Non-Hermitian Hamiltonian and Lamb shift in circular dielectric microcavity

TL;DR

This paper explores the effects of non-Hermitian Hamiltonians on the modes of circular dielectric microcavities, revealing the Lamb shift and its relation to system parameters through numerical and theoretical analysis.

Contribution

It introduces a non-Hermitian Hamiltonian framework to analyze normal and quasi-normal modes, highlighting the Lamb shift and mode-environment interactions in microcavities.

Findings

Identification of the Lamb shift due to environment coupling.

Relation between Lamb shift and angular momentum of whispering gallery modes.

Refractive index as a control parameter analogous to potential in Schrödinger equation.

Abstract

We study the normal modes and quasi-normal modes (QNMs) in circular dielectric microcavities through non-Hermitian Hamiltonian, which come from the modifications due to system-environment coupling. Differences between the two types of modes are studied in detail, including the existence of resonances tails. Numerical calculations of the eigenvalues reveal the Lamb shift in the microcavity due to its interaction with the environment. We also investigate relations between the Lamb shift and quantized angular momentum of the whispering gallery mode as well as the refractive index of the microcavity. For the latter, we make use of the similarity between the Helmholtz equation and the Schr\"{o}dinger equation, in which the refractive index can be treated as a control parameter of effective potential. Our result can be generalized to other open quantum systems with a potential term.