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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

翟淑琴, 康晓兰, 刘奎. 基于级联四波混频过程的量子导引. 物理学报, 2021, 70(16): 160301. doi: 10.7498/aps.70.20201981
引用本文: 翟淑琴, 康晓兰, 刘奎. 基于级联四波混频过程的量子导引. 物理学报, 2021, 70(16): 160301. doi: 10.7498/aps.70.20201981
Zhai Shu-Qin, Kang Xiao-Lan, Liu Kui. Quantum steering based on cascaded four-wave mixing processes. Acta Phys. Sin., 2021, 70(16): 160301. doi: 10.7498/aps.70.20201981
Citation: Zhai Shu-Qin, Kang Xiao-Lan, Liu Kui. Quantum steering based on cascaded four-wave mixing processes. Acta Phys. Sin., 2021, 70(16): 160301. doi: 10.7498/aps.70.20201981

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.
      PACS:
      03.65.Ud(Entanglement and quantum nonlocality)
      03.67.Mn(Entanglement measures, witnesses, and other characterizations)
      03.67.Hk(Quantum communication)
      42.50.Lc(Quantum fluctuations, quantum noise, and quantum jumps)
      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)

    在量子光学领域, 量子关联突破了经典力学局域性的限制, 能描绘远距离量子系统间的相互作用, 是量子信息处理过程中一种重要的资源. 量子非局域性有三种表现形式: 贝尔非局域性[1]、量子导引以及量子纠缠[2], 其中量子导引描绘的是共享一对纠缠粒子的两体系统中的一方通过对他手中的粒子进行测量可以推测另一方粒子所处的状态[3-5]. 量子导引是介于贝尔非局域与量子纠缠之间的一种特殊的量子关联, 具有天然的不对称性[6-10], 这对单端设备无关的安全量子通信[11-18]、量子秘密共享[19]等的实现是非常重要的.

    多组份量子导引可以通过分束器网络、光学频率梳系统以及非线性过程产生. 何琼毅研究小组[20,21]利用线性光学分束器网络实现了连续变量四模以及八模量子导引、利用光学频率梳系统实现了十六模量子导引[22]. 山西大学光电研究所利用两个非简并光学参量放大器(NOPA)产生了连续变量四组份方形cluster纠缠态光场, 并对输出模式间的导引特性以及单配性关系进行了实验验证[20], 他们在利用NOPA产生双模压缩态的基础上通过引入损耗, 实验验证了量子导引单向性的操控[23]. 产生量子导引的另一个有效的方法是基于铷原子的四波混频过程. 华东师大的李思谨[24]应用四波混频过程实现了多光束纠缠态的制备, 理论上应用四波混频实现了真空场中的三光束纠缠和四光束纠缠, 并且实验上应用级联四波混频过程制备出了三光束纠缠态以及四光束纠缠态. 四波混频效应具有强的非线性特性以及空间多模等特性[25], 在没有腔的情况下, 不发生模式选择, 可以产生空间多模双光束[26]. 此外, 原子介质可以作为一个量子存储器有效地存储窄带宽的纠缠光子[27-29]. 因此, 利用级联四波混频过程制备量子多光束纠缠态具有广阔的应用前景. 本文基于四波混频过程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, 分别实现了三组份以及五组份量子导引, 虽然后者与前者相比输出光束只增加了两个, 但丰富了量子导引的种类, 这对安全量子网络的构建具有重要的意义.

    本文的结构设计如下, 第二部分介绍了量子导引的判定依据, 第三部分基于设计的模型理论分析了输出光束的协方差矩阵关系式, 第四部分研究了两种模型输出光束间的导引特性并进行了分析, 第五部分是对本文研究工作的总结.

    通过量子态的协方差矩阵可以量化输出光模之间的导引特性. 两组份系统A (对应用户Alice)及B (对应用户Bob), 其中A中有m个模式, B中有n个模式, 可以用(mA+nB)来表示, 则A, B系统的高斯量子态的协方差矩阵可以表示为

    σAB=(ACCTB). (1)

    协方差矩阵中的矩阵元定义为 σij=ˆξiˆξj+ˆξjˆξi/2ˆξiˆξj, 其中ˆξi为光模的正交算符, 包括正交振幅和正交相位, 正交振幅分量定义为ˆX=ˆa+ˆa, 正交相位分量定义为ˆP=(ˆaˆa)/i, 其中ˆa, ˆa分别为光模的湮灭算符和产生算符.

    系统A(Alice)对系统B(Bob)的导引能力可以通过下式进行量化[30]:

    GAB(σAB)=max (2)

    式中 {{\boldsymbol{\sigma}} _{{\rm{AB}}}} 表示两体量子态的协方差矩阵, \bar \nu _j^{{\rm{AB}}\backslash {\rm{A}}} ( {j \!=\! 1, \cdots, {n_{\rm{B}}}} )表示矩阵{\bar {\sigma} _{{\rm{AB}}\backslash {\rm{A}}}} \!=\! {\boldsymbol{B}} \!-\! {{\boldsymbol{C}}^{\rm{T}}}{{\boldsymbol{A}}^{ - 1}}{\boldsymbol{C}}的辛本征值. 当{{\cal{G}}^{{\rm{A}} \to {\rm{B}}}}({{\boldsymbol{\sigma}} _{{\rm{AB}}}}) > 0时, 系统{\rm{A}}\left( {{\rm{Alice}}} \right)可以导引系统{\rm{B}}\left( {{\rm{Bob}}} \right). 同样, 系统{\rm{B}}\left( {{\rm{Bob}}} \right)对系统{\rm{A}}\left( {{\rm{Alice}}} \right)的导引能力可以通过式子

    {\cal{G}}^{{\rm{B}}\to {\rm{A}}}({{\boldsymbol{\sigma}} }_{\rm{AB}})= {\max\!\!\left\{\! {{0},\! -\!\!{\displaystyle \sum\limits _{j:{\overline{\nu }}_{j}^{{\rm{AB}}\backslash {\rm{B}}} \! < 1}\!{\rm{In}}({\overline{\nu }}_{j}^{{\rm{AB}}\backslash {\rm{B}}}{)}}}\!\! \right\}}

    来量化. 当{{\cal{G}}^{{\rm{A}} \to {\rm{B}}}}\!({{\boldsymbol{\sigma}} _{{\rm{AB}}}}) \!\! > 0, 且{{\cal{G}}^{{\rm{B}} \to {\rm{A}}}}({{\boldsymbol{\sigma}} _{{\rm{AB}}}}) > {\rm{0}}, 但{{\cal{G}}^{{\rm{A}} \to {\rm{B}}}}({{\boldsymbol{\sigma}} _{{\rm{AB}}}}) \ne {{\cal{G}}^{{\rm{B}} \to {\rm{A}}}}({{\boldsymbol{\sigma}} _{{\rm{AB}}}}), 此时, 系统A可以导引系统B, 系统B也可以导引系统A, 但两者间的导引能力不同, 体现出量子导引的不对称性; 当{{\cal{G}}^{{\rm{A}} \to {\rm{B}}}}({{\boldsymbol{\sigma}} _{{\rm{AB}}}}) > 0, {{\cal{G}}^{{\rm{B}} \to {\rm{A}}}}({{\boldsymbol{\sigma}} _{{\rm{AB}}}}) = 0, 系统A可以导引系统B, 但系统B不可以导引系统A, 或当{{\cal{G}}^{{\rm{B}} \to {\rm{A}}}}({{\boldsymbol{\sigma}} _{{\rm{AB}}}}) > {\rm{0}}, {{\cal{G}}^{{\rm{A}} \to {\rm{B}}}}({{\boldsymbol{\sigma}} _{{\rm{AB}}}}){\rm{ = }}0, 系统B可以导引系统A, 但系统A不可以导引系统B, 此时体现出量子导引的单向性.

    在四波混频过程中, 泵浦光和信号光注入到铷原子蒸汽池中, 在泵浦光的作用下发生四波混频效应产生一对纠缠关联光束, 这一过程满足能量守恒和动量守恒, 依据四波混频过程中的相互作用哈密顿量以及光模在海森堡表象下的运动方程, 可以写出输出光模与输入光模之间的关系式.

    图1(a)为两个四波混频串联级联产生三组份量子导引的方案图. 如图所示, 信号光和泵浦光注入到第一个铷原子蒸汽池中, 经过第一个四波混频过程后产生一对纠缠光束({\hat{A}}_{1}, {\hat{a}}_{\rm{s1}}), 第一个四波混频对应的输出输入光模满足下式:

    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; ${\hat a_{{\rm{v1}}}}$, ${\hat a_{{\rm{v2}}}}$, ${\hat a_{{\rm{v3}}}}$和${\hat a_{{\rm{v4}}}}$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\r\nFig. 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 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.
    {\hat a_{{\rm{s1}}}} = {G_1}{\hat a_{{\rm{s0}}}} + {g_1}{\hat a^\dagger }_{{\rm{v1}}},\;\;\;\;\;{\hat A_1} = {G_1}{\hat a_{{\rm{v1}}}} + {g_1}{\hat a^\dagger }_{{\rm{s0}}}, (3)

    其中{G_1}为第一个四波混频过程的振幅增益, {G_1}{g_1}满足关系式{g_1} = \sqrt {G_1^2 - 1} . 接着, {\hat a_{{\rm{s1}}}}作为第二个四波混频过程的信号光注入到第二个铷原子蒸汽池后产生一对纠缠光束({\hat{B}}_{1}, {\hat{C}}_{1}). 用同样的方法可以写出第二个四波混频过程所对应的输出光模{\hat{B}}_{1}, {\hat{C}}_{1}的表达式.

    最终产生的三个输出模式与输入模式的关系可以写作\hat {\boldsymbol{\xi}} _1^{{\rm{out}}} = {{\boldsymbol{U}}_1}\hat {\boldsymbol{\xi}} _1^{{\rm{in}}}, 其中

    \begin{split} & {\hat{{\boldsymbol{\xi}} }}_{1}^{\rm{out}} \equiv \Big({\hat{X}}_{1}^{A}, {\hat{P}}_{1}^{A}, {\hat{X}}_{1}^{B}, {\hat{P}}_{1}^{B}, {\hat{X}}_{1}^{C}, {\hat{P}}_{1}^{C}\Big)^{\rm{T}}, \\ & \hat {\boldsymbol{\xi}} _1^{{\rm{in}}} \equiv {\left( {{{\hat X}_{{\rm{s0}}}}, {{\hat P}_{{\rm{s0}}}}, {{\hat X}_{{\rm{v1}}}}, {{\hat P}_{{\rm{v1}}}}, {{\hat X}_{{\rm{v2}}}}, {{\hat P}_{{\rm{v2}}}}} \right)^{\rm{T}}}. \end{split}
    {{\boldsymbol{U}}_1} = \!\left(\!\! {\begin{array}{*{20}{c}} {{g_1}}&0&{{G_1}}&0&0&0 \\ 0&{ - {g_1}}&0&{{G_1}}&0&0 \\ {{g_2}{G_1}}&0&{{g_1}{g_2}}&0&{{G_2}}&0 \\ 0&{ - {g_2}{G_1}}&0&{{g_1}{g_2}}&0&{{G_2}} \\ {{G_1}{G_2}}&0&{{g_1}{G_2}}&0&{{g_2}}&0 \\ 0&{{G_1}{G_2}}&0&{ - {g_1}{G_2}}&0&{ - {g_2}} \end{array}} \!\! \right), (4)

    图1(b)为四个四波混频过程混合级联产生五组份量子导引的方案图. 如图所示, 信号光和泵浦光注入第一个铷原子蒸汽池中, 经过第一个四波混频过程后产生一对纠缠光束({\hat{a}}_{\rm{i1}}, {\hat{a}}_{\rm{s1}}), 它们又分别作为第二个和第三个四波混频过程的信号光注入, 产生两对纠缠光束({\hat{A}}_{2}, {\hat{B}}_{2})({\hat{C}}_{2}, {\hat{a}}_{\rm{s2}}), {\hat a_{{\rm{s2}}}}接着作为第四个四波混频过程的信号光注入到第四个铷原子蒸汽池后产生纠缠光束({\hat{D}}_{2}, {\hat{E}}_{2}). 最终产生五模输出({\hat{A}}_{2}, {\hat{B}}_{2}, {\hat{C}}_{2}, {\hat{D}}_{2}, {\hat{E}}_{2}), 基于上面的理论分析, 该模型输入输出模的关系可以写作\hat {\boldsymbol{\xi}} _2^{{\rm{out}}} = {{\boldsymbol{U}}_2}\hat {\boldsymbol{\xi}} _2^{{\rm{in}}}, 其中

    \begin{split} & {\hat{{\boldsymbol{\xi}} }}_{2}^{\rm{out}} \equiv \Big({\hat{X}}_{2}^{A}, {\hat{P}}_{2}^{A}, {\hat{X}}_{2}^{B}, {\hat{P}}_{2}^{B}, {\hat{X}}_{2}^{C}, {\hat{P}}_{2}^{C}, {\hat{X}}_{2}^{D}, {\hat{P}}_{2}^{D}, {\hat{X}}_{2}^{E}, {\hat{P}}_{2}^{E}\Big)^{\rm{T}},\\ & \hat {\boldsymbol{\xi}} _2^{{\rm{in}}} \equiv \Big( {{\hat X}_{{\rm{s0}}}}, {{\hat P}_{{\rm{s0}}}}, {{\hat X}_{{\rm{v1}}}}, {{\hat P}_{{\rm{v1}}}}, {{\hat X}_{{\rm{v2}}}}, {{\hat P}_{{\rm{v2}}}}, {{\hat X}_{{\rm{v3}}}}, {{\hat P}_{{\rm{v3}}}}, {{\hat X}_{{\rm{v4}}}}, {{\hat P}_{{\rm{v4}}}} \Big)^{\rm{T}}. \end{split}

    {{\boldsymbol{U}}_2} = \left( {\begin{array}{*{20}{c}} {{g_1}{G_2}}&0&{{G_1}{G_2}}&0&{{g_2}}&0&0&0&0&0 \\ 0&{ - {g_1}{G_2}}&0&{{G_1}{G_2}}&0&{ - {g_2}}&0&0&0&0 \\ {{g_1}{g_2}}&0&{{g_2}{G_1}}&0&{{G_2}}&0&0&0&0&0 \\ 0&{{g_1}{g_2}}&0&{ - {g_2}{G_1}}&0&{{G_2}}&0&0&0&0 \\ {{g_3}{G_1}}&0&{{g_1}{g_3}}&0&0&0&{{G_3}}&0&0&0 \\ 0&{ - {g_3}{G_1}}&0&{{g_1}{g_3}}&0&0&0&{{G_3}}&0&0 \\ {{g_4}{G_1}{G_3}}&0&{{g_1}{g_4}{G_3}}&0&0&0&{{g_3}{g_4}}&0&{{G_4}}&0 \\ 0&{ - {g_4}{G_1}{G_3}}&0&{{g_1}{g_4}{G_3}}&0&0&0&{{g_3}{g_4}}&0&{{G_4}} \\ {{G_1}{G_3}{G_4}}&0&{{g_1}{G_3}{G_4}}&0&0&0&{{g_3}{G_4}}&0&{{g_4}}&0 \\ 0&{{G_1}{G_3}{G_4}}&0&{ - {g_1}{G_3}{G_4}}&0&0&0&{ - {g_3}{G_4}}&0&{ - {g_4}} \end{array}} \right). (5)

    根据串联级联四波混频方案, 利用导引判据(2)可以研究不同输出模组合间的量子导引特性. 图2为(1 + 1)型导引, 在图2(a)中, 取{G_2} = 1.5, 研究了{A_1}, {C_1}之间以及{B_1}, {C_1}之间导引参数随{G_1}的变化, 在图中的整个取值范围内{B_1}, {C_1}之间的导引是双向不对称的, {A_1}, {C_1}之间的导引是单向的. 在图2(b)中, 取{G_1} = 1.5, 研究了{A_1}, {C_1}之间以及{B_1}, {C_1}之间的导引参数随{G_2}的变化. 由图2(b)可以看出, 当1 < {G_2} < 1.25时, 可以实现{A_1}, {C_1}之间不对称双向导引, 此时, {C_1} \to {B_1}单向导引; 当{G_2} > 1.25时, 可以实现{B_1}, {C_1}之间不对称双向导引, 此时, {C_1} \to {A_1}单向导引, 且{B_1}, {C_1}之间的导引参数随{G_2}的增大而增大, 这是因为增大第二个四波混频的振幅增益增强了{B_1}, {C_1}之间的量子关联; {A_1}, {C_1}之间的导引参数随{G_2}的增大而减小, 这是因为光束{C_1}是来自于第一个四波混频过程FWM1产生的信号光{\hat a_{{\rm{s1}}}}经过第二个四波混频过程FWM2放大产生的, 这一过程引入了额外的噪声使得{A_1}, {C_1}之间的关联减弱. 从图2(b)可以看出当{C_1}作为被导引的一方时, {{\cal{G}}^{{A_1} \to {C_1}}} > 0, {{\cal{G}}^{{B_1} \to {C_1}}} > 0不能同时成立, 即两个不同的系统不能同时导引第三个系统[31,32], 验证了量子导引的单配性关系.

    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\r\nFig. 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 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给出了(2 + 1)型及(1 + 2)型导引, 并比较了与(1 + 1)型导引的关联. 图3(a)图3(b){G_2} = 1.5时导引参数随{G_1}的变化关系, 可以看出, (2 + 1)型以及(1 + 2)型的导引能力大于(1 + 1)型的导引能力. 图3(c)图3(d){G_1} = 1.5时导引参数随{G_2}的变化关系, 同样, (2 + 1)型以及(1 + 2)型的导引能力均大于(1 + 1)型的导引能力, 这表明此种情况下多光束之间的联合量子导引优于两路单光束间的导引, 为多用户间量子网络秘密共享提供了基础.

    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\r\nFig. 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 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.

    基于方案二产生的五个输出模, 我们分析了不同模式组合与E{}_2之间的导引关系.

    两模联合对E{}_2的导引情况如图4所示. 图4(a)分析了(2 + 1)型导引, 并比较了与(1 + 1)型导引的关联, 将{C_2}{D_2} \to {E_2}, {C_2} \to {E_2}以及{D_2} \to {E_2}的导引作了对比. 可以看出, {{\cal{G}}^{{C_2}{D_2} \to {E_2}}} > {{\cal{G}}^{{C_2} \to {E_2}}}, {{\cal{G}}^{{C_2}{D_2} \to {E_2}}} > {{\cal{G}}^{{D_2} \to {E_2}}}, 且{{\cal{G}}^{{C_2} \to {E_2}}} > 0{{\cal{G}}^{{D_2} \to {E_2}}} > 0不能同时成立. 图4(b)分析了(2 + 1)型导引, 将不同的两模联合对{E_2}的导引情况作了对比. 可以看出, {{\cal{G}}^{{A_2}{C_2} \to {E_2}}} > 0{{\cal{G}}^{{B_2}{D_2} \to {E_2}}} > 0不能同时成立, {{\cal{G}}^{{A_2}{D_2} \to {E_2}}} > 0{{\cal{G}}^{{B_2}{C_2} \to {E_2}}} > 0不能同时成立, 验证了两个不同的系统不能同时导引第三个系统[31,32]. 从图4(b)同时可以看出, 经过一定的阈值条件后, {{\cal{G}}^{{A_2}{D_2} \to {E_2}}}, {{\cal{G}}^{{B_2}{D_2} \to {E_2}}}{G_4}的增大而增大, 这是因为它们的关联主要是来自于同一个四波混频FWM4产生的{D_2}{E_2}之间的关联, 因此增加第四个四波混频过程的振幅增益, 导引参数增大; {{\cal{G}}^{{A_2}{C_2} \to {E_2}}}, {{\cal{G}}^{{B_2}{C_2} \to {E_2}}}{G_4}的增大而减小, 直至零, 这是因为它们的关联主要是来自于{C_2}{E_2}的关联, 光束{E_2}是第三个四波混频过程FWM3产生的光束{\hat a_{{\rm{s2}}}}经过第四个四波混频过程FWM4放大产生的, 这一过程引入了噪声, 因此随着{G_4}的增大{C_2}{E_2}之间的关联变小.

    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}$\r\nFig. 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 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所示, 将{E_2}作为导引的一方, 研究了(1 +2)型导引, 并比较了与(1 + 1)型导引的关联. 图5(a)研究了{E_2} \to {A_2}{D_2}, {E_2} \to {A_2}以及{E_2} \to {D_2}的导引情况, 可以看出, 在整个{G_4}取值范围内, {{\cal{G}}^{{E_2} \to {A_2}}} = 0, 而{{\cal{G}}^{{E_2} \to {A_2}{D_2}}} > {{\cal{G}}^{{E_2} \to {D_2}}}, 说明{E_2}导引{A_2}{D_2}联合的能力大于对单方的导引能力. 图5(b)研究了{E_2} \to {A_2}{B_2}, {E_2} \to {A_2}以及{E_2} \to {B_2}的导引特性, 可以看出, {{\cal{G}}^{{E_2} \to {A_2}}} = 0{{\cal{G}}^{{E_2} \to {B_2}}} = 0, 但{{\cal{G}}^{{E_2} \to {A_2}{B_2}}} > 0, 意味着只有{A_2}, {B_2}互相协作才能接收到{E_2}传送的信息.

    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) ${E_2}$对${A_2}$, ${D_2}$的导引以及${E_2}$对$\left( {{A_2}{D_2}} \right)$联合的导引; (b) ${E_2}$对${A_2}$, ${B_2}$的导引以及${E_2}$对$\left( {{A_2}{B_2}} \right)$联合的导引\r\nFig. 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 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分析了(3 + 1)型, (1 + 3)型, (4 + 1)型以及(1 + 4)型的导引并比较了与(1 + 1)型的关联, 同样也可以看到多模联合对单模的导引能力大于单模对单模的导引能力. 图6(a)中, 在{G_4}的整个取值范围内, {{\cal{G}}^{{A_2}{C_2}{D_2} \to {E_2}}}{G_4}的增大而增大, 而在整个取值范围内{{\cal{G}}^{{A_2} \to {E_2}}} = 0; 当{G_4} \in \left( {1.0, 1.15} \right)时, {{\cal{G}}^{{C_2} \to {E_2}}} > 0; 当{G_4} \in \left( {1.35, 2.0} \right)时, {{\cal{G}}^{{D_2} \to {E_2}}} > 0; 当{G_4} \in \left( {1.15, 1.35} \right)时, 仅存在联合模式 ({A}_{2}{C}_{2}{D}_{2})对单模 ({E}_{2})的导引, 任何单独的模式都不能导引{E_2}({\cal{G}}^{{A}_{2}{C}_{2}{D}_{2}\to {E}_{2}} > 0,\; {\cal{G}}^{{A}_{2}\to {E}_{2}}=0, \;{\cal{G}}^{{C}_{2}\to {E}_{2}}=0, {\cal{G}}^{{D}_{2}\to {E}_{2}}= 0), 因此{G_4}在取值区间\left( {1.15, 1.35} \right), 只有{A_2}, {C_2}, {D_2}三者相互协作才能导引{E_2}. 图6(b)表明(1 + 3)型导引也大于单模间的导引. 图6(c)图6(d)中, (4 + 1)型以及(1 + 4)型的导引也大于单模对单模的导引. 在图6(c)中可以看到与图6(a)类似的现象, 此结果对多方联合量子安全通信网络的建立具有重要的参考价值.

    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)${E_2}$对${A_2}$, ${C_2}$, ${D_2}$的导引以及${E_2}$对$\left( {{A_2}{C_2}{D_2}} \right)$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)${E_2}$对${A_2}$, ${B_2}$, ${C_2}$, ${D_2}$的导引以及${E_2}$对$\left( {{A_2}{B_2}{C_2}{D_2}} \right)$联合的导引\r\nFig. 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 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分析了不同类型的多模联合对 {E_2} 的导引情况, 从图中可以看到 {{\cal{G}}^{{A_2}{B_2}{C_2}{D_2} \to {E_2}}} > {{\cal{G}}^{{A_2}{C_2}{D_2} \to {E_2}}} > {{\cal{G}}^{{B_2}{C_2}{D_2} \to {E_2}}} > {{\cal{G}}^{{C_2}{D_2} \to {E_2}}}, 即导引的一方联合的模式越多, 导引能力越强, 这说明四波混频过程产生的多光束存在着量子关联, 且出自同一个四波混频过程的两束光之间的量子关联大于不同四波混频过程产生的光束之间的量子关联. 这些导引特性对实际的安全量子通信具有重要的意义.

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

    本文基于四波混频过程提出了能够实现多组份量子导引的两种不同方案, 方案一基于串联级联四波混频过程实现三组份量子导引, 方案二通过混合级联四波混频过程实现五组份量子导引, 在实验可行性参数取值范围内对两种模型进行了理论分析, 通过对比发现: 两种方案都可以通过调节四波混频过程的振幅增益实现不对称双向量子导引以及单向量子导引; 方案一在整个取值范围内均存在两模联合与单模之间的导引以及单模与单模之间的导引, 方案二可以实现在某一取值范围内仅存在联合模式与单模之间的导引, 且通过增加四波混频的个数以及调节级联四波混频的结构可以产生更加丰富的量子网络结构, 这对于单方设备无关的安全量子通信、量子秘密共享等的实现具有一定的参考价值, 此研究结果为安全量子通信提供了可行性方案.

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    期刊类型引用(2)

    1. 桑苏玲. 单缀饰四波混频缀饰效应的频率控制. 激光杂志. 2022(04): 41-45 . 百度学术
    2. 杨鑫鑫,梁芸芸,杜培林,张静. 基于单腔光力学系统的连续变量量子导引研究. 量子光学学报. 2022(04): 311-319 . 百度学术

    其他类型引用(2)

  • 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.

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  • 期刊类型引用(2)

    1. 桑苏玲. 单缀饰四波混频缀饰效应的频率控制. 激光杂志. 2022(04): 41-45 . 百度学术
    2. 杨鑫鑫,梁芸芸,杜培林,张静. 基于单腔光力学系统的连续变量量子导引研究. 量子光学学报. 2022(04): 311-319 . 百度学术

    其他类型引用(2)

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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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