Revisiting the hierarchical construction of higher-order exceptional points

Abstract

Higher-order exceptional points in the spectrum of non-Hermitian Hamiltonians describing open quantum or wave systems have a variety of potential applications, in particular in optics and photonics. However, the experimental realization is notoriously difficult. Recently, Zhong et al. [Q. Zhong, J. Kou, \ifmmode \mbox{\c{S}}\else \c{S}\fi{}. K. \"Ozdemir, and R. El-Ganainy, Phys. Rev. Lett. 125, 203602 (2020)] have introduced a robust construction where a unidirectional coupling of two subsystems having exceptional points of the same order leads generically to a single exceptional point of twice the order. Here, we investigate this scheme in a different manner by exploiting the nilpotency of the traceless part of the involved Hamiltonians. We generalize the scheme and derive a simple formula for the spectral response strength of the composite system hosting a higher-order exceptional point. Its relation to the spectral response strengths of the subsystems is discussed. Moreover, we investigate nongeneric perturbations. The results are illustrated with an example.