Realization of geometric phase topology induced by multiple exceptional points

TL;DR

This work addresses the problem of realizing and classifying geometric-phase topology arising from multiple exceptional points (EPs) in a realistic photonic system. It employs a deformed elliptic microcavity to host three EPs in a two-dimensional parameter space and analyzes adiabatic encircling along loops $\Gamma_i$ to induce specific mode exchanges and geometric phases, including a $\pi$ phase where appropriate. The study demonstrates all five topological classes for three modes, detailing loop-dependent exchange sequences and phase accrual across four encircling loops, thereby connecting refined EP classifications to concrete photonic implementations. The findings broaden the practical landscape of non-Hermitian topology in optics and point to new avenues for designing devices that harness multi-EP geometry and associated phase structures.

Abstract

Non-Hermitian systems have Riemann surface structures of complex eigenvalues that admit singularities known as exceptional points. Combining with geometric phases of eigenstates gives rise to unique properties of non-Hermitian systems, and their classifications have been studied recently. However, the physical realizations of classes of the classifications have been relatively limited because a small number of modes and exceptional points are involved. In this work, we show in microcavities that all five classes [J.-W. Ryu, et al., Commun. Phys. 7, 109 (2024)] of three modes can emerge with three exceptional points. In demonstrations, we identified various combinations of exceptional points within a two-dimensional parameter space of a single microcavity and defined five distinct encircling loops based on three selected exceptional points. According to the classification, these loops facilitate different mode exchanges and the acquisition of additional geometric phases during the adiabatic encircling of exceptional points. Our results provide a broad description of the geometric phases-associated topology induced by multiple exceptional points in realistic physical systems.

Figures (5)

• Figure 1: (a) Dielectric microdisk system configuration with a deformed elliptic boundary shape as given in Eq. (\ref{['eq:system']}). (b) Three EP positions (A, B, and C) and four encircling loops, $\Gamma_1$, $\Gamma_2$, $\Gamma_3$, and $\Gamma_4$ examined. Black dash-dotted and green dashed lines represent real and imaginary branch cuts. Riemann surfaces of (c) real and (d) imaginary parts of complex eigenfrequencies of three eigenmodes. Yellow dashed lines denote associated real and imaginary branch cuts. Note that the eigenfrequencies in (c) and (d) are presented relative to the average value of the three modes for visual clarity: $\Delta nk_i(\varepsilon,n)R=nk_i(\varepsilon,n)R-\langle nk(\varepsilon,n)R \rangle$ where $\langle nk(\varepsilon,n)R \rangle=1/3\sum_{j=1}^3nk_j(\varepsilon,n)R$
• Figure 2: (a) The real parts of complex eigenfrequencies and (b) additional geometric phases when encircling an EP along the loop $\Gamma_1$. (c) shows Re[$\psi(\mathbf{r})$], the spatial distributions of the real part of eigenmodes normalized by max$\{$Re[$\psi(\mathbf{r})$]$\}$. The red and blue colors represent positive and negative values, respectively. The first mode changes into the second mode after the first encircling. The second mode changes into the first mode after the second encircling. Finally, the mode obtains additional geometric phase $\pi$. The third mode does not change in spite of encircling the EP.
• Figure 3: (a) The real parts of complex eigenfrequencies and (b) additional geometric phases when encircling a pair of EPs along the loop $\Gamma_2$. (c) shows Re[$\psi(\mathbf{r})$] normalized by max$\{$Re[$\psi(\mathbf{r})$]$\}$. Three modes do not change, but two modes obtain additional phases $\pi$, and one mode does not obtain additional phases.
• Figure 4: (a) The real parts of complex eigenfrequencies and (b) additional geometric phases when encircling two intersected EPs along the loop $\Gamma_3$. (c) shows Re[$\psi(\mathbf{r})$] normalized by max$\{$Re[$\psi(\mathbf{r})$]$\}$. The second mode changes into the first mode after the first encircling, and the first mode changes into the third mode after the second encircling. Finally, the third mode changes into the second mode after the third encircling. The mode does not obtain additional geometric phase.
• Figure 5: (a) The real parts of complex eigenfrequencies and (b) additional geometric phases when encircling three EPs along the loop $\Gamma_4$. (c) shows Re[$\psi(\mathbf{r})$] normalized by max$\{$Re[$\psi(\mathbf{r})$]$\}$. The second mode changes into the third mode after the first encircling, and the third mode changes into the second mode after the second encircling. The mode does not obtain the additional geometric phase. The first mode does not change but obtains the additional phase $\pi$.