Graph-theoretical approach to the eigenvalue spectrum of perturbed higher-order exceptional points

Abstract

Exceptional points are special degeneracy points in parameter space that can arise in (effective) non-Hermitian Hamiltonians describing open quantum and wave systems. At an nth-order exceptional point, n eigenvalues and the corresponding eigenvectors simultaneously coalesce. These coalescing eigenvalues typically exhibit a strong response to small perturbations that can be useful for sensor applications. A so-called generic perturbation with strength ε changes the eigenvalues proportional to the nth root of ε. A different eigenvalue behavior under perturbation is called nongeneric. An understanding of the behavior of the eigenvalues for various types of perturbations is desirable and also crucial for applications. We advocate a graph-theoretical perspective that contributes to the understanding of perturbative effects on the eigenvalue spectrum of higher-order exceptional points, i.e., n>2. Our approach is mathematically derived and the intuitive consequences elucidated. To highlight the relevance of nongeneric perturbations and to give a physical interpretation for their occurrence, we consider an illustrative example, a system of microrings coupled by a semi-infinite waveguide with an end mirror. Furthermore, the saturation effect occurring for cavity-selective sensing in such a system is naturally explained within the graph-theoretical picture.