Generalized Petermann factor of non-Hermitian systems at exceptional points

TL;DR

This work addresses how non-Hermitian mode nonorthogonality at exceptional points (EPs) qualitatively alters resonant response and noise, by generalizing the Petermann factor (PF) to EPs. It develops two geometrically grounded reference frameworks based on right and left eigenvectors, introducing EP Petermann factors $K_l^{(R)}$, $K_l^{(L)}$, and their geometric mean $K_l$, linked to the spectral response strength $\xi_l$ and its left/right counterparts $\xi^{(R)}_l$, $\xi^{(L)}_l$ via oblique projections onto generalized eigenspaces. The theory is anchored in Green's function formalism and Jordan (isomorphic) decompositions, and is validated with a toy $3\times3$ model and a realistic two-microring photonic system, where EP-enhanced response is accurately captured beyond conventional truncations. The results provide a unified framework for predicting spectral response and quantum-limited excess noise at EPs, with practical implications for sensing, photonics, and open-system design.

Abstract

The nonorthogonality of modes in open systems significantly modifies their resonant response, resulting in quantitative and qualitative deviations from Breit-Wigner resonance relations. For isolated resonances with a Lorentzian lineshape, the deviations amount to an enhancement of the resonance linewidth by the Petermann factor (PF), given by the overlap of left and right eigenmodes of the underlying effectively non-Hermitian Hamiltonian. The PF diverges at exceptional points (EPs), where resonance frequencies degenerate, and right and left eigenmodes are orthogonal to each other. This divergence signifies a qualitative departure from a Lorentzian lineshape, which has gained recent attention. In this work, we extend this concept to EPs, and describe how this EP PF manifests in a variety of physical scenarios. Firstly, we identify this PF in physical terms as an enhancement of the response of a system to external or parametric perturbations. Utilizing two natural orthogonally projected reference systems based on the right and left eigenvectors, we show that each choice carries a precise geometric interpretation that naturally extends the notion of the PF for isolated resonances to EPs. The two choices can be combined into an overall EP PF, which again can be expressed in purely geometric terms. Secondly, we illuminate the geometric mechanisms that determine the size of the EP PF, by considering the role of modes participating in the degeneracy and those that remain spectrally separated. Thirdly, we design a system to study the EP PF in a specific physical setup, consisting of two microrings coupled to a waveguide with embedded semitransparent mirrors. This example shows our approach yields a more accurate spectral response strength than conventional truncation. These results complete the description of systems at EPs in the same way as the original PF does for isolated resonances.

Figures (4)

• Figure 1: Illustration of the system consisting of two equal microrings coupled to an infinite waveguide, which contains two partial mirrors.
• Figure 2: (a) Complex frequencies $\Omega$ for the system shown in Fig. \ref{['fig:example']}. Results from FEM simulations are shown in thick blue dots. Black pluses are the eigenvalues of the effective Hamiltonian (\ref{['eq:Hexample']}). The parameters $\omega_\text{a}$ and $\omega_\text{b}$ are shown as thin black dots. (b) Relative intensity $I_{\text{rel}}$ of the modes in cavity a and cavity b. Blue thick bars are FEM simulations (cf. Fig. \ref{['fig:SystemModes']}) and thin black bars correspond to the eigenstates of the effective Hamiltonian (\ref{['eq:Hexample']}).
• Figure 3: Mode patterns from FEM simulations for the system of two waveguide-coupled microrings; cf. Fig. \ref{['fig:example']}. Shown is $|\psi(x,y)|^{1/2}$ to enhance the visibility of small intensities. In the waveguide are two semi-transparent mirrors: Between the cavities is a thin mirror with width $d_{\text{b}} = 0.005R$ and right to the cavities is a thick mirror with width $d_{\text{a}} = 0.06R$. Both mirrors are indicated by red rectangles.
• Figure 4: Mode patterns $|\phi_{1-4}(x,y)|^{1/2}$ for different configurations of a single cavity coupled to a waveguide with a mirror of width $d=0.06R$. The mirror is indicated by a red rectangle.