Research progress of parity-time symmetry and anti-symmetry

Abstract

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 ( \begin{document}${\cal {PT}}$\end{document} ) symmetry also have real energy spectra, and there is a novel non-Hermitian exceptional point between \begin{document}${\cal {PT}}$\end{document} -symmetric phase and \begin{document}${\cal {PT}} $\end{document} -symmetry-broken phase, which is unique to non-Hermitian systems. Recently, people have realized \begin{document}${\cal {PT}} $\end{document} symmetric and anti- \begin{document}${\cal {PT}}$\end{document} symmetric non-Hermitian Hamiltonians in various physical systems and demonstrated novel quantum phenomena, which not only deepened our understanding of the basic laws of quantum physics, but also promoted the breakthrough of application technology. This review will introduce the basic physical principles of \begin{document}${\cal {PT}} $\end{document} symmetry and anti- \begin{document}${\cal {PT}}$\end{document} symmetry, summarize the schemes to realize \begin{document}${\cal {PT}} $\end{document} symmetry and anti- \begin{document}${\cal {PT}} $\end{document} symmetry in optical and atomic systems systematically, including the observation of \begin{document}${\cal {PT}} $\end{document} -symmetry transitions by engineering time-periodic dissipation and coupling in ultracold atoms and single trapped ion, the realization of anti- \begin{document}${\cal {PT}} $\end{document} symmetry in dissipative optical system by indirect coupling, and realizing anti- \begin{document}${\cal {PT}} $\end{document} -symmetry through fast atomic coherent transmission in flying atoms. Finally, we review the research on precision sensing using non-Hermitian exceptional points of \begin{document}${\cal {PT}} $\end{document} -symmetric systems. Near the exceptional points, the eigenfrequency splitting follows an \begin{document}${\varepsilon }^{\tfrac{1}{N}}$\end{document} -dependence, where the \begin{document}$\varepsilon$\end{document} is the perturbation and \begin{document}$ N $\end{document} is the order of the exceptional point. We review the \begin{document}${\cal {PT}}$\end{document} -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.