Cavity Quasinormal Modes: Open-System Resonances

• Cavity quasinormal modes are natural resonances in open systems, defined by complex eigenfrequencies that signal both oscillation and decay.
• They manifest in electromagnetic resonators and gravitational settings, playing a key role in scattering, black hole ringdown, and cavity perturbation analyses.
• Their study bridges non-Hermitian spectral theory, numerical modal expansions, and quantum open-system quantization, offering practical insights for engineering and fundamental research.

Cavity quasinormal modes are the natural resonances of open cavities and cavity-like trapping regions. They are source-free solutions selected by radiative, reflective, or regularity conditions rather than by Hermitian confinement, and they therefore carry complex frequencies that encode both oscillation and decay. In electromagnetic resonators, they arise from Maxwell’s equations with outgoing-wave boundary conditions at infinity; in gravitational settings, analogous spectra appear when a black hole is enclosed by a mirror, when a wormhole throat and infinity define a resonant boundary-value problem, or when stable and unstable photon spheres create an effective cavity (Wu et al., 2024, Liu et al., 30 Dec 2025, Batic et al., 3 Apr 2025, Heidmann et al., 2023). The subject sits at the intersection of non-Hermitian spectral theory, wave scattering, numerical spectral methods, and quantum open-system theory.

1. Defining structure of cavity QNMs

For open electromagnetic cavities, the defining eigenproblem is the source-free Maxwell system

$$\nabla\times \mathbf{E}_m=i\tilde{\omega}_m\mu_0\mathbf{H}_m,\qquad \nabla\times \mathbf{H}_m=-i\tilde{\omega}_m\varepsilon(\mathbf{r},\tilde{\omega}_m)\mathbf{E}_m,$$

supplemented by outgoing-wave conditions as $|\mathbf r|\to\infty$. The eigenfrequency is complex,

$\tilde{\omega}_n=\omega_n-i\gamma_n,$

and the quality factor is

$$Q_n=-\frac{1}{2}\frac{\Re(\tilde{\omega}_n)}{\Im(\tilde{\omega}_n)}.$$

This complex spectrum is the defining signature of openness and non-Hermiticity (Wu et al., 2024).

In black-hole perturbation theory, the same resonance structure appears in a Schrödinger-like equation on the tortoise coordinate,

$\frac{d^2R}{dr_*^2}+\big(\omega^2-V(r)\big)R=0,$

with “purely outgoing” asymptotics

$R\sim e^{\pm i\omega r_*}\qquad (r_*\to\pm\infty).$

A key clarification is that these outgoing conditions are necessary but not sufficient to fully specify QNMs; the rigorous notion is given by poles of the analytically continued Green’s function, or equivalently by zeros of the Wronskian in one dimension (Macedo et al., 2024).

Literal cavity geometries replace one asymptotic radiation condition by a reflecting boundary. For charged Dirac perturbations on a Reissner–Nordström black hole enclosed by a mirror-like cavity at $r=r_m$, the horizon condition remains ingoing, while the mirror boundary is fixed by the vanishing energy flux principle. This yields two inequivalent Robin boundary conditions and therefore two spectral branches (Liu et al., 30 Dec 2025). In this sense, cavity QNMs are characterized not by a single universal boundary prescription, but by the physically appropriate leak-and-reflect structure of the open system under study.

2. Open electromagnetic cavities and non-Hermitian modal expansions

In micro- and nanoresonators, QNMs are the fundamental resonances that determine the intrinsic optical response of the structure. A central expansion is the scattered field decomposition

$$\mathbf{E}^S(\mathbf{r},\omega)=\sum_m \alpha_m(\omega)\,\tilde{\mathbf{E}}_m(\mathbf{r}),$$

with modal excitation coefficients evaluated from normalized QNM fields and the incident field. The same framework underlies time-domain reconstructions, local-density-of-states formulas, mode-hybridization analyses, and first-order perturbation theory (Wu et al., 2024).

Because outgoing solutions are continued to complex frequency, QNM fields diverge exponentially in open space. In electromagnetism this behavior is not treated as a pathology to be removed. For a compact resonator in free space, the far-field form

$$|r|^{-1}\exp\!\left[-i\tilde{\omega}\left(t-\frac{|r|}{c}\right)\right]$$

implies an exterior factor $e^{+\operatorname{Im}(\tilde{\omega})|r|/c}$ at fixed time, and since $|\mathbf r|\to\infty$0, the field grows exponentially with distance. The key physical point is that the growth is real and causal: before the outgoing signal reaches a given point, the field is zero, and the divergence is observable only within the causal light cone (Wu et al., 2024).

This exterior behavior has concrete consequences. In first-order cavity perturbation theory,

$|\mathbf r|\to\infty$1

a distant perturber may sample a large QNM field precisely because the open-system eigenfunction diverges outside the resonator. Likewise, dissipative coupling between remote resonators can strengthen with distance when evaluated on the complex QNM frequency, even though the real-frequency scattered field between distant bodies decays as $|\mathbf r|\to\infty$2. The literature therefore distinguishes a global normal-mode-like picture, which is complete inside compact resonators but incomplete outside, from a pointwise or spacetime-causal picture based on PML regularization or retarded-time mappings (Wu et al., 2024, Wu et al., 2024).

Normalization is correspondingly nontrivial. A standard route is PML regularization, in which the physical region and the PML domain are combined in a normalization integral such as

$|\mathbf r|\to\infty$3

This regularized setting makes QNM expansions computationally usable and analytically controlled for design, scattering reconstruction, and perturbative sensing (Wu et al., 2024).

3. Gravitational cavity realizations and cavity effects

The phrase “cavity QNM” is not restricted to laboratory resonators. In black-hole physics, the potential barrier and the horizon together act like a leaky resonant cavity, supporting modes that ring for a while before escaping or being absorbed (Zimmerman et al., 2014). More literal cavity configurations occur when a mirror boundary is imposed outside the hole. For charged Dirac fields on Reissner–Nordström backgrounds, the vanishing energy flux condition at the mirror produces two Robin boundary conditions,

$|\mathbf r|\to\infty$4

with a corresponding pair for $|\mathbf r|\to\infty$5. The resulting spectra display a hidden symmetry between the two boundary conditions, and in the large-$|\mathbf r|\to\infty$6 regime the paper identifies an anomalous decay pattern in which excited modes decay slower than the fundamental mode (Liu et al., 30 Dec 2025).

Wormholes provide a second cavity archetype. For a noncommutative geometry-inspired wormhole with throat $|\mathbf r|\to\infty$7, the perturbation problem is posed on $|\mathbf r|\to\infty$8 with regularity at the throat and purely outgoing behavior at infinity. Frobenius analysis gives exponents

$|\mathbf r|\to\infty$9

leading to two admissible throat behaviors: a finite perturbation at the throat or a perturbation that vanishes there. The asymptotic behavior is outgoing,

$\tilde{\omega}_n=\omega_n-i\gamma_n,$0

so the geometry acts like a resonant cavity with a reflecting inner structure at the throat and radiation to infinity. In the parameter range examined, no overdamped modes were found; all detected QNMs are oscillatory, and in the large-$\tilde{\omega}_n=\omega_n-i\gamma_n,$1 limit they approach the Schwarzschild wormhole spectrum (Batic et al., 3 Apr 2025).

A third realization is the cavity effect in horizonless compact objects. In topological stars of the second kind, an inner stable photon sphere at $\tilde{\omega}_n=\omega_n-i\gamma_n,$2 and an outer unstable photon sphere at $\tilde{\omega}_n=\omega_n-i\gamma_n,$3 create a genuine trapping region. The first few QNMs are “microstructure modes” localized near the inner stable orbit and have extremely small damping, while another family of black-hole-like modes is localized near the outer unstable photon sphere. Their real parts remain close to the corresponding black-hole values, but their imaginary parts are smaller because the smooth interior reflects part of the wave back toward the outer barrier. In the WKB description, the damping correction is suppressed by the tunneling factor $\tilde{\omega}_n=\omega_n-i\gamma_n,$4, making the cavity physics explicit (Heidmann et al., 2023).

These examples establish a broad structural principle: cavity QNMs arise whenever the wave dynamics combines leakage with reflection, trapping, or regularity at an inner boundary. The “cavity” may be a mirror, a wormhole throat, a stable photon-sphere region, or the black-hole potential barrier itself.

4. Spectral, geometric, and numerical formulations

A major line of development reformulates QNMs as regular eigenvalue problems on compactified domains. In the hyperboloidal approach, the apparent divergence of black-hole oscillations near the bifurcation sphere and spatial infinity is attributed to the geometry of standard time slices. One instead introduces a hyperboloidal time function

$\tilde{\omega}_n=\omega_n-i\gamma_n,$5

together with a rescaled radial function

$\tilde{\omega}_n=\omega_n-i\gamma_n,$6

so that the mode is regular at the future event horizon and future null infinity. The exterior region is compactified to a finite interval, and regularity at the endpoints replaces the manual imposition of ingoing and outgoing conditions. In this framework, QNMs become regular eigenfunctions on a compact domain. At the same time, not every bounded hyperboloidal solution is a QNM; numerically, the QNM eigenfunctions are distinguished by faster decay of spectral coefficients and by Gevrey-2 regularity (Macedo et al., 2024).

The noncommutative wormhole calculation provides a concrete spectral implementation of this philosophy. After factoring out the known throat and infinity behavior, the authors expand the regular remainder as

$\tilde{\omega}_n=\omega_n-i\gamma_n,$7

with Chebyshev polynomials on $\tilde{\omega}_n=\omega_n-i\gamma_n,$8, impose the equation at Chebyshev collocation points, and obtain a quadratic matrix eigenvalue problem

$\tilde{\omega}_n=\omega_n-i\gamma_n,$9

Physically relevant QNMs are those stable under increasing truncation order $$Q_n=-\frac{1}{2}\frac{\Re(\tilde{\omega}_n)}{\Im(\tilde{\omega}_n)}.$$0 and convergent under root tracking (Batic et al., 3 Apr 2025).

Geometric-optics methods provide a complementary asymptotic description. For the charged regular Hayward black hole, the eikonal QNM frequency is written as

$$Q_n=-\frac{1}{2}\frac{\Re(\tilde{\omega}_n)}{\Im(\tilde{\omega}_n)}.$$1

where $$Q_n=-\frac{1}{2}\frac{\Re(\tilde{\omega}_n)}{\Im(\tilde{\omega}_n)}.$$2 is the angular velocity of the unstable circular null geodesic and $$Q_n=-\frac{1}{2}\frac{\Re(\tilde{\omega}_n)}{\Im(\tilde{\omega}_n)}.$$3 is the Lyapunov exponent of the photon orbit. Closely related light-ring formulas were derived for black holes with quadrupole moment in Hartle–Thorne spacetimes, where the eikonal QNMs interpolate continuously to the Kerr result when the quadrupole takes the Kerr value (Lopez et al., 2018, Allahyari et al., 2018). These constructions isolate the barrier-top physics that controls black-hole-like cavity spectra in the high-angular-momentum regime.

Machine-learning methods have also been adapted to the boundary-value structure of QNMs. A feedforward neural network method writes a trial function as a boundary factor times a one-hidden-layer network, minimizes a collocation-based residual for

$$Q_n=-\frac{1}{2}\frac{\Re(\tilde{\omega}_n)}{\Im(\tilde{\omega}_n)}.$$4

and updates the eigenvalue through a Rayleigh-quotient-like expression. For 4D pure de Sitter and 5D Schwarzschild AdS, the resulting spectra were reported to be in good agreement with analytical or established numerical results, showing that nontraditional solvers can handle QNM boundary data when the ansatz is engineered to satisfy the required asymptotics (Övgün et al., 2019).

5. Spatial structure, coupled cavities, and quantized QNMs

QNM theory is not purely temporal. In Kerr ringdown, each linear gravitational QNM has a specific angular profile given by the spin-weighted spheroidal harmonic, and “black-hole cartography” reconstructs that shape directly from numerical relativity by fitting a common QNM time dependence across many spherical-harmonic components. For a selected mode $$Q_n=-\frac{1}{2}\frac{\Re(\tilde{\omega}_n)}{\Im(\tilde{\omega}_n)}.$$5, the reconstructed map is

$$Q_n=-\frac{1}{2}\frac{\Re(\tilde{\omega}_n)}{\Im(\tilde{\omega}_n)}.$$6

and its agreement with the perturbative prediction is quantified by a spatial mismatch. The same procedure extends to quadratic QNMs with frequencies $$Q_n=-\frac{1}{2}\frac{\Re(\tilde{\omega}_n)}{\Im(\tilde{\omega}_n)}.$$7, where the angular basis is not uniquely fixed by second-order perturbation theory. For the dominant quadratic mode sourced by $$Q_n=-\frac{1}{2}\frac{\Re(\tilde{\omega}_n)}{\Im(\tilde{\omega}_n)}.$$8, the reconstructed amplitude is not maximal on the spin axis but near a finite inclination, with $$Q_n=-\frac{1}{2}\frac{\Re(\tilde{\omega}_n)}{\Im(\tilde{\omega}_n)}.$$9 for remnant spins near $\frac{d^2R}{dr_*^2}+\big(\omega^2-V(r)\big)R=0,$0, and more generally $\frac{d^2R}{dr_*^2}+\big(\omega^2-V(r)\big)R=0,$1 (Dyer et al., 2024).

In quantum optics, open-cavity QNMs can be quantized, but their non-orthogonality survives in operator form. For hybrid metal–dielectric systems with Fano-like resonances, the field is expanded in QNM operators whose commutators are governed by an overlap matrix $\frac{d^2R}{dr_*^2}+\big(\omega^2-V(r)\big)R=0,$2, rather than by $\frac{d^2R}{dr_*^2}+\big(\omega^2-V(r)\big)R=0,$3. A Löwdin-type symmetrization,

$\frac{d^2R}{dr_*^2}+\big(\omega^2-V(r)\big)R=0,$4

restores bosonic commutators at the price of introducing off-diagonal coherent and dissipative couplings in the master equation. In this setting, dissipation-induced mode coupling is not a perturbative nuisance but an essential mechanism behind Fano interference, modified Fock-state populations, and output-field correlation functions that differ qualitatively from phenomenological dissipative Jaynes–Cummings models (Franke et al., 2020, Ren et al., 2021).

For spatially separated cavities, retardation becomes part of the modal structure. A multi-cavity QNM quantization scheme decomposes the field into separate cavity subsystems and a shared bath, and introduces a cavity separation parameter

$\frac{d^2R}{dr_*^2}+\big(\omega^2-V(r)\big)R=0,$5

with inter-cavity overlap scaling as $\frac{d^2R}{dr_*^2}+\big(\omega^2-V(r)\big)R=0,$6. In the examples studied, $\frac{d^2R}{dr_*^2}+\big(\omega^2-V(r)\big)R=0,$7 corresponds to overlap below about $\frac{d^2R}{dr_*^2}+\big(\omega^2-V(r)\big)R=0,$8, while $\frac{d^2R}{dr_*^2}+\big(\omega^2-V(r)\big)R=0,$9 corresponds to overlap below about $R\sim e^{\pm i\omega r_*}\qquad (r_*\to\pm\infty).$0. A subsequent time-dependent theory makes the separation between localized QNM excitations and an orthogonal bath of traveling photons explicit, so that emission from one cavity appears as the input field for another after the propagation delay $R\sim e^{\pm i\omega r_*}\qquad (r_*\to\pm\infty).$1 (Fuchs et al., 2024, Fuchs et al., 14 Apr 2026). This establishes a causal, finite-delay version of cavity-QNM dynamics appropriate for photonic networks and distributed cavity QED.

6. Conceptual issues, rigorous definitions, and broader significance

Several recurrent misconceptions have shaped the subject. One is that spatial divergence invalidates QNM physics outside the cavity. The electromagnetic literature rejects this: the divergence is the spatial counterpart of temporal damping, and its physical interpretation is controlled by causality rather than by square integrability on a constant-time slice (Wu et al., 2024). Another is that QNMs are exhausted by an asymptotic boundary-condition statement. In black-hole theory, the hyperboloidal and resolvent-based literature emphasizes that outgoing conditions alone do not define a resonance; the operative objects are poles of the Green’s function, zeros of the Wronskian, or eigenfunctions of a suitably defined generator on an appropriate Hilbert space (Macedo et al., 2024, Gajic et al., 2019).

The role of adjoint or dual objects is equally important. On de Sitter space, the existence of scalar QNMs has been proved for all spacetime dimensions $R\sim e^{\pm i\omega r_*}\qquad (r_*\to\pm\infty).$2 and for all scalar masses, with frequencies

$R\sim e^{\pm i\omega r_*}\qquad (r_*\to\pm\infty).$3

The amplitudes with which these modes contribute to field expansions are determined by dual resonant states—distributional solutions of the adjoint equation satisfying a generalized incoming condition at the horizon. Some QNMs appear only when the initial data do not vanish near the cosmological horizon, which clarifies earlier claims of “missing” modes (Hintz et al., 2021).

A related functional-analytic development appears for extremal Reissner–Nordström spacetimes. There, QNMs are interpreted as honest eigenfunctions of the generator of time translations acting on a Hilbert space of initial data that is Gevrey regular at infinity and at the event horizon. In that framework, the regularity QNF set is discrete in the sector $R\sim e^{\pm i\omega r_*}\qquad (r_*\to\pm\infty).$4, with no point spectrum in $R\sim e^{\pm i\omega r_*}\qquad (r_*\to\pm\infty).$5, and the construction is tied directly to a meromorphically continued resolvent (Gajic et al., 2019). This strengthens the view that cavity QNMs are best understood as resonances of an open evolution problem, not as ordinary normal modes with slightly modified losses.

Across photonics and gravitation, cavity QNMs now serve several distinct but connected purposes: they organize ringdown and spectroscopy; they provide reduced-order models for scattering, LDOS, and sensing; they expose spatial mode structure rather than only temporal decay; and, when quantized, they connect rigorous Maxwell solvers to open-system Hamiltonians with dissipation, retardation, and mode non-orthogonality built in (Dyer et al., 2024, Wu et al., 2024, Fuchs et al., 2024, Fuchs et al., 14 Apr 2026). The unifying theme is that a cavity QNM is not a bound mode of a closed domain, but a resonance of a leak-and-trap geometry whose complex spectrum encodes both storage and escape.