Non-Hermiticity and conservation of orthogonal relation in dielectric microcavity

TL;DR

This paper explores how orthogonality is preserved in the total Hermitian system when a non-Hermitian elliptical microcavity subsystem interacts with its environment, revealing processes that maintain overall Hermiticity despite non-Hermitian behavior.

Contribution

It demonstrates the conservation of orthogonality in a total Hermitian system containing a non-Hermitian microcavity subsystem, linking inner-outer part correlations to resonance phenomena.

Findings

Orthogonality is conserved in the total Hermitian system.

Correlation between inner and outer parts affects avoided resonance crossings.

Trade-off identified between self-energy differences and Lamb shifts.

Abstract

Non-Hermitian properties of open quantum systems and their applications have attracted much attention in recent years. While most of the studies focus on the characteristic nature of non-Hermitian systems, here we focus on the following issue: A non-Hermitian system can be a subsystem of a Hermitian system as one can clearly see in Feshbach projective operator (FPO) formalism. In this case, the orthogonality of the eigenvectors of the total (Hermitian) system must be sustained, despite the eigenvectors of the subsystem (non-Hermitian) satisfy the bi-orthogonal condition. Therefore, one can predict that there must exist some remarkable processes that relate the non-Hermitian subsystem and the rest part, and ultimately preserve the Hermiticity of the total system. In this paper, we study such processes in open elliptical microcavities. The inner part of the cavity is a non-Hermitian system, and the outer part is the coupled bath in FPO formalism. We investigate the correlation between the inner- and the outer-part behaviors associated with the avoided resonance crossings (ARCs), and analyze the results in terms of a trade-off between the relative difference of self-energies and collective Lamb shifts. These results come from the conservation of the orthogonality in the total Hermitian quantum system.