Progress on Chip-Based Spontaneous Four-Wave Mixing Quantum Light Sources
基于芯片的自发四波混频量子光源进展

For applications such as boson sampling [41] and quantum computing [2,3], it is necessary to generate spectrally pure single photons, which is crucial to achieve high visibility in multiphoton interference experiments.
对于玻色采样[41]和量子计算[2,3]等应用来说,生成光谱纯度高的单光子是必要的,这对于在多光子干涉实验中实现高可见度至关重要。

The two photon states generated by SFWM can be represented by Schmidt decomposition of signal photons and idler photon systems
基于 SFWM 产生的两个光子态可通过信号光子和对目光子系统的 Schmidt 分解来表示

Thus, the purity of daughter photons is defined by the reciprocal of the Schmidt number K [42]
因此,女儿光子的纯度由 Schmidt 数 K 的倒数定义[42]

where ${\widehat{\rho }}_{\mathrm{s}}={\mathrm{Tr}}_{\mathrm{i}}\widehat{\rho },{\widehat{\rho }}_{\mathrm{i}}={\mathrm{Tr}}_{\mathrm{s}}\widehat{\rho }$, and $\widehat{\rho }=\mid \mathrm{\Psi }\rangle \langle \mathrm{\Psi }\mid$ is the density matrix of two photon states.
其中 ${\widehat{\rho }}_{\mathrm{s}}={\mathrm{Tr}}_{\mathrm{i}}\widehat{\rho },{\widehat{\rho }}_{\mathrm{i}}={\mathrm{Tr}}_{\mathrm{s}}\widehat{\rho }$ ，而 $\widehat{\rho }=\mid \mathrm{\Psi }\rangle \langle \mathrm{\Psi }\mid$ 是两光子态的密度矩阵。

Therefore, in order to obtain completely pure photons, it can only be achieved in a single Schmidt mode, in which the two-photon state can be written as the product of the signal photon state and the idle frequency photon state
因此,为了获得完全纯净的光子,只能在单个施密特模式下实现,在该模式中,两光子态可以写为信号光子态和空闲频率光子态的乘积and amplitude function of the two-photon joint spectrum can be written as $F\left({\omega }_s,{\omega }_i\right)=f\left({\omega }_s\right)f\left({\omega }_i\right)$. Hence, the pure two-photon state can be achieved by making $F\left({\omega }_s,{\omega }_i\right)$ factorable.
双光子联合谱的振幅函数可以写为 $F\left({\omega }_s,{\omega }_i\right)=f\left({\omega }_s\right)f\left({\omega }_i\right)$ 。因此,通过使 $F\left({\omega }_s,{\omega }_i\right)$ 可分解,可以实现纯双光子态。

Several approaches can be adopted to improve the purity of two-photon states, including engineering an optimal cross-section of the waveguide and selecting appropriate pump wavelength and bandwidth [43]. It is also possible to use a resonant cavity structure for this purpose [44]. Due to the cavity enhancement, the SFWM signal and idler photons are naturally and independently distributed under a pulsed, broadband resonant pump [45]. With further optimization in the cavity coupling structure, spectral purity close to unit has been reported [46].
提高双光子态纯度的几种方法包括设计最优波导横截面以及选择合适的泵浦波长和带宽[43]。使用共振腔结构也可以实现这一目的[44]。由于共振腔增强效应,在脉冲宽带共振泵浦下,参数下转转换信号和关联光子自然地独立分布[45]。通过进一步优化腔耦合结构,已经报道获得接近单一的频谱纯度[46]。

In the absence of losses and noise photons, the count rate of signal (idler) photons is identical to the PGR as SFWM photons are generated in pairs. Taking signal photons as example (the case for idler photons is similar), the single counting rate can be expressed as [17]
在没有损耗和噪声光子的情况下,信号(闲置)光子的计数率与 SFWM 光子成对产生时的 PGR 相同。以信号光子为例(闲置光子的情况类似),单计数率可表示为[17]

where A₁ is a constant and I_sc is a double integral value (detailed process can be found in [17]). σ₀ is the filtering bandwidth of the signal photon, and σ_p is the bandwidth of the pump pulse. As indicated in Eq. 10, the single counting rate depends quadratically on the peak power of the pump. This dependence is useful to determine the severeness of the noise photons in the system, such as SpRS, which scales linearly with the bump power. In addition, the single counting rate is proportional to the ratio of filtering bandwidth to pump bandwidth. On the one hand, this is because as the filtering bandwidth increases, more signal photons are collected. On the other hand, as the pump bandwidth increases, the pulse width decreases, which reduces the interaction time for pump photons.
式中，A1 为常数，Isc 为双重积分值(详细过程见[17])。σ0 为信号光子的滤波带宽，σp 为泵浦脉冲的带宽。如式 10 所示，单计数率随泵浦峰值功率呈二次方依赖关系。这种依赖性有利于确定系统中噪声光子(如 SpRS)的严重程度,该噪声光子与泵浦功率呈线性关系。此外,单计数率与滤波带宽和泵浦带宽的比例成正比。一方面,这是因为滤波带宽增加时,可收集更多信号光子。另一方面,泵浦带宽增加时,脉冲宽度减小,从而减少了泵浦光子的相互作用时间。

A coincidence refers to an event that the signal and idler photon are simultaneously detected. The coincidence rate of an SFWM source can be expressed as [17]
巧合指的是信号光子和闲置光子同时检测到的事件。SFWM 源的巧合率可以表示为 [ 17]

It can be seen from Eqs. 10 and 11 that both single counting rate and coincidence counting rate exhibit a quadratic dependence on the pump power, which comes from the fact that SFWM requires annihilation of two pump photons. Figure 1C illustrates qualitatively the dependence of the single and coincidence counting rates on the pump power. This is an important difference from an SPDC photon pair source, whose single and coincidence counting rates are linearly dependent on its pump power, as shown in Fig. 1B. This difference can be understood from their respective photon pair generation processes. An SPDC process corresponds to annihilation of a parent photon and creation of a pair of daughter photons.
从等式 10 和 11 可以看出,单计数率和偶然计数率都呈现出对泵浦功率的二次依赖性,这是由于 SFWM 需要消耗两个泵浦光子。图 1C 定性地说明了单计数率和偶然计数率对泵浦功率的依赖关系。这与 SPDC 光子对源有重要区别,后者的单计数率和偶然计数率线性依赖于其泵浦功率,如图 1B 所示。这种差异可从它们各自的光子对生成过程理解。SPDC 过程对应于一个母光子的湮灭和一对女儿光子的产生。

The major difference between single counting rate and coincidence counting rate is the ratio between σ₀ and σ_p. Here, we consider two limit cases [17]. When σ₀ ≪ σ_p, it is equivalent to that the pump has a large frequency bandwidth, and the pump light at each frequency may produce photon pairs. Because of the narrow bandwidth of the filter, only the photon pairs produced by the pump light at a specific frequency can be detected, and the coincidence counting rate is linearly correlated to $\frac{{\sigma }_0^2}{{\sigma }_p^2}$. On the contrary, when σ₀ ≫ σ_p, the photon pairs generated by the pump light component of any frequency can be detected, so it is linearly correlated to $\frac{{\sigma }_0}{{\sigma }_p}$.
单次计数率和 coincidence 计数率之间的主要区别在于σ0 和σp 之间的比率。这里,我们考虑两个极限情况[17]。当σ0≪σp 时,相当于泵浦具有较大的频带宽度,每个频率的泵浦光都可以产生光子对。由于滤波器的带宽较窄,只能检测到特定频率泵浦光产生的光子对,coincidence 计数率与 $\frac{{\sigma }_0^2}{{\sigma }_p^2}$

Without noise photons, the coincidence counting rate in the experiment can be simplified as [47]
在没有噪声光子的情况下,实验中的巧合计数率可以简化为[47]

where μ is the number of photon pairs generated per pulse, which is proportional to (γP_pL)² according to Eq. 11. η_s and η_i are the collection efficiencies of signal photon and idler photon, respectively, and R is the repetition rate of pump pulse. From coincidence counting rate, we can obtain the on-chip PGR using
其中μ为每个脉冲产生的光子对数量,根据公式 11 与(γPpL)2 成正比。ηs 和η分别为信号光子和闲置光子的收集效率,R 为泵浦脉冲的重复率。从巧合计数率,我们可以使用

which represents the rate of photon pairs generated at unit power on chip. PGR allows a direct comparison of different sources’ abilities to generate photon pairs at an identical pump power.
这表示单位功率下芯片上产生的光子对的速率。PGR 允许对不同源产生光子对的能力在相同泵浦功率下进行直接比较。

In addition to single and coincidence counting rates, CAR is another important figure of merit to parametric conversion source. In the absence of background noise, for photon pair sources generated by spontaneous parametric processes, CAR = 1/μ [48], as shown in Fig. 1D. In order to avoid multiphoton pairs, the pump power is often set such that μ < 0.1. In the presence of channel loss and the detector dark count noise in the experiment, CAR can be expressed as
除了单独和撞击计数率之外，CAR 也是参数转换源的另一个重要性能指标。如图 1D 所示，在没有背景噪声的情况下，对于由自发参数过程产生的光子对源，CAR = 1/μ [ 48]。为了避免多光子对，通常会将泵浦功率设置为 μ < 0.1。在存在信道损耗和探测器暗计数噪声的实验中，CAR 可以表示为

where d_{s, i} is the dark count rate of detectors and τ is the full width at half maximum (FWHM) of coincidence count peak. Owing to optical losses and the detector dark counts, experimentally measured CAR values are typically lower than their theoretical expectations [49].
其中 ds, i 是探测器的暗计数率,τ是偶然计数峰的全宽半高值(FWHM)。由于光学损耗和探测器暗计数,实验测量的 CAR 值通常低于其理论预期值[49]。

When considering only the channel loss, according to Eq. 12, CAR can be expressed as
当仅考虑信道损失时,根据公式 12,CAR 可以表示为

According to Eq. 15, in the case of obtaining the same level of coincidence counting, CAR increases with the pulse repetition rate, which is ultimately determined by the timing resolution of the single-photon detectors (SPDs) [47]. Therefore, to evaluate the performance of an SFWM photon pair source under pulsed excitation, both the pump power and the repetition frequency must be taken into consideration.
根据等式 15,在获得相同水平的巧合计数的情况下,CAR 随脉冲重复率的增加而增加,这最终由单光子探测器(SPD)的时间分辨率决定[47]。因此,为了评估脉冲激励下 SFWM 光子对源的性能,必须同时考虑泵浦功率和重复频率。

Unlike SPDC relying upon second-order nonlinear effect, SFWM is a third-order nonlinear process that can occur in crystals also with center symmetry. As a result, a broader set of materials is suitable for generating photon pairs through SFWM. Currently, notable platforms include silicon on insulator (SOI), silicon nitride (Si ₃N₄), Hydex, AlGaAs, and several other III-V semiconductor materials [33,34]. Table 2 lists the relevant characteristics of commonly used materials.
与依赖于二阶非线性效应的 SPDC 不同,SFWM 是一种三阶非线性过程,也可以在具有中心对称性的晶体中发生。因此,通过 SFWM 产生光子对的材料范围更广。目前,值得注意的平台包括硅基绝缘体(SOI)、硅氮化物(Si 3N4)、Hydex、AlGaAs 以及其他几种 III-V 半导体材料[33,34]。表 2 列出了常用材料的相关特性。

Among the above listed materials, SOI platform is complementary metal oxide semiconductor (CMOS) compatible and has already had a vast library of developed optical components including filters [51–53], phase modulators [54,55], and beam splitters [56]. Naturally, it has become the most explored platform for SFWM photon pair generation. However, the generation rate is limited by both its relatively high waveguide propagation loss (about several dB/cm) [57] and, more detrimentally, nonlinear losses arising from two-photon absorption (TPA) and/or free carrier absorption (FCA) [58] that cause saturation at a modest pump power [37,59].
在上述材料中,SOI 平台与互补金属氧化物半导体(CMOS)兼容,并已经拥有了包括滤波器[51-53]、相位调制器[54,55]和光束分割器[56]在内的大量已开发的光学元件库。自然,它已成为 SFWM 光子对生成的最广泛研究的平台。然而,其生成速率受到相对较高的波导传播损耗(约几 dB/cm)[57]以及更为严重的由两光子吸收(TPA)和/或自由载流子吸收(FCA)引起的非线性损耗[58]的限制,这些损耗会在较低的泵浦功率下导致饱和[37,59]。

To suppress TPA and FCA effects, incorporation of organic materials into silicon waveguides was proposed, with a theoretical simulation showing that a silicon-organic hybrid waveguide allows a much higher saturation pump power than SOI [60]. However, this scheme is yet to be experimentally demonstrated.
为抑制 TPA 和 FCA 效应,将有机材料掺入硅波导被提出,理论模拟显示硅-有机杂化波导允许比 SOI 更高的饱和泵浦功率[ 60 ]。但是,这种方案尚未得到实验验证。

Silicon nitride is another CMOS-compatible material with low transmission loss. Its relatively high thermal stability makes it highly competitive for constructing microring devices with wavelength stability [61,62]. In addition, its large bandgap (238 nm) eliminates the effect of TPA at telecom wavelengths [62]. However, its relatively low third-order nonlinearity limits the brightness as an SFWM light source. Due to the spectral overlap between SpRS and SFWM in silicon nitride platform [63], a relatively high level of uncorrelated photons will mix into the generated photon pairs, resulting in an approximately one order of magnitude reduction in the measured CAR from the corresponding theoretical value [49]. Combining the characteristic of high nonlinearity and high bandgap, ultra-silicon-rich nitride (USRN) is newly proposed, which has a large third-order nonlinear coefficient [64,65], and can overcome nonlinear loss. However, its fabrication process is yet to be fully developed to reduce its presently considerable waveguide propagation losses [66].
氮化硅是另一种 CMOS 兼容材料,具有低传输损耗。其相对较高的热稳定性使其在构建具有波长稳定性的微环装置方面具有很强的竞争力[61,62]。此外,其较大的带隙(238 nm)消除了在电信波长下 TPA 的影响[62]。然而,其相对较低的三阶非线性限制了其作为 SFWM 光源的亮度。由于硅酰氮平台上 SpRS 和 SFWM 的光谱重叠[63],大量不相关的光子会混入产生的光子对中,导致测量的 CAR 比相应的理论值降低约一个数量级[49]。结合高非线性和高带隙的特性,新近提出的超硅富氮化物(USRN)具有较大的三阶非线性系数[64,65],可克服非线性损耗。然而,其制造工艺尚未完全成熟,以减少目前相当大的导波传播损耗[66]。

Quantum light sources based on CMOS-incompatible semiconductor materials have also been demonstrated, including InP [67], GaInP [68], and AlGaAs [69,70]. Among them, AlGaAs material has attracted the greatest attention due to its high third-order nonlinearity as well as its ability to mitigate the TPA impact by adjusting the Al concentration and thus the alloy bandgap [71]. Promisingly, an AlGaAs microring has been reported to achieve a high PGR of 20 GHz/mW², which is more than two orders of magnitude higher than the same structure of silicon. However, a system-level demonstration for quantum information processing remains elusive on this platform due to its CMOS incompatibility and lack of well-developed optical components.
基于 CMOS 不兼容半导体材料的量子光源也已得到了证明,包括 InP[67]、GaInP[68]和 AlGaAs[69,70]。其中,AlGaAs 材料由于其高三阶非线性以及通过调整铝浓度和合金带隙来缓解 TPA 影响的能力[71],受到了最大的关注。值得注意的是,一个 AlGaAs 微环谐振器已报告实现了 20 GHz/mW2 的高 PGR,这比同样结构的硅高两个数量级。然而,由于其与 CMOS 不兼容以及缺乏成熟的光学元件,在量子信息处理系统级演示方面仍然缺乏。

Lithium niobate (LiNbO₃) is a widely used material platform for integrated quantum photonics because of its high second-order nonlinearity and large electro-optic effect [72]. However, its modest third-order nonlinearity makes it unattractive to generate photon pairs through SFWM. So far, most applications uses its SPDC processes to generate photon pairs [24,50]. To combine advantages of different material platforms, heterogeneous integration has been proposed as a way forward [73–76]. It not only reduces costs but also achieves better performance compared to components based on individual materials [77,78].
氮化锂(LiNbO3)是一种广泛应用于集成量子光子学的材料平台,因其具有高二阶非线性和大电光效应[72]。然而,其适度的三阶非线性使其无法通过四波混频来发生光子对[24,50]。到目前为止,大多数应用都使用其 SPDC 过程来产生光子对[24,50]。为了结合不同材料平台的优势,异构集成被提出为一条前进的道路[73-76]。它不仅降低了成本,而且与基于单一材料的组件相比,还能获得更好的性能[77,78]。

On above material platforms, a variety of waveguide structures have been explored for photon pair generation. In this subsection, we review three most important structures: plain waveguide, microring, and photonic crystal (PhC) waveguide. Among them, microring and PhC structures possess desirable characteristics, e.g., narrow spectral bandwidth, that are not possible with plain waveguides [79].
在上述材料平台上,已探索了各种波导结构用于光子对生成。在这一小节中,我们回顾了三种最重要的结构:普通波导、微环和光子晶体(PhC)波导。其中,微环和 PhC 结构具有窄频带等理想特性,这是普通波导所无法实现的[ 79 ]。

As photons are generated in pairs, SPDC or SFWM sources can always be used to produce single photons through heralding, i.e., the detection of one photon indicates the presence of the other [18] (see Fig. 4A). The probability of obtaining heralding event by detecting this other photon is referred to as heralding efficiency, which can be expressed as [106]
由于光子是成对产生的,SPDC 或 SFWM 源始终可用于通过先报的方式产生单光子,即检测到一个光子表示另一个光子的存在[18](见图 4A)。通过检测这个另一个光子获得先报事件的概率称为先报效率,可以表示为[106]

where C_c is the coincidence counting rate and $S_C^{i,s}$ are respective single counting rates of the idler and signal photons, as defined previously in the “Characterization of Four-Wave Mixing Process” section.
其中 Cc 是巧合计数率, $S_C^{i,s}$ 为先前在"四波混频过程的表征"部分定义的信号光子和闲置光子的相应单计数率。

Ideally, heralding efficiency equals to the collection efficiency of signal (idler) photons. In practice, actual measured heralding efficiency is lower than the corresponding collection efficiency, because of the presence of noise counts arising from residual pump light, spontaneous Raman photons and detector dark counts, as well as spectral filtering loss [107]. As shown in Fig. 3, a portion of photon pairs have just one photon within the spectral filter windows and will not contribute to the coincidence. This filtering loss can be mitigated through engineering a source to generate narrowband photons [43], for example, microrings.
理想情况下,引导效率等于信号(闲置)光子的收集效率。实际上,实测的引导效率低于相应的收集效率,这是因为存在噪声计数,包括残余泵浦光、自发拉曼光子和探测器暗计数,以及频谱滤波损耗[107]。如图 3 所示,一部分光子对只有在频谱滤波窗内出现一个光子,不会贡献到同时性计数。这种滤波损耗可以通过设计一个产生窄带光子的光源来缓解[43],例如微环谐振器。

In additional to heralding efficiency, photon number purity 1 − g⁽²⁾(0) is another important parameter to evaluate the quality of a heralded single-photon source. This quantity can be experimentally measured using a setup shown in Fig. 4B. If there is just one photon in the heralded mode, only one of the two detectors behind the beam splitter will register a photon. Therefore, one can use the time-correlated single-photon counting (TCSPC) module to analyze the coincidence counts between detectors to measure the purity of a single photon, which can be expressed as [108]
除了预示效率,光子数纯度 1-g(2)(0)是评估启动单光子源质量的另一个重要参数。可以使用图 4B 所示的设置实验测量这一数量。如果启动模式中只有一个光子,则光束分光器后的两个探测器中只有一个会检测到光子。因此,可以使用时间相关单光子计数(TCSPC)模块分析探测器之间的同时计数,以测量单光子的纯度,表达为[108]

where C₁ is the counting rate in SPD 1, C₁₂₍₃₎ is the twofold coincidence counting rate between SPD 1 and SPD 2 (SPD 3), and C₁₂₃ is the threefold coincidence counting rate across all detectors. τ is the relative time delay between SPD 2 and SPD 3. When the measured g⁽²⁾(0) < 0.5, the source can be regarded as nonclassical.
其中 C1 是 SPD 1 的计数率,C12(3)是 SPD 1 和 SPD 2(SPD 3)之间的双重巧合计数率,C123 是所有探测器之间的三重巧合计数率。τ是 SPD 2 和 SPD 3 之间的相对时间延迟。当测量的 g(2)(0) < 0.5 时,该源可视为非经典。

As a single-photon source, it is desirable to keep both a low g⁽²⁾(0) and a high brightness. However, the g⁽²⁾(0) value increases with the source brightness for a heralded single-photon source. Because of the random nature of SFWM processes, simultaneous generation of two photon pairs is equally probable as two pairs generated at separate times. One approach to this problem is to employ either spatial [18] or temporal multiplexing [19,20] to combine weak single-photon sources into a brighter one without deterioration in the single-photon purity. Figure 5A shows a spatial multiplexing scheme based on PhC waveguides. Here, the pump light is split into N ways through a 1 × N beam splitter and excites N× identical SFWM devices, each of which generates photon pairs problematically. To ensure the purity of the multiplexed source, that is, to produce heralded single photons with the same wavelength, the filtering settings of all SFWM components must be set identical. Finally, photons in the heralded mode are actively routed into a single path through a low-loss optical switch. Temporal multiplexing is an alternative method to resolve the conflict between brightness and photon number purity. As shown in Fig. 5B, it increases the source brightness by actively routing heralded single photons of different temporal modes into a single time bin using optical switches and a fiber-based photon storage line.
作为一个单光子源,保持既低的 g(2)(0)又高的亮度是理想的。然而,对于一个预告单光子源,g(2)(0)值随着源亮度的增加而增加。由于 SFWM 过程的随机性,同时产生两对光子与分别产生两对光子同样可能。解决这个问题的一种方法是采用空间[18]或时间多路复用[19,20]来将弱的单光子源组合成一个更亮的单光子源,而不会降低单光子纯度。图 5A 显示了一种基于 PhC 导波管的空间多路复用方案。在这里,泵浦光通过 1×N 光分路器被分成 N 路,并激励 N 个相同的 SFWM 器件,每个器件都会产生成对的光子。为了确保多路复用源的纯度,即产生具有相同波长的预告单光子,所有 SFWM 组件的滤波设置必须相同。最后,预告模式中的光子通过低损耗光学开关被主动引导到单一通路。时间多路复用是解决亮度和光子数纯度冲突的另一种方法。如图 5B 所示,它通过使用光学开关和基于光纤的光子存储线将不同时间模式的预告单光子主动引导到单一时间窗,从而提高了源亮度。

Entangled photon pairs are an important resource in the field of quantum information and find a wide range of applications for quantum communication, quantum computing, quantum measurement, and quantum imaging [110]. Their generation via on-chip SFWM process has advantages in scalability, compactness, and reconfigurability [111]. So far, various degrees of freedom entanglements have been demonstrated, including polarization [112], time bin [113,114], frequency bin [79,115], and transverse mode [116].
纠缠光子对是量子信息领域的一项重要资源,在量子通信、量子计算、量子测量和量子成像等方面有广泛应用[110]。通过片上四波混频(SFWM)过程生成纠缠光子对具有可扩展性、紧凑性和可重构性等优势[111]。目前已经演示了包括偏振[112]、时间窒碍[113,114]、频率窒碍[79,115]和横向模式[116]在内的各种自由度的纠缠。

Polarization is the most commonly exploited degree of freedom in photonic entanglement systems. While an individual SFWM source cannot directly generate polarization entanglement like type I or II SPDC processes [117,118], it is possible to entangle photon pairs generated by a pair of identical chips sharing the same pump through on-chip polarization manipulation and subsequent coherent combining [112]. As shown in Fig. 6, when injecting a 45^∘ linearly polarized light, the generated two-photon state can be written as $\left|\psi ⟩\right.=1/\sqrt{2}\left(|\mathit{TE},\mathit{TE}\rangle {}_{s,i}+e^{\mathit{i\varphi }}\mathit{TM}\mathit{TM}\rangle {}_{s,i}\right)$. Because this structure utilizes the symmetry between two cascading waveguides, it eliminates the degradation of the polarization state caused by polarization drift and propagation loss of the waveguides [112].
偏振是光子纠缠系统中最常被利用的自由度。尽管单个 SFWM 源不能像 I 型或 II 型 SPDC 过程那样直接产生偏振纠缠[ 117, 118]，但通过芯片内偏振操纵和随后的相干组合[ 112]，有可能纠缠由两个相同芯片共享同一泵源产生的光子对。如图 6 所示，当注入 45°线性偏振光时，所生成的二光子态可写为 $\left|\psi ⟩\right.=1/\sqrt{2}\left(|\mathit{TE},\mathit{TE}\rangle {}_{s,i}+e^{\mathit{i\varphi }}\mathit{TM}\mathit{TM}\rangle {}_{s,i}\right)$ 。由于这种结构利用了两个级联波导之间的对称性，因此消除了由于波导偏振漂移和传播损耗导致的偏振状态退化[ 112]。

Time-bin entanglement is robust against environment disturbance owing to the fact that all photons have the same properties in all degrees of freedom other than time. This property makes it ideal for entanglement distribution experiments over long distances [100,119]. As energy-time entanglement can be conveniently generated using simple waveguide structures, it has been widely employed in practical implementations [120,121].
时间-bin 缠结对环境干扰是鲁棒的,因为所有光子在除时间之外的所有自由度上都具有相同的属性。这一性质使其非常适合用于长距离遥测分配实验[100,119]。由于能量-时间缠结可以使用简单的波导结构方便地生成,它已被广泛应用于实际实施中[120,121]。

Energy-time entangled photon pairs can be verified in two ways. The first is to verify entanglement by nonlocal dispersion compensation effects [100,122], using a setup shown in Fig. 7A. Due to the simultaneity of photons in each generated pair, the detected photon pairs will exhibit strong time correlation provided that they are under the same measurement basis [123].
能量-时间纠缠光子对可以通过两种方式进行验证。第一种是通过非局域色散补偿效应来验证纠缠[100,122]，使用图 7A 所示的设置。由于每对生成的光子具有同步性，在相同的测量基础下，检测到的光子对将表现出强时间相关性[123]。

The second verification method is through Franson interference experiments to detect violations of Bell’s inequality [124]. The principle of this setup is shown in Fig. 7B. The signal and idler photons generated by continuous light pumping pass through an asymmetric Mach–Zehnder interferometer (AMZI) and then detected by single-photon detectors. Due to the simultaneity of the generated photons, three coincidence peaks appear, where the coincidence peak of the zero time difference corresponds to both photons passing through the long arms ∣L, L〉_s,i or short arms ∣S, S〉_s,i of interferometer, and the two side peaks correspond to one photon passing through the short arm and the other through the long arm, ∣S, L〉_s,i and ∣L, S〉_s,i. Due to the indistinguishability of which arm the photon actually passes, a time-bin entangled state is post-selected at the central coincidence peak, which can be expressed as $\left|\psi ,⟩\right.=1/\sqrt{2}\left(|S,S\rangle {}_{s,i}+e^{-i\left({\varphi }_{s+i}\right)}LL\rangle {}_{s,i}\right),$ where ϕ_s+i is the phase introduced at the short arms of the AMZI.
第二种验证方法是通过 Franson 干涉实验来检测贝尔不等式的[124]违背。该设置的原理如图 7B 所示。连续光泵浦产生的信号和闲置光子通过不对称的迈赫-塞尔干涉仪(AMZI)然后被单光子探测器检测。由于产生的光子同时性,出现三个同态峰,其中零时间差的同态峰对应两个光子都通过干涉仪的长臂|L,L>s,i 或短臂|S,S>s,i,而两个旁峰对应一个光子通过短臂,另一个通过长臂|S,L>s,i 和|L,S>s,i。由于光子实际通过哪条臂不可辨别,在中央同态峰上事后选择了一个时间 bin 缠结态,可以表示为 e^(i(ϕs+i)),其中ϕs+i 是 AMZI 短臂引入的相位。

Beyond merely the preparation of entanglement source, the applications of on-chip SFWM have gradually become a hot research topic of current research [125–129]. Recently, relevant work has been reported to dynamically control the polarization [130] or phase [131] in the respective SFWM sources to encode different Bell states, which demonstrates its ability to encode through entangled Bell states. Entanglement-based QKD networks have also been realized with SFWM sources on silicon chips. For example, a reconfigurable entanglement distribution network has been demonstrated by integrating several SFWM sources on the same chip and utilizing the quantum interference between the states generated from them [132–134]. Additionally, applications such as multidimensional entanglement, multiple degrees of freedom entanglement, quantum teleportation, and entanglement swapping were also demonstrated using on-chip SFWM sources [125–129].
除了仅仅准备纠缠源之外,片上自发四波混频(SFWM)的应用逐渐成为当前研究的热点话题[125-129]。最近,有相关工作报告动态控制各自 SFWM 源的偏振[130]或相位[131]来编码不同的贝尔态,这展示了它编码通过纠缠贝尔态的能力。基于纠缠的量子密钥分配(QKD)网络也已通过硅芯片上的 SFWM 源实现。例如,通过在同一芯片上集成多个 SFWM 源并利用它们产生的状态之间的量子干涉,实现了可重构的纠缠分配网络[132-134]。此外,还使用片上 SFWM 源展示了多维纠缠、多自由度纠缠、量子隧穿和纠缠交换等应用[125-129]。

As described in the previous sections, many experiments have been conducted to prepare and distribute entangled states at various degrees of freedom through on-chip SFWM. However, integrating the preparation, manipulation, and measurement modules onto a single chip poses new challenges.
正如前几节所述，已经进行了许多实验来通过芯片上的 SFWM 制备和分发各种自由度的纠缠态。然而,将制备、操纵和测量模块集成到单个芯片上会面临新的挑战。

On-chip manipulation and measurement of entangled states require both high source brightness and integration of active optical components. Silicon platform, due to its CMOS compatibility, offers a wide availability of optical components for realizing large-scale programmable quantum photonic integrated circuits. On this frontier, tremendous experimental progress has been demonstrated, including Mach–Zehnder interferometer (MZI) [135], phase modulators [54,55], tunable wavelength filters [51–53], and single-photon detectors [136]. In particular, tunable filters based on coupled resonator optical waveguide (CROW) were demonstrated to offer over 96 dB of on-chip suppression of pumped light, exceeding the performance of commercially available filters [52]. Rapid development of these on-chip components offers prospects of large-scale integration of quantum photonic circuits [54,137].
芯片上操纵和测量纠缠态需要高亮度源和集成主动光学元件。由于 CMOS 兼容性,硅平台拥有实现大规模可编程量子光子集成电路所需的大量光学元件。在这个前沿,已经取得了巨大的实验进展,包括马赫-曾德尔干涉仪(MZI)[ 135 ]、相位调制器[ 54、55 ]、可调谐波长滤波器[ 51-53 ]和单光子探测器[ 136 ]。特别是,基于耦合共振腔光波导(CROW)的可调谐滤波器已被证明可以提供超过 96 dB 的芯片内泵浦光抑制,超过了商业滤波器的性能[ 52 ]。这些芯片组件的快速发展为量子光子电路的大规模集成提供了前景[ 54、137 ]。

The above developments make fully integrated applications on chip no longer just imaginations [35,110]. Figure 8 shows several examples of system-level on-chip applications, which all exploit the compactness of spiral waveguides for integration of multiple SFWM sources. Wang et al. [138] demonstrated the first chip-to-chip entanglement distribution using two spirals, and the outputs of which are combined together to form a polarization entanglement (see Fig. 8A). Figure 8B shows an implementation of multidimensional quantum entanglement on a single chip that integrated 16 identical spiral waveguides and over 550 optical components in total, demonstrating maturity and scalability of silicon quantum photonic technology [139]. In comparison with spirals, microring structures have natural advantages of low pump power requirement and high photon purity, making it suitable for systems with high filtering requirements. As shown in Fig. 8C, chip-to-chip quantum teleportation and entanglement swapping have been demonstrated by integrating an array of microring resonators followed by a programmable quantum photonic circuit [9]. Progressing further, interconnection between multiple chips has been recently demonstrated to prove the viability of large-scale quantum networks [140,141].
上述发展使完全集成的片上应用不再仅仅是想象[ 35, 110]。图 8 展示了几个系统级片上应用的例子,它们都利用螺旋形波导的紧凑性来集成多个 SFWM 源。王等人[ 138]演示了第一个使用两个螺旋的芯片到芯片纠缠分配,其输出被组合形成偏振纠缠(见图 8A)。图 8B 展示了在单片集成 16 个相同螺旋波导和 550 多个光学元件的多维量子纠缠实现,展示了硅量子光子技术的成熟性和可扩展性[ 139]。与螺旋相比,微环结构具有低泵浦功率要求和高光子纯度的天然优势,使其适用于有高滤波要求的系统。如图 8C 所示,通过集成一个微环谐振器阵列,并后接一个可编程的量子光子电路,已经实现了芯片到芯片的量子隧穿传输和纠缠交换[ 9]。进一步来说,多芯片之间的互连最近也已经被证明了大规模量子网络的可行性[ 140, 141]。

Along with rapid developments of quantum photonic integration technology, the number of integrated components on a single chip is growing exponentially, with a current record of 2,500 components for monolithic integration [142]. Integrated SPD is the last piece to realize fully integrated quantum information processing. The integration of SPD has been demonstrated on GaAs [143], Si [144], and Si₃N₄ [145] and can be comparable to the performance of non-integrated detectors.
随着量子光子集成技术的快速发展,单片上集成的组件数量呈指数级增长,目前最高记录为 2,500 个用于单片集成[142]。集成化的单光子探测器是实现完全集成量子信息处理的最后一环。单光子探测器已在 GaAs[143]、Si[144]和 Si3N4[145]上实现集成,性能可与非集成探测器相媲美。