ${E_2}$ to individual ${A_2}$, ${B_2}$ and the group of them.

    图 6  ${G_1} = {G_2} = {G_3} = 1.5$时, (3 + 1)型、(1 + 3)型、(4 + 1)型以及(1 + 4)型导引参数随${G_4}$的变化关系 (a)$\left( {{A_2}{C_2}{D_2}} \right)$联合导引${E_2}$以及${A_2}$, ${C_2}$, ${D_2}$单独对${E_2}$的导引; (b)${E_2}$${A_2}$, ${C_2}$, ${D_2}$的导引以及${E_2}$$\left( {{A_2}{C_2}{D_2}} \right)$联合的导引; (c)$\left( {{A_2}{B_2}{C_2}{D_2}} \right)$联合导引${E_2}$以及${A_2}$, ${B_2}$, ${C_2}$, ${D_2}$单独对${E_2}$的导引; (d)${E_2}$${A_2}$, ${B_2}$, ${C_2}$, ${D_2}$的导引以及${E_2}$$\left( {{A_2}{B_2}{C_2}{D_2}} \right)$联合的导引

    Fig. 6.  The (3 + 1)-type、(1 + 3)-type、(4 + 1)-type and(1 + 4)-type steering parameters versus with ${G_4}$ for fixed ${G_1} = 1.5$, ${G_2} = 1.5$, ${G_3} = 1.5$: (a)${E_2}$ can be steered by $\left( {{A_2}{C_2}{D_2}} \right)$ jointly and ${E_2}$ can be steered by ${C_2}$, ${D_2}$ individually; (b) the steering from mode ${E_2}$ to individual ${A_2}$, ${C_2}$, ${D_2}$ and the group of them; (c) ${E_2}$ can be steered by $\left( {{A_2}{B_2}{C_2}{D_2}} \right)$ jointly and ${E_2}$ can be steered by ${C_2}$, ${D_2}$ individually; (d) The steering from mode ${E_2}$ to individual ${A_2}$, ${B_2}$, ${C_2}$, ${D_2}$ and the group of them.

    图 7  ${G_1} = {G_2} = {G_3} = 1.5$时, 多模联合对${E_2}$的导引参数随${G_4}$变化的比较

    Fig. 7.  Comparison of steering parameters of multimode combination for ${E_2}$ versus with ${G_4}$ for fixed ${G_1} = 1.5$, ${G_2} = 1.5$, ${G_3} = 1.5$.

  • [1]

    Bell J S 1964 Physics 1 195Google Scholar

    [2]

    Horodecki R, Horodecki P, Horodecki M, Horodecki K 2009 Rev. Mod. Phys. 81 865Google Scholar

    [3]

    Jones S J, Wiseman H M, Doherty A C 2007 Phys. Rev. A 76 052116Google Scholar

    [4]

    Cavalcanti E G, Jones S J, Wiseman H M, Reid M D 2009 Phys. Rev. A 80 032112Google Scholar

    [5]

    Cavalcanti D, Skrzypczyk P 2017 Rep. Prog. Phys. 80 024001Google Scholar

    [6]

    Bowles J, Vértesi T, Quintino M T, Brunner N 2014 Phys. Rev. Lett. 112 200402Google Scholar

    [7]

    He Q Y, Gong Q H, Reid M D 2015 Phys. Rev. Lett. 114 060402Google Scholar

    [8]

    Händchen V, Eberle T, Steinlechner S, Samblowski A, Franz T, Werner R F, Schnabel R 2012 Nat. Photonics 6 596Google Scholar

    [9]

    Wollmann S, Walk N, Bennet A J, Wiseman H M, Pryde G J 2016 Phys. Rev. Lett. 116 160403Google Scholar

    [10]

    Sun K, Ye X J, Xu J S, Xu X Y, Tang J S, Wu Y C, Chen J L, Li C F, Guo G C 2016 Phys. Rev. Lett. 116 160404Google Scholar

    [11]

    Branciard C, Cavalcanti E G, Walborn S P, Scarani V, Wiseman H M 2012 Phys. Rev. A 85 010301(RGoogle Scholar

    [12]

    Walk N, Hosseini S, Geng J, Thearle O, Haw J Y, Armstrong S, Assad S M, Janousek J, Ralph T C, Symulet T, Wiseman H M, Lam P K 2016 Optica 3 634Google Scholar

    [13]

    Kogias I, Xiang Y, He Q Y, Adesso G 2017 Phys. Rev. A 95 012315Google Scholar

    [14]

    Reid M D 2013 Phys. Rev. A 88 062338Google Scholar

    [15]

    He Q Y, Rosales-Zárate L, Adesso G, Reid M D 2015 Phys. Rev. Lett. 115 180502Google Scholar

    [16]

    Chiu C Y, Lambert N, Liao T L, Nori F, Li C M 2016 npj Quantum Inf. 2 16020Google Scholar

    [17]

    Piani M, Watrous J 2015 Phys. Rev. Lett. 114 060404Google Scholar

    [18]

    Uola R, Costa A C S, Nguyen H C, Gühne O 2020 Rev. Mod. Phys. 92 015001Google Scholar

    [19]

    Xiang Y, Kogias I, Adesso G, He Q Y 2017 Phys. Rev. A 95 010101(RGoogle Scholar

    [20]

    Deng X W, Xiang Y, Tian C X, Adesso G, He Q Y, Gong Q H, Su X L, Xie C D, Peng K C 2017 Phys. Rev. Lett. 118 230501Google Scholar

    [21]

    Armstrong S, Meng W, Teh R Y, Gong Q H, He Q Y, Janousek J, Bachor H A, Reid M D, Ping K L 2015 Nat. Phys. 11 167Google Scholar

    [22]

    Cai Y, Xiang Y, Liu Y, He Q Y, Treps N 2020 Phys. Rev. Res. 2 032046(R)

    [23]

    Qin Z Z, Deng X W, Tian C X, Wang M H, Su X L, Xie C D, Peng K C 2017 Phys. Rev. A 95 052114Google Scholar

    [24]

    李思瑾 2020 博士学位论文(上海: 华东师范大学)

    Li S J 2020 Ph. D. Dissertation (Shanghai: East China Normal University) (in Chinese)

    [25]

    McCormick C F, Boyer V, Arimondo E, Lett P D 2007 Opt. Lett. 32 178Google Scholar

    [26]

    Boyer V, Marino A M, Lett P D 2008 Phys. Rev. Lett. 100 143601Google Scholar

    [27]

    Ding D S, Zhang W, Zhou Z Y, Shi S, Shi B S, Guo G C 2015 Nat. Photonics 9 332Google Scholar

    [28]

    Shu C, Guo X X, Chen P, Loy M M T, Du S W 2015 Phys. Rev. A 91 043820Google Scholar

    [29]

    Wang Y F, Li J F, Zhang S C, Su K Y, Zhou Y R, Liao K Y, Du S W, Yan H, Zhu S L 2019 Nat. Photonics 13 346Google Scholar

    [30]

    Kogias I, Lee A R, Ragy S, Adesso G 2015 Phys. Rev. Lett. 114 060403Google Scholar

    [31]

    Ji S W, Kim M S, Nha H 2015 J. Phys. A 48 135301Google Scholar

    [32]

    Reid M D 2013 Phys. Rev. A 88 062108Google Scholar

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出版历程
  • 收稿日期:  2020-11-24
  • 修回日期:  2021-04-09
  • 上网日期:  2021-06-07
  • 刊出日期:  2021-08-20

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