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

TL;DR

This paper introduces a graph-theoretical framework to analyze how eigenvalues of higher-order exceptional points in non-Hermitian systems respond to perturbations, aiding understanding of sensor applications and non-generic behaviors.

Contribution

It presents a novel graph-theoretical approach to study eigenvalue perturbations at higher-order exceptional points, including interpretation of non-generic effects and an illustrative microring system example.

Findings

Graph-theoretical perspective clarifies eigenvalue behavior under perturbations.

Non-generic perturbations are explained through the graph model.

Saturation effects in cavity sensing are interpreted within this framework.

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 n-th order exceptional point, n eigenvalues and the corresponding eigenvectors simultaneously coalesce. These coalescing eigenvalues typically exhibit a strong response to small perturbations which can be useful for sensor applications. A so-called generic perturbation with strength $\epsilon$ changes the eigenvalues proportional to the n-th root of $\epsilon$. A different eigenvalue behavior under perturbation is called non-generic. 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. To highlight the relevance of non-generic perturbations and to give an 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.