Opposite amplitude phase entropy responses at a non Hermitian avoided crossing

TL;DR

Non-Hermitian open resonators exhibit leakage, complex spectra, and biorthogonal modes, motivating a field-level diagnostic based on the joint statistics of local amplitude $A(r)$ and phase $\Phi(r)$ sampled with probability $P(r)\propto|\psi(r)|^2$. In an open elliptical microcavity at a strong avoided crossing, the study finds that $H(A)$ and $H(A|\Phi)$ dip while $H(\Phi)$ and $H(\Phi|A)$ peak, signaling amplitude localization and phase delocalization with strong amplitude–phase coupling captured by $I(A;\Phi)$. Introducing a coarse spatial label $\Pi$ and the co-information $I_3(A;\Phi;\Pi)$ reveals that the global coupling near the A.C. is strongly shaped by spatial heterogeneity across the cavity. The proposed framework provides a gauge-invariant, information-theoretic toolkit for identifying and comparing strong-interaction regimes in open resonators, with potential applications to phase-sensitive sensing, interferometry, and mode engineering in non-Hermitian photonics.

Abstract

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.

Figures (4)

• Figure 1: Avoided crossing and intensity-based spatial entropy. (a) $\Re(k)$ of two interacting resonances versus eccentricity $e$ (solid: mode 1; dashed: mode 2) shows a clear avoided crossing; the inset plots $\Im(k)$ in the same window. Intensity snapshots A--F illustrate strong hybridization near the A.C. (B,E) and exchange of modal character across it. (b) Shannon entropy $H_P$ (bits) of the normalized spatial density $p(\mathbf r)=|\psi(\mathbf r)|^2/\sum_{\mathbf r\in\Omega}|\psi(\mathbf r)|^2$ (cavity interior $\Omega$) peaks near the A.C., quantifying enhanced delocalization of the intensity distribution.
• Figure 2: Born-weighted amplitude--phase joint distributions across the A.C. Joint histograms show $p_{A,\Phi}(A,\Phi)$ from cavity-interior points weighted by $w(\mathbf r)=|\psi(\mathbf r)|^2$, with $A(\mathbf r)=|\psi(\mathbf r)|$ and $\Phi(\mathbf r)=\arg\psi(\mathbf r)\in[-\pi,\pi)$. Green/red walls are the marginals $p_A(A)$ and $p_\Phi(\Phi)$. Panels (a--c) follow points A--C on the upper branch and (d--f) points D--F on the lower branch in Fig. \ref{['Figure-1']}(a), with B and E at the A.C. At the A.C. (b,e), $p_A$ narrows while $p_\Phi$ broadens, anticipating a local minimum of $H(A)$, an increase of $H(\Phi)$, and a peak of $I(A;\Phi)$.
• Figure 3: Amplitude and phase entropies across the avoided crossing. (a) Amplitude entropy $H(A)$ and conditional amplitude entropy $H(A|\Phi)$ versus eccentricity $e$ for the two interacting modes. Both curves exhibit a clear dip in the avoided-crossing (A.C.) window. (b) Phase entropy $H(\Phi)$ and conditional phase entropy $H(\Phi|A)$ versus $e$. Both curves form a pronounced peak in the same A.C. window. In both sectors, conditioning substantially reduces the entropy, demonstrating strong statistical dependence between local amplitude and phase under Born-weighted sampling.
• Figure 4: Joint entropy and position-conditioned amplitude--phase dependence across the avoided crossing. (a) Joint entropy $H(A,\Phi)$ versus eccentricity $e$. (b,c) Global mutual information $I(A;\Phi)$ (solid) and position-conditioned mutual information $I(A;\Phi\mid \Pi)$ (dashed) versus $e$ for the two datasets. Insets plot the co-information $I_3(A;\Phi;\Pi)=I(A;\Phi)-I(A;\Phi\mid\Pi)$. All information measures use base-2 logarithms and are reported in bits.