Optical finite representation of the Lorentz group

TL;DR

This paper introduces a class of photonic lattices with finite-dimensional Lorentz group representations, utilizing $ ext{PT}$-symmetry to control their optical behavior as oscillators or amplifiers.

Contribution

It demonstrates how to construct finite-dimensional Lorentz group representations in photonic lattices using $ ext{PT}$-symmetry, linking group theory with optical device design.

Findings

Photonic lattices can realize finite-dimensional Lorentz group representations.

$ ext{PT}$-symmetry controls whether the device acts as an oscillator or amplifier.

Linear $ ext{PT}$-symmetric dimers are part of this class.

Abstract

We present a class of photonic lattices with an underlying symmetry given by a finite-dimensional representation of the 2+1D Lorentz group. In order to construct such a finite-dimensional representation of a non-compact group, we have to design a $\mathcal{PT}$-symmetric optical structure. Thus, the array of coupled waveguides may keep or break $\mathcal{PT}$-symmetry, leading to a device that behaves like an oscillator or directional amplifier, respectively. We show that the so-called linear $\mathcal{PT}$-symmetric dimer belongs to this class of photonic lattices.