Filter cavity design and length measurement for preparing Schrödinger cat state
Authors: Zhai Ze-Hui , Hao Wen-Jing , Liu Jian-Li , Duan Xi-Ya ,   School of Physics and Electronic Engineering, Shanxi University, Taiyuan 030006, China
Published: December 2020

Optical Schrödinger cat state is not only one of the basic elements of quantum mechanics, but also a pivotal resource of continuous-variable quantum information. The non-Gaussian operation in its preparation can also be a key technology in distilling continuous-variable squeezing and entanglement. In the experimental preparation, a small part of a beam of vacuum squeezing is separated and detected as the trigger of appearance of Schrödinger cat state. Filter operation in the trigger optical path is important since it affects dark counts of single photon detector, frequency mode matching of trigger mode and signal mode, and preparing rate of the Schrödinger cat state, etc. In this paper, we describe the design of optical filter in the trigger path and the measurement of the filter cavity length. According to the design, filter cavity length <inline-formula><tex-math id="M4">\begin{document}$ {l_{{\rm{FC}}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M4.png"/></alternatives></inline-formula> should satisfy <inline-formula><tex-math id="M5">\begin{document}$ {\rm{189}}\;{\text{μm}} > {l_{{\rm{FC}}}} > {\rm{119}}\;{\text{μm}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M5.png"/></alternatives></inline-formula>. This cavity length is too small to be measured with a ruler. To measure the cavity length, we introduce an optical method, in which Gouy phases of Hermite Gaussian transverse modes TEM<sub>00</sub> and TEM<sub>10</sub> are used. When the cavity length is scanned, resonant peaks and the corresponding scanning voltages are recorded. From theoretical derivation, the cavity length is related to the filter cavity piezo response to the scanning voltage <inline-formula><tex-math id="M6">\begin{document}$ {\varPsi '_{\rm{G}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M6.png"/></alternatives></inline-formula>, the slope rate of piezo scanning voltage <inline-formula><tex-math id="M7">\begin{document}$ U'$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M7.png"/></alternatives></inline-formula>, and the time distance between TEM<sub>00</sub> and TEM<sub>10</sub> resonant peaks <inline-formula><tex-math id="M8">\begin{document}$ \Delta t$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M8.png"/></alternatives></inline-formula>. The finally measured cavity length is <inline-formula><tex-math id="M9">\begin{document}${l_{{\rm{FC}}}} = ({\rm{141}} \pm 28)~{\text{μm}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M9.png"/></alternatives></inline-formula>, which satisfies the design requirement. The measurement error mainly originates from inaccurate fitting of <inline-formula><tex-math id="M10">\begin{document}$ {\varPsi '_{\rm{G}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M10.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ U'$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M11.png"/></alternatives></inline-formula>, and readout error of <inline-formula><tex-math id="M12">\begin{document}$ \Delta t$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M12.png"/></alternatives></inline-formula>. It is shown that the error of <inline-formula><tex-math id="M13">\begin{document}$ {\varPsi '_{\rm{G}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M13.png"/></alternatives></inline-formula> is dominant since less data are used in the curve fitting. The measurement error is expected to be reduced if much more data of piezo response to scanning voltage are collected and used to fit <inline-formula><tex-math id="M14">\begin{document}$ {\varPsi '_{\rm{G}}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20200589_M14.png"/></alternatives></inline-formula> with higher order polynomials. The proposed measurement method of short cavity length needs neither wide tuning laser nor any peculiar instrument, and does not depend on any dispersion property of the cavity, and hence it has a certain generality. It can be hopefully used in many other optical systems, such as cavity quantum electrodynamics, where ultrashort cavity plays a central role.