Narrowband biphoton generation near atomic resonance
原子共振附近的窄带双光子产生

Received April 16, 2008; accepted July 16, 2008; posted August 6, 2008 (Doc. ID 94869); published September 25, 2008
接收日期：2008 年 4 月 16 日;2008 年 7 月 16 日接受;发布于 2008 年 8 月 6 日（文档 ID 94869）;发布时间：2008 年 9 月 25 日

Generating nonclassical light offers a benchmark tool for fundamental research and potential applications in quantum optics. Conventionally, it has become a standard technique to produce nonclassical light through the nonlinear optical processes occurring in nonlinear crystals. We describe this process using cold atomic-gas media to generate such nonclassical light, especially focusing on narrowband biphoton generation. Compared with the standard procedure the new biphoton source has such properties as long coherence time, long coherence length, high spectral brightness, and high conversion efficiency. Although there exist two methodologies describing the physical process, we concentrate on the theoretical aspect of the entangled two-photon state produced from the four-wave mixing in a multilevel atomic ensemble using perturbation theory. We show that both linear and nonlinear optical responses to the generated fields play an important role in determining the biphoton waveform and, consequently, on the two-photon temporal correlation. There are two characteristic regimes determined by whether the linear or nonlinear coherence time is dominant. In addition, our model provides a clear physical picture that brings insight into understanding biphoton optics with this new source. We apply our model to recent work on generating narrowband (and even subnatural linewidth) paired photons using the technique of electromagnetically induced transparency and slow-light effect in cold atoms and find good agreement with experimental results. © 2008 Optical Society of America
产生非经典光为量子光学的基础研究和潜在应用提供了基准工具。传统上，通过非线性晶体中发生的非线性光学过程产生非经典光已成为一种标准技术。我们使用冷原子气体介质来描述这个过程来产生这种非经典光，特别是关注窄带双光子的产生。与标准程序相比，新型双光子源具有相干时间长、相干长度长、光谱亮度高、转换效率高等特性。尽管存在两种描述物理过程的方法，但我们专注于使用微扰理论在多级原子系综中由四波混合产生的纠缠双光子态的理论方面。我们表明，对生成场的线性和非线性光学响应在确定双光子波形方面起着重要作用，因此在确定双光子时间相关性方面也起着重要作用。有两个特征状态，由线性还是非线性相干时间占主导地位决定。此外，我们的模型提供了清晰的物理图片，为使用这个新光源理解双光子光学提供了见解。我们将我们的模型应用于最近使用电磁感应透明和冷原子中的慢光效应技术生成窄带（甚至亚自然线宽）配对光子的工作，并与实验结果非常吻合。© 2008 美国光学学会

OCIS codes: 270.0270, 190.4410, 190.4380.
OCIS 代码：270.0270、190.4410、190.4380。

Nonclassical light generation has attracted much attention over the last 40 years, partly because it not only provides a powerful probe for addressing fundamental issues of quantum theory such as complementarity, hidden variables, and other aspects central to the foundations of quantum mechanics [1], but also because it holds promise for many potential applications to quantum information processing [2], quantum computation and communication [3], quantum cryptography [4], quantum imaging [5], quantum lithography [6,7], and quantum metrology [8]. In particular, entangled photon pairs have already been established as a standard research tool in the field of quantum optics. Traditionally, paired photons are produced from spontaneous parametric downconversion (SPDC) $[9,10]$$[9,10]$[9,10][9,10], in which a strong pump laser drives the atomic oscillators in a noncentrosymmetric crystal into a nonlinear regime, and then two downconverted beams are radiated by these oscillators. The two downconverted photons from such a nonlinear process usually have very broad bandwidth (typically in the terahertz range) and very short coherence time (typically around a few picoseconds) so that their waveforms are not resolvable by existing single-photon detectors (which have a resolution around the nanosecond range). In addition, the short coherence length of the SPDC biphotons, roughly about $100\mu \mathrm{m}$$100\mu \mathrm{m}$100 mum100 \mu \mathrm{~m}, also limits their application in long-distance quantum communication. The optical properties of these
非经典光生成在过去 40 年中引起了广泛关注，部分原因是它不仅为解决量子理论的基本问题提供了强大的探针，例如互补性、隐藏变量和量子力学基础的其他核心方面 [1]，还因为它为量子信息处理的许多潜在应用带来了希望 [2]。 量子计算和通信 [3]、量子密码学 [4]、量子成像 [5]、量子光刻 [6,7] 和量子计量 [8]。特别是，纠缠光子对已经被确立为量子光学领域的标准研究工具。传统上，成对光子是由自发参数下转换 （SPDC） $[9,10]$$[9,10]$[9,10][9,10] 产生的，其中强泵浦激光器将非中心对称晶体中的原子振荡器驱动到非线性状态，然后这两个下转换光束由这些振荡器辐射。来自这种非线性过程的两个下转换光子通常具有非常宽的带宽（通常在太赫兹范围内）和非常短的相干时间（通常在几皮秒左右），因此它们的波形无法被现有的单光子探测器（其分辨率约为纳秒范围）分辨。此外，SPDC 双光子的短相干长度（大致约为 $100\mu \mathrm{m}$$100\mu \mathrm{m}$100 mum100 \mu \mathrm{~m} ）也限制了它们在长距离量子通信中的应用。这些的光学特性
wideband biphotons have been well studied both experimentally and theoretically for more than two decades [11-13].
宽带双光子在实验和理论上已经得到了二十多年的深入研究 [11-13]。

Narrow-bandwidth biphotons are ideal for a number of recently proposed protocols for long-distance quantum communication based on coherent interaction between single photons and atomic ensembles [14,15], which require absorbing biphotons efficiently and storing the entanglement. Other applications include the long-distance quantum state teleportation, which requires long temporal coherence time of paired photons [16]. Using cavityenhanced SPDC in nonlinear-crystal-paired photons with a bandwidth of about 10 MHz and a coherence time of about 50 ns have been reported [17-20]. However, the SPDC process is usually tuned to occur in the region of far-off-resonant atomic transitions, which limits the conversion efficiency and requires a large pump power. Moreover, in the far-off-resonant pumping the nonlinear parametric coupling coefficient can be treated as a constant, and the information about the material internal transition structure is unresolvable from the two-photon correlation measurement.
窄带宽双光子非常适合最近提出的许多基于单光子和原子集合之间相干相互作用的长距离量子通信协议[14,15]，这需要有效地吸收双光子并存储纠缠。其他应用包括长距离量子态隐形传态，这需要成对光子的长时间相干时间 [16]。据报道，在带宽约为 10 MHz、相干时间约为 50 ns 的非线性晶体配对光子中使用腔增强 SPDC [17-20]。然而，SPDC 过程通常被调整为发生在远共振原子跃迁区域，这限制了转换效率并需要大泵浦功率。此外，在远谐振泵浦中，非线性参数耦合系数可以被视为常数，并且有关材料内部过渡结构的信息无法从双光子相关测量中解析出来。

In this paper, we will review the theoretical aspect, especially based on our work using perturbation theory, on a new biphoton source that generates narrowband biphotons near atomic resonance in the four-wave mixing (FWM) process. The two most recently developed advanced technologies make the near- and on-resonance
在本文中，我们将回顾理论方面的理论方面，特别是基于我们使用微扰理论的工作，在四波混频 （FWM） 过程中在原子共振附近产生窄带双光子的新型双光子源。最近开发的两项先进技术实现了近共振和开谐振
nonlinear processes possible. The first is laser cooling and trapping of neutral atoms [21] Because cold atoms with a temperature of about $100\mu \mathrm{K}$$100\mu \mathrm{K}$100 muK100 \mu \mathrm{~K} have negligible Doppler broadening and a very long lifetime, their atomic hyperfine structures can be resolved without the need of a Doppler-free setup. The second is the electromagnetically induced transparency (EIT) [22-24], which not only can eliminate absorption on resonance but also can enhance nonlinear interactions at low light level [25-28]. Following the theoretical proposal in [14] and early experiments by Van der Wal et al. [29] and Kuzmich et al. [30], the generation of nonclassical correlated photon pairs by exploring the “writing-reading” procedure has attracted much research interest. However, the created paired photons are not maximally time-frequency entangled due to two separate operation processes. Using cold atomic-gas media and EIT-assisted FWM Balic et al. [31] and Kolchin et al. [32] at Stanford University were the first-to the best of our knowledge-to generate time-frequency entangled narrowband biphotons with cw driving fields in a double- $\mathrm{\Lambda }$$\mathrm{\Lambda }$Lambda\Lambda system. By controlling the EIT at a sufficiently high optical depth (OD) around 50, biphotons with a subnatural linewidth have been produced in a recent experiment, where the coherence time is up to about $1\mu \mathrm{s}$$1\mu \mathrm{s}$1mus1 \mu \mathrm{~s} in a two-dimensional magneto-optical trap (MOT) [33]. Thompson and his colleagues have also produced biphotons with a temporal coherence length of 100 ns by trapping cold atoms inside an optical cavity [34]. Du et al., by driving a two-level system into the nonlinear region, have also created paired photons with a correlation time of the same order of magnitude [35].
非线性过程。第一种是激光冷却和中性原子的俘获[21]，因为温度约为 $100\mu \mathrm{K}$$100\mu \mathrm{K}$100 muK100 \mu \mathrm{~K} 的冷原子的多普勒展宽可以忽略不计，而且寿命很长，所以它们的原子超精细结构可以在不需要无多普勒设置的情况下被解析。第二种是电磁感应透明性（EIT）[22-24]，它不仅可以消除共振时的吸收，还可以增强弱光下的非线性相互作用[25-28]。继 [14] 中的理论提出以及 Van der Wal 等人 [29] 和 Kuzmich 等人 [30] 的早期实验之后，通过探索“写入-阅读”过程生成非经典相关光子对引起了广泛的研究兴趣。然而，由于两个独立的操作过程，创建的成对光子并不是最大时频纠缠的。斯坦福大学的 Balic 等人 [31] 和 Kolchin 等人 [32] 使用冷原子气体介质和 EIT 辅助的 FWM [31] 率先在双 $\mathrm{\Lambda }$$\mathrm{\Lambda }$Lambda\Lambda 系统中生成了具有连续驱动场的时频纠缠窄带双光子。通过将 EIT 控制在 50 左右的足够高的光学深度 （OD） 上，在最近的一个实验中产生了具有亚自然线宽的双光子，其中相干时间在二维磁光阱 （MOT） 中达到约 $1\mu \mathrm{s}$$1\mu \mathrm{s}$1mus1 \mu \mathrm{~s} [33]。Thompson和他的同事们还通过将冷原子捕获在光腔内，产生了时间相干长度为100 ns的双光子[34]。Du et al.通过将一个两能级系统驱动到非线性区域，还创建了具有相同数量级相关时间的成对光子[35]。

To describe the physics behind narrowband biphoton generation from the EIT-based multilevel atomic system, two different approaches have been presented in the literature. One approach is to use the Heisenberg-Langevin method by solving the coupled field operator equations in the Heisenberg picture, and the results agree well with the experimental data in the two-photon correlation measurement $[36,37,39]$$[36,37,39]$[36,37,39][36,37,39]. The advantage of this method is that it provides a sophisticated calculation of the effects from the Langevin noises, the medium absorption, and the optical gain on the biphoton wave packet. Working in the interaction picture the other approach developed by Wen et al. [38,40-42] focuses on the description of the two-photon state vector after the nonlinear medium using perturbation theory. This state vector theory not only properly describes the experimental results but also offers a clear physical picture of biphoton generation mechanism. It is the intent of this paper to review and generalize this latter approach by taking into account both linear loss and gain and to provide more insights on the roles of nonlinear interaction and linear response to the biphoton amplitude. Using this state vector picture, we show that the two-photon wave function is a convolution of the nonlinear and linear optical responses. According to this convolution, the two-photon temporal correlation is considered in two regimes: damped Rabi oscillation and group delay.
为了描述基于 EIT 的多能级原子系统产生窄带双光子背后的物理学，文献中提出了两种不同的方法。一种方法是通过求解海森堡图中的耦合场算子方程来使用 Heisenberg-Langevin 方法，结果与双光子相关测量 $[36,37,39]$$[36,37,39]$[36,37,39][36,37,39] 中的实验数据非常吻合。这种方法的优点是，它可以复杂地计算朗之万噪声、介质吸收和光增益对双光子波包的影响。在交互图片中，温等[38,40-42]开发的另一种方法侧重于使用微扰理论描述非线性介质之后的双光子状态矢量。该状态向量理论不仅正确地描述了实验结果，而且还提供了双光子产生机制的清晰物理图景。本文的目的是通过考虑线性损耗和增益来回顾和概括后一种方法，并提供有关非线性相互作用和对双光子振幅的线性响应的作用的更多见解。使用这张状态矢量图片，我们表明双光子波函数是非线性和线性光学响应的卷积。根据这种卷积，双光子时间相关性在两种状态下考虑：阻尼 Rabi 振荡和群延迟。

The paper is organized as follows. In Section 2, using perturbation theory we lay out a general formalism of the two-photon state and biphoton wave packet (or amplitude) generated from the FWM parametric process. In Section 3, taking a four-level double- $\mathrm{\Lambda }$$\mathrm{\Lambda }$Lambda\Lambda atomic system as
本文的组织结构如下。在第 2 节中，我们使用微扰理论阐述了 FWM 参数化过程产生的双光子态和双光子波包（或振幅）的一般形式。在第 3 节中，以一个四能级双 $\mathrm{\Lambda }$$\mathrm{\Lambda }$Lambda\Lambda 原子系统为
an example, we show that the linear and nonlinear optical responses to the generated fields play an important role in determining the two-photon amplitude. By looking at the nonlinear susceptibility, we illustrate the mechanism of biphoton generation in such a system. To take into account the gain and loss, we have generalized previous work by extending the wavenumber to complex variables. In Section 4 we describe the damped Rabi oscillation regime where the biphoton amplitude is mainly characterized by the structure of the third-order nonlinear susceptibility. We show that the two-photon temporal correlation can exhibit damped and overdamped Rabi oscillations. In Section 5, we describe the group-delay regime where the biphoton wave packet is mainly determined by the phase matching due to the linear optical response. In Section 6, we examine the two-photon interference using biphotons generated from such a cold atomic-gas medium. We show that by manipulating the linear optical response, it is possible to switch the two-photon anti-bunching-like effect to a bunchinglike effect. Finally, we draw the conclusions and outlook in Section 7.
举个例子，我们表明对生成场的线性和非线性光学响应在确定双光子振幅方面起着重要作用。通过观察非线性磁化率，我们说明了这种系统中双光子产生的机制。为了考虑收益和损失，我们通过将 wavenumber 扩展到复变量来推广以前的工作。在第 4 节中，我们描述了阻尼 Rabi 振荡状态，其中双光子振幅主要由三阶非线性磁化率的结构来表征。我们表明，双光子时间相关性可以表现出阻尼和过阻尼的 Rabi 振荡。在第 5 节中，我们描述了群延迟机制，其中双光子波包主要由线性光学响应引起的相位匹配决定。在第 6 节中，我们研究了使用这种冷原子气体介质产生的双光子干涉。我们表明，通过操纵线性光学响应，可以将双光子反聚束效应切换到聚束效应。最后，我们在第 7 节中得出结论和展望。

A schematic of biphoton generation via a four-level double- $\mathrm{\Lambda }$$\mathrm{\Lambda }$Lambda\Lambda atomic system is shown in Fig. 1, where in the presence of a cw pump ( ${\omega }_p$${\omega }_p$omega_(p)\omega_{p} ) and coupling ( ${\omega }_c$${\omega }_c$omega_(c)\omega_{c} ) lasers, phase-matched, paired Stokes ( ${\omega }_s$${\omega }_s$omega_(s)\omega_{s} ) and anti-Stokes ( ${\omega }_{as}$${\omega }_{as}$omega_(as)\omega_{a s} ) photons are spontaneously produced from the FWM process in the low-gain limit. As shown in Fig. 1(a), the strong coupling laser forms a standard three-level $\mathrm{\Lambda }$$\mathrm{\Lambda }$Lambda\Lambda EIT scheme with the generated anti-Stokes field. Therefore, the role of the coupling laser here is that it not only assists the FWM nonlinear process but also creates a transparency window for the anti-Stokes photons with the slow-light effect. With the use of EIT it is possible to generate the anti-Stokes photons near atomic resonance $|1⟩$$|1⟩$|1:)|1\rangle $\to |3⟩$$\to |3⟩$rarr|3:)\rightarrow|3\rangle, which greatly enhances the efficiency of FWM process. The phase-matching condition allows both forward and backward generation configurations, as shown in Figs. 1(b) and 1©, respectively. In the backward generation configuration paired Stokes and anti-Stokes photons
图 1 显示了通过四能级双 $\mathrm{\Lambda }$$\mathrm{\Lambda }$Lambda\Lambda 原子系统产生双光子的示意图，其中在连续泵浦 （ ${\omega }_p$${\omega }_p$omega_(p)\omega_{p} ） 和耦合 （ ${\omega }_c$${\omega }_c$omega_(c)\omega_{c} ） 激光器存在下，相位匹配、成对的斯托克斯 （ ${\omega }_s$${\omega }_s$omega_(s)\omega_{s} ） 和反斯托克斯 （ ${\omega }_{as}$${\omega }_{as}$omega_(as)\omega_{a s} ） 光子在低增益极限内从 FWM 过程中自发产生。如图 1（a） 所示，强耦合激光器与生成的反斯托克斯场形成标准的三能级 $\mathrm{\Lambda }$$\mathrm{\Lambda }$Lambda\Lambda EIT 方案。因此，耦合激光器在这里的作用是，它不仅辅助 FWM 非线性过程，而且还为具有慢光效果的反斯托克斯光子创造了一个透明窗口。通过使用 EIT，可以在原子共振 $|1⟩$$|1⟩$|1:)|1\rangle 附近产生反斯托克斯光子 $\to |3⟩$$\to |3⟩$rarr|3:)\rightarrow|3\rangle ，这大大提高了 FWM 过程的效率。相位匹配条件允许正向和反向生成配置，分别如图 1（b） 和图 1© 所示。在倒代配置中，成对的斯托克斯光子和反斯托克斯光子

Fig. 1. (Color online) Biphoton generation via a four-level double- $\mathrm{\Lambda }$$\mathrm{\Lambda }$Lambda\Lambda atomic system. (a) The level structure, where in the presence of a cw pump ( ${\omega }_p$${\omega }_p$omega_(p)\omega_{p} ) and coupling ( ${\omega }_c$${\omega }_c$omega_(c)\omega_{c} ) beams, paired Stokes ( ${\omega }_s$${\omega }_s$omega_(s)\omega_{s} ), and anti-Stokes ( ${\omega }_{as}$${\omega }_{as}$omega_(as)\omega_{a s} ) photons are spontaneously created from the FWM processes in the low-gain regime. (b) The forward generation configuration. © The backward generation geometry.
图 1.（在线彩色）通过四能级双 $\mathrm{\Lambda }$$\mathrm{\Lambda }$Lambda\Lambda 原子系统产生双光子。（a） 水平结构，其中在连续泵 （ ${\omega }_p$${\omega }_p$omega_(p)\omega_{p} ） 和耦合 （ ${\omega }_c$${\omega }_c$omega_(c)\omega_{c} ） 光束存在的情况下，成对的斯托克斯 （ ${\omega }_s$${\omega }_s$omega_(s)\omega_{s} ） 和反斯托克斯 （ ${\omega }_{as}$${\omega }_{as}$omega_(as)\omega_{a s} ） 光子是从低增益状态下的 FWM 过程自发产生的。（b） 正向生成配置。© 向后生成几何体。
can prpagate collinearly with the pump and coupling beams 31,33$]$$]$]] Ind can also propagate in a right-angle geometry [32]. These flexible generation setups depend on whether thre relationship ${\overrightarrow{k}}_p+{\overrightarrow{k}}_c=0$${\overrightarrow{k}}_p+{\overrightarrow{k}}_c=0$vec(k)_(p)+ vec(k)_(c)=0\vec{k}_{p}+\vec{k}_{c}=0 is well satisfied, where ${\overrightarrow{k}}_{p,c}$${\overrightarrow{k}}_{p,c}$vec(k)_(p,c)\vec{k}_{p, c} are wave vectors of the input pump and coupling fields. One advantage for the backward generation geometry is that it allows us to easily separate the generated weak fields from the strong inputs.
可以与泵和耦合梁 31,33 $]$$]$]] Ind 一起以直角几何形状传播 [32]。这些灵活的生成设置取决于是否很好地满足 thre 关系 ${\overrightarrow{k}}_p+{\overrightarrow{k}}_c=0$${\overrightarrow{k}}_p+{\overrightarrow{k}}_c=0$vec(k)_(p)+ vec(k)_(c)=0\vec{k}_{p}+\vec{k}_{c}=0 ，其中 ${\overrightarrow{k}}_{p,c}$${\overrightarrow{k}}_{p,c}$vec(k)_(p,c)\vec{k}_{p, c} 是输入泵浦和耦合场的波矢。后代几何的一个优点是，它使我们能够轻松地将生成的弱场与强输入分开。

In the following discussions, we assume that the cold atomic-gas medium consists of identical four-level atoms prepared in their ground state $|1⟩$$|1⟩$|1:)|1\rangle [see Fig. 1(a)]. The idealized atoms or molecules are confined within a long, thin cylindrical volume with a length $L$$L$LL and atomic density $N$$N$NN. For simplicity, in this paper we will not take into account the Doppler broadening and polarization effects In the two-photon limit, the quantur Langevin noise introduces unpaired photons, which are not or interest nere and so are ignored. In addition, we concentrate on the twophoton temporal correlation.
在下面的讨论中，我们假设冷原子气体介质由以基态 $|1⟩$$|1⟩$|1:)|1\rangle 制备的相同四能级原子组成 [见图 1（a）]。理想化的原子或分子被限制在一个长而薄的圆柱形体积内，其长度 $L$$L$LL 和原子密度 $N$$N$NN .为简单起见，在本文中，我们不会考虑多普勒展宽和偏振效应 在双光子极限中，量子朗之万噪声引入了不成对的光子，这些光子不是或不感兴趣的光子，因此被忽略。此外，我们专注于双光子时间相关性。

We start with the geometries shown in Figs. 1(b) and 1©, where Stokes-anti-Stokes photon pairs propagate along the $z$$z$zz axis. We denote the electric field as $E$$E$EE $=\frac{1}{2}[E^{(+)}+E^{(-)}]=\frac{1}{2}[E^{(+)}+$$=\frac{1}{2}\left[E^{(+)}+E^{(-)}\right]=\frac{1}{2}\left[E^{(+)}+\right.$=(1)/(2)[E^((+))+E^((-))]=(1)/(2)[E^((+))+:}=\frac{1}{2}\left[E^{(+)}+E^{(-)}\right]=\frac{1}{2}\left[E^{(+)}+\right.c.c. $]$$]$]], where $E^{(+)}$$E^{(+)}$E^((+))E^{(+)}and $E^{(-)}$$E^{(-)}$E^((-))E^{(-)}are the positive- and negative-frequency parts. The input pump and coupling beams are assumed to be strong classical fields, so that
我们从图 1（b） 和 1© 中所示的几何结构开始，其中斯托克斯反斯托克斯光子对沿 $z$$z$zz 轴传播。我们将电场表示为 $E$$E$EE $=\frac{1}{2}[E^{(+)}+E^{(-)}]=\frac{1}{2}[E^{(+)}+$$=\frac{1}{2}\left[E^{(+)}+E^{(-)}\right]=\frac{1}{2}\left[E^{(+)}+\right.$=(1)/(2)[E^((+))+E^((-))]=(1)/(2)[E^((+))+:}=\frac{1}{2}\left[E^{(+)}+E^{(-)}\right]=\frac{1}{2}\left[E^{(+)}+\right. c.c. $]$$]$]] ，其中 $E^{(+)}$$E^{(+)}$E^((+))E^{(+)} 和 $E^{(-)}$$E^{(-)}$E^((-))E^{(-)} 是正频和负频部分。假设输入泵浦和耦合梁是强经典场，因此

where $k_{p,c}$$k_{p,c}$k_(p,c)k_{p, c} are pump and coupling field wavenumbers. The $\pm$$\pm$+-\pm sign in Eq. (1) stands for the forward and backward propagation geometries for the pump laser. The single-transverse-mode Stokes and anti-Stokes fields are taken as quantized,
其中 $k_{p,c}$$k_{p,c}$k_(p,c)k_{p, c} 是泵浦和耦合场波数。方程 （1） 中的 $\pm$$\pm$+-\pm 符号代表泵浦激光器的正向和反向传播几何。单横模 Stokes 场和反 Stokes 场被视为量子化的，

where $k_{s,as}$$k_{s,as}$k_(s,as)k_{s, a s} are wavenumbers of Stokes and anti-Stokes photons, and $A$$A$AA is the single-mode cross-section area. The annihilation operators ${\hat{a}}_s\left({\omega }_s\right)$${\widehat{a}}_s\left({\omega }_s\right)$hat(a)_(s)(omega_(s))\hat{a}_{s}\left(\omega_{s}\right) and ${\hat{a}}_{as}\left({\omega }_{as}\right)$${\widehat{a}}_{as}\left({\omega }_{as}\right)$hat(a)_(as)(omega_(as))\hat{a}_{a s}\left(\omega_{a s}\right) satisfy the commutation relation
其中 $k_{s,as}$$k_{s,as}$k_(s,as)k_{s, a s} 是斯托克斯光子和反斯托克斯光子的波数，是 $A$$A$AA 单模横截面积。湮灭运算符 ${\hat{a}}_s\left({\omega }_s\right)$${\widehat{a}}_s\left({\omega }_s\right)$hat(a)_(s)(omega_(s))\hat{a}_{s}\left(\omega_{s}\right) 和 ${\hat{a}}_{as}\left({\omega }_{as}\right)$${\widehat{a}}_{as}\left({\omega }_{as}\right)$hat(a)_(as)(omega_(as))\hat{a}_{a s}\left(\omega_{a s}\right) 满足换向关系

In the interaction picture the effective interaction Hamiltonian for the FWM parametric process takes the form [38-42]
在交互图片中，FWM 参数化过程的有效交互哈密顿量采用 [38-42] 的形式 [38-42]

where H.c. means the Hermitian conjugate. ${\hat{E}}_{as}^{(-)}$${\widehat{E}}_{as}^{(-)}$hat(E)_(as)^((-))\hat{E}_{a s}^{(-)}and ${\hat{E}}_s^{(-)}$${\widehat{E}}_s^{(-)}$hat(E)_(s)^((-))\hat{E}_{s}^{(-)} are the Hermitian conjugates of quantum-field operators ${\hat{E}}_{as}^{(+)}$${\widehat{E}}_{as}^{(+)}$hat(E)_(as)^((+))\hat{E}_{a s}^{(+)}and ${\hat{E}}_s^{(+)}$${\widehat{E}}_s^{(+)}$hat(E)_(s)^((+))\hat{E}_{s}^{(+)}, respectively. ${\chi }^{(3)}$${\chi }^{(3)}$chi^((3))\chi^{(3)} is the third-order nonlinear susceptibility to the Stokes (or anti-Stokes) field defined by the nonlinear polarizability ${\hat{P}}_{s,as}^{(3)(+)}={\epsilon }_0{\chi }^{(3)}{E}_p^{(+)}{E}_c^{(+)}{\hat{E}}_{as,s}^{(-)}$${\widehat{P}}_{s,as}^{(3)(+)}={\epsilon }_0{\chi }^{(3)}{E}_p^{(+)}{E}_c^{(+)}{\widehat{E}}_{as,s}^{(-)}$hat(P)_(s,as)^((3)(+))=epsi_(0)chi^((3))E_(p)^((+))E_(c)^((+)) hat(E)_(as,s)^((-))\hat{P}_{s, a s}^{(3)(+)}=\varepsilon_{0} \chi^{(3)} E_{p}^{(+)} E_{c}^{(+)} \hat{E}_{a s, s}^{(-)}
其中 H.c. 表示厄米特共轭。 ${\hat{E}}_{as}^{(-)}$${\widehat{E}}_{as}^{(-)}$hat(E)_(as)^((-))\hat{E}_{a s}^{(-)} 和 ${\hat{E}}_s^{(-)}$${\widehat{E}}_s^{(-)}$hat(E)_(s)^((-))\hat{E}_{s}^{(-)} 分别是量子场算子 ${\hat{E}}_{as}^{(+)}$${\widehat{E}}_{as}^{(+)}$hat(E)_(as)^((+))\hat{E}_{a s}^{(+)} 和 ${\hat{E}}_s^{(+)}$${\widehat{E}}_s^{(+)}$hat(E)_(s)^((+))\hat{E}_{s}^{(+)} 的 Hermitian 共轭。 ${\chi }^{(3)}$${\chi }^{(3)}$chi^((3))\chi^{(3)} 是由非线性极化率 ${\hat{P}}_{s,as}^{(3)(+)}={\epsilon }_0{\chi }^{(3)}{E}_p^{(+)}{E}_c^{(+)}{\hat{E}}_{as,s}^{(-)}$${\widehat{P}}_{s,as}^{(3)(+)}={\epsilon }_0{\chi }^{(3)}{E}_p^{(+)}{E}_c^{(+)}{\widehat{E}}_{as,s}^{(-)}$hat(P)_(s,as)^((3)(+))=epsi_(0)chi^((3))E_(p)^((+))E_(c)^((+)) hat(E)_(as,s)^((-))\hat{P}_{s, a s}^{(3)(+)}=\varepsilon_{0} \chi^{(3)} E_{p}^{(+)} E_{c}^{(+)} \hat{E}_{a s, s}^{(-)} 定义的斯托克斯场（或反斯托克斯场）的三阶非线性磁化率
[43]. With the use of Eqs. (1) and (2), after the $z$$z$zz integration we rewrite the Hamiltonian (4) as
[43]. 使用 Eqs.（1） 和 （2） 中， $z$$z$zz 在积分之后，我们将哈密顿量 （4） 重写为

where $\mathrm{\Delta }k=k_{as}\pm k_s-(k_c\pm k_p)$$\mathrm{\Delta }k=k_{as}\pm k_s-\left(k_c\pm k_p\right)$Delta k=k_(as)+-k_(s)-(k_(c)+-k_(p))\Delta k=k_{a s} \pm k_{s}-\left(k_{c} \pm k_{p}\right) is the phase mismatching for the forward (+) and backward ( - ) configurations. When the pump and coupling fields are incident with an angle respect to the $z$$z$zz axis, the phase mismatching can be generalized as $\mathrm{\Delta }k=({\overrightarrow{k}}_{as}+{\overrightarrow{k}}_s-{\overrightarrow{k}}_c-{\overrightarrow{k}}_p)\cdot \hat{z}$$\mathrm{\Delta }k=\left({\overrightarrow{k}}_{as}+{\overrightarrow{k}}_s-{\overrightarrow{k}}_c-{\overrightarrow{k}}_p\right)\cdot \widehat{z}$Delta k=( vec(k)_(as)+ vec(k)_(s)- vec(k)_(c)- vec(k)_(p))* hat(z)\Delta k=\left(\vec{k}_{a s}+\vec{k}_{s}-\vec{k}_{c}-\vec{k}_{p}\right) \cdot \hat{z}, where $\hat{z}$$\widehat{z}$hat(z)\hat{z} is the unit vector along the $+z$$+z$+z+z axis. The integral over the length $L$$L$LL gives a sinc function $\mathrm{sinc}(\mathrm{\Delta }kL/2)$$\mathrm{sinc}(\mathrm{\Delta }kL/2)$sinc(Delta kL//2)\operatorname{sinc}(\Delta k L / 2), which determines the two-photon natural spectrum width. $\kappa ({\omega }_{as},{\omega }_s)$$\kappa \left({\omega }_{as},{\omega }_s\right)$kappa(omega_(as),omega_(s))\kappa\left(\omega_{a s}, \omega_{s}\right) $=-i\left(\sqrt{{\varpi }_{as}{\varpi }_s}/2c\right){\chi }^{(3)}({\omega }_{as},{\omega }_s)E_p{E}_c$$=-i\left(\sqrt{{\varpi }_{as}{\varpi }_s}/2c\right){\chi }^{(3)}\left({\omega }_{as},{\omega }_s\right)E_p{E}_c$=-i(sqrt(ϖ_(as)ϖ_(s))//2c)chi^((3))(omega_(as),omega_(s))E_(p)E_(c)=-i\left(\sqrt{\varpi_{a s} \varpi_{s}} / 2 c\right) \chi^{(3)}\left(\omega_{a s}, \omega_{s}\right) E_{p} E_{c} is the nonlinear parametric coupling coefficient, where ${\varpi }_{as}$${\varpi }_{as}$ϖ_(as)\varpi_{a s} and ${\varpi }_s$${\varpi }_s$ϖ_(s)\varpi_{s} are central frequencies of the anti-Stokes and Stokes fields, respectively.
其中 $\mathrm{\Delta }k=k_{as}\pm k_s-(k_c\pm k_p)$$\mathrm{\Delta }k=k_{as}\pm k_s-\left(k_c\pm k_p\right)$Delta k=k_(as)+-k_(s)-(k_(c)+-k_(p))\Delta k=k_{a s} \pm k_{s}-\left(k_{c} \pm k_{p}\right) 是正向 （+） 和向后 （ - ） 配置的相位不匹配。当泵场和耦合场以相对于 $z$$z$zz 轴的角度入射时，相位失配可以推广为 $\mathrm{\Delta }k=({\overrightarrow{k}}_{as}+{\overrightarrow{k}}_s-{\overrightarrow{k}}_c-{\overrightarrow{k}}_p)\cdot \hat{z}$$\mathrm{\Delta }k=\left({\overrightarrow{k}}_{as}+{\overrightarrow{k}}_s-{\overrightarrow{k}}_c-{\overrightarrow{k}}_p\right)\cdot \widehat{z}$Delta k=( vec(k)_(as)+ vec(k)_(s)- vec(k)_(c)- vec(k)_(p))* hat(z)\Delta k=\left(\vec{k}_{a s}+\vec{k}_{s}-\vec{k}_{c}-\vec{k}_{p}\right) \cdot \hat{z} ，其中 $\hat{z}$$\widehat{z}$hat(z)\hat{z} 是沿轴的 $+z$$+z$+z+z 单位向量。沿长度 $L$$L$LL 的积分得到一个 sinc 函数 $\mathrm{sinc}(\mathrm{\Delta }kL/2)$$\mathrm{sinc}(\mathrm{\Delta }kL/2)$sinc(Delta kL//2)\operatorname{sinc}(\Delta k L / 2) ，它决定了双光子自然光谱宽度。 $\kappa ({\omega }_{as},{\omega }_s)$$\kappa \left({\omega }_{as},{\omega }_s\right)$kappa(omega_(as),omega_(s))\kappa\left(\omega_{a s}, \omega_{s}\right) $=-i\left(\sqrt{{\varpi }_{as}{\varpi }_s}/2c\right){\chi }^{(3)}({\omega }_{as},{\omega }_s)E_p{E}_c$$=-i\left(\sqrt{{\varpi }_{as}{\varpi }_s}/2c\right){\chi }^{(3)}\left({\omega }_{as},{\omega }_s\right)E_p{E}_c$=-i(sqrt(ϖ_(as)ϖ_(s))//2c)chi^((3))(omega_(as),omega_(s))E_(p)E_(c)=-i\left(\sqrt{\varpi_{a s} \varpi_{s}} / 2 c\right) \chi^{(3)}\left(\omega_{a s}, \omega_{s}\right) E_{p} E_{c} 是非线性参数耦合系数，其中 ${\varpi }_{as}$${\varpi }_{as}$ϖ_(as)\varpi_{a s} 和 ${\varpi }_s$${\varpi }_s$ϖ_(s)\varpi_{s} 分别是反斯托克斯场和斯托克斯场的中心频率。

Perturbation theory gives the photon state at the output surface(s) approximately as a linear superposition of $|0⟩+|\mathrm{\Psi }⟩$$|0⟩+|\mathrm{\Psi }⟩$|0:)+|Psi:)|0\rangle+|\Psi\rangle, where $|0⟩$$|0⟩$|0:)|0\rangle is the vacuum state. The two-photon (biphoton) state $|\mathrm{\Psi }⟩$$|\mathrm{\Psi }⟩$|Psi:)|\Psi\rangle can be expressed as
微扰理论给出输出表面的光子状态大约是 的 $|0⟩+|\mathrm{\Psi }⟩$$|0⟩+|\mathrm{\Psi }⟩$|0:)+|Psi:)|0\rangle+|\Psi\rangle 线性叠加，其中 $|0⟩$$|0⟩$|0:)|0\rangle 是真空状态。双光子 （biphoton） 状态 $|\mathrm{\Psi }⟩$$|\mathrm{\Psi }⟩$|Psi:)|\Psi\rangle 可以表示为

Because the vacuum is not detectable, from now on we will ignore it and only consider the two-photon part. Higher-order terms with four, six, etc. photons are negligible for low-power continuous pumping. Using the Hamiltonian (5) in Eq. (6), the time integral yields a $\delta$$\delta$delta\delta function, $2\pi \delta ({\omega }_c+{\omega }_p-{\omega }_{as}-{\omega }_s)$$2\pi \delta \left({\omega }_c+{\omega }_p-{\omega }_{as}-{\omega }_s\right)$2pi delta(omega_(c)+omega_(p)-omega_(as)-omega_(s))2 \pi \delta\left(\omega_{c}+\omega_{p}-\omega_{a s}-\omega_{s}\right), which expresses the energy conservation for the process. If $L$$L$LL is infinite, then the sinc function $L\mathrm{sinc}(\mathrm{\Delta }kL/2)$$L\mathrm{sinc}(\mathrm{\Delta }kL/2)$L sinc(Delta kL//2)L \operatorname{sinc}(\Delta k L / 2) in Hamiltonian (5) becomes a $\delta$$\delta$delta\delta function, $\delta (\mathrm{\Delta }k)$$\delta (\mathrm{\Delta }k)$delta(Delta k)\delta(\Delta k). In this case, the conditions
因为真空是无法检测到的，所以从现在开始我们将忽略它，只考虑双光子部分。具有 4 个、6 个等光子的高阶项对于低功率连续泵浦来说可以忽略不计。使用方程 （6） 中的哈密顿量 （5），时间积分得到一个 $\delta$$\delta$delta\delta 函数 $2\pi \delta ({\omega }_c+{\omega }_p-{\omega }_{as}-{\omega }_s)$$2\pi \delta \left({\omega }_c+{\omega }_p-{\omega }_{as}-{\omega }_s\right)$2pi delta(omega_(c)+omega_(p)-omega_(as)-omega_(s))2 \pi \delta\left(\omega_{c}+\omega_{p}-\omega_{a s}-\omega_{s}\right) ，该函数表示该过程的能量守恒。如果 $L$$L$LL 是无限的，则哈密顿量 （5） 中的 sinc 函数 $L\mathrm{sinc}(\mathrm{\Delta }kL/2)$$L\mathrm{sinc}(\mathrm{\Delta }kL/2)$L sinc(Delta kL//2)L \operatorname{sinc}(\Delta k L / 2) 变为 $\delta$$\delta$delta\delta 函数 $\delta (\mathrm{\Delta }k)$$\delta (\mathrm{\Delta }k)$delta(Delta k)\delta(\Delta k) 。在这种情况下，条件