$PT$-symmetric invisible defects and confluent Darboux-Crum transformations

TL;DR

This paper demonstrates how confluent Darboux-Crum transformations can design PT-symmetric optical systems with invisible defects that trap bound states, revealing new spectral and supersymmetric properties.

Contribution

It introduces a novel application of confluent Darboux-Crum transformations to create PT-symmetric optical systems with invisible defects and analyzes their spectral and supersymmetric features.

Findings

Defects can be completely invisible with wave functions asymptotically matching undistorted systems.

Bound states in the continuum are confined with power law decay.

Spectral properties are linked to Lax-Novikov integrals and supersymmetry.

Abstract

We show that confluent Darboux-Crum transformations with emergent Jordan states are an effective tool for the design of optical systems governed by the Helmholtz equation under the paraxial approximation. The construction of generic, asymptotically real and periodic, $PT$-symmetric systems with local complex periodicity defects is discussed in detail. We show how the decay rate of the defect is related with the energy of the bound state trapped by the defect. In particular, the bound states in the continuum are confined by the periodicity defects with power law decay. We show that these defects possess complete invisibility; the wave functions of the system coincide asymptotically with the wave functions of the undistorted setting. The general results are illustrated with explicit examples of reflectionless models and systems with one spectral gap. We show that the spectral properties of the studied models are reflected by Lax-Novikov-type integrals of motion and associated supersymmetric structures of bosonized and exotic nature.