Non-Hermitian $\mathrm{sl}(3, \mathbb{C})$ three-mode couplers

TL;DR

This paper introduces a comprehensive algebraic framework for analyzing non-Hermitian three-mode couplers using $ ext{sl}(3, ext{C})$ algebra, enabling classification, exact dynamics, and insights into exceptional points relevant for classical and quantum photonics.

Contribution

It develops the first explicit $ ext{sl}(3, ext{C})$ algebraic model for three-mode non-Hermitian couplers, allowing exact solutions and classification of exceptional points.

Findings

Exact diagonalization and geometric phase analysis of three-mode couplers.

Identification of exceptional point structures within $ ext{PT}$-symmetric and cyclic families.

Application to a lossy three-leg beam splitter revealing propagation dynamics.

Abstract

Photonic systems with exceptional points, where eigenvalues and corresponding eigenstates coalesce, have attracted interest due to their topological features and enhanced sensitivity to external perturbations. Non-Hermitian mode-coupling matrices provide a tractable analytic framework to model gain, loss, and chirality across optical, electronic, and mechanical platforms without the complexity of full open-system dynamics. Exceptional points define their spectral topology, and enable applications in mode control, amplification, and sensing. Yet $N$-mode couplers, the minimal setting for $N$th-order exceptional points, are often studied in specific designs that overlook their algebraic structure. We introduce a general $\mathrm{sl}(N,\mathbb{C})$ framework for arbitrary $N$-mode couplers in classical and quantum regimes, and develop it explicitly for $N=3$. This case admits algebraic diagonalization, where a propagation-dependent gauge aligns local and dynamical spectra and reveals the geometric phase connecting adiabatic and exact propagation. An exact Wei--Norman propagator captures the full dynamics and makes crossing exceptional points explicit. Our framework enables classification of coupler families. We study the family spanning $\mathcal{PT}$-symmetric and non-Hermitian cyclic couplers, where two exceptional points of order three lie within a continuum of exceptional points of order two, ruling out pure encircling. As an application, we study these exceptional points for a lossy three-leg beam splitter and reveal its propagation dynamics as a function of initial states, such as Fock and NOON states. Our approach provides a systematic route to analyze non-Hermitian mode couplers and guide design in classical and quantum platforms.