Research progress of parity-time symmetry and anti-symmetry
Authors: Tang Yuan-Jiang , Liang Chao , Liu Yong-Chun ,   State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China,   Frontier Science Center for Quantum Information, Beijing 100084, China
Published: August 2022

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