Revisiting the optical $PT$-symmetric dimer

TL;DR

This paper reviews the optical $ ext{PT}$-symmetric dimer, explores its relation to larger waveguide systems, and introduces a non-Hermitian Ehrenfest theorem for describing light propagation in such non-Hermitian couplers.

Contribution

It extends the understanding of optical $ ext{PT}$-symmetric dimers by linking them to Lorentz group representations and develops a non-Hermitian Ehrenfest theorem for mode coupling analysis.

Findings

The dimer is the smallest system in a class of $N$-waveguide couplers related to Lorentz symmetry.

A formulation based on a non-Hermitian Ehrenfest theorem describes light propagation in non-Hermitian waveguides.

The work bridges symmetry principles with practical optical device modeling.

Abstract

Optics has proved a fertile ground for the experimental simulation of quantum mechanics. Most recently, optical realizations of $\mathcal{PT}$-symmetric quantum mechanics have been shown, both theoretically and experimentally, opening the door to international efforts aiming at the design of practical optical devices exploiting this symmetry. Here, we focus on the optical $\mathcal{PT}$-symmetric dimer, a two-waveguide coupler were the materials show symmetric effective gain and loss, and provide a review of the linear and nonlinear optical realizations from a symmetry based point of view. We go beyond a simple review of the literature and show that the dimer is just the smallest of a class of planar $N$-waveguide couplers that are the optical realization of Lorentz group in 2+1 dimensions. Furthermore, we provide a formulation to describe light propagation through waveguide couplers described by non-Hermitian mode coupling matrices based on a non-Hermitian generalization of Ehrenfest theorem.