Higher-order exceptional points unveiled by nilpotence and mathematical induction

TL;DR

The paper introduces nilpotence as a constructive design principle for higher-order exceptional points (HEPs) and an inductive doubling scheme to realize EPs of arbitrarily large order. By requiring a nilpotent effective Hamiltonian $H$ with $H^m=0$, all eigenvalues vanish and the EP order equals the largest Jordan block size, enabling systematic EP design via Jordan form analysis. The authors then prove an inductive theorem that combines a system with EP$_N$ and its inverted copy to produce a symmetric EP$_{2N}$, scalable to $N2^h$ through repeated doublings, and they demonstrate this with photonic lattices, achieving EP$_3$ with 3-vector chirality, EP$_6$ (passive) and EP$_7$ (minimally active) with distinctive transmission features, and EP$_{14}$ and beyond. These high-order EPs exhibit pronounced phenomena, including directional radiation, induced transparency, narrow linewidths, enhanced transmittance, and amplified spontaneous emission, with resolvent analyses supporting the observed behavior. The framework offers a scalable, symmetry-agnostic path to EP-based sensing, control, and potentially nonlinear extensions across photonics and other physical platforms.

Abstract

Non-Hermitian systems can have peculiar degeneracies of eigenstates called exceptional points (EPs). An EP of $n$ degenerate states is said to have order $n$, and higher-order EPs (HEPs) with $n \ge 3$ exhibit intrinsic order-scaling responses potentially applied to superior sensing and state control. However, traditional eigenvalue-based searches for HEPs are facing fundamental limitations in terms of complexity and implementation. Here, we propose a design paradigm for HEPs based on a simple property for matrices termed nilpotence and concise inductive procedure. The nilpotence guarantees a HEP with desired order and helps divide the problem. Our inductive scheme repeatedly extends a system and doubles its EP order, starting with a known design. Based on the nilpotence, we systematically design photonic cavity arrays operating at chiral, passive, and active HEPs with $n = 3, 6, 7$ and show their peculiar directional radiation, induced transparency, and enhanced transmittance and spontaneous emission, respectively. We inductively find lattice systems with diverging EP order originating from a well-known $2 \times 2$ parity-time-symmetric Hamiltonian. We also extend the active HEP system with $n = 7$ to another with $n = 14$ and have further magnified responses. Our work pushes the investigation and application of HEPs to previously unexplored regimes in various physical systems.

Figures (6)

• Figure 1: EPs, JNF, and nilpotence of a system matrix $\hat{\rm H}$. (a) The JNF $\hat{\rm J}_{\rm H}$ is a block-diagonal similarity transformation of $\hat{\rm H}$ comprising specific semi-diagonal square matrices called JCs. Here, $\omega_j$ and $d_j$ ($j = 1, 2, \dots, u \in \mathbb{N}$) are the eigenvalue and dimension (size) of the $j$th JC, which is denoted as $\hat{\rm J}(\omega_j, d_j)$. A JC $\hat{\rm J}(\omega_{\rm EP}, n)$ in $\hat{\rm J}_{\rm H}$ means that $\hat{\rm H}$ has an EP$n$ with an eigenvalue $\omega = \omega_{\rm EP}$. (b) When $\hat{\rm H}$ has nilpotence for an index of $m$, it follows that all eigenvalues of $\hat{\rm H}$ are null, and that $m$ is the largest dimension of $\hat{\rm H}$'s JCs. Therefore, it is a sufficient condition for the existence of an EP$m$ with $\omega = 0$ of $\hat{\rm H}$, and this can be used for the design of HEP systems.
• Figure 2: Demonstration of an EP3 with 3-vector chirality by using coupled PhC cavities in an air-suspended slab. (a) System schematic. Each cavity has an active region whose complex refractive index is $n_j$ ($j = 1, 2, 3$). Two couples of air holes on its ends are displaced outwards by $s_{j, 1}$ and $s_{j, 2}$ for the variation in its resonance frequency with its radiation loss restricted. The rows of holes colored yellow (magenta) have an radius of $R_1$ ($R_2$), so that the evanescent coupling between cavities 1 and 2 (2 and 3) is controlled. (b) Real and (c) imaginary parts of the eigenfrequencies $\Omega$ for the optimized system. Here, the imaginary parts of the active medium indices are varied as ${\rm Im} \, (n_1, n_2, n_3) = (0.00502, -0.00871, 0.00366) \times c_{\rm Im}$. The TE-like eigenmodes come close to the EP3 when $c_{\rm Im} = 1$. Markers: simulation result. Black lines: analytic fitting based on Eq. (\ref{['eq:H3S_EP3']}) for $(a, b) = (-111.1, 9.744) \ {\rm GHz}$ with linearly introduced on-site imaginary potential. (d) Snapshots of magnetic fields $H_z(x, y) \ (z = 0)$ for the eigenmode nearest to the EP3 indicating the chiral state $\bm{A}(t) \propto (1, e^{i 2 \pi/3}, e^{i 4 \pi/3})^{\mathrm{T}} e^{i \Omega_{\rm EP} t}$. (e) Reciprocal Fourier intensity of transverse electric fields $|E_y (k_x, k_y)|^2 \ (z = 0)$ within the light cone for the same state. The most intense peak residing on $k_x = 0, \ k_y < 0$ shows directional radiation with a polar angle of $-16.0\tcdegree$ from the $k_z$ axis.
• Figure 3: Photonic designs and responses of passive and active HEPs. (a) System of coupled ring resonators and waveguides involving a passive EP6, and (b) another for a minimally active EP7. Evanescent and waveguide-based dissipative couplings between cavities are all $-\mu < 0$ and $i\gamma \ (\gamma > 0)$, respectively. Each waveguide also introduces an out-coupling loss $i \gamma$ of each cavity coupled. In (a), cavities 3 and 4 have resonance frequency detuning $\pm \delta$. In (b), only cavity 5 is active and its loss $2 i \gamma$ is partially compensated by a gain $-i g \ (g > 0)$. Arrows indicate the light propagation directions considered and hence determine the input and output of the waveguide ports and the directions of responding cavity modes. Complex eigenfrequency detuning of the (c) passive and (d) active systems for $\mu = \gamma$. The EP6 and EP7 are found when $\delta = \mu = \gamma$ and $g = \mu = \gamma$. (e) Internal and (f) external spectral transmission responses for the passive cavities, where $T_{lj}$ is the ratio of power transmission from port $j$ to $l$. While there is no steady transmission inside the cavities for the EP frequency or $\omega_s = 0$, the rejection signal through waveguide 2 ($T_{22}$) exhibits an induced transparency peak fit by the LF6 (orange curve). (g) Transmission ratios for the active system. $T_{51}$ is strictly described as the LF7. $T_{43}$ and $T_{33}$ show 16- and 25-fold enhancement of the transmission power at the EP7. (h) Spectral transfer functions $(S_3, S_4)$ from excitation intensity (coupled spontaneous emission) of cavity 5 to flux outputs at ports 3 and 4. Their peak enhancements are 16- and 9-fold, compared to the case of large evanescent coupling or Hermitian limit ($\mu \rightarrow \infty$) equivalent to the sole Lorentzian response of cavity 5.
• Figure 4: Inductive theorem for raising the EP order and its application. (a) Illustration of the procedure. The extended system $\hat{\rm H}'$ has diagonal blocks based on the source Hamiltonian $\hat{\rm H}$ and its inverted copy $\hat{\rm H}_{\rm I}= \hat{\rm R} \hat{\rm H} \hat{\rm R}$, and their boundary elements are updated as $[\hat{\rm H}']_{NN} = [\hat{\rm H}]_{NN} + A$, $[\hat{\rm H}']_{N(N+1)} = [\hat{\rm H}']_{(N+1)N} = B$, $[\hat{\rm H}']_{(N+1)(N+1)} = [\hat{\rm H}_{\rm I}]_{11} - A$, where $A, B \ne 0$ and $A^2 + B^2 = 0$. (b) A series of one-dimensional HEP systems with exponentially raised order based on a $2 \times 2$ PT-symmetric Hamiltonian. We can generate instances with limitlessly large EP order by repeating the process in (a). (c) A 14-cavity system designed by using Theor. 1 for $\hat{\rm H} = \hat{\rm H}_{\rm 7A}$ in Eq. (\ref{['eq:H7A']}) and $A \propto i$, $B \propto -1$. It has an EP14 when $\mu = g = \gamma$. (d) Spectral transfer functions $\{S_{l,j}\}$ from excitation intensities of cavities $j = 5, 10$ to output fluxes at ports $l = 3,4,5,6$. The upper part of the array takes over responses of $\hat{\rm H}_{\rm 7A}$, and spontaneous emission coupled to cavity 5 can be enhanced by 20 dB at $\omega_s = 0$. However, the responses within the lower part are mostly suppressed at the EP, as the outputs based on local emission in cavity 10 tend to be frequency-split due to the evanescent coupling involving cavity 8.
• Figure S1: Transmission response dynamics of the systems with HEPs. Blue and red curves: normalized inverse Fourier transforms $\mathcal{F}^{-1}$ of the characteristic transmission spectra $T_{22}$ and $T_{51}$ for the passive EP6 and active EP7, respectively. Green curve: exponential decay $e^{-\gamma t}$ of the single cavity mode with a loss $\gamma$. Dashed lines mark $\pm 1/e$ for evaluating time constants of the dynamics. The negative initial value of $\mathcal{F}^{-1}[T_{22}](t)$ comes from the system's instant rejection, and the response peculiar to the EP6 is limited to the weak positive tail. In contrast, $\mathcal{F}^{-1}[T_{51}](t)$ exhibits both a slow onset of damping and large time constant of $\approx 4.92/\gamma$, which signify a delayed internal response fully attributed to the active EP7.