Clifford and quadratic composite operators with applications to non-Hermitian physics

Abstract

A variety of physical phenomena, such as amplification, absorption, and radiation, can be effectively described using non-Hermitian operators. However, the introduction of non-uniform non-Hermiticity can lead to the formation of exceptional points in a system’s spectrum, where two or more eigenvalues become degenerate and their associated eigenvectors coalesce causing the underlying operator or matrix to become defective. We explore extensions of the Clifford and quadratic ϵ-pseudospectrum, previously defined for Hermitian operators. We provide a framework for finding approximate joint eigenvectors of a d-tuple of Hermitian operators A and non-Hermitian operators B, and show that their Clifford and quadratic ϵ-pseudospectra are still well-defined despite any non-normality. We prove that the non-Hermitian quadratic gap is local with respect to the probe location when there are perturbations to one or more of the underlying operators. This framework enables the exploration of non-Hermitian physical systems’ ϵ-pseudospectra, including but not limited to photonic systems.