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

TL;DR

This work addresses how to systematically analyze non-Hermitian $N$-mode couplers and their higher-order exceptional points using a Lie-algebraic approach. It develops a general $gl(N,\\mathbb{C})$ framework and reduces to traceless $sl(N,\\mathbb{C})$ via a propagation-dependent gauge, with a bosonic $N$-mode representation for quantum dynamics. For $N=3$, the authors introduce the isospin–hypercharge basis, construct an exact Wei–Norman propagator, and reveal a geometric holonomy that distinguishes adiabatic from exact propagation within this algebraic setting. They classify spectral regimes and EPs in $PT$-symmetric and non-Hermitian cyclic couplers and illustrate the theory on a lossy three-leg beam splitter, providing a versatile route to design and analyze non-Hermitian photonic and quantum devices. $gl(N,\\mathbb{C})$, $sl(3,\\mathbb{C})$, $EP$, Wei–Norman, holonomy, bosonic embedding, spectral discriminant

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.

Figures (9)

• Figure 1: Weight diagrams for the totally symmetric representations $(n,0)$ in the $(I_{0},Y)$ basis, shown for (a) $n=1$ and (b) $n=2$.
• Figure 2: Schematic of a single, lossy waveguide beam splitter with three waveguides excited with $N$ indistinguishable photons prepared in the state $\ket{ n - n_{2} - n_{3}, n_{2}, n_{3}}$. In the Schwinger representation, it can be mapped to a kagome lattice Sankar2024p021046. The coupling between adjacent “modes” is given by matrix elements of $\bm{I}_{\pm}$, $\bm{U}_{\pm}$ and $\bm{V}_{\pm}$.
• Figure 3: Spectral structure of the non-Hermitian coupler family in Eq. (\ref{['eq:Example']}). (a) Sign of the discriminant with $\Delta>0$ in blue, $\Delta<0$ in white, $\Delta=0$ in black for EP2, and red for EP3. (b) Real and (c) imaginary part of the spectral Riemann surface with branches in red, cream, and blue.
• Figure 4: Dynamics of the non-Hermitian coupler family in Eq. \ref{['eq:Example']} for the initial classical field $\bm{E}(0) = \left\{1, 0, 0 \right\}$. (a)–(e) Classical field intensities and (f)–(j) their renormalization, showing (a,f) compact periodic dynamics for $\Delta>0$ with $\kappa_{1} / \gamma = 3/2$, $\kappa_{2} / \gamma = 7/2$; (b,g) non-compact hyperbolic dynamics for $\Delta<0$ with $\kappa_{1} / \gamma = \kappa_{2} / \gamma = 3/2$; (c,h) quadratic algebraic dynamics at EP3 with $\Delta=0$, $\kappa_{1} / \gamma = 1 / \sqrt{2}$, $\kappa_{2} / \gamma = 0$; (d,i) and (e,j) linear dynamics at EP2 with $\Delta=0$, $\kappa_{1} / \gamma = 3/2$, $\kappa_{2} / \gamma \in \{0.6718,2.3882\}$. Solid lines in red, black, and blue correspond to the intensities of the field amplitudes $E_{1}(z)$, $E_{2}(z)$, and $E_{3}(z)$, respectively.
• Figure 5: Propagation-dependent dynamics for three clockwise turns along a circular loop in the $(\kappa_{1},\kappa_{2})$ plane centered at EP3. (a) Sign of the discriminant with the trajectory. (b)–(d) Projection into the instantaneous right eigenbasis with $|c_{j}(z)|^{2}$ in cyan, yellow, and magenta for the first, second, and third eigenvector. (e)–(g) Field intensities and (h)–(j) their renormalization in red, black, and blue, as in Fig. \ref{['fig:Figure3']}.