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

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>n</a:mi><a:mi>th</a:mi></a:mrow></a:math>-order exceptional point, <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>n</b:mi></b:math> 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>ε</c:mi></c:math> changes the eigenvalues proportional to the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>n</d:mi><d:mi>th</d:mi></d:mrow></d:math> root of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>ε</e:mi></e:math>. 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., <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>n</f:mi><f:mo>&gt;</f:mo><f:mn>2</f:mn></f:mrow></f:math>. 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.