Use of Equivalent Hermitian Hamiltonian for $PT$-Symmetric Sinusoidal Optical Lattices

TL;DR

This paper demonstrates how to analyze $PT$-symmetric sinusoidal optical lattices using an equivalent Hermitian Hamiltonian, simplifying the study of their band structure and beam dynamics through the Mathieu equation.

Contribution

It introduces a method to transform non-Hermitian $PT$-symmetric optical lattices into an equivalent Hermitian problem, facilitating analysis of their properties.

Findings

Eigenvalue problem reduces to Mathieu equation

Beam propagation can be analyzed via stationary states

Birefringence diminishes in interpolated potentials

Abstract

We show how the band structure and beam dynamics of non-Hermitian $PT$-symmetric sinusoidal optical lattices can be approached from the point of view of the equivalent Hermitian problem, obtained by an analytic continuation in the transverse spatial variable $x$. In this latter problem the eigenvalue equation reduces to the Mathieu equation, whose eigenfunctions and properties have been well studied. That being the case, the beam propagation, which parallels the time-development of the wave-function in quantum mechanics, can be calculated using the equivalent of the method of stationary states. We also discuss a model potential that interpolates between a sinusoidal and periodic square well potential, showing that some of the striking properties of the sinusoidal potential, in particular birefringence, become much less prominent as one goes away from the sinusoidal case.