Experimental Observation of the Topological Structure of Exceptional Points

Abstract

We report on a microwave cavity experiment where exceptional points (EPs), which are square root singularities of the eigenvalues as function of a complex interaction parameter, are encircled in the laboratory. The real and imaginary parts of an eigenvalue are given by the frequency and width of a resonance and the eigenvectors by the field distributions. Repulsion of eigenvalues--always associated with EPs--implies frequency anticrossing (crossing) whenever width crossing (anticrossing) is present. The eigenvalues and eigenvectors are interchanged while encircling an EP, but one of the eigenvectors undergoes a sign change which can be discerned in the field patterns.

Relativistic supersymmetric quantum mechanics based on Klein–Gordon equation

“…Marginally, let us point out the possible methodical relevance of the present regularization of the level crossing at ε 0 = 0 for an improvement of our future understanding of some models in non-quantum domains of physics where the very similar "exceptional-point" singularities have been observed inside the intervals of variability of relevant parameters [18,23].…”

“…Such a feature is characteristic for the non-diagonalizable (usually called Jordan-block) limits of non-Hermitian operators [17]. In the spectra of differential operators these points are also known as "Bender-Wu singularities" [24], as the points of an "unavoided level-crossing" [8] or simply as "exceptional points" [18]. In our present paper, their properties will only be derived via a limiting transition ε 0 → 0 + from the regular domain.…”

“…During a limiting transition ε 0 → 0 + the two smallest eigenvalues E (±) 0 do merge and vanish. Under the name of an exceptional point (EP, [18]) the energy E 0 = 0 still represents a valid bound state because after one abbreviates…”

Coalescence of exceptional points and phase diagrams for one-dimensionalPT-symmetric photonic crystals

“…In a system with many coupled modes, the emergence of multiple EPs and their interactions can occur under system parameter variation [25][26][27]. Their interactions and the associated topological properties have been studied in both microwave cavities [25,26] and acoustics systems [27]. For systems with continuous spectra, various PT -symmetric photonic crystals (PCs) have been considered by studying their complex band structures.…”

“…For example, a system with asymmetric loss is equivalent to a PT -symmetric system with a background of uniform loss, and so no gain is required to achieve an EP [3,[21][22][23][24]. In a system with many coupled modes, the emergence of multiple EPs and their interactions can occur under system parameter variation [25][26][27]. Their interactions and the associated topological properties have been studied in both microwave cavities [25,26] and acoustics systems [27].…”