Continuous variable polarization entanglement in microwave domain

As a widely utilized information carrier, polarization microwave shows plenty of merits. Quantum microwave is booming gradually due to the development of superconducting technology, which makes it a promising potential to apply quantum entanglement to polarization microwave. In this paper, we introduce the concept of continuous variable polarization entanglement. Meanwhile, a scheme of polarization entanglement in microwave domain is proposed and simulated. The detail derivations are given and discussed. Polarization entangled microwaves are prepared by combining quadrature entangled signals and strong coherent signals on polarization beam splitters, and quadrature entangled signals are prepared by utilizing Josephson mixer. In order to probe the polarization entanglement between output signals, inseparability of Stokes vectors &lt;inline-formula&gt;&lt;tex-math id="M12"&gt;\begin{document}$I({\hat S_1},{\hat S_2})$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M12.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M12.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M13"&gt;\begin{document}$I({\hat S_2},{\hat S_3})$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M13.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M13.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, is analyzed in 100 MHz operation bandwidth of Josephson mixer. The relation between inseparability &lt;i&gt;I&lt;/i&gt; and squeezing degree &lt;i&gt;r&lt;/i&gt; and between inseparability &lt;i&gt;I&lt;/i&gt; and amplitude ratio &lt;i&gt;Q&lt;/i&gt; are analyzed respectively. The results show that &lt;inline-formula&gt;&lt;tex-math id="M14"&gt;\begin{document}$I({\hat S_1},{\hat S_2})$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M14.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M14.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is sensitive to the variation of &lt;i&gt;Q&lt;/i&gt;, while &lt;inline-formula&gt;&lt;tex-math id="M15"&gt;\begin{document}$I({\hat S_2},{\hat S_3})$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M15.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M15.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is sensitive to the change of &lt;i&gt;r&lt;/i&gt;. The physical reasons for these results are explored and discussed. Apart from these, &lt;inline-formula&gt;&lt;tex-math id="M16"&gt;\begin{document}$I({\hat S_1},{\hat S_2})$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M16.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M16.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; remains its value above 1 under the condition in this paper, but on the contrary, &lt;inline-formula&gt;&lt;tex-math id="M17"&gt;\begin{document}$I({\hat S_2},{\hat S_3})$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M17.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M17.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; keeps its value well below 1. It proves that &lt;inline-formula&gt;&lt;tex-math id="M18"&gt;\begin{document}${\hat S_2}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M18.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M18.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M19"&gt;\begin{document}${\hat S_3}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M19.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M19.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; of Stokes vectors are inseparable from each other, thus output signals &lt;inline-formula&gt;&lt;tex-math id="M20"&gt;\begin{document}${\hat E_a}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M20.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M20.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M21"&gt;\begin{document}${\hat E_b}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M21.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M21.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; of our scheme exhibit bipartite entanglement. The best entanglement appears nearly at about 70 MHz, at this point the minimum &lt;inline-formula&gt;&lt;tex-math id="M22"&gt;\begin{document}$I({\hat S_2},{\hat S_3})$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M22.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="6-20181911_M22.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; value is 0.25.