Generalized Petermann factor of non-Hermitian systems at exceptional points

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 a quantitative enhancement of the resonance linewidth, given by the Petermann factor, which is determined by the overlap of left and right eigenmodes of the underlying effectively non-Hermitian Hamiltonian. The Petermann factor diverges at exceptional points (EPs), where complex resonance frequencies become degenerate, and right and left eigenmodes are orthogonal to each other. This divergence signifies a qualitative departure from a Lorentzian lineshape, which has moved into the focus of recent attention. In this work, we develop the analog of the Petermann factor for this qualitatively modified response at an EP, and describe how this EP Petermann factor manifests in a variety of physical scenarios. First, we identify this analog 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 Petermann factor for isolated resonances to EPs. The two choices can be combined into an overall EP Petermann factor, which again can be expressed in purely geometric terms. Second, we illuminate the geometric mechanisms that determine the size of the EP Petermann factor, by considering the role of modes participating in the degeneracy and those that remain spectrally separated from it. This also leads to a systematic description to evaluate this factor. Third, we design a system that allows us to study the EP Petermann factor in a specific physical setup, consisting of two microrings coupled to a waveguide with embedded semitransparent mirrors. With this example, we demonstrate that our approach gives more accurate results for the spectral response strength than conventional truncation schemes. These results complete the description of systems operating at exceptional points in the same way as the original Petermann factor does for isolated resonances, and pave the way to the design of open systems with unconventional spectral response, be it in emission or in response to static and dynamic perturbations.