基于1/4波片的腔增强自发参量下转换过程中双折射效应的补偿

贺海; 杨鹏飞; 张鹏飞; 李刚; 张天才

doi:10.7498/aps.72.20230422

摘要

Abstract

作者及机构信息

Authors and contacts

文章全文

1. 引　言
2. 利用QWP补偿双折射效应的理论分析
3. 实验装置
4. 实验结果
5. 结　论

参考文献

摘要

腔增强的光学自发参量下转换是量子光学中产生量子光场的基本方法之一, 然而下转换过程往往受到非线性晶体的双折射效应的影响. 特别是在利用II类准相位匹配非线性晶体产生双光子对的过程中, 晶体的双折射效应使得信号光和闲置光不能同时起振. 本文提出并验证了一种利用1/4波片补偿信号光和闲置光的光程的方法, 在保证较小内腔损耗及良好的调节自由度的情况下, 以相对简洁的装置实现信号光和闲置光的双共振.

关键词:

Single-photon source is an essential element in quantum information processing, and extensively used in the proof-in-principle demonstration in quantum physics, quantum imaging, quantum cryptography, etc. Considering the operating temperature and system complexity, it is a favorable option to choose spontaneous parametric down-conversion (SPDC) combined with the enhancement effect of a cavity. When generating significant single-photon source via the cavity-enhanced type-II spontaneous parametric down-conversion method, there appears inevitable birefringence effect which obviously influences the resonance condition. In order to compensate for birefringence effect, different approaches have been used such as introducing compensating crystal, placing a half-wave plate, tuning the temperature of the nonlinear crystal, customized conjoined double-cavity structure, and cluster effect. In this work, two quarter-wave plates, with an angle of 45° between the optical axis and the crystal axis, are placed in the cavity to ensure the double resonance of signal photon and idler photon. In the process, the signal photon and idler photon generated simultaneously have different polarizations perpendicular to each other through the type-II nonlinear crystal. Considering horizontally polarized photon, its polarization is changed into left circular polarization by the first quarter-wave plate and then returns as vertical polarization. After traversing a long optical path, it shifts to right circular polarization through the second quarter-wave plate. When the photon passes through the same quarter-wave plate again, the polarization state is originally converted into horizontal polarization state. Then the photon completes a round-trip. The other photon with vertical polarization experiences the same process. As a result, the signal photon and idler photon travel identical optical path. The general explanation is described by the Jones matrices, with the emphasis on the transformation of the polarizations of photons. This method can effectively compensate for birefringence effect, achieving double resonance by using a relatively simple device under the condition of smaller intra-cavity loss and more flexible for adjustment. The signal (idler) photon has a sub-natural linewidth of

$1.01( 1.08 )\;{\rm{MHz}}$ , demonstrating the feasibility of the proposed technique. This introduced compensating method paves the way to the realization of single-photon quantum source applied to the research of single-photon-single-atom quantum information processing, quantum interface and quantum network node with a single cesium atom confined in the strongly coupled cavity quantum electrodynamics system.

Keywords:

cavity-enhanced spontaneous parametric down-conversion /
birefringence effect /
double resonance /
heralded single-photon source

PACS:

42.65.Lm	(Parametric down conversion and production of entangled photons)
42.50.-p	(Quantum optics)
14.70.Bh	(Photons)
03.67.-a	(Quantum information)

作者及机构信息

1.
山西大学, 量子光学与光量子器件国家重点实验室, 太原　030006

2.
山西大学, 极端光学省部共建协同创新中心, 太原　030006

通信作者: 李刚, gangli@sxu.edu.cn ; 张天才, tczhang@sxu.edu.cn

基金项目: 国家自然科学基金(批准号: 11974223, 11974225)和山西省“1331 工程”重点学科建设基金资助的课题.

Authors and contacts

1.
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China

2.
Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China

Corresponding author: Li Gang, gangli@sxu.edu.cn ; Zhang Tian-Cai, tczhang@sxu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11974223, 11974225) and the Fund for Shanxi 1331 Project Key Subjects Construction, China.

文章全文

1. 引　言

自发参量下转换(spontaneous parametric down-conversion, SPDC)^[1-3]是制备单光子源、压缩光等量子光源的基本手段之一, 在量子光学、量子模拟和量子信息处理中有着广泛的应用^[4-11]. SPDC过程中采用光学谐振腔的增强效应^[12-18]不但能够提高量子光源的产率, 而且能够压窄量子光源的线宽^[19]. 在利用II类准相位匹配非线性晶体的腔增强SPDC过程产生双光子对的实验中, 产生的光子对(信号光和闲置光)偏振相互垂直, 在晶体中的折射率不同, 从而影响光学腔对于信号光和闲置光的双共振. 为了在特定波长实现光学腔的双共振条件, 增加双光子对的产率, 通常采取腔内放置补偿晶体^[12]或1/2波片^[14]、调节晶体温度^[20]、设计结合型腔^[21]和簇效应^[22]等方法来补偿晶体的双折射效应. 本文提出一种在SPDC腔中放置2个1/4波片(quarter-wave plate, QWP)的方法实现信号光与闲置光的双共振. 2个QWP分别放置于非线性晶体两侧, 其光轴与晶体主轴夹角为 ${45^ \circ }$ , 光学腔输出的信号光子和闲置光子偏振分别为固定的左旋和右旋. 此方法能够使非线性晶体工作在最佳温度, 腔型结构简洁, 易于操作, 可用于预告式单光子源的制备, 研究单光子-单原子的相互作用.

2. 利用QWP补偿双折射效应的理论分析

理论设计图如图1所示, 在装有II类准相位匹配非线性晶体如周期性极化磷酸氧钛钾(periodically poled KTiOPO₄, PPKTP)的SPDC腔中, 放置2个光轴与晶体主轴夹角为45°的QWPs, 用于补偿信号光(signal photon, s)和闲置光(idler photon, i)由于偏振垂直造成的双折射效应, 实现双共振. 图1(a)和图1(b)分别展示了下转换腔中水平(H)偏振(红虚线箭头)和竖直(V)偏振 (黑虚线箭头)的光子在腔内完成一次自再现的偏振变化. 图1(a)中, SPDC产生的H偏振的光子经QWP1后变换为左旋(L)圆偏振光, 经反射镜M2反射再次经过QWP1后变为V偏振. V偏振的光子经QWP2后变为右旋(R)圆偏振光, 经M4反射再次经过QWP2时变为H偏振, 最后以H偏振再次通过PPKTP晶体, 完成一次循环. 图1(b)为V偏振的光子的情况, 与H偏振光类似. 综上, 偏振相互垂直的光子对在腔内完成一次循环经历了相同的光程, 补偿了双折射效应并实现了光子对的双共振. M4为输出耦合镜, 出射光子利用QWP和偏振分光棱镜(polarization beam splitter, PBS)能够将偏振相互垂直的信号光和闲置光完全分开.

$图 1 置有QWPs的SPDC腔中光子的偏振变化(PZT, 压电陶瓷)　(a) SPDC产生的水平偏振光子(红虚线箭头)的情况; (b) 竖直偏振光子(黑虚线箭头)的情况. 其中, 蓝实线箭头和红实线箭头分别表示波长为426 nm和852 nm的激光, R和L分别表示右旋圆偏振光和左旋圆偏振光, k, s, p分别表示光波矢(z轴)、光场垂直分量(y轴)和平行分量(x轴)\r\nFig. 1. Polarization transformations of photons in SPDC cavity with QWPs. Photon with an initial polarization of H (red dashed arrow) in (a) or V (black dashed arrow) in (b) generated from SPDC. Blue and red solid arrows stand for 426 nm and 852 nm laser respectively, R (L) shows right (left) circular polarization, k is wave vector (z axis), s is perpendicular part (y axis) and p represents parallel part (x axis). PZT represents piezoelectric transducer.$

图 1 置有QWPs的SPDC腔中光子的偏振变化(PZT, 压电陶瓷)　(a) SPDC产生的水平偏振光子(红虚线箭头)的情况; (b) 竖直偏振光子(黑虚线箭头)的情况. 其中, 蓝实线箭头和红实线箭头分别表示波长为426 nm和852 nm的激光, R和L分别表示右旋圆偏振光和左旋圆偏振光, k, s, p分别表示光波矢(z轴)、光场垂直分量(y轴)和平行分量(x轴)

Fig. 1. Polarization transformations of photons in SPDC cavity with QWPs. Photon with an initial polarization of H (red dashed arrow) in (a) or V (black dashed arrow) in (b) generated from SPDC. Blue and red solid arrows stand for 426 nm and 852 nm laser respectively, R (L) shows right (left) circular polarization, k is wave vector (z axis), s is perpendicular part (y axis) and p represents parallel part (x axis). PZT represents piezoelectric transducer.

下载: 全尺寸图片幻灯片

利用琼斯矩阵^[23]分析信号光子和闲置光子在腔内循环的偏振变换. SPDC产生的光子在腔内循环1次的琼斯矩阵表示为

$\begin{split} {\boldsymbol{M}} =\;& {{\boldsymbol{M}}_{{\text{non}}}}{{\boldsymbol{M}}_1}{{\boldsymbol{M}}_3}{{\boldsymbol{M}}^{\text{r}}}({\alpha _2},{{\text{π }} /2})\\ \;&\cdot{{\boldsymbol{M}}_4}{\boldsymbol{M}}({\alpha _2},{{\text{π }}/2}){{\boldsymbol{M}}_3}{{\boldsymbol{M}}_1}{{\boldsymbol{M}}_{{\text{non}}}} \\ &\cdot{{\boldsymbol{M}}^{\text{r}}}({\alpha _1},{{\text{π }}/2}){{\boldsymbol{M}}_2}{\boldsymbol{M}}({\alpha _1},{{\text{π }}/2}) \end{split} \text{, }$

(1)

其中, 平面反射镜的琼斯矩阵 ${{\boldsymbol{M}}_l} = {r_l}\left( \begin{array}{cc} { - 1}&0 \\ 0&1 \end{array} \right)$ ( ${r_l}$ 为振幅反射率, $l = 1, 2, 3, 4$ ). 非线性晶体的琼斯矩阵 ${{\boldsymbol{M}}_{{\text{non}}}} = \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&{\exp ( - {\text{i}}\delta )} \end{array}} \right)$ , $\delta$ 为晶体相位偏移量. QWP的琼斯矩阵 ${\boldsymbol{M}}({\alpha _i}, {{\text{π }}/2}) = \left( {\begin{array}{*{20}{c}} {{{\cos }^2}{\alpha _i} - {\rm i}{{\sin }^2}{\alpha _i}}& {(1 + {\rm i})\cos {\alpha _i}\sin {\alpha _i}} \\ {(1 + {\rm i})\cos {\alpha _i}\sin {\alpha _i}}& {{{\sin }^2}{\alpha _i} - {\rm i}{{\cos }^2}{\alpha _i}} \end{array}} \right)$ ( ${\alpha _i}$ 为主轴与x轴夹角, $i = 1, 2$ ), 光路反向^[24]时有

${\boldsymbol M}^{\text r} ({\alpha _i}, \text{π}/2) = \left(\begin{array}{cc} 1&0 \\ 0&{ - 1} \end{array}\right){\boldsymbol M} ({\alpha _i}, {\text{π}/2})\left(\begin{array}{cc} 1&0 \\ 0&{ - 1} \end{array} \right).$

本文设计选择 ${\alpha _1} = {\alpha _2} = {\text{π}}/4$ . 解本征方程 ${\boldsymbol{M}}{E_j} = {\xi _j}{E_j}\left( {j = 1, 2} \right)$ 得到相应的本征偏振态. 以本征偏振态的输入光场在腔内循环一周偏振保持不变, 即能够实现自再现. 本征值 ${\xi _j} = {r_j}{{\text{e}}^{{\text{i}}{\theta _j}}}$ 为复数, 其中 ${r_j}$ 为振幅, ${\theta _j}$ 为相位. 当2个本征值的振幅、相位均相等时即说明双折射效应得到有效补偿; 本征矢量E_j即腔内2个相互垂直的纵模.

本文首先模拟分析了调整QWP时光学SPDC腔的本征值变化, 如图2所示. 其中红线和蓝线对应了2个不同的本征值. 取 $\delta = 0.5$ , ${\alpha _2} = {45^ \circ }$ 并改变QWP1时的分析结果如图2(a)和(b); ${\alpha _1} = {45^ \circ }$ 并改变QWP2时的分析结果如图2(c)和(d), ${\alpha _2} = 0.23{\text{π }}$ 并改变QWP1时的分析结果如图2(e)和(f). 结合本征值的振幅和相位可以看出, 只有当2个波片角度都为 ${45^ \circ }$ 或 ${135^ \circ }$ 时, 2个本征值完全相等, 可以成功地补偿双折射效应, 实现双共振; 当其中一个QWP的角度不等于 ${45^ \circ }$ 或 ${135^ \circ }$ 时, 无论怎样调节另一个QWP角度, 无法同时满足本征值的振幅及相位均相等的条件, 不足以实现双共振的补偿.

$图 2 2个QWPs不同角度下的本征值. ${\alpha _2} = {45^ \circ }$并改变QWP1时本征值的振幅(a)和相位(b); ${\alpha _1} = {45^ \circ }$并改变QWP2时本征值的振幅(c)和相位(d); ${\alpha _2} = 0.23{\text{π }}$并改变QWP1时本征值的振幅(e)和相位(f). 其中红线和蓝线对应了2个不同的本征值\r\nFig. 2. Eigenvalues with different degrees of two QWPs. Amplitude in (a) of eigenvalues and phase in (b) of the eigenvalues on the condition of ${\alpha _2} = {45^ \circ }$ and different degrees of QWP1; amplitude in (c) and the phase in (d) with ${\alpha _1} = {45^ \circ }$ and different degrees of QWP2; amplitude in (e) and the phase in (f) with ${\alpha _2} = 0.23{\text{π }}$ and different degrees of QWP1. Red line is about one of eigenvalues, while the blue line is about the other.$

图 2 2个QWPs不同角度下的本征值.

${\alpha _2} = {45^ \circ }$

并改变QWP1时本征值的振幅(a)和相位(b);

${\alpha _1} = {45^ \circ }$

并改变QWP2时本征值的振幅(c)和相位(d);

${\alpha _2} = 0.23{\text{π }}$

并改变QWP1时本征值的振幅(e)和相位(f). 其中红线和蓝线对应了2个不同的本征值

Fig. 2. Eigenvalues with different degrees of two QWPs. Amplitude in (a) of eigenvalues and phase in (b) of the eigenvalues on the condition of

${\alpha _2} = {45^ \circ }$

and different degrees of QWP1; amplitude in (c) and the phase in (d) with

${\alpha _1} = {45^ \circ }$

and different degrees of QWP2; amplitude in (e) and the phase in (f) with

${\alpha _2} = 0.23{\text{π }}$

and different degrees of QWP1. Red line is about one of eigenvalues, while the blue line is about the other.

下载: 全尺寸图片幻灯片

3. 实验装置

图3展示了采用所述方法补偿双折射效应的实验装置图. 实验中波长为852 nm的激光 (Toptica TA pro 850)注入倍频腔产生426 nm 的倍频光. 倍频腔前的1/2波片(half-wave plate, HWP)用于调节进腔光的偏振. 倍频产生的426 nm光作为泵浦光, 依次经过滤波片(filter, Thorlabs FESH0500)过滤852 nm的基频光的残留成分、经透镜模式匹配后, 注入SPDC腔. SPDC腔为折叠型Fabry-Perot腔, 由4面反射镜组成, 物理腔长约为0.65 m, 自由光谱区(free spectral range, FSR)约为224.7 MHz. 其中, M1和M2在852 nm处反射率R大于99.99%, 曲率半径为100 mm, 在晶体处产生最佳腰斑来保证足够高的转换效率; M3为反射率99.99% (@852 nm)的平面镜; 平面镜M4同样为平面镜并作为输出耦合镜, 其反射率为98% (@852 nm); 4个腔镜在426 nm处的透射率均大于99%. 一块温度精确可控的尺寸为 $1{\text{ mm}} \times 2{\text{ mm}} \times 10{\text{ mm}}$ 的II类PPKTP晶体(raicol crystals)置于腔内. 用于补偿双折射效应的2个QWPs安装在高精度旋转安装座(Thorlabs PRM05/M)上, 角度调节精度为10^'. SPDC腔的腔镜、晶体支架和波片架等均安置在一块殷钢底座上, 增加系统的整体稳定性. 通过压电陶瓷(piezoelectric transducer, PZT)扫描SPDC腔腔长, 并利用Hansch-Couilland方法^[25]锁定. 其中, 频率远离下转换光场且已通过频率链^[26]锁定的840 nm辅助激光经长通滤波片(long-pass filter, LPF, Semrock ${\text{TLP01-887-25}} \times {\text{36}}$ )反射后耦合进腔, 出射光经双色片(dichroic mirror, DM)透射后被探测器收集产生鉴频信号, 进而锁定腔长. SPDC腔产生的偏振相互垂直的光子对经LPF和3个长度(反射率)分别为3 mm (R = 95%), 5.56 mm (R = 92%), 11.12 mm (R = 92%)的级联标准具进行模式过滤, 被QWP和PBS分离后分别耦合进单模光纤进行探测. 2片带宽为3.2 nm的干涉滤波片(Semrock LL01-852-12.5)置于光纤耦合头前, 用于隔离背景噪声. 实验中利用单光子计数模块(single-photon counting modules, SPCMs, Excelitas SPCM-850-60-FC)进行信号探测, SPCMs的输出信号由时间数字转换器(time to digital converter, TDC, Swabian Instruments Time Tagger 20)进行数据分析和处理. 其中使用的SPCM的时间分辨率的参考值为350 ps, 当2个SPCMs进行符合计数测量时, 符合的时间分辨率为 $\sqrt 2 \times 350 \approx 500{\text{ ps}}$ .

$图 3 使用QWP补偿双折射效应的装置图. 852 nm的激光(红色)通过倍频腔(Doubler)产生腔增强的SPDC所需的426 nm (蓝色)的泵浦光. SPDC腔腔长由波长远离光子对的840 nm的辅助光(紫色)进行锁定. SPDC产生的偏振相互垂直的光子对耦合进光纤进行后续实验\r\nFig. 3. Experimental apparatus about birefringence compensating utilizing two QWPs. Laser with the wavelength of 852 nm in red color is sent into the doubler cavity. Generated frequency-doubling light at 426 nm (in blue color) is filtered and coupled to the SPDC cavity after lens-transforming. Length of SPDC cavity is stabilized by the 840 nm auxiliary light. Signal and idler photons generated from SPDC cavity filtered by cascaded etalons are split on a PBS and coupled to multi-fibers for further processing. Doubler, second-harmonic generation cavity.$

图 3 使用QWP补偿双折射效应的装置图. 852 nm的激光(红色)通过倍频腔(Doubler)产生腔增强的SPDC所需的426 nm (蓝色)的泵浦光. SPDC腔腔长由波长远离光子对的840 nm的辅助光(紫色)进行锁定. SPDC产生的偏振相互垂直的光子对耦合进光纤进行后续实验

Fig. 3. Experimental apparatus about birefringence compensating utilizing two QWPs. Laser with the wavelength of 852 nm in red color is sent into the doubler cavity. Generated frequency-doubling light at 426 nm (in blue color) is filtered and coupled to the SPDC cavity after lens-transforming. Length of SPDC cavity is stabilized by the 840 nm auxiliary light. Signal and idler photons generated from SPDC cavity filtered by cascaded etalons are split on a PBS and coupled to multi-fibers for further processing. Doubler, second-harmonic generation cavity.

下载: 全尺寸图片幻灯片

4. 实验结果

通过观测光子对的互关联函数, 可以精细调节两波片角度进而有效地补偿双折射效应, 也可用来表征补偿效果. 其中, 光子对之间的互关联函数表示在t时刻探测到信号光子, 在延时 $\tau$ 后的 $t + \tau$ 时刻探测到闲置光子的概率^[27,28]. 值得注意的是, 实际测量过程中只有当探测系统有限的分辨时间率 ${\tau _{\text{D}}}$ 远小于光子的腔内循环时间 ${t_{{\text{rt}}}}$ 时, 互关联函数才能呈现由于多纵模造成的以 ${t_{{\text{rt}}}}$ 为时间间隔的梳状结构.

实验首先探究了腔内2个QWPs的角度对互关联函数的影响. 如图4所示, 固定QWP2的角度 ${\beta _0}$ 约为 ${45^ \circ }$ , 测量QWP1角度 ${\alpha _0}$ 在 ${45^ \circ }$ 附近变化时的互关联函数. 为了更加直观地分析, 插图展示了腔内2个本征矢量对应纵模的透射峰. 将2个透射峰的重叠程度定义为

$图 4 保持QWP2的角度约$ {45^ \circ } $, QWP1不同角度(重合程度)下的互关联函数　(a) $ {\alpha _0} + 80' $(72.8%); (b) $ {\alpha _0} $ (42.1%); (c) $ {\alpha _0} - 80' $ (25.0%); (d) $ {\alpha _0} - 160' $ (14.6%). 插图为相应腔内2个本征值所对应纵模的透射峰\r\nFig. 4. Cross-correlation function with different degrees of QWP1 (overlap) on the condition of the degree of QWP2 approximately equal to $ {45^ \circ } $: (a) $ {\alpha _0} + 80' $ (72.8%); (b) $ {\alpha _0} $(42.1%); (c) $ {\alpha _0} - 80' $ (25.0%); (d) $ {\alpha _0} - 160' $ (14.6%). Insets are corresponding transmission spectra for the two different orthogonal modes.$

图 4 保持QWP2的角度约

${45^ \circ }$

, QWP1不同角度(重合程度)下的互关联函数　(a)

${\alpha _0} + 80'$

(72.8%); (b)

${\alpha _0}$

(42.1%); (c)

${\alpha _0} - 80'$

(25.0%); (d)

${\alpha _0} - 160'$

(14.6%). 插图为相应腔内2个本征值所对应纵模的透射峰

Fig. 4. Cross-correlation function with different degrees of QWP1 (overlap) on the condition of the degree of QWP2 approximately equal to

${45^ \circ }$

: (a)

${\alpha _0} + 80'$

(72.8%); (b)

${\alpha _0}$

(42.1%); (c)

${\alpha _0} - 80'$

(25.0%); (d)

${\alpha _0} - 160'$

(14.6%). Insets are corresponding transmission spectra for the two different orthogonal modes.

下载: 全尺寸图片幻灯片

$\eta = \displaystyle\frac{{{{\left| {\displaystyle\int {E_1^ * {E_2}{\text{d}}A} } \right|}^2}}}{{\displaystyle\int {{{\left| {{E_1}} \right|}^2}{\text{d}}A \times \int {{{\left| {{E_2}} \right|}^2}{\text{d}}A} } }},$

其中 ${E_1}(A), {E_2}(A)$ 分别为2个纵模在空间A的电场分布. 测量过程中, 选择泵浦光功率为 $860\;{\text{μW}}$ , Time Tagger 20的时间分辨率设置为远小于 ${t_{{\text{rt}}}}$ 的75 ps. 如图4所示, 对比QWP1在不同角度(重合程度)下的互关联函数状态: (a) ${\alpha _0} + 80'$ (72.8%), (b) ${\alpha _0}$ (42.1%), (c) ${\alpha _0} - 80'$ (25.0%), (d) ${\alpha _0} - 160'$ (14.6%), 可以发现QWP1为 ${\alpha _0} + 80'$ 时补偿效果相对较好. 距离双共振的最佳角度越远, 双折射效应对光学腔共振条件的影响越大, 2个纵模的重合程度越小, 梳状信号的包络出现振荡.

将QWP1的角度固定为相对较好的 ${\alpha _0} + 80'$ , 改变QWP2的角度为 ${\beta _0} - 40'$ (60.9%), ${\beta _0} + 40'$ (90.0%), ${\beta _0} + 80'$ (35.2%), ${\beta _0} + 120'$ (23.8%), 并测量对应的互关联函数, 结果如图5所示. 可以发现QWP2处于 ${\beta _0} + 40'$ 时, 2个波片重合程度较好. 由实验结果可以看出, 互关联函数对QWP角度的依赖性非常敏感, 只有当2个QWP角度均为 ${45^ \circ }$ 时才足够补偿偏振相互垂直的纵模的双折射效应, 实现双共振. 参考互关联函数随QWPs角度的变化状态, 多次精细调节将腔内QWP的角度调到最佳值, 有效地完成对双折射效应的补偿.

$图 5 QWP1的角度固定为相对较好的$ {\alpha _0} + 80' $时QWP2在不同角度(重合程度)下的互关联函数　(a) $ {\beta _0} - 40' $ (60.9%); (b) $ {\beta _0} + 40' $ (90.0%); (c) $ {\beta _0} + 80' $ (35.2%); (d) $ {\beta _0} + 120' $ (23.8%)\r\nFig. 5. Cross-correlation function with different degrees of QWP2 (overlap) on the condition of QWP1 equal to $ {\alpha _{\text{0}}}{\text{ + 80'}} $: (a) $ {\beta _0} - 40' $ (60.9%); (b) $ {\beta _0} + 40' $ (90.0%); (c) $ {\beta _0} + 80' $ (35.2%); (d) $ {\beta _0} + 120' $ (23.8%).$

图 5 QWP1的角度固定为相对较好的

${\alpha _0} + 80'$

时QWP2在不同角度(重合程度)下的互关联函数　(a)

${\beta _0} - 40'$

(60.9%); (b)

${\beta _0} + 40'$

(90.0%); (c)

${\beta _0} + 80'$

(35.2%); (d)

${\beta _0} + 120'$

(23.8%)

Fig. 5. Cross-correlation function with different degrees of QWP2 (overlap) on the condition of QWP1 equal to

${\alpha _{\text{0}}}{\text{ + 80'}}$

: (a)

${\beta _0} - 40'$

(60.9%); (b)

${\beta _0} + 40'$

(90.0%); (c)

${\beta _0} + 80'$

(35.2%); (d)

${\beta _0} + 120'$

(23.8%).

下载: 全尺寸图片幻灯片

互关联函数可以用来衡量补偿效果. 在双共振情况下, 泵浦功率为1.26 mW、测量了时间分辨率分别为100 ps和4.4 ns时的互关联函数, 如图6所示. 图6(a), 互关联函数其归一化最大值随着泵浦功率增大而减小, 光子对之间的聚束效应逐渐减弱^[29,30]. 使用简化的理论模型 $H\left( \tau \right){{\text{e}}^{ - 2{\text{π }}{\gamma _{\text{s}}}\tau }} + H\left( { - \tau } \right){{\text{e}}^{2{\text{π }}{\gamma _{\text{i}}}\tau }}$ 对图6(b)中测量的数据点进行拟合^[17], 其中 $H\left( \tau \right)$ 为郝维赛德阶跃函数, ${\gamma _{\text{s}}}, {\gamma _{\text{i}}}$ 分别表示信号光和闲置光在腔中的衰减率. 拟合得到信号光(蓝色)和闲置光(红色)的腔衰减率分别为 $( 1.58 \pm 0.01 ){\text{ MHz}}$ 和 $\left( {1.69 \pm 0.01} \right){\text{ MHz}}$ , 进而光子的线宽为 $(1.01 \pm 0.01){\text{ MHz}}$ 和 $(1.08 \pm 0.01){\text{ MHz}}$ ^[19]. 比较光子对线宽的差异, 可以发现这一方法把双折射效应补偿得很好.

$图 6 不同时间分辨率条件下的互关联函数　(a) 100 ps; (b) 4.4 ns, 蓝色曲线(数据点)和红色曲线(数据点)分别表示信号光和闲置光的拟合曲线(数据)\r\nFig. 6. Cross-correlation function with different resolution time: (a) 100 ps; (b) 4.4 ns. The blue curve (points) and red curve (points) are representative fitting curves (experimental data) of signal and idler photons respectively.$

图 6 不同时间分辨率条件下的互关联函数　(a) 100 ps; (b) 4.4 ns, 蓝色曲线(数据点)和红色曲线(数据点)分别表示信号光和闲置光的拟合曲线(数据)

Fig. 6. Cross-correlation function with different resolution time: (a) 100 ps; (b) 4.4 ns. The blue curve (points) and red curve (points) are representative fitting curves (experimental data) of signal and idler photons respectively.

下载: 全尺寸图片幻灯片

5. 结　论

本文利用与非线性晶体主轴夹角为 ${45^ \circ }$ 的2个1/4波片来补偿双折射效应, 实现了SPDC腔中线宽分别为 $(1.01 \pm 0.01){\text{ MHz}}$ 和 $(1.08 \pm 0.01){\text{ MHz}}$ 的信号光和闲置光的双共振. 此方法在成功补偿双折射效应的同时, 引入的内腔损耗较小, 装置相对简洁, 保证非线性晶体工作在最佳温度, 亮度损耗较小. 这一方法能够应用于预告式单光子源的产生.

参考文献

[1]	Perumangatt C, Lohrmann A, Ling A 2020 Phys. Rev. A 102 012404 Google Scholar
[2]	Tang J S, Tang L, Wu H D, Wu Y, Sun H, Zhang H, Li T, Lu Y Q, Xiao M, Xia K Y 2021 Phys. Rev. Appl. 15 064020 Google Scholar
[3]	杨宏恩, 韦联福 2019 物理学报 68 234202 Google Scholar Yang H E, Wei L F 2019 Acta Phys. Sin. 68 234202 Google Scholar
[4]	Wang B X, Tao M J, Ai Q, Xin T, Lambert N, Ruan D, Cheng Y C, Nori F, Deng F G, and Long G L 2018 NPJ Quantum Inf. 4 52 Google Scholar
[5]	Yang B, Sun H, Ott R, Wang H Y, Zache T V, Halimeh J C, Yuan Z S, Hauke P, Pan J W 2020 Nature 587 392 Google Scholar
[6]	Piro N, Rohde F, Schuck C, Almendros M, Huwer J, Ghosh J, Haase A, Hennrich M, Dubin F, and Eschner J 2011 Nat. Phys. 7 17 Google Scholar
[7]	Pittman T P, Shih Y H, Strekalov D V, Sergienko A V 1995 Phys. Rev. A 52 3429 Google Scholar
[8]	Zhang H, Jin X M, Yang J, Dai H N, Yang S J, Zhao T M, Rui J, He Y, Jiang X, Yang F, Pan G S, Yuan Z S, Deng Y J, Chen Z B, Bao X H, Chen S, Zhao B, Pan J W 2011 Nat. Photonics 5 628 Google Scholar
[9]	Yin J, Cao Y, Li Y H, Liao S K, Zhang L, Ren J G, Cai W Q, Liu W Y, Li S, Dai H, Li G B, Lu Q M, Gong Y H, Xu Y, Li S L, Li F Z, Yin Y Y, Jiang Z Q, Li M, Jia J J, Ren G, He D, Zhou Y L, Zhang X X, Wang N, Chang X, Zhu Z C, Liu N L, Chen Y A, Lu C Y, Shu R, Peng C Z, Wang J Y, Pan J W 2017 Science 356 1140 Google Scholar
[10]	Broome M A, Fedrizzi A, Lanyon B P, Kassal I, Guzik A A, White A G 2010 Phys. Rev. Lett. 104 153602 Google Scholar
[11]	Esposito C, Barros M R, Hernandez A D, Carvacho G, Colandrea F D, Barboza R, Cardano F, Spagnolo N, Marrucci L, Sciarrino F 2022 NPJ Quantum Inf. 8 34 Google Scholar
[12]	Scholz M, Koch L, Benson O 2009 Phys. Rev. Lett. 102 063603 Google Scholar
[13]	Zhou Z Y, Ding D S, Li Y, Wang F Y, Shi B S 2014 J. Opt. Soc. Am. B 31 128 Google Scholar
[14]	Rambach M, Nikolova A, Weinhold T J, White A G 2016 APL Photonics 1 096101 Google Scholar
[15]	Niizeki K, Ikeda K, Zheng M Y, Xie X P, Okamura K, Takei N, Namekata N, Inoue S, Kosaka H, Horikiri T 2018 Appl. Phys. Express 11 042801 Google Scholar
[16]	Liu J J, Liu J C, Yu P, Zhan G Q 2020 APL Photonics 5 066105 Google Scholar
[17]	Moqanaki A, Maaas F, Walther P 2019 APL Photonics 4 090804 Google Scholar
[18]	Tsai P J, Chen Y C 2018 Quantum Sci. Technol. 3 034005 Google Scholar
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图 1 置有QWPs的SPDC腔中光子的偏振变化(PZT, 压电陶瓷)　(a) SPDC产生的水平偏振光子(红虚线箭头)的情况; (b) 竖直偏振光子(黑虚线箭头)的情况. 其中, 蓝实线箭头和红实线箭头分别表示波长为426 nm和852 nm的激光, R和L分别表示右旋圆偏振光和左旋圆偏振光, k, s, p分别表示光波矢(z轴)、光场垂直分量(y轴)和平行分量(x轴)

Fig. 1. Polarization transformations of photons in SPDC cavity with QWPs. Photon with an initial polarization of H (red dashed arrow) in (a) or V (black dashed arrow) in (b) generated from SPDC. Blue and red solid arrows stand for 426 nm and 852 nm laser respectively, R (L) shows right (left) circular polarization, k is wave vector (z axis), s is perpendicular part (y axis) and p represents parallel part (x axis). PZT represents piezoelectric transducer.

下载: 全尺寸图片幻灯片

图 2 2个QWPs不同角度下的本征值. ${\alpha _2} = {45^ \circ }$ 并改变QWP1时本征值的振幅(a)和相位(b); ${\alpha _1} = {45^ \circ }$ 并改变QWP2时本征值的振幅(c)和相位(d); ${\alpha _2} = 0.23{\text{π }}$ 并改变QWP1时本征值的振幅(e)和相位(f). 其中红线和蓝线对应了2个不同的本征值

Fig. 2. Eigenvalues with different degrees of two QWPs. Amplitude in (a) of eigenvalues and phase in (b) of the eigenvalues on the condition of ${\alpha _2} = {45^ \circ }$ and different degrees of QWP1; amplitude in (c) and the phase in (d) with ${\alpha _1} = {45^ \circ }$ and different degrees of QWP2; amplitude in (e) and the phase in (f) with ${\alpha _2} = 0.23{\text{π }}$ and different degrees of QWP1. Red line is about one of eigenvalues, while the blue line is about the other.

下载: 全尺寸图片幻灯片

图 3 使用QWP补偿双折射效应的装置图. 852 nm的激光(红色)通过倍频腔(Doubler)产生腔增强的SPDC所需的426 nm (蓝色)的泵浦光. SPDC腔腔长由波长远离光子对的840 nm的辅助光(紫色)进行锁定. SPDC产生的偏振相互垂直的光子对耦合进光纤进行后续实验

Fig. 3. Experimental apparatus about birefringence compensating utilizing two QWPs. Laser with the wavelength of 852 nm in red color is sent into the doubler cavity. Generated frequency-doubling light at 426 nm (in blue color) is filtered and coupled to the SPDC cavity after lens-transforming. Length of SPDC cavity is stabilized by the 840 nm auxiliary light. Signal and idler photons generated from SPDC cavity filtered by cascaded etalons are split on a PBS and coupled to multi-fibers for further processing. Doubler, second-harmonic generation cavity.

下载: 全尺寸图片幻灯片

图 4 保持QWP2的角度约 ${45^ \circ }$ , QWP1不同角度(重合程度)下的互关联函数　(a) ${\alpha _0} + 80'$ (72.8%); (b) ${\alpha _0}$ (42.1%); (c) ${\alpha _0} - 80'$ (25.0%); (d) ${\alpha _0} - 160'$ (14.6%). 插图为相应腔内2个本征值所对应纵模的透射峰

Fig. 4. Cross-correlation function with different degrees of QWP1 (overlap) on the condition of the degree of QWP2 approximately equal to ${45^ \circ }$ : (a) ${\alpha _0} + 80'$ (72.8%); (b) ${\alpha _0}$ (42.1%); (c) ${\alpha _0} - 80'$ (25.0%); (d) ${\alpha _0} - 160'$ (14.6%). Insets are corresponding transmission spectra for the two different orthogonal modes.

下载: 全尺寸图片幻灯片

图 5 QWP1的角度固定为相对较好的 ${\alpha _0} + 80'$ 时QWP2在不同角度(重合程度)下的互关联函数　(a) ${\beta _0} - 40'$ (60.9%); (b) ${\beta _0} + 40'$ (90.0%); (c) ${\beta _0} + 80'$ (35.2%); (d) ${\beta _0} + 120'$ (23.8%)

Fig. 5. Cross-correlation function with different degrees of QWP2 (overlap) on the condition of QWP1 equal to ${\alpha _{\text{0}}}{\text{ + 80'}}$ : (a) ${\beta _0} - 40'$ (60.9%); (b) ${\beta _0} + 40'$ (90.0%); (c) ${\beta _0} + 80'$ (35.2%); (d) ${\beta _0} + 120'$ (23.8%).

下载: 全尺寸图片幻灯片

图 6 不同时间分辨率条件下的互关联函数　(a) 100 ps; (b) 4.4 ns, 蓝色曲线(数据点)和红色曲线(数据点)分别表示信号光和闲置光的拟合曲线(数据)

Fig. 6. Cross-correlation function with different resolution time: (a) 100 ps; (b) 4.4 ns. The blue curve (points) and red curve (points) are representative fitting curves (experimental data) of signal and idler photons respectively.

下载: 全尺寸图片幻灯片

[1]	Perumangatt C, Lohrmann A, Ling A 2020 Phys. Rev. A 102 012404 Google Scholar
[2]	Tang J S, Tang L, Wu H D, Wu Y, Sun H, Zhang H, Li T, Lu Y Q, Xiao M, Xia K Y 2021 Phys. Rev. Appl. 15 064020 Google Scholar
[3]	杨宏恩, 韦联福 2019 物理学报 68 234202 Google Scholar Yang H E, Wei L F 2019 Acta Phys. Sin. 68 234202 Google Scholar
[4]	Wang B X, Tao M J, Ai Q, Xin T, Lambert N, Ruan D, Cheng Y C, Nori F, Deng F G, and Long G L 2018 NPJ Quantum Inf. 4 52 Google Scholar
[5]	Yang B, Sun H, Ott R, Wang H Y, Zache T V, Halimeh J C, Yuan Z S, Hauke P, Pan J W 2020 Nature 587 392 Google Scholar
[6]	Piro N, Rohde F, Schuck C, Almendros M, Huwer J, Ghosh J, Haase A, Hennrich M, Dubin F, and Eschner J 2011 Nat. Phys. 7 17 Google Scholar
[7]	Pittman T P, Shih Y H, Strekalov D V, Sergienko A V 1995 Phys. Rev. A 52 3429 Google Scholar
[8]	Zhang H, Jin X M, Yang J, Dai H N, Yang S J, Zhao T M, Rui J, He Y, Jiang X, Yang F, Pan G S, Yuan Z S, Deng Y J, Chen Z B, Bao X H, Chen S, Zhao B, Pan J W 2011 Nat. Photonics 5 628 Google Scholar
[9]	Yin J, Cao Y, Li Y H, Liao S K, Zhang L, Ren J G, Cai W Q, Liu W Y, Li S, Dai H, Li G B, Lu Q M, Gong Y H, Xu Y, Li S L, Li F Z, Yin Y Y, Jiang Z Q, Li M, Jia J J, Ren G, He D, Zhou Y L, Zhang X X, Wang N, Chang X, Zhu Z C, Liu N L, Chen Y A, Lu C Y, Shu R, Peng C Z, Wang J Y, Pan J W 2017 Science 356 1140 Google Scholar
[10]	Broome M A, Fedrizzi A, Lanyon B P, Kassal I, Guzik A A, White A G 2010 Phys. Rev. Lett. 104 153602 Google Scholar
[11]	Esposito C, Barros M R, Hernandez A D, Carvacho G, Colandrea F D, Barboza R, Cardano F, Spagnolo N, Marrucci L, Sciarrino F 2022 NPJ Quantum Inf. 8 34 Google Scholar
[12]	Scholz M, Koch L, Benson O 2009 Phys. Rev. Lett. 102 063603 Google Scholar
[13]	Zhou Z Y, Ding D S, Li Y, Wang F Y, Shi B S 2014 J. Opt. Soc. Am. B 31 128 Google Scholar
[14]	Rambach M, Nikolova A, Weinhold T J, White A G 2016 APL Photonics 1 096101 Google Scholar
[15]	Niizeki K, Ikeda K, Zheng M Y, Xie X P, Okamura K, Takei N, Namekata N, Inoue S, Kosaka H, Horikiri T 2018 Appl. Phys. Express 11 042801 Google Scholar
[16]	Liu J J, Liu J C, Yu P, Zhan G Q 2020 APL Photonics 5 066105 Google Scholar
[17]	Moqanaki A, Maaas F, Walther P 2019 APL Photonics 4 090804 Google Scholar
[18]	Tsai P J, Chen Y C 2018 Quantum Sci. Technol. 3 034005 Google Scholar
[19]	Ou Z Y, Lu Y J 1999 Phys. Rev. Lett. 83 2556 Google Scholar
[20]	Tian L, Li S J, Yuan H X, Wang H 2016 J. Phys. Soc. Japan 85 124403 Google Scholar
[21]	Wang J, Huang Y F, Zhang C, Cui J M, Zhou Z Y, Liu B H, Zhou Z Q, Tang J S, Li C F, Guo G C 2018 Phys. Rev. Appl. 10 054036 Google Scholar
[22]	Chuu C S, Yin G Y, Harris S E 2012 Appl. Phys. Lett. 101 051108 Google Scholar
[23]	吕百达 2003 激光光学(第三版) (北京: 高等教育出版社) 第422—426页 Lv B D 2003 Laser Optics (3rd Ed.) (Beijing: Higher Education Press) pp422–426 (in Chinese)
[24]	Bhandari R 2008 Opt. Lett. 33 854 Google Scholar
[25]	Hansch T W, Couilland B 1980 Opt. Commun. 35 441 Google Scholar
[26]	刘鑫鑫 2013 硕士学位论文 (太原: 山西大学) Liu X X 2013 M. S. Thesis (Taiyuan: Shanxi University) (in Chinese)
[27]	Scholz M, Koch L, Benson O 2009 Opt. Commun. 282 3518 Google Scholar
[28]	李岩 2016 博士学位论文 (合肥: 中国科学技术大学) Li Y 2016 Ph. D. Dissertation (Hefei: University of Science and Technology of China) (in Chinese)
[29]	Bocquillon E, Couteau C, Razavi M, Laflamme R, Weihs G 2009 Phys. Rev. A 79 035801 Google Scholar
[30]	Wahl M, Rohlicke T, Rahn J H, Erdmann R, Kell G, Ahlrichs A, Kernbach M, Schell A W, Benson O 2013 Rev. Sci. Instrum. 84 043102 Google Scholar

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基于1/4波片的腔增强自发参量下转换过程中双折射效应的补偿