Opposite amplitude phase entropy responses at a non Hermitian avoided crossing

Avoided crossings (A.C.) in open resonators arise from non-Hermitian mode interaction, where leakage produces complex spectra and biorthogonal eigenmodes. Intensity-based entropies are robust markers of mode mixing but discard the phase structure of the complex field. Here we introduce a field-level information-theoretic analysis based on the joint statistics of local amplitude and phase under Born-weighted sampling on the cavity grid. For an open elliptical microcavity in the strong-interaction A.C. regime, we find a distinctive sector-resolved response: amplitude statistics tighten while phase statistics broaden maximally at the mixing point, and conditioning reveals strong amplitude-phase dependence. By introducing a coarse position label and the associated co-information, we further show that the enhancement of global amplitude-phase coupling is strongly shaped by spatial heterogeneity across the cavity.

💡 Research Summary

This paper presents a novel information-theoretic framework for analyzing non-Hermitian mode interactions in open resonators, specifically focusing on the avoided crossing (A.C.) regime. In open systems, energy leakage leads to complex eigenvalues and biorthogonal eigenmodes. While intensity-based entropies are robust markers of mode mixing, they discard the crucial phase information inherent to the complex field. To address this, the authors propose a field-level analysis based on the joint statistics of local amplitude and phase.

The core methodology employs “Born-weighted sampling” on the cavity grid. Each interior grid point of an open elliptical dielectric microcavity is treated as a sample point, with a probability of being sampled proportional to the field intensity |ψ(r)|² at that point. On this statistical sample space, the local amplitude A(r)=|ψ(r)| and phase Φ(r)=arg ψ(r) are defined as random variables. From their joint probability distribution p_AΦ(A,Φ), standard information measures—such as marginal entropies H(A) and H(Φ), joint entropy H(A,Φ), and mutual information I(A;Φ)—are computed to quantify the statistical relationship between amplitude and phase across parameter variations.

The system studied is a two-dimensional elliptical microcavity supporting a pair of modes undergoing a strong-interaction avoided crossing as the ellipticity is tuned. Spectral analysis shows the characteristic repulsion in the real part of the complex wavenumber (frequency) and a crossing in the imaginary part (loss rate). As expected, the intensity-based spatial entropy H_P peaks at the A.C., indicating maximal delocalization and hybridization of the mode patterns.

The key finding is a distinctive, sector-resolved response in the amplitude-phase statistics at the A.C. point. Contrary to the intensity entropy, the marginal entropy of the amplitude H(A) exhibits a pronounced decrease, meaning the amplitude distribution becomes tighter and more concentrated. Conversely, the marginal entropy of the phase H(Φ) shows a clear increase, indicating that the phase distribution broadens and becomes more disordered. This opposite behavior highlights a fundamental aspect of non-Hermitian mode mixing not accessible via intensity alone. Furthermore, the mutual information I(A;Φ) between amplitude and phase peaks sharply at the A.C., revealing that the two variables become maximally statistically dependent during strong mode interaction.

To dissect the origin of this enhanced global amplitude-phase coupling, the authors introduce an additional coarse-grained position label variable (L), partitioning the cavity into spatial sectors. They then employ co-information (multivariate mutual information) to analyze the interplay between Amplitude (A), Phase (Φ), and Location (L). This analysis demonstrates that a significant portion of the observed peak in I(A;Φ) is not due to simple local correlation but is mediated and amplified by spatial heterogeneity across the cavity. Different spatial regions contribute differently to the joint statistics, and their mixing enhances the overall dependence measured globally.

In summary, this work moves beyond intensity-only diagnostics by establishing amplitude-phase information measures as a sensitive probe of non-Hermitian mode interaction. The discovery of opposite entropy responses (tightening amplitude vs. broadening phase) and the role of spatial heterogeneity in shaping their mutual dependence provide a deeper, more nuanced understanding of avoided crossings in open systems. This framework offers a new lens for characterizing non-Hermitian physics, with potential applications in sensing at exceptional points and analyzing complex wave systems.