Gilmore-Perelomov symmetry based approach to photonic lattices

TL;DR

This paper introduces a symmetry-based framework for analyzing electromagnetic propagation in photonic lattices, leveraging group theory and coherent states to derive key properties for lattices with specific symmetries.

Contribution

It presents a novel approach using Gilmore-Perelomov coherent states to analyze photonic lattices with $SU(2)$, $SU(1,1)$, and Heisenberg-Weyl symmetries, generalizing to any lattice with a dynamical group.

Findings

Derived dispersion relations for symmetric photonic lattices

Constructed normal states and impulse functions using symmetry methods

Unified approach applicable to various symmetry groups in photonic systems

Abstract

We revisit electromagnetic field propagation through tight-binding arrays of coupled photonic waveguides, with properties independent of the propagation distance, and recast it as a symmetry problem. We focus our analysis on photonic lattices with underlying symmetries given by three well-known groups, $SU(2)$, $SU(1,1)$ and Heisenberg-Weyl, to show that disperssion relations, normal states and impulse functions can be constructed following a Gilmore-Perelomov coherent state approach. Furthermore, this symmetry based approach can be followed for each an every lattice with an underlying symmetry given by a dynamical group.