Geometry of the Field-Moment Spaces for Quadratic Bosonic Systems: Diabolically Degenerated Exceptional Points on Complex $k$-Polytopes

Abstract

$k$-Polytopes are a generalization of polyhedra in $k$ dimensions. Here, we show that complex $k$-polytopes naturally emerge in the higher-order field moments spaces of quadratic bosonic systems, thus revealing their geometric character. In particular, a complex-valued evolution matrix, governing the dynamics of $k$th-order field moments of a bosonic dimer, can describe a complex $k$-dimensional hypercube. The existence of such $k$-polytopes is accompanied by the presence of high-order diabolic points (DPs). Interestingly, when the field-moment space additionally exhibits exceptional points (EPs), the formation of $k$-polytopes may lead to the emergence of diabolically degenerated EPs, due to the interplay between DPs and EPs. Such intriguing spectral properties of complex polytopes may enable constructing photonic lattice systems with similar spectral features in real space. Our results can be exploited in various quantum protocols based on EPs, paving a new direction of research in this field.Comment: 9 page