Avoided Mode Crossing (AMX)

• Avoided Mode Crossing (AMX) is the phenomenon where coupled spectral modes repel near degeneracy, resulting in hybridized eigenstates and a finite gap determined by the coupling strength.
• It arises from symmetry breaking and mode coupling mechanisms, with clear manifestations in both closed systems (energy gaps) and open systems (linewidth exchange and non-orthogonality).
• In nonlinear and multimode configurations, AMX enables controlled dispersion tuning, radiation management, and dynamic spectral reorganization across optical, microwave, and plasmonic platforms.

Avoided Mode Crossing (AMX), also called an avoided crossing (AC) and, in some subliteratures, avoided mode crossing (AMC), denotes the repulsion of two spectral branches when modes that would otherwise become degenerate are coupled. In its minimal form, AMX is the lifting of a nominal crossing by a finite off-diagonal interaction, producing hybridized eigenmodes and a finite minimum gap; in open systems, the same phenomenon occurs for complex eigenfrequencies or wavenumbers and is accompanied by linewidth exchange, non-orthogonality, and loss redistribution (Park et al., 3 Feb 2026, D'Aguanno et al., 2017, Stern et al., 2019).

1. Minimal structure and spectral signatures

The canonical AMX description is a two-mode problem. For resonances with uncoupled frequencies $\omega_1$ and $\omega_2$ and real coupling $g$, one writes

$$H=\begin{pmatrix} \omega_1 & g\ g & \omega_2 \end{pmatrix}, \qquad \omega_\pm=\frac{\omega_1+\omega_2}{2}\pm\sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$

At the nominal degeneracy, the gap is finite and equals $2|g|$; the eigenmodes are hybridized combinations of the uncoupled states (D'Aguanno et al., 2017, Kolomenskii et al., 2010).

Open systems require a non-Hermitian generalization. In open resonators, coupling to the radiative environment produces complex mode energies $\varepsilon_j=\nu_j-i\gamma_j$ and an effective Hamiltonian

$$H(e)= \begin{pmatrix} \varepsilon_1(e) & v\ v & \varepsilon_2(e) \end{pmatrix}, \qquad E_\pm(e)=\frac{\varepsilon_1(e)+\varepsilon_2(e)}{2}\pm \sqrt{\left[\frac{\varepsilon_1(e)-\varepsilon_2(e)}{2}\right]^2+v^2}.$$

In the strong-interaction regime, defined by $2v>|\Im(\varepsilon_1)-\Im(\varepsilon_2)|$, the real parts repel while the imaginary parts cross; in the weak-interaction regime, the opposite occurs. The exceptional point is the transition case $Z(e)=0$, where both eigenvalues and eigenmodes coalesce (Park et al., 3 Feb 2026).

AMX may be resolved either as an energy gap or as a momentum gap. In plasmonic gratings, a coupled-mode formulation in frequency produces an $\omega$-gap at fixed wavevector, while a propagation-constant formulation produces a $\omega_2$0-gap at fixed frequency. The latter is characterized by

$\omega_2$1

with $\omega_2$2 at $\omega_2$3 (Kolomenskii et al., 2010).

2. Symmetry, topology, and coupling mechanisms

AMX is frequently the spectral manifestation of broken or tuned symmetry. In cylindrical microwave cavities for axion haloscopes, longitudinal symmetry breaking rather than transverse symmetry breaking is the mechanism for AMC. End-cap gaps, rod tilt, and other axial inhomogeneities couple nominally orthogonal TM and TE modes; near the closest approach the modes hybridize and the gap scales quasi-linearly with perturbation size, with the empirical relation $\omega_2$4 for small rod-end gaps (Stern et al., 2019).

In optical lattices, symmetry can protect an exact crossing until a weak perturbation converts it into AMX. In the optical chequerboard lattice, perfect $\omega_2$5 symmetry produces a triply degenerate point at $\omega_2$6 involving the 2nd, 3rd, and 4th Bloch bands. Weakly broken $\omega_2$7 symmetry lifts this degeneracy and opens small but robust gaps, notably $\omega_2$8 and $\omega_2$9, while the condensate changes from an approximately $g$0-invariant zero-momentum state to a finite-momentum state with reduced $g$1 symmetry when tuned across the avoided crossing (Ölschläger et al., 2011).

Parity selection rules govern AMX and its radiative consequences in leaky-mode photonic lattices. For same-parity guided modes in symmetric slabs, the far-field phases permit Friedrich-Wintgen cancellation and a true BIC can accompany the AMX; in asymmetric slabs, the same mechanism yields quasi-BICs with finite $g$2 because upward and downward radiation cannot both be canceled. For different-parity coupling in asymmetric slabs, the result is a unidirectional-BIC rather than a true BIC, with reported power ratios $g$3 up to $g$4 dB (Lee et al., 2020).

Certain platforms render AMX effectively intrinsic. In X-cut lithium niobate microrings, the optical axis lies in the resonator plane, so the TE mode experiences an azimuthally varying effective index

$g$5

while the TM mode remains approximately ordinary, $g$6. The resulting TE-TM phase matching at specific azimuths forces polarization conversion and makes the avoided crossing unavoidable (Tan et al., 2024).

3. Open and non-Hermitian AMX

In open resonators, AMX is not exhausted by level repulsion. Radiation leakage makes the spectrum complex and the eigenmodes biorthogonal, so the crossing region reorganizes both amplitude and phase structure. In an open elliptical microcavity computed by BEM with outgoing Green’s functions, the strong-interaction AC window occurs for eccentricity $g$7–$g$8, where two resonances show an avoided crossing in $g$9 together with correlated variation in $$H=\begin{pmatrix} \omega_1 & g\ g & \omega_2 \end{pmatrix}, \qquad \omega_\pm=\frac{\omega_1+\omega_2}{2}\pm\sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$0 and strong real-space hybridization of $$H=\begin{pmatrix} \omega_1 & g\ g & \omega_2 \end{pmatrix}, \qquad \omega_\pm=\frac{\omega_1+\omega_2}{2}\pm\sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$1 (Park et al., 3 Feb 2026).

Intensity-only diagnostics detect some of this restructuring. The spatial Shannon entropy

$$H=\begin{pmatrix} \omega_1 & g\ g & \omega_2 \end{pmatrix}, \qquad \omega_\pm=\frac{\omega_1+\omega_2}{2}\pm\sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$2

peaks in the AC window, signaling delocalization of intensity. However, intensity discards the phase structure of the complex field and cannot resolve the amplitude-phase coupling intrinsic to non-Hermitian mixing (Park et al., 3 Feb 2026).

Field-level information measures sharpen the diagnosis. Writing $$H=\begin{pmatrix} \omega_1 & g\ g & \omega_2 \end{pmatrix}, \qquad \omega_\pm=\frac{\omega_1+\omega_2}{2}\pm\sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$3 and sampling the cavity interior with Born weights $$H=\begin{pmatrix} \omega_1 & g\ g & \omega_2 \end{pmatrix}, \qquad \omega_\pm=\frac{\omega_1+\omega_2}{2}\pm\sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$4, one obtains a joint distribution for amplitude and phase. At the mixing point, the amplitude marginal sharpens while the phase marginal broadens; correspondingly, $$H=\begin{pmatrix} \omega_1 & g\ g & \omega_2 \end{pmatrix}, \qquad \omega_\pm=\frac{\omega_1+\omega_2}{2}\pm\sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$5 exhibits a dip and $$H=\begin{pmatrix} \omega_1 & g\ g & \omega_2 \end{pmatrix}, \qquad \omega_\pm=\frac{\omega_1+\omega_2}{2}\pm\sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$6 a peak. The conditional entropies behave similarly, with $$H=\begin{pmatrix} \omega_1 & g\ g & \omega_2 \end{pmatrix}, \qquad \omega_\pm=\frac{\omega_1+\omega_2}{2}\pm\sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$7 and $$H=\begin{pmatrix} \omega_1 & g\ g & \omega_2 \end{pmatrix}, \qquad \omega_\pm=\frac{\omega_1+\omega_2}{2}\pm\sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$8 near the avoided crossing, indicating strong amplitude-phase dependence. The mutual information

$$H=\begin{pmatrix} \omega_1 & g\ g & \omega_2 \end{pmatrix}, \qquad \omega_\pm=\frac{\omega_1+\omega_2}{2}\pm\sqrt{\left(\frac{\omega_1-\omega_2}{2}\right)^2+g^2}.$$9

increases in the interaction window, and the co-information

$2|g|$0

is positive near the AC when a coarse position label $2|g|$1 is introduced, showing that the enhanced global dependence is strongly shaped by spatial heterogeneity (Park et al., 3 Feb 2026).

A complementary quadrature-space formulation reaches a closely related conclusion. For quasi-normal modes represented as $2|g|$2, a covariance-aligned quadrature frame yields weighted histograms for $2|g|$3 and Shannon-type measures on the joint distribution. Near the AC, $2|g|$4 and the joint entropy $2|g|$5 peak, and the mutual information $2|g|$6 also peaks; the ratio $2|g|$7 reaches approximately $2|g|$8, so roughly half of the total quadrature-space entropy is attributable to inter-quadrature correlation (Park et al., 24 Sep 2025).

4. Linear realizations across wave systems

In periodic gold nanostructures, AMX arises from Bragg-mediated coupling of counter-propagating surface plasmon modes. For one sample, a clear energy gap of about $2|g|$9 meV appears near normal incidence, with gap edges at $\varepsilon_j=\nu_j-i\gamma_j$0 nm and $\varepsilon_j=\nu_j-i\gamma_j$1 nm; for another, the higher-frequency branch forms a momentum gap with $\varepsilon_j=\nu_j-i\gamma_j$2. The transition is tied to a dark-to-bright mode switch and to changes in the sign of the coupling parameter $\varepsilon_j=\nu_j-i\gamma_j$3, so the system can display $\varepsilon_j=\nu_j-i\gamma_j$4-gap, $\varepsilon_j=\nu_j-i\gamma_j$5-gap, or mixed behavior (Kolomenskii et al., 2010).

In whispering-gallery sapphire resonators, the two states involved in AMX are the spin-polarized photon states $\varepsilon_j=\nu_j-i\gamma_j$6 and $\varepsilon_j=\nu_j-i\gamma_j$7 of a single WGM doublet. Gyrotropy induced by dilute Fe$\varepsilon_j=\nu_j-i\gamma_j$8 impurities under an external magnetic field tunes the detuning and coupling between these helicity states, producing an avoided crossing with minimal splitting near $\varepsilon_j=\nu_j-i\gamma_j$9 mT. The reported coupling and linewidth are $$H(e)= \begin{pmatrix} \varepsilon_1(e) & v\ v & \varepsilon_2(e) \end{pmatrix}, \qquad E_\pm(e)=\frac{\varepsilon_1(e)+\varepsilon_2(e)}{2}\pm \sqrt{\left[\frac{\varepsilon_1(e)-\varepsilon_2(e)}{2}\right]^2+v^2}.$$0 kHz and $$H(e)= \begin{pmatrix} \varepsilon_1(e) & v\ v & \varepsilon_2(e) \end{pmatrix}, \qquad E_\pm(e)=\frac{\varepsilon_1(e)+\varepsilon_2(e)}{2}\pm \sqrt{\left[\frac{\varepsilon_1(e)-\varepsilon_2(e)}{2}\right]^2+v^2}.$$1 Hz, giving $$H(e)= \begin{pmatrix} \varepsilon_1(e) & v\ v & \varepsilon_2(e) \end{pmatrix}, \qquad E_\pm(e)=\frac{\varepsilon_1(e)+\varepsilon_2(e)}{2}\pm \sqrt{\left[\frac{\varepsilon_1(e)-\varepsilon_2(e)}{2}\right]^2+v^2}.$$2 and a resolvable strong-coupling AMX (Goryachev et al., 2013).

In cylindrical haloscope cavities, the experimentally relevant consequence is mode hybridization of the tunable TM$$H(e)= \begin{pmatrix} \varepsilon_1(e) & v\ v & \varepsilon_2(e) \end{pmatrix}, \qquad E_\pm(e)=\frac{\varepsilon_1(e)+\varepsilon_2(e)}{2}\pm \sqrt{\left[\frac{\varepsilon_1(e)-\varepsilon_2(e)}{2}\right]^2+v^2}.$$3 search mode with nearby TE/TM spectators. At the point of closest approach, the interacting hybrid modes have equal and reduced form factors; for the case $$H(e)= \begin{pmatrix} \varepsilon_1(e) & v\ v & \varepsilon_2(e) \end{pmatrix}, \qquad E_\pm(e)=\frac{\varepsilon_1(e)+\varepsilon_2(e)}{2}\pm \sqrt{\left[\frac{\varepsilon_1(e)-\varepsilon_2(e)}{2}\right]^2+v^2}.$$4, the reported value is $$H(e)= \begin{pmatrix} \varepsilon_1(e) & v\ v & \varepsilon_2(e) \end{pmatrix}, \qquad E_\pm(e)=\frac{\varepsilon_1(e)+\varepsilon_2(e)}{2}\pm \sqrt{\left[\frac{\varepsilon_1(e)-\varepsilon_2(e)}{2}\right]^2+v^2}.$$5 for both hybrids. The resulting spectral holes reduce usable scan coverage and motivate stringent suppression of end gaps and rod tilt (Stern et al., 2019).

Leaky-mode photonic lattices illustrate the close relation between AMX and continuum-coupled radiation control. In symmetric lattices, same-parity guided-mode resonances generate Friedrich-Wintgen BICs with $$H(e)= \begin{pmatrix} \varepsilon_1(e) & v\ v & \varepsilon_2(e) \end{pmatrix}, \qquad E_\pm(e)=\frac{\varepsilon_1(e)+\varepsilon_2(e)}{2}\pm \sqrt{\left[\frac{\varepsilon_1(e)-\varepsilon_2(e)}{2}\right]^2+v^2}.$$6 exceeding $$H(e)= \begin{pmatrix} \varepsilon_1(e) & v\ v & \varepsilon_2(e) \end{pmatrix}, \qquad E_\pm(e)=\frac{\varepsilon_1(e)+\varepsilon_2(e)}{2}\pm \sqrt{\left[\frac{\varepsilon_1(e)-\varepsilon_2(e)}{2}\right]^2+v^2}.$$7 at the avoided crossing. In asymmetric lattices, the corresponding quasi-BICs saturate below $$H(e)= \begin{pmatrix} \varepsilon_1(e) & v\ v & \varepsilon_2(e) \end{pmatrix}, \qquad E_\pm(e)=\frac{\varepsilon_1(e)+\varepsilon_2(e)}{2}\pm \sqrt{\left[\frac{\varepsilon_1(e)-\varepsilon_2(e)}{2}\right]^2+v^2}.$$8, while different-parity couplings generate unidirectional-BICs with strongly asymmetric radiation rather than complete trapping (Lee et al., 2020).

5. Driven, nonlinear, and multimode AMX

AMX can be created dynamically rather than merely tuned through. In bi- and multilayer graphene, coherent mid-infrared driving of the IR-active $$H(e)= \begin{pmatrix} \varepsilon_1(e) & v\ v & \varepsilon_2(e) \end{pmatrix}, \qquad E_\pm(e)=\frac{\varepsilon_1(e)+\varepsilon_2(e)}{2}\pm \sqrt{\left[\frac{\varepsilon_1(e)-\varepsilon_2(e)}{2}\right]^2+v^2}.$$9 phonon at approximately $2v>|\Im(\varepsilon_1)-\Im(\varepsilon_2)|$0 meV activates the cubic anharmonic coupling $2v>|\Im(\varepsilon_1)-\Im(\varepsilon_2)|$1 to the Raman-active $2v>|\Im(\varepsilon_1)-\Im(\varepsilon_2)|$2 mode. The two degenerate oscillators hybridize into vibronic states $2v>|\Im(\varepsilon_1)-\Im(\varepsilon_2)|$3 and $2v>|\Im(\varepsilon_1)-\Im(\varepsilon_2)|$4, with a fluence-dependent splitting $2v>|\Im(\varepsilon_1)-\Im(\varepsilon_2)|$5 and linewidth modification $2v>|\Im(\varepsilon_1)-\Im(\varepsilon_2)|$6, both scaling with $2v>|\Im(\varepsilon_1)-\Im(\varepsilon_2)|$7. Transient Raman spectroscopy shows line splitting, sharpening, and enhanced lifetimes; equilibrium phonon lifetimes of about $2v>|\Im(\varepsilon_1)-\Im(\varepsilon_2)|$8 fs become a driven-state persistence of several picoseconds (Buchenau et al., 2023).

In Kerr microresonators, AMX can be represented either in the hybrid-mode basis or as a localized dispersion defect in a modified Lugiato-Lefever equation. In the coupled-LLE treatment, guided-mode coupling near the AMX creates two hybrid branches with opposite dispersion sign, enabling bright solitons and broadband combs when both branches are pumped with suitable powers and detunings. A deterministic route proceeds from periodic patterns to hyperparametric oscillations and then to bright solitons, without a chaotic intermediary, provided the anomalous branch dominates and the cross-phase modulation remains strong (D'Aguanno et al., 2017).

A complementary Kerr-cavity treatment models AMX as a $2v>|\Im(\varepsilon_1)-\Im(\varepsilon_2)|$9-like defect in spectral dispersion, with position $Z(e)=0$0 and strength $Z(e)=0$1. The AMX term acts as an effective narrowband secondary pump and can stabilize otherwise unstable soliton crystals or pin Turing patterns that seed them. For the parameter set used in that study, the pump is fixed at $Z(e)=0$2, below the reported thresholds $Z(e)=0$3 and $Z(e)=0$4, and deterministic perfect soliton crystals occur when $Z(e)=0$5 does not exceed the roll count of the first Turing pattern; for $Z(e)=0$6, PSCs always form with periodicity equal to $Z(e)=0$7 (Silvestri et al., 2 Jun 2026).

In $Z(e)=0$8 nanowaveguides, AMX between two second-harmonic modes is incorporated explicitly as a linear coupling $Z(e)=0$9 in a three-mode model. For an X-cut lithium niobate ridge waveguide, the fitted value $\omega$0 reproduces the hybrid second-harmonic dispersions near $\omega$1 nm and supports avoided-crossing solitons with characteristic pedestals in the pulse tails and pronounced spectral peaks, arising from resonant coupling to the linear hybrid modes (Rowe et al., 2021).

6. Inference, control, and limitations

AMX is also a diagnostic tool for evolving complex media. In proto-neutron stars, a linear perturbation analysis shows that the gravitational-wave ramp-up track corresponds to the $\omega$2 $\omega$3 mode in the early phase and to the $\omega$4 $\omega$5 mode later on, with the exchange mediated by an avoided crossing around $\omega$6 s after core bounce. The spectrogram ridge rises from approximately $\omega$7 Hz to approximately $\omega$8 kHz over $\omega$9–$\omega_2$00 s, and the fitted relation between frequency and the square root of the PNS average density exploits the AMX-mediated continuity of that track (Sotani et al., 2020).

Because AMX is highly sensitive to the symmetry-breaking channel that creates it, device design often reduces to coupling management. In haloscope cavities, longitudinal symmetry must be preserved as closely as possible; minimizing $\omega_2$01 and rod tilt suppresses spectral holes and form-factor loss (Stern et al., 2019). In X-cut lithium niobate microrings, TE-TM AMX positions can be predicted with the reported 2D equivalent model, which avoids full 3D anisotropic simulation while reproducing the hybridization and the local distortion of $\omega_2$02 that suppresses one sideband in nondegenerate four-wave mixing (Tan et al., 2024).

Field-level AMX diagnostics have their own assumptions. The amplitude-phase analysis of open microcavities requires access to the complex field, relies on histogram estimators whose quantitative values depend on binning and sampling density, and uses Born weighting that emphasizes high-intensity regions while underweighting subtle phase structure at low amplitude. The quadrature-space approach is similarly estimator-based, though the reported weighted and unweighted constructions show the same qualitative AC trends (Park et al., 3 Feb 2026, Park et al., 24 Sep 2025).

A plausible implication is that AMX should be treated less as a single spectral motif than as a family of coupling-controlled reorganizations whose observable content depends on symmetry, openness, nonlinear feedback, and the choice of diagnostic. Across photonic, plasmonic, solid-state, microwave, and astrophysical settings, the recurring invariants are the lifted degeneracy, the hybridization of eigenstates, and the appearance of a control-parameter scale at which mode identity is most strongly redistributed (Park et al., 3 Feb 2026, Buchenau et al., 2023, Sotani et al., 2020).