Nonlocal PDEs and Quantum Optics: Bound States and Resonances

TL;DR

This paper investigates the spectral properties of nonlocal PDEs arising in quantum optics, specifically bound states and resonances, providing conditions for their existence and asymptotic behaviors with numerical validation.

Contribution

It introduces new conditions for bound state existence and derives asymptotic formulas for eigenvalues in quantum optical models involving nonlocal PDEs.

Findings

Necessary and sufficient conditions for bound states

Upper bound on the number of bound states

Asymptotic formulas for eigenvalues

Abstract

We consider the quantum optics of a single photon interacting with a system of two level atoms. This leads to the study of a nonlinear eigenproblem for a system of nonlocal partial differential equations. Two classes of solutions to these equations are studied. Bound states correspond to negative eigenvalues and resonances to eigenvalues with positive real parts. We have found necessary and sufficient conditions for the existence of bound states, along with an upper bound on the number of such states. We have also considered the eigenproblem for atomic models with small high contrast inclusions. In this setting, we have derived asymptotic formulas for the eigenvalues. Our results are illustrated with numerical computations.