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Raman protocol-based quantum memories

Shi Bao-Sen Ding Dong-Sheng Zhang Wei Li En-Ze

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Raman protocol-based quantum memories

Shi Bao-Sen, Ding Dong-Sheng, Zhang Wei, Li En-Ze
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  • Quantum memories are indispensable for quantum communication and quantum computation, which are able to realize the storage and retrieval of a quantum state such as a single photon, entanglement, or a squeezed state on demand. Among those memories realized by different protocols, the Raman quantum memory has advantages in its broadband and high-speed properties, resulting in huge potential applications in quantum network and quantum computation. However, the realization of Raman quantum memory for a true single photon and photonic entanglement is a challenging job. In this review, after briefly introducing the main benchmarks for quantum memories, showing the state of the art, we focus on the review of the experimental progress recently achieved in storing the quantum state by Raman scheme in our group. We believe that all achievements reviewed are very hopeful in building up a high-speed quantum network.
      Corresponding author: Shi Bao-Sen, drshi@ustc.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61435011, 61275115, 61525504) and Anhui Initiative in Quantum Information Technologies, China (Grant No. AHY020200).
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  • 图 1  拉曼存储方案能级简图, 其中$\left| g \right\rangle $, $\left| s \right\rangle $$\left| e \right\rangle $分别表示基态、亚稳态和激发态; $\varDelta $表示失谐量

    Figure 1.  Simplified energy level diagram for Raman protocol. $\left| g \right\rangle $, $\left|s\right\rangle $ and $\left| e \right\rangle $ correspond to ground, metastable and excited state respectively; $\varDelta $ represents detuning.

    图 2  简化的能级图和实验装置图[29] (a)简化的SRS能级图, 态$\left| 1 \right\rangle $$\left| 2 \right\rangle $分别对应于85Rb原子的两个亚稳态能级5S1/2 (F = 3)和5S1/2(F = 2), $\left| 3 \right\rangle $$\left| 4 \right\rangle $分别对应激发态能级5P3/2(F' = 3)和5P1/2(F' = 3); 泵浦光1由外腔二极管激光器(DL100, Toptica) 产生, 波长为795 nm, 与原子跃迁5S1/2(F = 3)→5P1/2(F' = 3)蓝失谐值70 MHz; 泵浦光2来自另一个波长为780 nm的外腔二极管激光器(DL100, Toptica), 对应5S1/2(F = 2)→5P3/2(F' = 3)的原子跃迁; 泵浦光1和泵浦光2被调制成脉冲模式, 脉冲宽度分别为50和160 ns, 上升沿为30 ns; 在存储过程中泵浦1和泵浦2脉冲之间的延迟时间被设置为260 ns; 泵浦1和泵浦2的激光功率分别为0.5和4 mW; 控制光来自于与泵浦光1相同的激光器, 也对应于原子跃迁5S1/2(F = 3)到5P1/2(F' = 3), 并蓝失谐值70 MHz, 功率为12 mW; (b)实验装置简化图, MOT 2中信号1的束腰为63 ${\text{μm}}$(MOT, 磁光阱; FC, 光纤耦合器; SLM, 空间光调制器; PBS, 偏振分束器; $\lambda$/2, 半波片)

    Figure 2.  Simplified energy level diagram and experimental setup[29]. (a) Simplified energy level diagram of the SRS. The states $\left| 1 \right\rangle $ and $\left| 2 \right\rangle $ correspond to two metastable levels 5S1/2(F = 3) and 5S1/2(F = 2) of 85Rb atom respectively, $\left| 3 \right\rangle $ and $\left| 4 \right\rangle $ are the excited levels of 5P3/2(F' = 3) and 5P1/2(F' = 3) respectively. The pump 1 laser is from an external-cavity diode laser (DL100, Toptica) with the wavelength of 795 nm, and is blue-detuned to the atomic transition of 5S1/2(F = 3)→5P1/2(F' = 3) with a value of 70 MHz. The pump 2 laser is from another external-cavity diode laser (DL100, Toptica) with the wavelength of 780 nm which couples the atomic transition of 5S1/2(F = 2)→5P3/2(F' = 3). The pump 1 and pump 2 are modulated into pulse modes with a width of 50 and 160 ns respectively, and a rising edge of 30 ns. The delayed time between the pump 1 pulse and the pump 2 pulse is programmed to be 260 ns for the process of storage. The powers of pump 1 and pump 2 are 0.5 and 4 mW respectively. The coupling laser is from the same laser with pump 1 and is also blue-detuned to atomic transition of 5S1/2(F = 3)→5P1/2(F' = 3) with a value of 70 MHz, its power is about 12 mW. (b) Simplified diagram depicting the storage of entanglement of OAM state. The waist of signal 1 at MOT 2 was 63 ${\text{μm}}$. MOT, magneto-optical trap; FC, fibre coupler; SLM, spatial light modulator; PBS, polarisation beam splitter; $\lambda$/2, half-wave plate.

    图 3  存储过程中的交叉相关函数${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$[58] (a)信号光子1和信号光子2之间的交叉相关函数${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$, 泵浦1和泵浦2之间延迟时间为260 ns; (b), (c)和(d)是信号光子2与读出的信号光子1之间的时间相关函数${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right),$ 信号光子1的存储时间分别为100, 150和200 ns; (e)在没有输入信号1至MOT2的情况下收集的噪音; 所有数据均为原始数据, 无噪声校正

    Figure 3.  Cross-correlated function of ${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$ in the process of storage[58]: (a) Cross-correlated function ${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$ between signal 1 and signal 2 photons with a delayed time of 260 ns between pump 1 and pump 2; (b), (c) and (d) were the time-correlated function ${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$ between signal 2 photon and the retrieval signal 1 photon with storage time of 100, 150 and 200 ns respectively; (e) the collected noise without the input signal. The signal 1 acted as trigger photon, and the signal 2 acted as stop signal. All data were raw, without noise correction.

    图 4  (a) 存储前反斯托克斯和斯托克斯光子之间的符合; (b)存储后斯托克斯光子和恢复反斯托克斯光子之间的符合; (c)噪声符合[58]

    Figure 4.  (a) Coincidence between the anti-Stokes and Stokes photons without storage; (b) coincidence between the Stokes and retrieved anti-Stokes photons; (c) coincidence from noise [58].

    图 5  (a), (b)存储前后单光子失谐200 MHz的反斯托克斯光子与斯托克斯光子的符合计数; (c)噪声记录[58]

    Figure 5.  (a), (b) Coincidence between the anti-Stokes and the Stokes photons with a single photon detuning of +200 MHz before/after storage; (c) the recorded noise[58].

    图 6  存储前后重构的密度矩阵[29] (a)用于重构密度矩阵的OAM态; (b)/(d)和(c)/(e)分别是存储前/后重构密度矩阵的实部和虚部, 背景噪声已被减去; 其中(b)和(c)中每组测量时间为500 s, (d)和(e)中每组测量时间为1000 s; L/R分别表示OAM为1/−1.

    Figure 6.  Reconstructed density matrices before and after storage[29]: (a) Four OAM states for reconstructing density matrix; (b)/(d) and (c)/(e) are the real and imaginary parts of the reconstructed density matrix of the state before/after storage respectively. The background noise has been subtracted. The background noise was estimated by repeating the experiment without input signal 1 photon to MOT 2. The measurement time for each data was 500 s in (b) and (c) and 1000 s in (d) and (e). L/R represents OAM = 1/−1.

    图 7  实验装置[30], 其中透镜L1和L2用于将信号1的相位结构映射到MOT2的中心; 利用L3, L4和L5将位于MOT1中心的信号2的相位结构映射到SLM2的表面; L6和L7用于将信号2的OAM模式耦合到C2中; 图的右侧部分用于将信号1存储在MOT2, 并在读出后将其耦合到C1; C, 光纤耦合器; M, 反射镜; L, 透镜

    Figure 7.  Experimental setup[30]. Lenses L1 and L2 are used to focus signal 1 on the centre of MOT 2. L3, L4, and L5 are used to focus the phase structure of signal 2 on the center of MOT 1 onto the surface of SLM 2. L6 and L7 are used to couple OAM mode of signal 2 to C2. There is an asymmetric optical path for coupling signal 1 into C1 in right frame of figure. C, fiber coupler; M, mirror; L, lens.

    图 8  构造的三维密度矩阵[30] (a)和(b)分别是存储前的实部和虚部; (c)和(d)是存储后的实部和虚部

    Figure 8.  Constructed density matrix of three-dimensional entanglement[30]. Panels (a) and (b) are the real and imaginary parts before storage; panels (c) and (d) that after storage.

    图 9  (a) 和(b)是存储前后的可见度之和[30]

    Figure 9.  (a) and (b) are the sum of visibilities before and after storage[30].

    图 10  (a) 简化的能级图, 用于产生和存储偏振纠缠和单光子的生成、存储的时序; P1是泵浦光1, P2是泵浦光2; (b)简化的实验装置; L和R是MOT A中的两个SRS过程, H和V分别代水平极化和垂直极化, P1和P2分别为来自两个声光调制器的25 ns ($\Delta t $)和160 ns的调制脉冲; MOT, 磁光阱; FC, 光纤耦合器; PBS, 偏振分束器; $\lambda$/2, 半波片; $\lambda$/4, 四分之一波片; S, Stokes光子; As, 反-Stokes光子; D1, D2和D3分别是单光子探测器1, 2和3 (PerkinElmer SPCM-AQR-15-FC); PD, 自制光电探测器; PZT, 压电陶瓷; U和D分别是输入到MOT B中的上和下光模式; P, 半波板; $\theta$, 插入相位板的相位[58]

    Figure 10.  (a) Simplified energy level diagram used to generate and store the polarization entanglement and the time sequence for the generation, storage and retrieval sequence of a single photon. P1 is pump 1 and P2 is pump 2. (b) Ssimplified setup depicting the storage of the polarization entanglement. L and R are the two SRS processes in MOT A. H and V are the horizontal and vertical polarizations, respectively. P1 and P2 are the modulated pulses with 25 ns ($\Delta t $) and 160 ns from two acoustic optic modulators, respectively. MOT, magneto-optical trap; FC, fibre coupler; PBS, polarization beam splitter; $\lambda$/2, half-wave plate; $\lambda$/4, quarter-wave plate; S, Stokes photon; As, anti-Stokes photon; D1, D2 and D3 are single photon detectors 1, 2 and 3, respectively (PerkinElmer SPCM-AQR-15-FC); PD, home-made photoelectric detector; PZT, piezoelectric transducer; U and D are the up- and down-optical modes input into MOT B, respectively; P, half-wave plate; $\theta$, the phase of the inserted phase plate [58].

    图 11  (a) 和(b) 分别为存储前后探测器D3探测到的Stokes光子与探测器D1 (圆形数据)和探测器D2 (三角形数据)分别探测到的反Stokes光子之间的符合计数; 实线是拟合曲线; 所有实验数据为原始数据, 没有进行误差校正; 误差为 ± 1的标准差[58]

    Figure 11.  (a) and (b) Coincidence between the Stokes photon detected by detector D3 and the anti-Stokes photon detected by detector D1 (circular data) and detector D2 (triangular data), respectively, with a different phase before/after storage. The solid lines are the fitted lines. All of the experimental data are raw data without error corrections. The error bars are ± 1 standard deviation [58].

    图 12  (a) 存储前输入态密度矩阵; (b)存储后输出态的密度矩阵; 所有的实验数据都是原始数据, 没有进行任何误差修正[58]

    Figure 12.  (a) Density matrices of the input state before storage; (b) the output state after storage. All of the experimental data here are raw data without any error corrections[58].

    图 13  (a)/(c)和(b)/(d)分别为输入/输出的重构密度矩阵的实部和虚部; 所有实验数据为原始数据, 无误差校正[58]

    Figure 13.  (a)/(c) and (b)/(d) Reconstructed real and imaginary parts of the input/output density matrix, respectively. The density matrices were reconstructed with losses. All of the experimental data are raw data without error corrections[58].

    图 14  多个DOF超纠缠的产生和存储[59] (a)实验装置简化图; (b)能级图和时间序列; 二维极化纠缠((c), (d))和三维OAM纠缠((e), (f))的构造密度矩阵的实部, 其中(c), (e)对应存储前; (d), (f))对应于存储后.

    Figure 14.  Generation and storage of entanglement in multiple DOFs [59]: (a) Simplified experimental set-up; (b) energy diagram and time sequence; the real parts of the constructed density matrices for the two-dimensional polarization entanglement ((c), (d)) and the three-dimensional OAM entanglement ((e), (f)), before ((c), (e)) and after ((d), (f)) storage.

    图 15  重构的杂化纠缠的密度矩阵实部[59] (a)存储前; (b)存储后

    Figure 15.  Real parts of the constructed density matrices for hybrid entanglement: (a) Before storage; (b) after storage [59].

    图 16  杂化纠缠的双光子关联干涉曲线[59] (a)存储前; (b)存储后; 误差由泊松统计估计, 表示为 ± s.d; 所有数据均为原始数据, 没有进行误差纠正

    Figure 16.  Interference curves of the two-photon correlations for hybrid entanglement [59]: (a) Before storage; (b) after storage. The error bars are estimated from Poisson statistics and represent as ± s.d. All the data are raw and not subjected to noise correction.

    表 1  存储前后的${\bar p_{ij}}$以及$\bar C$

    Table 1.  Values of ${\bar p_{ij}}$ and $\bar C$ before and after storage.

    ${\bar \rho _{{\rm{input}}}}$${\bar \rho _{{\rm{output}}}}$
    ${\bar p_{00}}$0.990393 ± 0.000060.998166 ± 0.000008
    ${\bar p_{10}}$(4.59 ± 0.03) × 10−3(9.64 ± 0.04) × 10−4
    ${\bar p_{01}}$(5.04 ± 0.03) × 10−3(8.71 ± 0.04) × 10−4
    ${\bar p_{11}}$(1.6 ± 0.2) × 10−6(5 ± 5) × 10−8
    $\bar C$(5.8 ± 0.2) × 10−3(1.2 ± 0.4) × 10−3
    DownLoad: CSV
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  • Received Date:  17 December 2018
  • Accepted Date:  06 January 2019
  • Available Online:  01 February 2019
  • Published Online:  05 February 2019

Raman protocol-based quantum memories

    Corresponding author: Shi Bao-Sen, drshi@ustc.edu.cn
  • 1. Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
  • 2. Synergetic Innovation Center of Quantum Information & Quantum Physics, University of Science and Technology of China, Hefei 230026, China

Abstract: Quantum memories are indispensable for quantum communication and quantum computation, which are able to realize the storage and retrieval of a quantum state such as a single photon, entanglement, or a squeezed state on demand. Among those memories realized by different protocols, the Raman quantum memory has advantages in its broadband and high-speed properties, resulting in huge potential applications in quantum network and quantum computation. However, the realization of Raman quantum memory for a true single photon and photonic entanglement is a challenging job. In this review, after briefly introducing the main benchmarks for quantum memories, showing the state of the art, we focus on the review of the experimental progress recently achieved in storing the quantum state by Raman scheme in our group. We believe that all achievements reviewed are very hopeful in building up a high-speed quantum network.

    • 量子信息科学(QIS)是利用量子力学基本原理完成信息处理任务的交叉学科. QIS有两个重要研究目标: 一是实现通用量子计算机, 并利用它快速高效地解决某些经典计算机难以解决的复杂问题; 二是实现远程量子通信和量子密码. 量子通信可以实现远程用户之间的量子态传输, 而量子密码技术可以帮助两个远程用户以绝对安全的方式进行信息交流. 量子通信特别是量子密码技术的发展非常迅速, 目前也最接近于实际应用. 在过去的几十年里, QIS受到了公众的广泛关注, 并取得了巨大的进步. 目前基于光纤系统的量子密钥分配可以在约100 km的距离用于商业系统; 2016年中国发射了第一颗量子通信卫星, 并基于它开展科学研究, 取得了一些重大进展, 如成功地实现了纠缠光子从太空到地球地面站之间的分发; 中国还基于可信量子中继技术在北京和上海之间建立了量子密钥分配线路; 此外包括谷歌、IBM、英特尔、微软等在内的多家大型跨国公司已开始在量子计算等领域投入巨资, 用于量子计算机的研发等. 由此可见, 量子信息研究是当今世界最热门的研究领域之一.

      QIS的发展离不开量子存储器. 量子存储器是可以实现按照需要存储/读出诸如单光子、纠缠或者压缩态等非经典量子态的系统, 是实现量子通信网络和量子计算机必不可少的核心器件. 如果信息载体是处于可见到近红外范围的光子, 则该量子存储器可称之为光量子存储器. 在量子计算机中, 量子存储器帮助我们在计算过程中寄存和同步各种量子态. 而在量子通信网络中可以利用量子存储器建立量子中继器, 从而解决量子通信过程中量子态的保真度随着通信距离的增加而呈现出指数下降的问题. 众所周知, 光信号在光纤等信道中传输的过程中不可避免地发生衰减, 严重制约了量子态可传输的距离, 进而造成通信时间的大幅度增加. 例如在光纤通信系统中, 假设信号源以1 GHz的速率在通信窗口(衰减最小的工作波长)发送光子, 则在相距500 km的通信双方之间传输一个qubit的信息所需时间是秒的量级. 如果通信距离拓展到1000 km, 则每3000年才能成功传送一个qubit信息[1]. 为了克服信号光在信道中传输的指数衰减问题, 需要借助类似于经典通信体系使用的基于信号放大的中继器. 然而量子不可克隆原理告诉我们不能对量子信号进行无噪声放大, 因而经典的中继模式不能应用于量子通信系统, 迫使人们去寻求一种新的方法来解决这一瓶颈问题. 1998年, Briegel等[2]提出了一种解决信道衰减问题的量子中继器方案, 利用该方案可以将信号衰减从原来的指数衰减转变为多项式衰减, 进而使得长距离量子通信成为可能. 比如同样是1000 km的传输距离(在光纤中), 1 GHz的信号光子发送速率, 利用优化的量子中继方案可以将成功传送1个qubit信息的效率提高到秒量级[1]. 构成量子中继器的核心器件就是量子存储器. 此外, 量子存储器还可以用来同步线性量子计算中不同qubit的到达时间[3]、提高量子逻辑门[4]的成功概率、实现量子计量学和磁测量[5]、以及检测单个光子[6]等问题. 因此, 世界上有许多研究组都在积极开展量子存储的实验和理论研究.

    • 量子存储器的性能主要由以下6个指标衡量. 1)存储保真度. 保真度定量地表征读出态与输入态之间的相似程度. 两个纯态$\left| \phi \right\rangle $$\left|\varphi \right\rangle $之间的保真度可以根据公式$F = \left\langle { \phi} | {\varphi } \right\rangle $计算. 保真度是表征存储器是否可以工作在量子区域的一个重要判据, 当信息载体是真正的单光子时, 只有当存储保真度大于2/3时该存储器才可以工作在量子区域. 而当存储的是纠缠态时则存储保真度应大于85.4%[7]. 2)存储效率. 它由读出信号和输入信号之间的能量比表示. 对于单光子存储而言, 它意味着从存储器读出光子的概率. 只有在存储效率大于90%的条件下使用量子中继器传输信息才能比直接传输具有更高的效率. 3)按需读出(on-demand). 量子存储器按照需要读出信号的概率, 即在存储发生后可以确定存储时间的可能性. 4)存储时间或时间带宽积. 存储时间是量子态保存在存储器中的时间, 是衡量一个存储器丢失所存信息的时间. 存储器中的消相干效应会缩短存储时间, 也影响存储效率和保真度. 信息所需的存储时间取决于所使用的通信协议和实际的传输需求. 一个更具包容性的度量是存储时间-存储带宽积, 它定量地描述了在量子存储器消相干之前可以完成的逻辑操作的数量. 5)合适的工作波长. 量子中继器的实现首先需要在不同的节点之间建立量子纠缠, 而后通过将来自相邻节点的两个光子在节点中间实现Bell测量来拓展通信距离, 因此信息光子的波长最好处于通信窗口波段, 存储器也应该工作于通信波段, 这样长距离通信所要求的中继器数量将大大减少, 然而这很难实现. 例如在著名的DLCZ (Duan-Lukin-Cirac-Zoller)方案中[8], 由于缺乏可工作于通信波段的存储介质, 导致绝大多数该类存储器工作于可见波段, 这就造成存储器与信息载体之间工作波长不匹配的问题. 可以用两种不同的方法来解决这个问题, 一种是利用原子系综中[9]或非线性晶体中的频率转换[10]; 另一种是如参考文献[7]中所指出的从一开始就避免这一问题, 即通过制备特殊的纠缠光子对, 其中一个光子位于通信波长, 可进行长距离传输, 而另一个光子的波长与光量子存储器兼容[11-14]. 6)多模容量. 量子通信的速度随着可并行处理的信息载体数量的增加而增加, 因此存储器内可并行存储的信号数量也是非常重要的. 不得不说, 并非所有的性能指标都能在一个单一的物理系统中同时得到满足, 但可以使用不同的存储协议在不同的物理系统中分别实现.

    • 用于量子存储的介质多种多样, 根据该介质是由单个粒子还是大量粒子组成的系综可以将介质大体分为两类. 在单粒子介质中, 一个自然的候选者是处于共振腔内的单个原子, 在该体系中目前报道的最高量子比特的存储效率为9%, 存储时间为180 ${\text{μ}}{\rm s}$[15]; 另一个候选者是单个囚禁离子, 最近的重要进展包括远程离子之间的量子隐形传态[16]和光子偏振态与单个离子内态之间的相互转移[17]; 第三种选择是金刚石中的氮空位(NV)色心, 单光子与NV色心电子自旋之间的纠缠已被证明[18], 并且最近已经实现了两个远程NV色心之间的纠缠[19]. 量子点也被认为是量子存储器的合适候选者[20,21]. 在粒子系综体系中, 通过激光冷却与囚禁得到的中性冷原子系综是目前最成熟的系统之一, 也是最早被用于光存储的介质. 通过激光冷却与囚禁技术可以使纯碱金属原子气体具有较低的温度, 从磁光阱(MOT)中的几个毫开到偶极阱中的微开, 甚至到玻色-爱因斯坦凝聚物(BEC)中的纳开. 由于原子温度低, 运动速度小, 因而由原子之间的碰撞和原子扩散引起的消相干效应较弱, 又由于冷原子具有超精细结构, 因而可以实现较长时间的信息存储. 同时原子系综光学厚度(OD)大, 因而可以实现效率高、保真度高的信息存储. 由此可以看出冷原子系综是一种理想的存储介质, 可以用来实现各种存储协议. 此外, 原子系统具有光谱一致性, 这对利用光在远程量子节点之间建立链接至关重要. 2010年, 文献[9]报道了基于冷原子系综实现100 ms的量子存储实验. 文献[22]利用动态解耦技术, 将弱经典光的存储时间提高到16 s. 最近山西大学报道了弱相干光的存储研究, 存储时间达到毫秒量级[23]. 中国台湾清华大学的研究组基于电磁感应透明(EIT)方案实现了高效的相干光存储, 存储效率达到93%[24]. 香港科技大学的研究小组用冷原子系综作为量子存储器, 实现了利用另一个冷原子系综中的自发四波混频(SFWM)产生的单光子的存储, 效率和保真度分别为49%和96%[25]. 最近, 华南师范大学的研究组将真实单光子的存储效率提高到60%以上[26]. 此外, 中国科学技术大学量子信息重点实验室实现了携带轨道角动量(OAM)的真实单光子和三维OAM光量子态的存储[27,28], 并在此基础上进一步通过量子存储在两个冷原子系综之间建立了OAM纠缠[29-31]. 热原子气体也可以实现类似于冷原子系综的各种量子存储. 相比于冷原子系综, 热原子体系具有系统简便易行的特点, 特别适合于可扩展的量子网路系统. 文献[32]报道了在热原子系综中实现的高效光存储, 存储效率为87%. 同样, 热原子系综也可以用于存储空间多模信息[33-36]. 稀土掺杂固体具有较长的消相干时间、较强的多模存储能力, 因而也是一种不错的量子存储介质. 在这一研究方向上澳大利亚的一个研究组报道了151Eu3+:Y2SiO5晶体中长达370 min ± 60 min的相干时间[37], 中国科学技术大学基于原子频率梳实现了100个模式的量子存储[38], 加拿大Calgary大学的研究者实现了26个频率模式的量子存储[39]等.

    • 实现光子存储的协议有很多种, 包括EIT[40-42]、远失谐双光子共振跃迁[43-45]、可控可逆非均匀展宽[46,47]、光子回波[48,49]、原子频率梳[50]等. 其中远失谐双光子共振跃迁协议又称为拉曼方案, 该方案与EIT方案使用的能级结构相同, 是一个由三个能级组成的$\Lambda$结构(图1). 与EIT方案不同的是拉曼方案采用单光子失谐而双光子共振的构型, 并在双光子共振附近产生一个虚激发能级, 同时通过增大介质的有效OD或增加控制激光的强度来提高虚激发能级的能带. 该存储协议具有存储短时间脉冲的能力, 因而可以实现高速量子存储. 此外, 由于控制激光的单光子失谐是可变的, 并且对非均匀展宽不敏感, 因而拉曼量子存储器可以工作在很大的频率范围内. 所有这些性质表明, 拉曼方案在量子通信网络和量子计算中具有巨大的应用潜力. 这也是近年来我们利用拉曼协议研究量子存储的主要动机和原因.

      Simplified energy level diagram for Raman protocol. <inline-formula><span class=$\left| g \right\rangle $, $\left|s\right\rangle $ and $\left| e \right\rangle $ correspond to ground, metastable and excited state respectively; $\varDelta $ represents detuning." />

      Figure 1.  Simplified energy level diagram for Raman protocol. $\left| g \right\rangle $, $\left|s\right\rangle $ and $\left| e \right\rangle $ correspond to ground, metastable and excited state respectively; $\varDelta $ represents detuning.

      国际上牛津大学的Walmsley研究组[44,45]最早在实验上实现了拉曼方案, 他们在热原子系综中实现了一个纳秒光脉冲的存储, 其存储时间带宽积为5000. 尽管这些工作中的信息载体都是弱衰减相干光, 但对于实现拉曼量子存储器具有重要参考价值. 最近, 此研究组试图在热原子系综中存储由非线性晶体中的自发参量下转换产生的单光子[51]. 然而, 正如作者指出的, 伴随存储过程产生的不可避免的SFWM噪声限制了它的实际应用, 因而该存储器不能工作在量子区域[51]. 此外人们还在分子系综中实现了THz光子存储[52]以及金刚石中的THz光脉冲存储[53]. 这些结果进一步证明了拉曼协议在宽带和高速存储中的潜力. 2015年, Walmsley研究组[54]又一次尝试用一个光学腔内的热原子系综来存储单光子水平的信号, 他们发现利用腔的抑制效应可以大大降低SFWM产生的噪声, 因而声称他们的存储器可以工作于量子区域. 不幸的是, 他们的存储信号仍然是衰减的相干光. 2018年, 此小组[55]采用了一种称为双光子非共振级联吸收的方案来实现高速无噪音光量子存储器, 他们在热原子系综中成功地实现了低噪声、大带宽(GHz)单光子的存储. 上海交通大学的Dou等[56]则实现了基于室温原子系综的宽带非共振DLCZ量子存储器. 最近, 华东师范大学的Guo等[57]报道了一个基于光学控制的高性能拉曼量子存储器, 他们在一个热原子系综中实现了存储效率为82.6%的10 ns微弱光脉冲存储. 虽然在过去十多年中拉曼协议的实验研究取得了显著的进展, 但绝大多数存储实验都是在弱相干光下完成的, 而关于利用拉曼协议存储真正单光子的报道很少. 此外, 虽然基于各种存储协议的光子存储取得了很大进展, 但只有一小部分存储器实现了光子偏振、路径、时间域以及时间-能量纠缠的量子存储, 并且到2014年底前为止, 还没有利用拉曼量子存储器实现光子偏振纠缠或其他自由度(DOF)纠缠如OAM存储的实验报道, 实际上构建这样一个量子存储是很困难的. 从2014年开始, 我们研究组开展了利用拉曼协议实现量子存储的实验研究, 几年来取得了系列研究成果: 实现了真正单光子、光子偏振纠缠、OAM纠缠和多DOF纠缠的存储[27-31,58,59], 下文将详细介绍这些进展.

      本文的主要内容如下: 在简要介绍量子存储器的主要指标, 以及量子存储器特别是拉曼量子存储器的发展现状后, 重点评述本研究组利用拉曼方案实现量子存储的系列实验进展, 包括真实单光子、二维和高维空间中OAM纠缠态的存储, 以及由OAM和其他DOF组成的超纠缠和混合纠缠的量子存储, 最后给出小结. 我们认为这些研究成果对于研究高速大带宽量子网络具有重要参考价值.

    2.   基于拉曼协议的量子存储
    • 图2是实验装置简图. 我们通过一个二维磁光阱(MOT1)[60]制备了雪茄形的碱金属铷85(85Rb)冷原子系综, 而后利用泵浦1通过自发拉曼散射(SRS)制备了一个波长在795 nm的反斯托克斯光子(信号光子1), 随后将产生的信号光子1传送到第二个磁光阱MOT2中, 借助一束正交偏振控制脉冲激光对信号光子1进行拉曼存储. 通过这种方法, 建立了两个MOT之间的非经典关联. 在读出MOT2中的信号光子1后, 用泵浦激光2将MOT1中原子系综的集体自旋激发态映射为信号光子2, 进而测量了两个信号光子之间的互相关函数${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$, 结果如图3所示. 信号光子1和信号光子2之间的时域相关性可以通过检验Cauchy-Schwarz不等式[40]是否被违反进行证明. 通常经典光满足不等式$R = \displaystyle\frac{{{{[{g_{{\rm{s1, s2}}}}(\tau )]}^2}}}{{{g_{{\rm{s1, s1}}}}(0){g_{{\rm{s2, s2}}}}(0)}} \leqslant 1$. 如果R > 1, 则光是非经典的. 其中gs1,s2, gs1,s1(0)和gs2,s2(0)分别是光子归一化的二阶互相关和自相关系数. 实验表明, 当信号光子1的存储时间为150 ns时, gs1,s2$\left( \tau \right)$具有最大值24. 同时考虑到gs1,s1(0) = gs2,s2(0) ≈ 2 (信号1和信号2的光子具有典型的热光场统计特性), 此时R = 144, 远大于1, 强烈地违反了Cauchy-Schwarz不等式, 从而清晰地证明了储存过程中非经典关联的保持.

      Simplified energy level diagram and experimental setup<sup>[<span class=29]. (a) Simplified energy level diagram of the SRS. The states $\left| 1 \right\rangle $ and $\left| 2 \right\rangle $ correspond to two metastable levels 5S1/2(F = 3) and 5S1/2(F = 2) of 85Rb atom respectively, $\left| 3 \right\rangle $ and $\left| 4 \right\rangle $ are the excited levels of 5P3/2(F' = 3) and 5P1/2(F' = 3) respectively. The pump 1 laser is from an external-cavity diode laser (DL100, Toptica) with the wavelength of 795 nm, and is blue-detuned to the atomic transition of 5S1/2(F = 3)→5P1/2(F' = 3) with a value of 70 MHz. The pump 2 laser is from another external-cavity diode laser (DL100, Toptica) with the wavelength of 780 nm which couples the atomic transition of 5S1/2(F = 2)→5P3/2(F' = 3). The pump 1 and pump 2 are modulated into pulse modes with a width of 50 and 160 ns respectively, and a rising edge of 30 ns. The delayed time between the pump 1 pulse and the pump 2 pulse is programmed to be 260 ns for the process of storage. The powers of pump 1 and pump 2 are 0.5 and 4 mW respectively. The coupling laser is from the same laser with pump 1 and is also blue-detuned to atomic transition of 5S1/2(F = 3)→5P1/2(F' = 3) with a value of 70 MHz, its power is about 12 mW. (b) Simplified diagram depicting the storage of entanglement of OAM state. The waist of signal 1 at MOT 2 was 63 ${\text{μm}}$. MOT, magneto-optical trap; FC, fibre coupler; SLM, spatial light modulator; PBS, polarisation beam splitter; $\lambda$/2, half-wave plate." />

      Figure 2.  Simplified energy level diagram and experimental setup[29]. (a) Simplified energy level diagram of the SRS. The states $\left| 1 \right\rangle $ and $\left| 2 \right\rangle $ correspond to two metastable levels 5S1/2(F = 3) and 5S1/2(F = 2) of 85Rb atom respectively, $\left| 3 \right\rangle $ and $\left| 4 \right\rangle $ are the excited levels of 5P3/2(F' = 3) and 5P1/2(F' = 3) respectively. The pump 1 laser is from an external-cavity diode laser (DL100, Toptica) with the wavelength of 795 nm, and is blue-detuned to the atomic transition of 5S1/2(F = 3)→5P1/2(F' = 3) with a value of 70 MHz. The pump 2 laser is from another external-cavity diode laser (DL100, Toptica) with the wavelength of 780 nm which couples the atomic transition of 5S1/2(F = 2)→5P3/2(F' = 3). The pump 1 and pump 2 are modulated into pulse modes with a width of 50 and 160 ns respectively, and a rising edge of 30 ns. The delayed time between the pump 1 pulse and the pump 2 pulse is programmed to be 260 ns for the process of storage. The powers of pump 1 and pump 2 are 0.5 and 4 mW respectively. The coupling laser is from the same laser with pump 1 and is also blue-detuned to atomic transition of 5S1/2(F = 3)→5P1/2(F' = 3) with a value of 70 MHz, its power is about 12 mW. (b) Simplified diagram depicting the storage of entanglement of OAM state. The waist of signal 1 at MOT 2 was 63 ${\text{μm}}$. MOT, magneto-optical trap; FC, fibre coupler; SLM, spatial light modulator; PBS, polarisation beam splitter; $\lambda$/2, half-wave plate.

      Cross-correlated function of <inline-formula><span class=${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$ in the process of storage[58]: (a) Cross-correlated function ${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$ between signal 1 and signal 2 photons with a delayed time of 260 ns between pump 1 and pump 2; (b), (c) and (d) were the time-correlated function ${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$ between signal 2 photon and the retrieval signal 1 photon with storage time of 100, 150 and 200 ns respectively; (e) the collected noise without the input signal. The signal 1 acted as trigger photon, and the signal 2 acted as stop signal. All data were raw, without noise correction." />

      Figure 3.  Cross-correlated function of ${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$ in the process of storage[58]: (a) Cross-correlated function ${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$ between signal 1 and signal 2 photons with a delayed time of 260 ns between pump 1 and pump 2; (b), (c) and (d) were the time-correlated function ${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$ between signal 2 photon and the retrieval signal 1 photon with storage time of 100, 150 and 200 ns respectively; (e) the collected noise without the input signal. The signal 1 acted as trigger photon, and the signal 2 acted as stop signal. All data were raw, without noise correction.

      随后通过测量信号光子1的条件自相关参数gs1;s1/s2(t)来检验存储过程中单光子特性是否保持. 一个严格的单光子gs1;s1/s2(t) = 0, 一个双光子态gs1;s1/s2(t) = 0.5. 而对于经典场, gs1;s1/s2(t) ≥ 1. 其中${g_{{\rm{s1}};{\rm{s1}}/{\rm{s2}}}}\left( t \right) = \displaystyle\frac{{{P_1}{P_{123}}}}{{{P_{12}}{P_{13}}}}$, P1是信号光子2的计数, P12P13是信号光子2和两个分离信号光子1之间的二重符合计数, P123是三重符合计数. 因此gs1;s1/s2(t) < 0.5显示了单光子特性. 实验得到存储前gs1;s1/s2(t)为0.074 ± 0.012, 在150 ns的长时间存储后为0.29 ± 0.02, 从而证明了存储后信号光子1的单光子特性保持不变. 实验结果清晰地证明了在冷原子系综中可以通过拉曼方案成功地存储真正的单光子, 实验存储效率为26.7%.

      在上述实验中, 信号光子1的带宽和蓝失谐分别为20和70 MHz. 随后, 通过任意函数发生器调制泵浦光1, 进一步增加了信号光子1的带宽. 受实验控制系统分辨率的限制, 信号光子1的最小脉冲宽度约为7 ns (约140 MHz). 图4给出了信号光子1的存储结果, 其中图4(a)表示输入信号, 图4(b)是存储后的结果, 图4(c)是噪声(信号光子1被阻挡). 存储效率为10.3%, 读出的信号光子1和信号光子2之间的二阶互相关(${g_{{\rm{s}}1,{\rm{s}}2}}\left( \tau \right)$)约为13.6, 表明存储后它们之间仍然存在非经典相关性. 如果能提高实验控制系统的时间分辨率, 则可以实现一个脉宽更小的脉冲(< 7 ns)的存储.

      (a) Coincidence between the anti-Stokes and Stokes photons without storage; (b) coincidence between the Stokes and retrieved anti-Stokes photons; (c) coincidence from noise <sup>[<span class=58]." />

      Figure 4.  (a) Coincidence between the anti-Stokes and Stokes photons without storage; (b) coincidence between the Stokes and retrieved anti-Stokes photons; (c) coincidence from noise [58].

      我们还进一步研究了大失谐条件下的存储性能. 通过改变泵浦光1和控制激光的频率, 使它们与原子跃迁$\left| 1 \right\rangle \to \left| 4 \right\rangle $的失谐为200 MHz, 因此产生的信号光子1也与原子跃迁$\left| 2 \right\rangle \to \left| 4 \right\rangle $失谐200 MHz. 为了在单光子水平上进行拉曼存储, 需要尽可能地增加控制激光的功率. 在存储过程中, 泵浦激光1的脉宽为50 ns, 控制激光的功率为110 mW, 束腰为2 mm. 同时, 在滤波系统中插入了自制的F-P腔滤波器, 最终消光比约为109 : 1, 足以消除控制激光的散射噪声. 实验结果如图5所示, 其中图5(a)为存储前信号光子1与信号光子2之间的符合, 图5(b)为存储后的数据, 图5(c)是噪声. 存储后的gAS, S$\left( \tau \right)$为5.6, 如果使用更多的滤波器来降低噪声, 那么信噪比就会得到提高.

      (a), (b) Coincidence between the anti-Stokes and the Stokes photons with a single photon detuning of +200 MHz before/after storage; (c) the recorded noise<sup>[<span class=58]." />

      Figure 5.  (a), (b) Coincidence between the anti-Stokes and the Stokes photons with a single photon detuning of +200 MHz before/after storage; (c) the recorded noise[58].

    • 带有OAM的光子其波前是一个螺旋面[61,62]. 由于OAM空间固有的无限维特性[63-65], 如果将光子编码在OAM空间则可以大幅度提高光子的信息携带量[66], 因此, 基于OAM的量子信息处理也成为近年来的研究热点. 基于OAM量子网络的建立涉及到OAM纠缠光子与物质之间的相干相互作用, 因此存储OAM纠缠态对于建立基于OAM量子网络至关重要. 接下来分别介绍近年来本研究组基于拉曼存储方案实现OAM纠缠存储方面的研究进展. 实验装置与图2相同, 首先利用SRS建立了反斯托克斯光子信号1与MOT1中原子集体自旋激发态之间的OAM纠缠, 其量子态可由$\left| \psi \right\rangle = \sum\limits_{m = - \infty }^{m = \infty } {{c_m}{{\left| m \right\rangle }_{{\rm{s}}1}} \otimes {{\left| { - m} \right\rangle }_{{\rm{a}}2}}} $表示[67,68], 其中下标s1和a2分别标记信号光子1和MOT1中原子系综的集体自旋激发态, ${\left| {{c_m}} \right|^2}$是激发概率, $\left| m \right\rangle $是量子数为m的OAM本征模, 系统的初始线性动量和角动量为零. 由于SRS过程中动量守恒, 因此反斯托克斯光子和原子自旋激发态的总角动量为零, 从而建立了它们之间的OAM关联. 通过拉曼方案将信号光子1存储在MOT2原子系综中, 从而建立了两个原子系综之间的OAM纠缠. 随后通过将两个原子自旋激发态分别映射到两个光子(信号1和信号2光子), 并检查它们的纠缠度来证明原子自旋激发态之间的OAM纠缠. 首先, 实验证明了在一个二维子空间中OAM纠缠的存储(子空间由$\left| m \right\rangle $$\left| { - m} \right\rangle $基构成), 此时光子纠缠态为$ \left| \varPsi \right\rangle ={1 / {\sqrt 2 }}\left[{\left| m \right\rangle \left| m \right\rangle + }\right.$$\left.{\left| {-m} \right\rangle \left| {-m} \right\rangle } \right]$. 测量了信号光子2与读出信号光子1之间的时间相关函数, 并根据测量的符合计数率重构了密度矩阵[69](如图6所示). 通过与理想密度矩阵${\rho}_ {\rm{ideal}}$相比较, 计算出密度矩阵的保真度为84.6% ± 2.6%, 并与存储前重建的密度矩阵${\rho}_ {\rm{input}}$进行比较, 存储前密度矩阵的保真度为90.3% ± 0.8%.

      Reconstructed density matrices before and after storage<sup>[<span class=29]: (a) Four OAM states for reconstructing density matrix; (b)/(d) and (c)/(e) are the real and imaginary parts of the reconstructed density matrix of the state before/after storage respectively. The background noise has been subtracted. The background noise was estimated by repeating the experiment without input signal 1 photon to MOT 2. The measurement time for each data was 500 s in (b) and (c) and 1000 s in (d) and (e). L/R represents OAM = 1/−1." />

      Figure 6.  Reconstructed density matrices before and after storage[29]: (a) Four OAM states for reconstructing density matrix; (b)/(d) and (c)/(e) are the real and imaginary parts of the reconstructed density matrix of the state before/after storage respectively. The background noise has been subtracted. The background noise was estimated by repeating the experiment without input signal 1 photon to MOT 2. The measurement time for each data was 500 s in (b) and (c) and 1000 s in (d) and (e). L/R represents OAM = 1/−1.

      通过检测贝尔不等式可以进一步刻画存储后的纠缠度[70], 其表达式为

      其中$\theta_ {\rm{A}}$, $\theta_ {\rm{B}}$图6(a)中定义的SLMs表面的相位分布角. $E({\theta _{\rm{A}}}, {\theta _{\rm{B}}})$可以根据特定方位角的符合计数率来计算,

      实验中选择$\theta_ {\rm{A}}$ = 0, $\theta_ {\rm{B}}={\text π}/8$, $\theta_ {\rm{A}}'={\text π}/4$, $\theta_ {\rm{B}}'=3{\text π}/8$. 存储前S = 2.48 ± 0.04, 存储后S = 2.41 ± 0.06 (不减噪声时存储前S = 2.16 ± 0.04, 存储后S = 2.10 ± 0.06). 当S值大于2时就违反了这个不等式, 而不等式的违反则意味着光子之间存在纠缠. 显然实验结果清楚地说明了存储过程中OAM纠缠的保存.

    • 实现不同量子存储器之间的高维纠缠不是二维纠缠存储的简单扩展, 实验难度很大, 特别是在高维纠缠度的证明、纠缠维数的确定等方面存在着诸多挑战. 我们克服了诸多难题, 实现了两个量子存储器之间高维OAM纠缠的建立, 实验装置如图7所示. 首先在MOT 1的冷原子系综中利用SRS过程建立光子和集体自旋激发态之间的高维OAM纠缠, 然后用拉曼协议将这个光子存入到另一个冷原子系综中, 这样就在两个原子系综之间建立了高维OAM纠缠, 表示形式与前文相同, 为$\left| {\psi '} \right\rangle = \sum\limits_{m = - \infty }^{m = \infty } {{o_m}{{\left| { - m} \right\rangle }_{{\rm{a1}}}} \otimes {{\left| { - m} \right\rangle }_{{\rm{a2}}}}} $, 其中${\left| {{o_m}} \right|^2}$是不同模式m的概率, 下标a1/a2分别是指MOT 1/2中的原子集体自旋激发态. 通过将两个系综中的自旋激发态映射到两个光子(信号1和2的光子), 并对它们的纠缠度进行检验, 从而间接检验原子自旋激发态之间的纠缠. 首先验证一个三维纠缠的量子存储, 将信号光子1和2分别投影到9个不同的态$\left| {\psi _{1 - 9}} \right\rangle $ (分别对应态$\left| L \right\rangle $, $\left| G \right\rangle $, $\left| R \right\rangle $, $(\left| G \right\rangle +\left| L \right\rangle)/2^{1/2}$, $\left(| G \right\rangle\!+\!\left| R \right\rangle)/2^{1/2}$, $(\left| G \right\rangle\!+\!{\rm{i}}\left| L \right\rangle)/2^{1/2}$, $(\left| G \right\rangle\!-\!{\rm {i}} \left| R \right\rangle)/2^{1/2} $, $(\left| L \right\rangle + \left| R \right\rangle)/2^{1/2} $, $(\left| L \right\rangle + {\rm {i}}\left| R \right\rangle)/2^{1/2}$, 其中$\left| L \right\rangle $, $\left| G \right\rangle $$\left| R \right\rangle $分别对应OAM为$ 1\hbar $, 0和$ - 1\hbar $), 进而重构出存储前(图8(a)图8(b))和存储后(图8(c)图8(d))的密度矩阵, 并计算出纠缠度为83.9% ± 2.9%.

      Experimental setup<sup>[<span class=30]. Lenses L1 and L2 are used to focus signal 1 on the centre of MOT 2. L3, L4, and L5 are used to focus the phase structure of signal 2 on the center of MOT 1 onto the surface of SLM 2. L6 and L7 are used to couple OAM mode of signal 2 to C2. There is an asymmetric optical path for coupling signal 1 into C1 in right frame of figure. C, fiber coupler; M, mirror; L, lens." />

      Figure 7.  Experimental setup[30]. Lenses L1 and L2 are used to focus signal 1 on the centre of MOT 2. L3, L4, and L5 are used to focus the phase structure of signal 2 on the center of MOT 1 onto the surface of SLM 2. L6 and L7 are used to couple OAM mode of signal 2 to C2. There is an asymmetric optical path for coupling signal 1 into C1 in right frame of figure. C, fiber coupler; M, mirror; L, lens.

      Constructed density matrix of three-dimensional entanglement<sup>[<span class=30]. Panels (a) and (b) are the real and imaginary parts before storage; panels (c) and (d) that after storage." />

      Figure 8.  Constructed density matrix of three-dimensional entanglement[30]. Panels (a) and (b) are the real and imaginary parts before storage; panels (c) and (d) that after storage.

      其次, 考虑高维(d > 3)纠缠态存储. 理论上可以重构高维纠缠的密度矩阵来刻画纠缠性质, 但是由于重构d维纠缠态密度矩阵所需要的数据量非常庞大(d4), 因而使得密度矩阵的重构变得不切实际, 因此必须寻找另外一种方法来描述纠缠. 原则上可以利用三种方法来检验一个系统是否处于高维纠缠: 1)使用整个空间的无偏基态[70,71]; 2)直接检验高维不等式[72,73]; 3)违背一个比二维空间允许的更强的Bell不等式, 从而间接证明在更高维中存在纠缠. 这里使用方法3来表征高维纠缠, 使用纠缠witness[74,75]来证明是否存在高维纠缠, 并使用维数witness[76-78]来表征纠缠的维数. 为此需要在包括对角线/反对角线、左旋/右旋、水平/垂直三组偏振上进行相关性测量, 此时需要测量的数据量相比于重构密度矩阵减少3d(d−1)[79]. 纠缠度和维数的witness可以分别在每个2 × 2的子空间中由可见度$M = {V_x} + {V_y}$$N = {V_x} + {V_y} + {V_z}$计算, 其中可见度定义为${V_i} = \left| {\left\langle {{\sigma _i} \otimes {\sigma _i}} \right\rangle } \right|$, $i = x, y, z$, 这里, ${\sigma _x}, {\sigma _y}, {\sigma _z}$分别表示上述三个相互正交基. 对于一个处于d维子空间中的可分离态, (d−1)维最大纠缠态和单态的乘积使得可见度之和为最大值. 由于二维子空间中纠缠所能允许的最大可见度为2 ($M = {V_x} + {V_y} = 2, {V_x} = {V_y} = 1$), 因此对于一个(d−1)维纠缠所允许的最大可见度计算为(d−1)(d−2). 剩余可分离态的最大可见度为(d−1)[76], 由此给出了一个高维纠缠的最大界限为

      如果系统存在d维纠缠, 则应该违反最大边界Md. 对于一个包含m = 2, 1, 0, −1组成的量子态, 最大边界M4 = 9. 实验测量得到的存储前和存储后的$M' $值分别为9.30 ± 0.06和9.19 ± 0.06, 结果表明这两个远距离的原子系综之间至少存在一个四维纠缠.

      我们还通过对每个基的可见度N进行求和来计算W, 进而确定高维纠缠的维数. 实验测量得到的可见度N图9(a)图9(b)表示, 分别对应于存储前和存储后的数据. 维度witness的值Wd [80]由以下公式给出:

      (a) and (b) are the sum of visibilities before and after storage<sup>[<span class=30]." />

      Figure 9.  (a) and (b) are the sum of visibilities before and after storage[30].

      其中D是测量中的OAM模式数. 如果W > Wd则证明至少存在d + 1维纠缠. 实验中测量得到的模式数为11 (m = −5—5); 实验得到的输入态的W为123.9 ± 0.8, 输出态的W为112.8 ± 0.8, 分别违反了输入d = 7时边界值为121和输出d = 6边界值为110, 证明两个原子系综之间存在七维纠缠.

      每个光子的可编码的维数可以简单地用这样的公式来估计, $w\left( z \right) = \sqrt {m + 1} {w_0}\left( z \right)$, 其中w(z)是原子系综中心OAM值为m的光的束腰, w0(z)是高斯光的束腰. 实验中w0(z)约为100 ${\text{μ}}{\rm {m}}$, MOT2中的原子系综半径约为1 mm, 因此m约为100, 即系统中可存储的每个光子的最大的OAM维数为200. 当然, 这个数目还受到其他因素的限制, 如原子系综的菲涅耳数和OD、信号场的角度和MOT2中的控制光等.

    • 下面介绍利用拉曼方案实现的两个与偏振DOF相关的存储实验: 1)单光子路径与偏振纠缠存储; 2)在两个冷原子系综中的偏振纠缠存储. 实验装置如图10所示, 利用一个原子系综作为非线性介质来制备单光子或双光子纠缠态, 使用另一个原子系综作为存储介质.

      (a) Simplified energy level diagram used to generate and store the polarization entanglement and the time sequence for the generation, storage and retrieval sequence of a single photon. P1 is pump 1 and P2 is pump 2. (b) Ssimplified setup depicting the storage of the polarization entanglement. L and R are the two SRS processes in MOT A. H and V are the horizontal and vertical polarizations, respectively. P1 and P2 are the modulated pulses with 25 ns (<inline-formula><span class=$\Delta t $) and 160 ns from two acoustic optic modulators, respectively. MOT, magneto-optical trap; FC, fibre coupler; PBS, polarization beam splitter; $\lambda$/2, half-wave plate; $\lambda$/4, quarter-wave plate; S, Stokes photon; As, anti-Stokes photon; D1, D2 and D3 are single photon detectors 1, 2 and 3, respectively (PerkinElmer SPCM-AQR-15-FC); PD, home-made photoelectric detector; PZT, piezoelectric transducer; U and D are the up- and down-optical modes input into MOT B, respectively; P, half-wave plate; $\theta$, the phase of the inserted phase plate [58]." />

      Figure 10.  (a) Simplified energy level diagram used to generate and store the polarization entanglement and the time sequence for the generation, storage and retrieval sequence of a single photon. P1 is pump 1 and P2 is pump 2. (b) Ssimplified setup depicting the storage of the polarization entanglement. L and R are the two SRS processes in MOT A. H and V are the horizontal and vertical polarizations, respectively. P1 and P2 are the modulated pulses with 25 ns ($\Delta t $) and 160 ns from two acoustic optic modulators, respectively. MOT, magneto-optical trap; FC, fibre coupler; PBS, polarization beam splitter; $\lambda$/2, half-wave plate; $\lambda$/4, quarter-wave plate; S, Stokes photon; As, anti-Stokes photon; D1, D2 and D3 are single photon detectors 1, 2 and 3, respectively (PerkinElmer SPCM-AQR-15-FC); PD, home-made photoelectric detector; PZT, piezoelectric transducer; U and D are the up- and down-optical modes input into MOT B, respectively; P, half-wave plate; $\theta$, the phase of the inserted phase plate [58].

    • 首先我们实现了标记单光子路径-偏振纠缠的存储, 该标记单光子直接由图10(b)中标记为L路径中的SRS过程产生(R路径阻塞). 经过偏振分束器(PBS)后, 将由SRS产生的反斯托克斯光子分为两束(U和D), 并经过两条路径射入MOT B中. 路径U和D由一个特殊Sagnac干涉仪和一个PBS组成. 这两条光路在MOT B中的原子云中完全重合. 频率失谐为+70 MHz的控制激光以与两条光路U和D相同的角度射入MOT B中. 控制激光功率为22 mW, 束腰2 mm. 通过使用特殊的Sagnac干涉仪后得到单光子态为

      其中$\left| H \right\rangle $$\left| V \right\rangle $分别表示水平偏振和垂直偏振, $\theta_1$为路径U和D之间的相位差, 在实验中设置为零. (5)式描述了单光子在路径和偏振纠缠的混合纠缠态.

      利用基$\left| {{n_ {\rm{U}}},{m_ {\rm{D}}}} \right\rangle $重构约化密度矩阵[81] (其中{n, m} = {0, 1})来检验存储前后$\left| {{\psi _1}} \right\rangle $的纠缠特性. 这里的密度矩阵${{\rho}}$可表示为

      其中pij是在模式Uk下找到i光子并且在模式Dk下找到j光子的概率, k代表输入和输出模式, 如表1所列, dV(p01 + p10)/2, 表示$\left| {{1_ {\rm{U}}}{0_ {\rm{D}}}} \right\rangle $$\left| {{0_ {\rm{U}}}{1_ {\rm{D}}}} \right\rangle $之间的相干, 以及P = p00 + p10 + p01 + p11. V是模式U和D之间的干涉可见度, 通过记录探测器D3和D1, 以及D3和D2相对于相位($\theta$)的符合计数来计算, 如图11所示, 其中半波片的光轴设置为22.5°方向. 重构后的密度矩阵如图12所示, 密度矩阵(${{\rho}}$)的纠缠性质以concurrence表征: $C = \displaystyle\frac{1}{P}\max (0, 2\left| d \right| - 2\sqrt {{p_{00}}{p_{11}}} )$. C在0—1之间代表态从可分离到最大纠缠. 在这个实验中, 存储前后的C的计算结果分别为Cinput = (5.8 ± 0.2) × 10−3Coutput = (1.2 ± 0.4) × 10−3, 其中包括了自制的三个传输效率为30%的Fabry-Perot腔以及耦合效率为50%的光纤所带来的损耗. 测得存储过程中的量子传输效率

      ${\bar \rho _{{\rm{input}}}}$${\bar \rho _{{\rm{output}}}}$
      ${\bar p_{00}}$0.990393 ± 0.000060.998166 ± 0.000008
      ${\bar p_{10}}$(4.59 ± 0.03) × 10−3(9.64 ± 0.04) × 10−4
      ${\bar p_{01}}$(5.04 ± 0.03) × 10−3(8.71 ± 0.04) × 10−4
      ${\bar p_{11}}$(1.6 ± 0.2) × 10−6(5 ± 5) × 10−8
      $\bar C$(5.8 ± 0.2) × 10−3(1.2 ± 0.4) × 10−3

      Table 1.  Values of ${\bar p_{ij}}$ and $\bar C$ before and after storage.

      (a) and (b) Coincidence between the Stokes photon detected by detector D3 and the anti-Stokes photon detected by detector D1 (circular data) and detector D2 (triangular data), respectively, with a different phase before/after storage. The solid lines are the fitted lines. All of the experimental data are raw data without error corrections. The error bars are ± 1 standard deviation <sup>[<span class=58]." />

      Figure 11.  (a) and (b) Coincidence between the Stokes photon detected by detector D3 and the anti-Stokes photon detected by detector D1 (circular data) and detector D2 (triangular data), respectively, with a different phase before/after storage. The solid lines are the fitted lines. All of the experimental data are raw data without error corrections. The error bars are ± 1 standard deviation [58].

      (a) Density matrices of the input state before storage; (b) the output state after storage. All of the experimental data here are raw data without any error corrections<sup>[<span class=58]." />

      Figure 12.  (a) Density matrices of the input state before storage; (b) the output state after storage. All of the experimental data here are raw data without any error corrections[58].

      $\eta$ = Coutput/Cinput≈20.9% ± 7.7%.

    • 接下来介绍两个冷原子系综中的光子偏振纠缠存储. 由马赫-曾德尔干涉仪中两束波长为795 nm的泵浦光驱动的两个对称SRS过程(L和R)相干叠加, 在MOT A中制备了反斯托克斯光子与原子系综集体自旋激发态之间的偏振纠缠, 纠缠态可以描述为$\left|\psi\right\rangle = \displaystyle\frac{1}{{\sqrt 2 }}(\left|L \right\rangle \left|H \right\rangle + {{\rm{e}}^{{\rm{i}}{\phi }}}\left|R\right\rangle \left|V \right\rangle)$, 其中第一项指代系综L中的SRS过程, 第二项表示系综R中的SRS过程, $\left|H \right\rangle (\left|V \right\rangle )$表示反斯托克斯光子的水平(垂直)偏振, $\left|L \right\rangle(\left. R \right\rangle )$表示原子系综L(R)中的集体自旋激发态, $\phi $是两个泵浦光路径差造成的两个反斯托克斯光子间的相位差, 在实验中将其设置为零. 我们主动锁定了干涉仪, 然后将反斯托克斯光子发送到MOT B中并利用拉曼协议实现存储, 从而在MOT A和MOT B中的原子集体自旋激发态之间建立了纠缠$ \left| {{\psi _{{\rm{aa}}}}} \right\rangle =$$\displaystyle\frac{1}{{\sqrt 2 }}({\left| U \right\rangle _{\rm{A}} }{\left| L \right\rangle _{\rm{B}}} + {\left| D \right\rangle _{\rm{A}}}{\left| R \right\rangle _{\rm{B}}})$. 在将MOT B中原子的集体自旋激发态读出为反斯托克斯光子后, 进而读出MOT A中的原子集体自旋激发态为斯托克斯光子, 从而在光子之间建立了偏振纠缠$ \left| {{\psi _2}} \right\rangle =$$\displaystyle\frac{1}{{\sqrt 2 }}(\left| H \right\rangle \left| V \right\rangle + \left| V \right\rangle \left| H \right\rangle )$. 在此过程中, MOT B中的态存储时间应该小于MOT A中的态存储时间, 从而保证纠缠的存储. 通过检测反斯托克斯光子和斯托克斯光子之间的纠缠证明了两个系综中存在偏振纠缠. 对纠缠态的密度矩阵进行重构, 结果如图13所示, 其中图13(a)图13(b)分别给出了输入态的重构密度矩阵(${\rho}_ {\rm{input}}$)的实部和虚部, 与理想密度矩阵(${\rho}_ {\rm{ideal}}$)比较后, 计算得到重构密度矩阵保真度(F1)为89.3% ± 1.7%. 图13(c)图13(d)分别给出了经过存储后密度矩阵的实部和虚部, 计算得到输出的保真度(F2)为85.0% ± 3.4%.

      (a)/(c) and (b)/(d) Reconstructed real and imaginary parts of the input/output density matrix, respectively. The density matrices were reconstructed with losses. All of the experimental data are raw data without error corrections<sup>[<span class=58]." />

      Figure 13.  (a)/(c) and (b)/(d) Reconstructed real and imaginary parts of the input/output density matrix, respectively. The density matrices were reconstructed with losses. All of the experimental data are raw data without error corrections[58].

      通过检验(1)式中CHSH不等式的违反情况可以表征存储前和存储后的纠缠性质, 其中$\theta_ {\rm{A}}$$\theta_ {\rm{S}}$分别为反斯托克斯光子和斯托克斯光子传播路径中插入的半波片的角度值, 其中$\theta_ {\rm{A}}$ = 0, $\theta_ {\rm{S}}={\text π}/8$, $\theta_ {\rm{A}}'={\text π}/4$$\theta_ {\rm{S}}'=3 {\text π}/8$. 计算的S值在存储前为2.40 ± 0.04, 存储后为2.26 ± 0.08, 表明存储过程中纠缠得到保持.

    • 作为信息载体, 光子不仅可以被纠缠在一个DOF中, 例如偏振、空间模式、时间bin和路径等, 也可以同时在多个DOF中存在纠缠, 例如超纠缠[82,83], 或者在不同的DOF之间相互纠缠, 例如杂化纠缠[84]. 多个DOF的纠缠相比单个DOF的纠缠存在许多优点, 例如, 多DOF超纠缠可以实现更有效的Bell测量, 使得超越传统线性光学阈值的超密集编码成为可能, 此外还存在很多其他应用. 同时, 多个DOF的纠缠可以利用不同DOF的优点, 例如, 纠缠在偏振或时间域的光子可以通过光纤有效传输, 而编码在OAM DOF的光子则提高了信道容量[66,85-88]. 因此, 在提高信道容量和改善网络的兼容性方面利用多DOF的纠缠具有巨大潜力. 然而构建这种量子网络需要量子存储来建立不同网络节点之间的纠缠. 最近, 文献[89]报道了在固体存储器中实现由偏振和时间bin DOF组成的$2 \otimes 2$超纠缠的量子存储. 在存储器之间建立多DOF的纠缠将是迈向高兼容量子网络的关键一步, 然而到目前为止尚无这方面的实验报道. 2016年, 我们小组实现了两个独立原子系综之间多DOF超纠缠和不同DOF之间杂化纠缠的量子存储[59].

      实验装置如图14所示. 在实验中, 由原子自旋激发态与单光子之间的偏振纠缠以及自旋激发态与单光子之间的OAM纠缠组成的超纠缠是通过MOT A中的SRS过程直接产生的, 生成的态(未归一化)可以表示为

      Generation and storage of entanglement in multiple DOFs <sup>[<span class=59]: (a) Simplified experimental set-up; (b) energy diagram and time sequence; the real parts of the constructed density matrices for the two-dimensional polarization entanglement ((c), (d)) and the three-dimensional OAM entanglement ((e), (f)), before ((c), (e)) and after ((d), (f)) storage." />

      Figure 14.  Generation and storage of entanglement in multiple DOFs [59]: (a) Simplified experimental set-up; (b) energy diagram and time sequence; the real parts of the constructed density matrices for the two-dimensional polarization entanglement ((c), (d)) and the three-dimensional OAM entanglement ((e), (f)), before ((c), (e)) and after ((d), (f)) storage.

      其中, $\left| {{D_ {\rm{A}}}} \right\rangle $$\left| {{U_ {\rm{A}}}} \right\rangle $分别表示MOT-A中路径U和D的自旋激发态; $\left| {{H_{{\rm{S}}1}}} \right\rangle $$\left| {{V_{{\rm{S}}1}}} \right\rangle $分别表示信号光子1的水平极化和垂直偏振; $\theta_1$为路径U和D的相位差, 实验中将其设为零; $\left| { - {m_ {\rm{A}}}} \right\rangle $为MOT A中量子数为−m的OAM对应的本征模, $\left| {{m_{ {\rm{S}}1}}} \right\rangle $为信号光子1的OAM量子数m. |cm|2为激发概率. 这里m的取值从−1到1, 通常cm = c−m.

      借助于马赫-曾德尔干涉仪, 将产生的信号光子1送到并存储在MOT B中的原子系综, 从而在MOT A和MOT B的自旋激发态之间建立了包含路径和OAM的超纠缠:

      其中$\left| {{N_ {\rm{B}}}} \right\rangle $$\left| {{M_ {\rm{B}}}} \right\rangle$分别表示MOT B中路径N和M对应的自旋波; $\alpha = {c_{m = 0}}{\rm{/}}{c_{m = 1}}$, ${c_{m = 1}}={c_{m = - 1}}$. $\left| L \right\rangle $, $\left| G \right\rangle $$\left| R \right\rangle $分别表示$\left| -1 \right\rangle $, $\left| 0 \right\rangle $$\left| 1 \right\rangle $. 在MOT B中对信号1进行100 ns存储, 在MOT A中将自旋激发态存储200 ns后, 将这两种自旋激发态分别读出为信号光子1和2, 由此产生的光子-光子态为

      为了刻画纠缠, 分别构建了偏振DOF和OAM DOF光子纠缠的密度矩阵, 如图14所示. 根据密度矩阵计算得到二维偏振纠缠的存储保真度为89.7% ± 3.8%. 存储前和存储后CHSH不等式参数分别为SP = 2.6 ± 0.03和SP = 2.51 ± 0.05 (未进行任何噪声纠正). 对于三维OAM纠缠, 对应的保真度为91.1% ± 4.5%.

      从(9)式可以出两个DOF的纠缠是相互独立的, 因此要充分证明超纠缠需要对每个DOF进行独立测量或者进行多DOF的联合测量. 然而由于实验中光子计数率较低以及不同DOF之间存在存储效率差异, 导致联合测量需要很长的测量时间, 由此使得采用包含多个DOF的联合测量的可能性降低, 因此我们采用了独立测量每个DOF纠缠的方法. 然而由于实验元件是与偏振相关的, 所以OAM纠缠度的测量与偏振相关. 为了达到证明的完备性, 实验中采用检验不同OAM DOF子空间中的偏振纠缠来检验其独立性.

      多DOF组成的另外一种重要形式是杂化纠缠, 即在多个DOF之间存在相互纠缠. 通过对实验装置的微小改变, 可以在MOT A中产生信号光子1与原子自旋激发态之间的偏振-路径杂化纠缠. 之后利用特殊设计的Sagnac干涉仪将光子偏振信息转换为OAM信息. 通过这种方法, 在光子的OAM DOF和不同路径的原子自旋激发态之间建立了杂化纠缠, 这个态表示为$\left| {{\psi _{{\text{1-hybrid}}}}} \right\rangle $. 进而将信号光子1输入并存储囚禁在MOT B的原子系综中, 从而在两个分立的原子系综中建立了两个自旋激发态之间的杂化纠缠, 这个态表示为$\left| {{\psi _{{\text{2-hybrid}}}}} \right\rangle $.

      为了刻画分立原子系综之间杂化纠缠的性质, 将原子自旋波激发态映射到光子态$\left(\left| {\psi {'_{{\text{2-hybrid}}}}} \right\rangle \right)$上. 通过密度矩阵重构, 得到密度矩阵$\left| {\psi {'_{{\text{1-hybrid}}}}} \right\rangle $$\left| {\psi {'_{{\text{2-hybrid}}}}} \right\rangle $, 分别对应图15(a)图15(b)(仅为实部). 通过比较$\left| {\psi {'_{{\text{1-hybrid}}}}} \right\rangle $$\left| {\psi {'_{{\text{2-hybrid}}}}} \right\rangle $的密度矩阵, 计算出纠缠的保真度为93.6% ± 1.4%.

      Real parts of the constructed density matrices for hybrid entanglement: (a) Before storage; (b) after storage <sup>[<span class=59]." />

      Figure 15.  Real parts of the constructed density matrices for hybrid entanglement: (a) Before storage; (b) after storage [59].

      同时测量了双光子干涉来进一步表征杂化纠缠的特性. 图16(a)图16(b)分别表示存储前后的干涉曲线. 存储前平均可见度为93.1%, 存储后平均可见度为84.6%. 这两个可见度值都大于违背Bell不等式的阈值70.7%, 充分证明了两个原子系综之间确实存在杂化纠缠.

      Interference curves of the two-photon correlations for hybrid entanglement <sup>[<span class=59]: (a) Before storage; (b) after storage. The error bars are estimated from Poisson statistics and represent as ± s.d. All the data are raw and not subjected to noise correction." />

      Figure 16.  Interference curves of the two-photon correlations for hybrid entanglement [59]: (a) Before storage; (b) after storage. The error bars are estimated from Poisson statistics and represent as ± s.d. All the data are raw and not subjected to noise correction.

    3.   结 语
    • 我们首次采用拉曼方案实现了真单光子、OAM纠缠、偏振纠缠以及多DOF超纠缠和杂化纠缠的存储. 因为利用拉曼协议可以存储具有大带宽的信息载体, 从而能够以更高的时钟速率处理量子信息, 因而拉曼存储是量子信息领域的一个研究热点. 我们取得的这些进展无疑迈出了构建高速量子网络的重要一步. 尽管如此, 真正实现实用化的拉曼量子存储器还需要走漫长的路, 还需要在诸多方面进行改进、优化和提高. 虽然我们的存储器比使用EIT方案的存储器工作带宽增加了一个数量级, 但目前所存储的光子带宽仍然不太宽. 限制拉曼存储带宽的基本因素主要由基态和存储态的间隔决定(在这里指85Rb原子的5S1/2 (F = 3)和5S1/2 (F = 2)两个态). 如果带宽太大, 例如, 大于基态和存储态之间间隔大小的一半, 则反斯托克斯光子或控制光就有可能与另外一个产生耦合, 从而与其他$\Lambda$结构相竞争, 进而降低存储的保真度和效率. 存储带宽同样还受功率、失谐、耦合光的带宽以及系综OD等因素的影响. 由于绝热演化的要求, 较大的存储带宽需要较大的失谐量, 因而要求失谐量要大于带宽, 这就需要用较强的耦合光功率或者较大的OD来补偿由于较大失谐造成的较低的光子和介质之间的相互作用. 因此, 在实际应用中, 存储性能取决于整个系统各种参数的优化设计.

      另外, 在以上所报道的实验中存储效率并不高, 存储时间也很短, 为此, 必须对系统进行优化改进. 在前向读出结构中存储效率主要取决于读出光子的再吸收情况, 理论预测的最大效率仅可以达到约60%. 而利用反向读出结构则可以克服这一缺点, 效率可以超过90%. 另外通过补偿磁场或者利用磁场不敏感态, 同时利用光晶格减小原子运动, 可以提高存储时间, 使之达到毫秒甚至几百毫秒的存储时间. 此外, 动态解耦方法也可以用来提高存储时间.

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