Cascaded Hong-Ou-Mandel interference of entangled photon pairs and its application in multiple delay parameters measurement
Authors: Zhai Yi-Wei , Dong Rui-Fang , Quan Run-Ai , Xiang Xiao , Liu Tao , Zhang Shou-Gang ,   School of Electrical and Control Engineering, Shaanxi University of Science & Technology, Xi’an 710021, China,   Key Laboratory of Time and Frequency Primary Standards, National Time Service Center, Chinese Academy of Sciences, Xi’an 710600, China,   School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China
Published: June 2021

<sec> The Hong-Ou-Mandel (HOM) interferometer using entangled photon source possesses important applications in quantum precision measurement and relevant areas. In this paper, a simultaneous measurement scheme of multiple independent delay parameters based on a cascaded HOM interferometer is proposed. The cascaded HOM interferometer is composed of <inline-formula><tex-math id="M3">\begin{document}$ n $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M3.png"/></alternatives></inline-formula> concatenated 50∶50 beam splitters and independent delay parameters <inline-formula><tex-math id="M4">\begin{document}$ {\tau }_{1} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M4.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ {\tau }_{2} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M5.png"/></alternatives></inline-formula>, ···, <inline-formula><tex-math id="M6">\begin{document}$ {\tau }_{n} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M6.png"/></alternatives></inline-formula>. The numbers <inline-formula><tex-math id="M7">\begin{document}$ n=1, 2\;\mathrm{a}\mathrm{n}\mathrm{d}\;3 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M7.png"/></alternatives></inline-formula> refer to the standard HOM interferometer, the second-cascaded HOM interferometer, and the third-cascaded HOM interferometer, respectively. Through the theoretical study of the cascaded HOM interference effect based on frequency entangled photon pairs, it can be concluded that there is a corresponding relationship between the dip position and the independent delay parameter in the second-order quantum interferogram. In the standard HOM interferometer, there is a dip in the second-order quantum interferogram, which can realize the measurement of delay parameter <inline-formula><tex-math id="M8">\begin{document}$ {\tau }_{1} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M8.png"/></alternatives></inline-formula>. In the second-cascaded HOM interferometer, there are two symmetrical dips in the second-order quantum interferogram, which can realize the simultaneous measurement of two independent delay parameters <inline-formula><tex-math id="M9">\begin{document}$ {\tau }_{1} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M9.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ {\tau }_{2} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M10.png"/></alternatives></inline-formula>. By analogy, in the third-cascaded HOM interferometer, there are six symmetrical dips in the second-order quantum interferogram, which can realize the simultaneous measurement of three independent delay parameters <inline-formula><tex-math id="M11">\begin{document}$ {\tau }_{1} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M11.png"/></alternatives></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ {\tau }_{2} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M12.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ {\tau }_{3} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M13.png"/></alternatives></inline-formula>. Therefore, multiple independent delay parameters can be measured simultaneously based on a cascaded HOM interferometer. </sec><sec> In the experiment, the second-cascaded HOM interferometer based on frequency entangled photon source is built. The second-order quantum interferogram of the second-cascaded HOM interferometer is obtained by the coincidence measurement device. Two independent delay parameters <inline-formula><tex-math id="M14">\begin{document}$ {\tau }_{1} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M14.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ {\tau }_{2} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M15.png"/></alternatives></inline-formula> are measured simultaneously by recording the positions of two symmetrical dips, which are in good agreement with the theoretical results. At an averaging time of 3000 s, the measurement accuracy of two delay parameters <inline-formula><tex-math id="M16">\begin{document}$ {\tau }_{1} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M16.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ {\tau }_{2} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="12-20210071_M17.png"/></alternatives></inline-formula> can reach 109 and 98 fs, respectively. These results lay a foundation for extending the applications of HOM interferometer in multi-parameter quantum systems. </sec>