Confluent Crum-Darboux transformations in Dirac Hamiltonians with $PT$-symmetric Bragg gratings

TL;DR

This paper extends the confluent Crum-Darboux transformation to Dirac Hamiltonians with PT-symmetry, enabling the construction of exactly solvable optical systems with balanced gain and loss.

Contribution

It introduces a higher-order confluent Crum-Darboux transformation for Dirac equations with PT-symmetry, providing explicit solutions for complex optical configurations.

Findings

Derived a multi-parametric class of exactly solvable PT-symmetric optical systems.

Showed that PT-symmetry can induce localization of electric fields.

Extended mathematical tools for designing optical systems with gain and loss balance.

Abstract

We consider optical systems where propagation of light can be described by a Dirac-like equation with $PT$-symmetric Hamiltonian. In order to construct exactly solvable configurations, we extend the confluent Crum-Darboux transformation for the one-dimensional Dirac equation. The properties of the associated intertwining operators are discussed and the explicit form for higher-order transformations is presented. We utilize the results to derive a multi-parametric class of exactly solvable systems where the balanced gain and loss represented by the $PT$-symmetric refractive index can imply localization of the electric field in the material.