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薄膜铌酸锂光学芯片因低损耗、高非线性系数及高电光调制带宽等特性, 有望成为开展集成光学量子信息研究的理想实验平台. 然而, 到目前为止, 基于薄膜铌酸锂的单光子源普遍采用周期性极化准相位匹配技术, 该技术要求精确地制备电极并对铌酸锂波导进行周期性极化, 工艺复杂且对加工精度要求较高. 本文提出了一种基于模式色散相位匹配的薄膜铌酸锂单光子源器件. 该器件无需制作电极, 具备加工简便和集成度更高的优势, 同时单光子产率可达
3.8×10⁷ /(s·mW), 能够满足光学量子信息处理的需求. 此器件有望替代传统准相位匹配单光子源, 进一步推动基于薄膜铌酸锂芯片的光学量子信息研究的发展.In the domain of integrated quantum photonics, the burgeoning superiority of lithium niobate’s second-order nonlinearity in electro-optic modulation makes thin-film lithium niobate a leading quantum photonic platform after silicon. To date, single-photon sources using thin-film lithium niobate has mainly adopted periodic polarization quasi-phase matching technology, which requires the preparation of complex electrodes for domain inversion in the waveguide to realize quasi-phase matching. This method inevitably introduces complexity, such as complex processing methods, enlarged polarization regions, and compromised integration density. With the development of quantum information technology, the ever-increasing degree of integration constantly creates new demands. Consequently, the development of a streamlined, high-efficiency quantum light source on a lithium niobate platform is a pressing issue. In this study, we propose a novel thin-film lithium niobate parametric down-conversion single-photon source based on mode dispersion phase matching theory. The strategy is different from conventional strategies that utilize periodic polarization to generate single-photon sources in thin-film lithium niobate devices. In contrast to traditional quasi-phase matching techniques that utilize the phase matching between pump fundamental mode light and parametric fundamental mode light, our method employs the phase matching between the pump light’s higher-order mode and the parametric light’s fundamental mode. The pump light’s higher-order mode is obtained by designing an asymmetric directional coupler. The device’s single-photon yield can attain3.8×107 /(s·mW), satisfying the requirements for optical quantum information processing. This innovative solution is expected to replace the traditional quasi-phase-matching single-photon sources, thus further promoting the study of optical quantum information based on thin-film lithium niobate chips.1. 引 言
量子信息技术作为一种将量子力学原理应用于信息科学的新兴技术, 有潜力引发通信和计算等领域的科技革命[1]. 光子因其抗干扰、退相干时间长和自由度丰富等特性, 已成为量子信息研究的理想物理载体之一[2]. 近年来, 光学量子信息领域取得了诸多重要成果, 如长距离量子密钥分发[3]、高保真度量子隐形传态[4]、多光子纠缠态制备[5]、光子量子模拟器[6]与玻色采样机[7]等. 传统光学量子信息实验通常采用分立光学元件构建光路, 但具有尺寸大、扩展性低、稳定性差等缺点, 限制了更复杂的量子操作和处理. 近年来, 基于集成光学芯片的量子信息研究得到了越来越多的关注与发展. 与分立光学器件相比, 集成光学芯片体积小、集成度高, 具有优越的可扩展性和可调控性, 便于实现复杂的量子操作与处理[8].
近年来, 集成光学量子信息研究取得了诸多关键进展, 如高维量子态制备[9]、任意两量子比特处理器[10]及干涉网络实现等[11–16]. 多数研究基于硅平台, 波导损耗和电光调制器速率表现欠佳[17]. 相较之下, 薄膜铌酸锂(thin film lithium niobate, TFLN)具备较低波导损耗、优越非线性系数和电光调制带宽[18]等优势, 可实现低损耗光学线路[19]、高效单光子源[20]以及高速电光调制器[21], 已成为量子信息研究最具潜力的集成光学平台之一. 然而, 薄膜铌酸锂平台在制备单光子源时通常采用准相位匹配法[22–25], 工艺复杂, 对精度要求高且极化区域占据大量面积, 限制了芯片尺寸.
鉴于光学量子信息技术对集成化水平要求不断提高, 本文设计了一种基于模式色散相位匹配的TFLN单光子源器件. 该器件与其他TFLN波导结构同时制备, 无需畴反转等后处理操作, 具有简化工艺和提高集成度的优势. 计算结果表明, 器件单光子产率可达
3.8×107 /(s·mW). 本文研究的单光子源器件有望取代传统准相位匹配单光子源, 推动基于TFLN芯片的光学量子信息研究发展.2. 模式色散相位匹配单光子源
以下阐述基于模式色散相位匹配的单光子源设计, 器件结构如图1所示. 整体结构包含4部分: 1) Y型分束器将入射的775 nm
TE0 (fundamental transverse electric field,TE0 )模式的泵浦光一分为二; 2)上下两个非对称定向耦合器分别将上下路径的775 nmTE0 模式光转换为TE2 模式光; 3)两段特定宽度波导使775 nmTE2 光与1550 nmTE0 光具有相同有效折射率, 从而实现模式色散相位匹配, 使上下路径发生参量下转换并产生|2,0⟩ 和|0,2⟩ 两光子叠加态; 4)电热移相器与2 × 2定向耦合器组合, 通过移相器调节相位将两光子态制备为1√2(|2,0⟩−|0,2⟩) . 随后, 定向耦合器实现反向Hong-Ou-Mandel干涉, 使上下路径各得到一个单光子.![图 1 波导结构截面图及单光子源器件结构示意图. 1)为Y分束器; 2)为非对称定向耦合器; 3)为模式色散相位匹配光子对源; 4)为上臂带有一个电热移相器的2 × 2定向耦合器. 建模采用的铌酸锂波导结构是侧壁倾角θ = 60°, h = 0.36 μm的条形波导, 材料模型为一致生长的铌酸锂[26]\r\nFig. 1. Cross-sectional diagram of the waveguide structure and schematic illustration of the single-photon source device. 1) Y-splitter. 2) Asymmetric directional coupler. 3) Mode dispersion phase-matched photon pair source. 4) 2 × 2 directional coupler in the upper arm with an electrothermal phase shifter. The modeled lithium niobate waveguide structure has a sidewall angle θ = 60° and a height h = 0.36 μm, with the material model being uniformly grown lithium niobate[26]](/fileWLXB/journal/article/wlxb/2023/15/PIC/15-20230743-1.png)
图 1 波导结构截面图及单光子源器件结构示意图. 1)为Y分束器; 2)为非对称定向耦合器; 3)为模式色散相位匹配光子对源; 4)为上臂带有一个电热移相器的2 × 2定向耦合器. 建模采用的铌酸锂波导结构是侧壁倾角θ = 60°, h = 0.36 μm的条形波导, 材料模型为一致生长的铌酸锂[26]Fig. 1. Cross-sectional diagram of the waveguide structure and schematic illustration of the single-photon source device. 1) Y-splitter. 2) Asymmetric directional coupler. 3) Mode dispersion phase-matched photon pair source. 4) 2 × 2 directional coupler in the upper arm with an electrothermal phase shifter. The modeled lithium niobate waveguide structure has a sidewall angle θ = 60° and a height h = 0.36 μm, with the material model being uniformly grown lithium niobate[26]如前所述, 关键器件包括第2)部分的非对称定向耦合器和第3)部分的模式色散相位匹配光子对源. 非对称定向耦合器负责将泵浦
TE0 转化成TE2 , 而模式色散相位匹配光子对源则利用转换得到的TE2 泵浦光与TE0 参量光进行相位匹配, 从而产生双光子. 以下将对这两个关键器件进行详细说明.2.1 非对称定向耦合器
非对称定向耦合器旨在将
TE0 泵浦光转化成TE2 , 进而用于后续相位匹配. 如图1的第2)部分所示, 非对称定向耦合器包含支持TE0 的窄波导(波导宽度为w1 )和支持TE2 的宽波导(波导宽度为w2 ), 两者底部间距为g. 在波长为775 nm时, 窄波导中TE0 与宽波导中TE2 模式光的有效折射率随波导宽度的变化如图2(a)所示. 根据模式耦合理论[27,28], 只有在两种模式的有效折射率相等时,TE0 才能转换为TE2 . 由图2(a)可知, 当w1 固定时,w2 也随之确定. 实现完全模式转换所需的耦合长度与波导宽度和间距有关, 宽度和间距越大, 耦合长度越长. 为了使器件更加紧凑且考虑当前工艺水平, 选择两波导底端间距g = 0,w1=0.3μm ,w2=1.43μm 的结构对775 nm的TE0 模进行模式转换. 经计算可知, 当两波导耦合长度为102.55 μm时, 775 nm的TE0 可完全转换为TE2 .
图 2 (a) 775 nm泵浦光在TE0 TE1 TE2 Fig. 2. (a) Effective refractive index variation curves for 775 nm pump light inTE0 TE1 TE2 具体而言, 当
w1 为0.3 μm,w2 为1.43 μm时,TE0 和TE2 模式光的有效折射率相等. 当TE0 从宽度为w1 的窄波导左端进入后, 在相互作用区域, 由于相位匹配,TE0 和TE2 模式会进行能量交换, 使TE2 模式光在宽波导的右端出现. 根据耦合模理论, 在相互作用区域中, 由窄波导中的TE0 转换为宽波导中的TE2 所需的耦合长度Lc由以下公式给出:Lc=λ2|nodd−neven|=λ2Δn, (1) 其中, λ表示入射泵浦光的工作波长(本例中λ = 775 nm),
nodd 和neven 分别代表非对称定向耦合器中TE光的奇、偶超模的有效折射率. 模式的有效折射率由MODE软件计算得出, 同时采用FDTD软件模拟器件的光场响应. 经计算, 耦合长度Lc = 102.55 μm. 模式转换过程中的模拟光场分布和光谱响应如图2(b)所示. 观察结果显示, 在波长为775 nm时,TE0 到TE2 模式的转换效率达到95.9%, 扣除损耗部分,TE0 几乎完全转换为TE2 .2.2 模式色散相位匹配光子对源
模式色散相位匹配光子对源的目标是通过自发参量下转换(spontaneous parametric down conversion, SPDC)过程, 将775 nm的
TE2 模式光转换为1550 nm的TE0 模式光. 为实现此过程, 两个TE0 模式参量光子在传播方向上的波矢β1 和β2 , 与TE2 模式泵浦光子沿传播方向的波矢β3 需满足相位匹配条件, 即β3−(β1+β2)=0 (其中βi=2πλiNi ,Ni 是模式的有效折射率), 此时1550 nm的信号光与775 nm的泵浦光的有效折射率应该相等[29]. 如图3(a)所示, 1550 nm的TE0 与775 nm的TE0 和TE1 的有效折射率曲线均不存在交点, 只与775 nmTE2 的有效折射率曲线相交于w3 = 0.661 μm处. 这说明对于具有特定倾角和高度的波导结构, 满足相位匹配条件的波导宽度具有唯一性. 在满足相位匹配条件的波导结构中, 775 nm的TE2 模式光可通过SPDC过程转换为1550 nm的TE0 模式光.
图 3 (a) 775 nm的TE0 TE1 TE2 TE0 Fig. 3. (a) Variation curves of effective refractive index for 775 nmTE0 TE1 TE2 TE0 接下来计算光子对的亮度. 经计算, 1550 nm
TE0 与775 nmTE2 的有效折射率曲线在波导宽度w3 = 0.661 µm处存在交点, 满足相位匹配条件. 假设SPDC过程中信号光的角频率为ω1 , 闲频光的角频率为ω2 , 泵浦光的角频率为ω3 , 则产生的参量光输出功率P可由以下公式表示[29,30] (详细的推导见附录A):P=4ℏω1c|κ|2P3L3/23λ1√|∂n1/∂λ|, (2) 式中
P3 是泵浦光功率, n1是信号光的群折射率, λ1是信号光波长. κ是非线性耦合系数, 具体形式为κ=ε02√(2ω)22(Nω)2N2ω(μ0ε0)3/2d20ξ2S0, (3) 其中,
ε0 和μ0 分别是真空中的介电常数和磁导率; ω是角频率(ω3=2ω=2ω1=2ω2 );Nω 是简并自发参量下转换过程中信号光(闲频光)的有效折射率;N2ω 是泵浦光的有效折射率;d0 是非线性系数;S0≡(S21S3)1/3 是有效模场面积,Si(i=1,3) 是泵浦光或参量光的模场面积,Si=(∫all|Ei|2dxdz)3|∫χ(2)|Ei|2Eidxdz|2 (i=1,3); ξ是
TE2 泵浦光和TE0 参量光的空间模场重叠因子,ξ=∫χ(2)|E1z|2E3zdxdz|∫χ(2)|E1|2E1dxdz|2/3|∫χ(2)|E3|2E3dxdz|1/3 (i=1,3). 这里的
∫all 和∫χ(2) 分别表示在全空间和铌酸锂材料的二维积分,E1z 是参量TE0 的电场E1 (x, z)的z分量,E3z 是TE2 电场E3 (x, z)的z分量.为了获取计算所需的具体参数值, 首先用MODE软件模拟了波导宽度w3 = 0.661 µm时, 775 nm的
TE2 模式光以及1550 nm的TE0 模式光, 得到两种模式下光场的有效折射率Nω=N2ω=1.697 . 然后提取两种模式光在xz平面的光场分布, 并对其进行积分和归一化处理, 得到泵浦光在775 nm的TE2 模式的模场面积S3=0.294×10−12m2 , 参量光在1550 nm的TE0 模式的模场面积S1=0.520×10−12m2 . 接着计算两个光场的有效模场面积和模场重叠因子, 得到S0=0.430×10−12m2 ,ξ=0.022 . 接下来提取参量TE0 模式光在1550 nm波长附近群折射率随波长的变化, 得到√|∂n1/∂λ|=1.827×103m−1/2 . 最后在(4)式中取d33=27pm/V , 就得到了归一化的非线性耦合系数κ=0.233W−1/2⋅cm−1 .下面计算(2)式中的输出参量光功率. 取泵浦
TE2 模式光功率P3=1×10−3W , 相互作用长度L = 1×10−2 m. 于是得到所有输出光子的总功率为P=9.779×10−12 W, 这相对TE2 泵浦光功率的自发参量下转换效率η=P/P3=9.779×10−9 , 接着可以得到单位时间光子流P/ℏω1(2)=7.630×107 /(s·mW). 光子对产生率G=Pℏω1(2)12=3.815×107 pairs/(s·mW), 图3(b)是参量光的光谱图, 其中光谱的半高宽W为6.6 nm, 于是可以得到模式色散相位匹配光子对源的双光子亮度:B=G/W=5.816×106 pairs/(s·mW·nm).2.3 分析与讨论
首先, 分析波导宽度、波导侧边倾角以及温度对图1中第2)部分的模式色散相位匹配光子对源的影响. 如图4(a)所示, 波导宽度每变化1 nm, 相位匹配波长移动约0.784 nm. 图4(b)显示, 波导倾角每偏离1°, 相位匹配波长移动约8.675 nm. 图4(c)给出了0—100 ℃范围内温度对相位匹配波长的影响, 结果表明温度每变化1 ℃时, 相位匹配波长变化约0.017 nm. 综合上述分析, 本文结构对波导侧边倾角的变化最敏感, 波导宽度变化对相位匹配波长的影响较小, 而温度对相位匹配几乎无影响.
图 4 (a)相位匹配波长随波导宽度的变化; (b)相位匹配波长随波导侧边倾角θ的变化; (c)相位匹配波长随温度的变化Fig. 4. (a) Variation of phase-matching wavelength with waveguide width; (b) variation of phase-matching wavelength with waveguide sidewall angle θ; (c) variation of phase-matching wavelength with temperature接下来, 计算不同高度条形波导结构中满足模式色散相位匹配所需的波导宽度(如图5所示). 从图5的两个折射率曲面可以观察到, 特定波长的模式色散相位匹配仅在特定波导结构中才能实现. 当波导高度超过某一阈值后, 775 nm的
TE2 泵浦光与1550 nm的TE0 参量光的有效折射率曲线无交点, 因此无法满足相位匹配条件. 图中绿色交点表明, 对于特定高度的条形波导, 实现相位匹配所需的波导宽度具有唯一性.
图 5 波导侧边倾角为60°的条形波导在高度为300—600 nm范围内、不同波导宽度情况下, 775 nmTE2 TE0 Fig. 5. Effective refractive index scan data for 775 nmTE2 TE0 3. 总 结
本文提出了一种基于模式色散相位匹配的薄膜铌酸锂单光子源器件. 该器件利用泵浦光的高阶模式与参量光的基模实现完美的相位匹配, 相较于传统的准相位匹配方法, 本文方法不需要畴反转等后处理操作, 具有工艺简单、集成度更高的优势. 该器件有望替代传统准相位匹配单光子源, 进一步推动基于薄膜铌酸锂芯片的光学量子信息研究的发展.
附录A 相关公式推导
设SPDC过程中信号光的角频率为
ω1 , 闲频光的角频率为ω2 , 泵浦光的角频率为ω3 , 传播常数用β表示, β定义为沿光场传播方向(y方向)的波矢, 于是相位失配参量表示为2Δ=β3−(β1+β2). 耦合模方程表示为[29]
ddyA1(y)=ik1A3(y)A2(y)∗ei2Δy, ddyA2(y)=ik2A3(y)A1(y)∗ei2Δy, ddyA3(y)=ik3A1(y)A2(y)e−i2Δy. 耦合系数定义为
κ1=ω1ε0d02∬ \kappa_{2}^{}=\frac{\omega_{2} \varepsilon_{0}{d}_{0}}{2} \iint\left[{\boldsymbol{E}}_{2}(x, z)\right]^{*}{\boldsymbol{E}}_{3}(x, z)[{\boldsymbol{E}}_{1}(x, z)]^{*} {\rm{d}} x {\rm{d}} z ,\tag{A6} \kappa_{3}^{}=\frac{\omega_{3} \varepsilon_{0}{d}_{0}}{2} \iint\left[{\boldsymbol{E}}_{3}(x, z)\right]^{*}{\boldsymbol{E}}_{2}(x, z){\boldsymbol{E}}_{1}(x, z) {\rm{d}} x {\rm{d}} z . \tag{A7} 耦合系数之间有如下关系:
\kappa_{1}/\omega_{1}=\kappa_{2}/\omega_{2}=\kappa_{3}/\omega_{3}. \tag{A8} 考虑无泵浦消耗的近似, 可得信号光和闲频光满足耦合模方程(A5)和(A6)[30,31]:
\frac{{\rm{d}}}{{\rm{d}} y} a_{1}(y) ={\rm{i}}\kappa A_{3} a_{2}^{\dagger}(y){\rm{e}}^{{\rm{i}}2\Delta y}, \tag{A9} \frac{{\rm{d}}}{{\rm{d}} y} a_{2}(y) ={\rm{i}}\kappa A_{3} a_{1}^{\dagger}(y){\rm{e}}^{{\rm{i}}2\Delta y}, \tag{A10} 式中
A_{3} 是功率归一化的泵浦光振幅; κ是耦合系数, 且\kappa= (\omega_{1}\omega_{2}/\omega_{3}^{2})^{1/2}K_{3}^{\ast} ,K_{3} 的具体形式为[29]K_{3}=\varepsilon_{0}\sqrt{\frac{(2 \omega)^{2}}{2\left(N^{\omega}\right)^{2} N^{2 \omega}}\left(\frac{\mu_{0}}{\varepsilon_{0}}\right)^{3 / 2} \frac{d_{\rm{eff}}^{2}}{S_{\rm{eff}}}}, \tag{A11} 其中
d_{\rm{eff}}=\frac{\sqrt{S_{\rm{eff}}} \displaystyle\iint[E^{2 \omega}]^{*} d_{0}[E^{\omega}]^{2} {\rm{d}}x{\rm{d}}z}{\sqrt{\displaystyle\iint|E^{2 \omega}|^{2} {\rm{d}}x{\rm{d}}z} \displaystyle\iint|E^{\omega}|^{2}{\rm{d}}x{\rm{d}}z}, \tag{A12} S_{\rm{eff}}=\frac{\displaystyle\iint\left| E^{2 \omega}\right|^{2} {\rm{d}} x {\rm{d}} z\left[\displaystyle\iint\left| E^{\omega}\right|^{2} {\rm{d}} x {\rm{d}} z\right]^{2}}{\left[\displaystyle\iint\left[E^{2 \omega}\right]^{*}\left[E^{\omega}\right]^{2} {\rm{d}} x {\rm{d}} z\right]^{2}}.\tag{A13} d_{\rm{eff}}^{2}/S_{\rm{eff}} 进一步可以化简成:\begin{split} \;& \frac{d_{\rm{eff}}^{2}}{S_{\rm{eff}}}=d_{0}^{2}\frac{\left[ \displaystyle\iint[E^{2 \omega}]^{*} [E^{\omega}]^{2} {\rm{d}} x {\rm{d}} z\right]^{2}}{{\displaystyle\iint|E^{2 \omega}|^{2} {\rm{d}} x {\rm{d}} z} \left[\displaystyle\iint|E^{\omega}|^{2} {\rm{d}} x {\rm{d}} z\right]^{2}} \\ =\;&d_{0}^{2} \frac{\frac{\left[\displaystyle\iint[E^{2 \omega}]^{*} [E^{\omega}]^{2} {\rm{d}} x {\rm{d}} z\right]^{2}}{\bigg|\displaystyle\iint|E^{2\omega}|^{2}E^{2\omega} {\rm{d}} x {\rm{d}} z \bigg|^{\frac{2}{3}} \bigg|\displaystyle\iint|E^{\omega}|^{2}E^{\omega} {\rm{d}} x {\rm{d}} z\bigg|^{\frac{4}{3}}}}{\frac{{\displaystyle\iint|E^{2 \omega}|^{2} {\rm{d}} x {\rm{d}} z} \left[\displaystyle\iint|E^{\omega}|^{2} {\rm{d}} x {\rm{d}} z\right]^{2}}{\left|\displaystyle\iint|E^{2\omega}|^{2}E^{2\omega} {\rm{d}} x {\rm{d}} z\right|^{\frac{2}{3}} \bigg|\displaystyle\iint|E^{\omega}|^{2}E^{\omega} {\rm{d}} x {\rm{d}} z\bigg|^{\frac{4}{3}}}}, \\\end{split} \tag{A14} 令
\begin{split} & \xi =\frac{\displaystyle\iint\left[E^{2 \omega}\right]^{*} \left[E^{\omega}\right]^{2} {\rm{d}} x {\rm{d}} z}{\left|\displaystyle\iint|E^{2\omega}|^{2}E^{2\omega} {\rm{d}} x {\rm{d}} z\right|^{\frac{1}{3}}\left|\displaystyle\iint|E^{\omega}|^{2}E^{\omega} {\rm{d}} x {\rm{d}} z\right|^{\frac{2}{3}}},\\ & S_{{\rm{0}}}= \frac{{\displaystyle\iint|E^{2 \omega}|^{2} {\rm{d}} x {\rm{d}} z} \left[\displaystyle\iint|E^{\omega}|^{2} {\rm{d}} x {\rm{d}} z\right]^{2}}{\left|\displaystyle\iint|E^{2\omega}|^{2}E^{2\omega} {\rm{d}} x {\rm{d}} z\right|^{\frac{2}{3}}\left|\displaystyle\iint|E^{\omega}|^{2}E^{\omega} {\rm{d}} x {\rm{d}} z\right|^{\frac{4}{3}}}, \end{split} ξ表示空间模场重叠因子,
S_{\rm{0}} 表示泵浦光和参量光的有效模场面积. 于是K_{3} 简化为K_{3}=\varepsilon_{0}\sqrt{\frac{(2 \omega)^{2}}{2\left(N^{\omega}\right)^{2} N^{2 \omega}}\left(\frac{\mu_{0}}{\varepsilon_{0}}\right)^{3 / 2} \frac{d_{0}^{2} \xi^{2}}{S_{\rm{0}}}}. \tag{A15} 由于在简并自发参量下转换中相位失配参量
2\varDelta=0 时2\omega_{3}=\omega_{1}=\omega_{2} , 所以有\kappa=\frac{\varepsilon_{0}}{2}\sqrt{\frac{(2 \omega)^{2}}{2\left(N^{\omega}\right)^{2} N^{2 \omega}}\left(\frac{\mu_{0}}{\varepsilon_{0}}\right)^{3 / 2} \frac{d_{0}^{2} \xi^{2}}{S_{\rm{0}}}}. \tag{A16} 利用
{\rm{e}}^{[{\rm{i}}(\beta_{i}+\varDelta)y-\omega_{i}t]}(i=1, 2) , 可将方程(A9)和(A10)写为[30,31]\frac{{\rm{d}}}{{\rm{d}} y} a_{1}(y) =\varGamma {\rm{e}}^{{\rm{i}} \varphi} a_{2}^{\dagger}(y)-{\rm{i}} \varDelta a_{1}(y),\tag{A17} \frac{{\rm{d}}}{{\rm{d}} y} a_{2}(y) =\varGamma {\rm{e}}^{{\rm{i}} \varphi} a_{1}^{\dagger}(y)-{\rm{i}} \varDelta a_{2}(y), \tag{A18} \varGamma {\rm{e}}^{{\rm{i}} \varphi} = {\rm{i}} \kappa A_{3}, \tag{A19} 其中φ是泵浦光的相位,
\varGamma=|\kappa| P_{3}^{1 / 2} . 参量下转换的光子是由真空涨落产生的非相干光, 输入信号光和闲频光的振幅为a_{1}(y)=a_{1}(0),\; a_{2}(y)=a_{2}(0) , 在y = 0时可以得到线性差分方程(A17)和(A18)的解:\begin{split}a_{1}(y)= \;&[\cosh \gamma y-({\rm{i}} \varDelta / \gamma) \sinh \gamma y] a_{1}(0)\\ &+[(\varGamma / \gamma) {\rm{e}}^{{\rm{i}} \varphi} \sinh \gamma y] a_{2}^{\dagger}(0), \end{split}\tag{A20} \begin{split}a_{2}(y)= \;&[(\varGamma / \gamma) {\rm{e}}^{{\rm{i}} \varphi} \sinh \gamma y] a_{1}^{\dagger}(0)\\ &+[\cosh \gamma y-({\rm{i}} \varDelta / \gamma)\sinh \gamma y] a_{2}(0), \end{split}\tag{A21} \gamma \; = \sqrt{\varGamma^{2}-\varDelta^{2}}, \tag{A22} 其中Γ是与耦合系数κ和泵浦光功率有关的实变量
\varGamma= |\kappa| P_{3}^{1 / 2} . 于是根据文献[30], 光子数的期望值可以表示为\begin{split} \langle N_{1} \rangle =\langle N_{2}\rangle &=\langle{a_{2}^{\dagger}(L)}{a_{1}(L)}\rangle {|{\varGamma/\gamma}\sinh \gamma L|}^{2} \\ \; &\approx\varGamma^{2}L^{2}[\sin(\Delta L)/(\Delta L)]^{2}. \end{split}\tag{A23} 由于不同的光子对之间没有相关性, 所以输出光功率是由各光子对功率的简单求和得到. 单位时间内流动的信号光子数表示为
\left \langle N_{1}\right \rangle ({\rm{d}}\omega_{1}/2\pi) , 不同信号光频率间隔表示为\Delta \omega_{1} . 因此, 信号光功率P表示为\begin{split} &\; P=h\omega_{1}\int \left \langle N_{1} \right \rangle ({\rm{d}}\omega_{1}/2\pi ) \\ \; &\approx h\omega_{1}\int \varGamma^{2}L^{2}[{\rm{sin}}(\Delta L)/(\Delta L)]^{2}({\rm{d}}\omega_{1}/2\pi ).\end{split}\tag{A24} 在简并SPDC中, 相位匹配适用于相同偏振和中心频率的信号光和闲频光(
\omega_{10}= \omega_{20} = \omega_{3}/2 ), 由(A1)式定义的相位失配参数Δ与\omega_{1} 关于精确的相位匹配频率\omega_{3}/2 的偏差有关:2 \varDelta=-g\left(\omega_{1}-\omega_{3} / 2\right)^{2}, \quad g=\left[\partial^{2} \beta / \partial \omega^{2}\right]_{\omega_{3}/ 2}. \tag{A25} (A24)式表示在全部的
\omega_{1} > \omega_{2}, \;\omega_{2} >\omega_{1} 范围进行积分, 因此信号光和闲频光是不可区分的, 通过使用\displaystyle\int ( \sin x^{2}/x^{2})^{2} {\rm{d}} x =(4/3)\pi^{1/2} , 可以得到总的输出功率的表达式:P=\frac{4 \hbar \omega_{1} c|\kappa|^{2} P_{3} L^{3 / 2}}{3 \lambda_{1} \sqrt{\left|\partial n_{1} / \partial \lambda\right|}},\tag{A26} 于是可以计算转化效率
\eta=P/P_{3} , 光子对产生率{{G}}= P/(2\hbar \omega_{1}) , 以及对应光谱半高宽的光子对亮度{{B}}={{G}}/ {{W}} .[1] Dowling J P, Milburn G J 2003 Philos. Trans. A. Math. Phys. Eng. Sci. 361 1655
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-
图 1 波导结构截面图及单光子源器件结构示意图. 1)为Y分束器; 2)为非对称定向耦合器; 3)为模式色散相位匹配光子对源; 4)为上臂带有一个电热移相器的2 × 2定向耦合器. 建模采用的铌酸锂波导结构是侧壁倾角θ = 60°, h = 0.36 μm的条形波导, 材料模型为一致生长的铌酸锂[26]
Fig. 1. Cross-sectional diagram of the waveguide structure and schematic illustration of the single-photon source device. 1) Y-splitter. 2) Asymmetric directional coupler. 3) Mode dispersion phase-matched photon pair source. 4) 2 × 2 directional coupler in the upper arm with an electrothermal phase shifter. The modeled lithium niobate waveguide structure has a sidewall angle θ = 60° and a height h = 0.36 μm, with the material model being uniformly grown lithium niobate[26]
图 2 (a) 775 nm泵浦光在
{\rm{TE}}_0 ,{\rm{TE}}_1 ,{\rm{TE}}_2 模式下有效折射率随波导宽度的变化; (b)非对称定向耦合器的自由光谱范围以及模拟光场分布图Fig. 2. (a) Effective refractive index variation curves for 775 nm pump light in
{\rm{TE}}_0 ,{\rm{TE}}_1 and{\rm{TE}}_2 modes as a function of waveguide width; (b) free spectral range and simulated optical field distribution diagram of asymmetric directional coupler
图 3 (a) 775 nm的
{\rm{TE}}_0 ,{\rm{TE}}_1 ,{\rm{TE}}_2 模式光和1550 nm的{\rm{TE}}_0 模式光的有效折射率随波导宽度的变化; (b)输出参量光光谱图Fig. 3. (a) Variation curves of effective refractive index for 775 nm
{\rm{TE}}_0 ,{\rm{TE}}_1 ,{\rm{TE}}_2 mode light and 1550 nm{\rm{TE}}_0 mode light as a function of waveguide width; (b) output parametric light spectrum diagram
图 4 (a)相位匹配波长随波导宽度的变化; (b)相位匹配波长随波导侧边倾角θ的变化; (c)相位匹配波长随温度的变化
Fig. 4. (a) Variation of phase-matching wavelength with waveguide width; (b) variation of phase-matching wavelength with waveguide sidewall angle θ; (c) variation of phase-matching wavelength with temperature
图 5 波导侧边倾角为60°的条形波导在高度为300—600 nm范围内、不同波导宽度情况下, 775 nm
{\rm{TE}}_2 泵浦光与1550 nm{\rm{TE}}_0 参量光的有效折射率扫描数据Fig. 5. Effective refractive index scan data for 775 nm
{\rm{TE}}_2 pump light and 1550 nm{\rm{TE}}_0 parametric light in the case of different waveguide widths for strip waveguides with a sidewall angle of 60° and heights ranging from 300 nm to 600 nm -
[1] Dowling J P, Milburn G J 2003 Philos. Trans. A. Math. Phys. Eng. Sci. 361 1655
Google Scholar
[2] Flamini F, Spagnolo N, Sciarrino F 2019 Rep. Prog. Phys. 82 016001
Google Scholar
[3] Tang Y L, Yin H L, Chen S J, Liu Y, Zhang W J, Jiang X, Zhang L, Wang J, You L X, Guan J Y 2014 Phys. Rev. Lett. 113 190501
Google Scholar
[4] Ren J G, Xu P, Yong H L, Zhang L, Liao S K, Yin J, Liu W Y, Cai W Q, Yang M, Li L 2017 Nature 549 70
Google Scholar
[5] Chi Y, Huang J, Zhang Z, Mao J, Zhou Z, Chen X, Zhai C, Bao J, Dai T, Yuan H 2022 Nat. Commun. 13 1166
Google Scholar
[6] Lloyd S 1996 Science 273 1073
Google Scholar
[7] Zhong H S, Deng Y H, Qin J, Wang H, Chen M C, Peng L C, Luo Y H, Wu D, Gong S Q, Su H 2021 Phys. Rev. Lett. 127 180502
Google Scholar
[8] Wang J, Sciarrino F, Laing A, Thompson M G 2020 Nat. Photonics 14 273
Google Scholar
[9] Wang J, Paesani S, Ding Y, Santagati R, Skrzypczyk P, Salavrakos A, Tura J, Augusiak R, Mančinska L, Bacco D 2018 Science 360 285
Google Scholar
[10] Qiang X G, Zhou X Q, Wang J W, Wilkes C M, Loke T, O'Gara S, Kling L, Marshall G D, Santagati R, Ralph T C 2018 Nat. Photonics 12 534
Google Scholar
[11] Politi A, Matthews J C, O'brien J L 2009 Science 325 1221
Google Scholar
[12] Peruzzo A, Lobino M, Matthews J C, Matsuda N, Politi A, Poulios K, Zhou X Q, Lahini Y, Ismail N, Wörhoff K 2010 Science 329 1500
Google Scholar
[13] Laing A, Peruzzo A, Politi A, Verde M R, Halder M, Ralph T C, Thompson M G, O'Brien J L 2010 Appl. Phys. Lett. 97 211109
Google Scholar
[14] Shadbolt P J, Verde M R, Peruzzo A, Politi A, Laing A, Lobino M, Matthews J C, Thompson M G, O'Brien J L 2012 Nat. Photonics 6 45
Google Scholar
[15] Gerrits T, Thomas Peter N, Gates J C, Lita A E, Metcalf B J, Calkins B, Tomlin N A, Fox A E, Linares A L, Spring J B 2011 Phys. Rev. A 84 060301
Google Scholar
[16] Carolan J, Harrold C, Sparrow C, Martín-López E, Russell N J, Silverstone J W, Shadbolt P J, Matsuda N, Oguma M, Itoh M 2015 Science 349 711
Google Scholar
[17] Kuyken B, Leo F, Clemmen S, Dave U, Van Laer R, Ideguchi T, Zhao H, Liu X, Safioui J, Coen S 2017 Nanophotonics 6 377
Google Scholar
[18] Alibart O, D'Auria V, De Micheli M, Doutre F, Kaiser F, LabontéL, Lunghi T, Picholle É, Tanzilli S 2016 J. Opt. 18 104001
Google Scholar
[19] Zhang M, Wang C, Cheng R, Shams-Ansari A, Lončar M 2017 Optica 4 1536
Google Scholar
[20] Jin H, Liu F, Xu P, Xia J, Zhong M, Yuan Y, Zhou J, Gong Y, Wang W, Zhu S 2014 Phys. Rev. Lett. 113 103601
Google Scholar
[21] Wang C, Zhang M, Chen X, Bertrand M, Shams-Ansari A, Chandrasekhar S, Winzer P, Lončar M 2018 Nature 562 101
Google Scholar
[22] Elkus B S, Abdelsalam K, Rao A, Velev V, Fathpour S, Kumar P, Kanter G S 2019 Opt. Express 27 38521
Google Scholar
[23] Javid U A, Ling J, Staffa J, Li M, He Y, Lin Q 2021 Phys. Rev. Lett. 127 183601
Google Scholar
[24] Zhao J, Ma C, Rüsing M, Mookherjea S 2020 Phys. Rev. Lett. 124 163603
Google Scholar
[25] 张晨涛, 石小涛, 朱文新, 朱金龙, 郝向英, 金锐博 2022 物理学报 71 204201
Google Scholar
Zhang C T, Shi X T, Zhu W X, Zhu J L, Hao X Y, Jin R B 2022 Acta Phys. Sin. 71 204201
Google Scholar
[26] Zelmon D E, Small D L, Jundt D 1997 J. Opt. Soc. Am. B 14 3319
Google Scholar
[27] Donnelly J P, Haus H A, Molter L A 1988 J. Lightwave. Technol. 6 257
Google Scholar
[28] Chrostowski L, Hochberg M 2015 Silicon Photonics Design: from Devices to Systems (Cambridge: Cambridge University Press) pp92–95
[29] Suhara T, Fujimura M 2003 Waveguide Nonlinear-optic Devices (Vol. 11) (New York: Springer Science & Business Media) pp41, 42
[30] Suhara T 2009 Laser. Photonics Rev. 3 370
Google Scholar
[31] Suhara T, Kintaka H 2005 IEEE J. Quantum. Electron. 41 1203
Google Scholar
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