Non-Hermitian propagation in Glauber-Fock optical lattices

TL;DR

This paper explores how non-unitary transformations applied to Hermitian optical systems, specifically Glauber-Fock lattices, lead to non-Hermitian Hamiltonians that model open, energy-exchanging waveguide arrays, with exact solutions provided.

Contribution

It introduces a non-unitary SUSY transformation framework for Glauber-Fock lattices, deriving exact solutions for the resulting non-Hermitian Hamiltonians.

Findings

Exact analytical solutions for non-Hermitian Hamiltonians derived from Glauber-Fock lattices.

Non-Hermitian models represent open systems exchanging energy with the environment.

Transformation connects Hermitian and non-Hermitian Schrödinger problems explicitly.

Abstract

The effect of a non-unitary transformation on an initial Hermitian operator is studied. The initial (Hermitian) optical system is a Glauber-Fock optical lattice. The resulting non-Hermitian Hamiltonian models an anisotropic (Glauber-Fock) waveguide array of the Hatano-Nelson-type. Several cases are analyzed and exact analytical solutions for both the Hermitian and non-Hermitian Schr\"odinger problems are given, as they are simply connected. Indeed, such transformation can be regarded as a non-unitary Supersymmetric (SUSY) transformation and the resulting non-Hermitian Hamiltonian can be considered as representing an open system that interchanges energy with the environment.