Research progress of parity-time symmetry and anti-symmetry

In standard quantum mechanics, the Hamiltonian describing physical system is generally Hermitian, so as to ensure that has real energy spectra and system’s evolution unitary. recent years, it been found non-Hermitian Hamiltonians with parity-time (<inline-formula><tex-math id="Z-20220827100836">\begin{document}${\cal {PT}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="17-20221323_Z-20220827100836.jpg"/><graphic xlink:href="17-20221323_Z-20220827100836.png"/></alternatives></inline-formula>) symmetry also have spectra, there a novel exceptional point between <inline-formula><tex-math id="Z-20220827100841">\begin{document}${\cal xlink:href="17-20221323_Z-20220827100841.jpg"/><graphic xlink:href="17-20221323_Z-20220827100841.png"/></alternatives></inline-formula>-symmetric phase id="Z-20220827100925">\begin{document}${\cal {PT}} $\end{document}</tex-math><alternatives><graphic xlink:href="17-20221323_Z-20220827100925.jpg"/><graphic xlink:href="17-20221323_Z-20220827100925.png"/></alternatives></inline-formula>-symmetry-broken phase, which unique systems. Recently, people realized id="Z-20220827100930">\begin{document}${\cal xlink:href="17-20221323_Z-20220827100930.jpg"/><graphic xlink:href="17-20221323_Z-20220827100930.png"/></alternatives></inline-formula> symmetric anti-<inline-formula><tex-math id="Z-20220827100849">\begin{document}${\cal xlink:href="17-20221323_Z-20220827100849.jpg"/><graphic xlink:href="17-20221323_Z-20220827100849.png"/></alternatives></inline-formula> in various systems demonstrated phenomena, not only deepened our understanding of basic laws physics, but promoted breakthrough application technology. This review will introduce principles id="Z-20220827100901">\begin{document}${\cal xlink:href="17-20221323_Z-20220827100901.jpg"/><graphic xlink:href="17-20221323_Z-20220827100901.png"/></alternatives></inline-formula> id="Z-20220827100845">\begin{document}${\cal xlink:href="17-20221323_Z-20220827100845.jpg"/><graphic xlink:href="17-20221323_Z-20220827100845.png"/></alternatives></inline-formula> symmetry, summarize schemes realize id="Z-20220827100915">\begin{document}${\cal xlink:href="17-20221323_Z-20220827100915.jpg"/><graphic xlink:href="17-20221323_Z-20220827100915.png"/></alternatives></inline-formula> id="Z-20220827100954">\begin{document}${\cal xlink:href="17-20221323_Z-20220827100954.jpg"/><graphic xlink:href="17-20221323_Z-20220827100954.png"/></alternatives></inline-formula> optical atomic systematically, including observation id="Z-20220905142649">\begin{document}${\cal xlink:href="17-20221323_Z-20220905142649.jpg"/><graphic xlink:href="17-20221323_Z-20220905142649.png"/></alternatives></inline-formula>-symmetry transitions by engineering time-periodic dissipation coupling ultracold atoms single trapped ion, realization id="Z-20220827100856">\begin{document}${\cal xlink:href="17-20221323_Z-20220827100856.jpg"/><graphic xlink:href="17-20221323_Z-20220827100856.png"/></alternatives></inline-formula> dissipative indirect coupling, realizing id="Z-20220827100947">\begin{document}${\cal xlink:href="17-20221323_Z-20220827100947.jpg"/><graphic xlink:href="17-20221323_Z-20220827100947.png"/></alternatives></inline-formula>-symmetry through fast coherent transmission flying atoms. Finally, we research on precision sensing using points id="Z-20220827100911">\begin{document}${\cal xlink:href="17-20221323_Z-20220827100911.jpg"/><graphic xlink:href="17-20221323_Z-20220827100911.png"/></alternatives></inline-formula>-symmetric Near points, eigenfrequency splitting follows an id="M1">\begin{document}${\varepsilon }^{\tfrac{1}{N}}$\end{document}</tex-math><alternatives><graphic xlink:href="17-20221323_M1.jpg"/><graphic xlink:href="17-20221323_M1.png"/></alternatives></inline-formula>-dependence, where id="M2">\begin{document}$\varepsilon$\end{document}</tex-math><alternatives><graphic xlink:href="17-20221323_M2.jpg"/><graphic xlink:href="17-20221323_M2.png"/></alternatives></inline-formula> perturbation id="M3">\begin{document}$ N xlink:href="17-20221323_M3.jpg"/><graphic xlink:href="17-20221323_M3.png"/></alternatives></inline-formula> order point. We id="Z-20220827100907">\begin{document}${\cal xlink:href="17-20221323_Z-20220827100907.jpg"/><graphic xlink:href="17-20221323_Z-20220827100907.png"/></alternatives></inline-formula>-symmetric composed three equidistant micro-ring cavities enhanced sensitivity at third-order points. addition, debate whether exceptional-point sensors can improve signal-to-noise ratio when considering noise, current development sensors, still open challenging question.