Perturbing scattering resonances in non-Hermitian systems: a generalized Wigner-Smith operator formulation

TL;DR

The paper addresses resonance shifts in open non-Hermitian systems by introducing a generalized Wigner-Smith operator framework. It derives a pole-shift formula $Δω_p = i Δα Res_{ω=ω_p} tr(Q_α) = -Δα \frac{Res_{ω=ω_p} tr(Q_α)}{Res_{ω=ω_p} tr(Q_ω)}$ that links perturbations to shifts in pole positions, and connects this to traditional cavity perturbation theory in the high-Q limit. A volume-integral formulation $Q_ξ = A^{-1} B_ξ$ is developed, with $A$ and $B_ξ$ expressed as energy-balance integrals, enabling physically transparent interpretation in terms of stored and dissipated energy. The theory is validated on complex photonic networks and extended to pumping scenarios, demonstrating selective control of resonance positions and potential sensing applications in non-Hermitian photonics.

Abstract

Resonances of open non-Hermitian systems are associated with the poles of the system scattering matrix. Perturbations of the system cause these poles to shift in the complex frequency plane. In this work, we introduce a novel method for calculating shifts in scattering matrix poles using generalized Wigner-Smith operators. We link our method to traditional cavity perturbation theory and validate its effectiveness through application to complex photonic networks. Our findings underscore the versatility of generalized Wigner-Smith operators for analyzing a broad spectrum of resonant systems and provides new insight into resonant properties of non-Hermitian systems.

Figures (5)

• Figure 1: An example random network. Purple nodes and links are 'internal', while green 'external' elements allow light to enter and exit the network. Red link ($\Omega_{\alpha}$) and highlighted links (2 and 3) correspond to those perturbed in numerical experiments described in the main text.
• Figure 2: (Main panel) Complex $\omega$ plane showing the trajectories of the network scattering matrix poles upon perturbation of $\Omega_{\alpha}$. Curves (solid for $\Delta n>0$ and dashed $\Delta n<0$) were traced (white) by numerically solving $\det(\mathbf{S}^{-1}) = 0$ and (red) from Eq. (\ref{['eq:pole-shift-main']}). White crosses (dots) denote the (un)tracked poles. Numbered poles are those described in the numerical pumping experiment. White dashed boxes depict regions shown in (a)--(d).
• Figure 3: (left) Imaginary parts of selected poles as different network links are pumped (see Figure \ref{['fig:network']}). Colors and line styles distinguish different modes and pumping methods respectively. Spatial intensity profiles of modes 2 (top right) and 3 (bottom right).
• Figure S1: Geometry of an internal network link.
• Figure S2: Geometry of an internal network node.