Higher-order exceptional points in a non-reciprocal waveguide beam splitter

TL;DR

The paper addresses realizing higher-order exceptional points (EPs) in photonic systems without heavy dissipation by introducing a non-reciprocal, lossy two-mode waveguide beam splitter. Using a Schwinger (SU(2)) mapping, it derives the spectrum with eigenvalues $\lambda_r = (\omega_0 - i \Gamma/2) N \pm r \sqrt{4 \nu \nu' - \Gamma^2}$ and shows that with $\nu = \nu_0(1+\eta)$ and $\nu' = \nu_0(1-\eta)$, EPs occur for $\Gamma \le 2\nu_0\sqrt{1-\eta^2}$ and an $(N+1)$-fold EP at $\Gamma_c = 2\nu_0\sqrt{1-\eta^2}$; in the limit $\eta \to 1$ EPs can arise without dissipation. The authors then analyze NOON-state dynamics under activated non-reciprocity using the Wei–Norman method, obtaining analytic expressions for the evolution functions $f_z(z)$ and $f_\pm(z)$, and show how non-reciprocity lowers the EP threshold and shapes mode occupation, including suppression or realization of HOM-like interference near EPs. These results reveal a versatile route to robust EP-based sensing and interferometry in open quantum systems, with potential extensions to nonlinear or time-dependent regimes.

Abstract

Non-Hermitian systems have attracted significant interest because of their intriguing and useful properties, including exceptional points (EPs), where eigenvalues and the corresponding eigenstates of non-Hermitian operators become degenerate. In particular, quantum photonic systems with EPs exhibit an enhanced sensitivity to external perturbations, which increases with the order of the EP. As a result, higher-order EPs hold significant potential for advanced sensing applications, but they are challenging to achieve due to stringent symmetry requirements. In this work, we study the dynamics of a generalized lossy waveguide beam splitter with asymmetric coupling by introducing non-reciprocity as a tunable parameter to achieve higher-order EPs even without dissipation. Using the Schwinger representation, we analytically derive eigenvalues and numerically demonstrate the formation of EPs. Moreover, we analyze the evolution of NOON states under activated non-reciprocity, highlighting its impact on quantum systems. Our results open new pathways for realizing higher-order EPs in non-reciprocal open quantum systems.

Figures (7)

• Figure 1: Schematic of a non-reciprocal single, lossy, 2-leg waveguide beam splitter excited with $N$ indistinguishable photons prepared in the state $\ket{N-m,m}=\ket{N-m}_a\ket{m}_b$ ($0 \le m \le N)$, where $a$ represents the neutral (white) waveguide and $b$ the lossy (red) waveguide.
• Figure 2: Real and Imaginary parts of the eigenvalues $\lambda_r$ of the Hamiltonian in Eq. \ref{['Hamiltoniansu(2)']} (with $\omega_{0}=\nu_{0}=1\,\text{cm}^{-1}$ in Eqs. \ref{['Hamiltoniansu(2)']}, \ref{['param_nu']}) vs. the dissipation constant $\Gamma$ for $N=4$ with (a) $\eta=0$ (reciprocal), (b) $\eta=0.8$ and (c) $\eta=1$ (unidirectional). The system reaches the exceptional point (EP of order five) at $\Gamma_{c}=2\nu_{0}(1-\eta^2)^{\frac{1}{2}}$.
• Figure 3: Flow of eigenvalues $\lambda_r$ of the Hamiltonian in Eq. \ref{['Hamiltoniansu(2)']} for $N=4$ as a function of the non-reciprocity parameter $\eta$ with dissipation values (a) $\Gamma=0$ (lossless), (b) $\Gamma=1.7$ and (c) $\Gamma=2\nu_{0}$. The system reaches the higher-order exceptional points (EPs of order five) by adjusting $\eta$ to the critical value $\eta_{c}=\sqrt{4\nu_{0}^{2}-\Gamma^{2}}/2\nu_{0}$.
• Figure 4: Progression of the spin-projection components $\hat{J}_{x,y,z}$ for $N+1$ eigenmodes $\ket{\lambda_{r}}$ where we consider $N=4$ photons. In our diagrams we choose the following values of the dissipation $\Gamma$ and non-reciprocity $\eta$: (a) $\Gamma=0.75~\Gamma_{c}$, $\eta=0$, (b) $\Gamma=\Gamma_{c}$, $\eta=0$, (c) $\Gamma=1.5~\Gamma_{c}$, $\eta=0$, (d) $\Gamma=0.75~\Gamma_{c}$, $\eta=\eta_c=0.661$, (e) $\Gamma=\Gamma_c$, $\eta=\eta_c$, (f) $\Gamma=1.5~\Gamma_{c}$, $\eta=\eta_c$, (g) $\Gamma=0.75~\Gamma_c$, $\eta=1$, (h) $\Gamma=\Gamma_c$, $\eta=1$, (i) $\Gamma=1.5~\Gamma_c$, $\eta=1$.
• Figure 5: Mode occupation dynamics for $N=4$ photons for different dissipation and non-reciprocity. (a) $\Gamma=0.25~\Gamma_c, \eta=0$, (b) $\Gamma=\Gamma_c, \eta=0$, (c) $\Gamma=2\Gamma_c, \eta=0$, (d) $\Gamma=0.25~\Gamma_c, \eta=0.5$, (e) $\Gamma=\Gamma_c, \eta=0.5$, (f) $\Gamma=2~\Gamma_c, \eta=0.5$, (g) $\Gamma=0.25~\Gamma_c, \eta=1$, (h) $\Gamma=\Gamma_c, \eta=1$ (i) $\Gamma=2~\Gamma_c, \eta=1$.