Analysis of multiple exceptional points related to three interacting eigenmodes in a non-Hermitian Hamiltonian

We have investigated the exceptional points (EPs) which are degeneracies of a non-Hermitian Hamiltonian, in the case that three modes are interacting with each other. Even though the parametric evolution of the modes cannot be uniquely determined when encircling more than two EPs once, we can recover the initial configuration of the modes by encircling two EPs three times or three EPs twice. We confirm our expectation by numerically calculating the modes of an open quantum system, two dielectric microdisks, and a $3\times 3$ matrix model.

Figure 1

Schematic diagrams describing evolutions of eigenmodes in various situations of encircling EPs. (a) Encircling a single EP in parameter space. Two parameters are continuously varied so as to enclose the loop denoted as the black circle. The blue line attached to the EP represents where the avoided level crossing occurs. The evolution of eigenmode is described by the holonomy matrix $\mathbf{M}\left(1\right)$. (b) Two eigenmodes are exchanged by applying $\mathbf{M}\left(1\right)$ once. (c) Encircling two EPs. (d) Three eigenmodes are rearranged by successive application of two different $\mathbf{M}\left(1\right)$'s, namely, ${\mathbf{M}}_{ij}$ and ${\mathbf{M}}_{ji}$, resulting in shifting the eigenmodes, i.e., $(i,j,k)\to (k,i,j)$. (e) Encircling three EPs. (f) Three eigenmodes are rearranged by applying successive application of three different $\mathbf{M}\left(1\right)$'s, resulting in simple exchange of two eigenmodes, i.e., $(i,j,k)\to (i,k,j)$.

Figure 2

(a) The evolution of real eigenvalues of complex wave numbers during the parametric variation along the closed loop enclosing two EPs, namely, ${\mathrm{EP}}_1$ and ${\mathrm{EP}}_2$ following the path $A\to B\to \cdots \to L\to A$, which reveals the structure of Riemann surfaces. The closed loop is formed by a trapezoid with four corners, namely, $(0.750,0.260)$, $(0.950,0.600)$, $(0.950,0.700)$, and $(0.750,0.360)$ in the parameter space $(r,d)$ which corresponds to $(0.2\alpha +0.750,0.34\alpha +0.1\beta +0.260)$. Two dots and the attached dashed lines represent EPs and the associated branch cuts, respectively. The level crossings and the avoided level crossing are pointed out by encircling the dashed and the solid circles, respectively. The avoided crossings between the $W_2$ and $W_2^{\prime }$ modes are denoted as $M$ and $N$ (see the text for details). (b) The shape of the eigenmodes for the eigenvalues of $A$–$L$ designated in (a) in coordinate space.

Figure 3

The energy surface of the real parts of the eigenvalues $\lambda$ of Eq. (6) ($z$ axis) as varying two external parameters ($\alpha$,$\beta$). Three EPs and the corresponding branch lines are represented as three dots and the attached dashed lines, respectively.