Quadratic Quasinormal Mode Ratios

• Quadratic QNM ratios are defined as amplitude relationships quantifying the nonlinear contributions that couple linear modes in dissipative resonant systems.
• They emerge from second-order perturbative expansions and are governed by selection rules, including parity and angular momentum constraints.
• Advanced techniques such as spectral decompositions and hyperboloidal slicing enable the computation of these ratios, enhancing gravitational wave spectroscopy and resonance modeling.

Quadratic quasinormal mode ratios quantify the nonlinear contributions to the spectrum and waveform of quasinormal modes (QNMs) that arise in dissipative systems supporting resonances with complex frequencies under outgoing boundary conditions. In gravitational physics, these ratios provide a bridge between linear perturbation theory—yielding the familiar set of QNMs typically governing black hole ringdown—and the next-order, quadratic, nonlinear couplings that emerge in second-order perturbative expansions. Similar quadratic correlations and corrections appear in quantum, optical, and condensed matter resonances modeled with open boundary conditions, and are now increasingly central to ringdown modeling, black hole spectroscopy, Lorentzian scattering theory, and nanophotonics.

1. Fundamental Principles and Definitions

Quadratic QNM ratios generally refer to the amplitude—and sometimes spectral—relationships between second-order (quadratic) QNMs and their linear parents. In gravitational systems, the second-order perturbative expansion of Einstein's equations on a black hole background yields source terms quadratic in the linear fields (e.g., in the Weyl scalars, metric master variables, or the Hertz potential). The resulting quadratic QNMs can be characterized both by their unique complex frequencies—typically sums and/or differences of parent QNM frequencies—and by amplitude ratios defined as

$$\mathcal{R} = \frac{\mathcal{A}^{(2)}}{\mathcal{A}^{(1)}_1 \mathcal{A}^{(1)}_2}$$

where $\mathcal{A}^{(2)}$ denotes the second-order (quadratic) mode amplitude at asymptotic infinity and $\mathcal{A}^{(1)}_{1,2}$ the amplitudes of the parent linear QNMs (Bucciotti et al., 9 May 2024).

These ratios encode not just the efficiency of nonlinear mode generation (quadratic excitability), but also the phase correlation, parity dependence, and angular selection properties of the system. In open optical systems, response and dissipation can be similarly decomposed into modal expansions where quadratic QNM ratios control the contributions to observables quadratic in the fields, such as radiated power and absorption (Betz et al., 2022).

2. Analytic Structures and Appearance of Quadratic Relationships

Several classes of model potentials and physically relevant systems are used to derive quadratic relationships and ratios analytically:

• Exactly Solvable Potentials: For 1D potentials such as the symmetric/asymmetric double delta, rectangular barrier, and especially the Eckart (Rosen–Morse/Morse–Feshbach) family, analytic expressions for QNFs display not only the familiar "offset + in (gap)" structure, but (for the Eckart case) the quadratic structure in the mode index $n$,

$$k_{QNF,+}(n) = i \frac{(2n+1) \pm \sqrt{1 - \frac{8mV_0 a^2}{\hbar^2}}}{2a} + \ldots$$

The QNFs appear as roots of quadratic equations in $k$ or $\omega$, and their spacing and corrections can be non-algebraic (logarithmic or higher-order) depending on the branch of the analytic solution (e.g., via the Comtet expansion for the Lambert $W$ function) (Boonserm et al., 2010).

• Highly Damped Modes and Quantization Conditions: In spacetimes such as Schwarzschild–de Sitter, highly damped QNFs often obey polynomial quantization conditions of degree $2(p_+ + p_-)$, with powers determined by the rationality of the ratio of the surface gravities at the horizons,

$$z^{2(p_+ + p_-)} - z^{2p_+} - z^{2p_-} - 4 \cos(\pi \alpha_+) \cos(\pi \alpha_-) z^{p_+ + p_-} + 1 = 0$$

The polynomial's degree and structure dictate the multiplicity, spacing, and periodic organization of the QNFs (Skakala et al., 2010).

• Coupled and Perturbatively Corrected Potentials: When corrections to GR potentials or inclusion of off-diagonal couplings (e.g., in effective field theory or modified gravity) are treated perturbatively, quadratic corrections in the form of second-order frequency shifts or mixing terms appear systematically in the QNM frequencies:

$$\omega \approx \omega_0 + \alpha^{(k)} d_{(k)} + \alpha^{(k)} \alpha^{(s)} e_{(ks)} + \ldots$$

where the $e_{(ks)}$ coefficients capture quadratic responses to the perturbation (McManus et al., 2019).

3. Selection Rules and Parity Dependence

Quadratic QNM ratios are not unconstrained; their excitation is subject to angular momentum and parity selection rules:

• Angular Wigner 3j Symbol: The quadratic source involves angular products of spherical (or spheroidal) harmonics; thus, the associated 3j symbol vanishes if, for example, $\ell_1 = \ell_2$, $m_1 = m_2$ and $\ell+\ell_1+\ell_2$ is odd, forbidding those quadratic transitions (Bucciotti et al., 20 Jun 2024).
• Parity/Exchange Antisymmetry: The normalized quadratic amplitude ratio is antisymmetric under exchange of the two linear modes if $\ell+\ell_1+\ell_2$ is odd. For identical linear modes (including parity/polarization label $\kappa_1 = \kappa_2$), this antisymmetry enforces exact cancellation and thus vanishing of the quadratic amplitude for that coupling (Bucciotti et al., 20 Jun 2024).
• Parity-Controlled Dependence: The quadratic amplitude ratio is not a universal number but depends on the relative amplitude of the even and odd parity content (or, equivalently, on the mirror-to-regular ratio of linear parent polarization). This dependence captures a memory of the physical progenitor (symmetry of the merger or environmental asymmetry) (Bourg et al., 16 May 2024, Bourg et al., 10 Mar 2025).

4. Methodological Advances and Computability

The extraction and computation of quadratic QNM ratios require careful handling of boundary conditions, regularization, and numerical decomposition:

• Regularization: For both open optical resonators and gravitational ringdowns, naive use of bare QNMs can yield divergent or nonphysical results outside the region of interest. Regularized QNMs—constructed via Dyson/residue representations or complex contour integrations—restore physical interpretation by properly accounting for the radiative and continuum spectrum and removing diverging spatial behavior (Franke et al., 2022, Betz et al., 2022, Bucciotti et al., 9 May 2024).
• Laplace Transform and Hyperboloidal Framework: Advanced techniques combine Laplace transforms along hyperboloidal (null or compactified) slices in the spacetime, rendering the boundary-value problem well-posed on the entire domain from horizon to infinity and allowing clean extraction of the residues (amplitudes) at QNM and QQNM poles (Bourg et al., 10 Mar 2025, Bourg et al., 16 May 2024).
• Spectral and Leaver Methods: Many results rest on frequency-domain spectral decompositions (including Leaver's continued fraction algorithm), with master equations constructed for both the linear and quadratic sectors, and source terms extended to products of linear eigenfunctions and their derivatives (Ma et al., 27 Jan 2024, Bucciotti et al., 20 Jun 2024, Khera et al., 18 Oct 2024).

5. Scaling, Universality, and Eikonal Limit

The amplitude ratios and the scaling behavior of quadratic QNMs in the large-$\ell$ (eikonal) regime test the robustness of both perturbative theory and universality conjectures:

• In the "eikonal" limit (large $\ell$), it was previously conjectured that the quadratic ratio for the $\ell \times \ell \rightarrow 2\ell$ channel would show quadratic growth (e.g., as $\ell^2/27$ (Bucciotti et al., 29 Jan 2025)). However, robust numerical investigations using both time- and frequency-domain methods have shown this ratio saturates, i.e., $\mathcal{R}^{2\ell}_{\ell \ell \times \ell \ell}$ approaches a finite limit, in contrast to Penrose–limit–based analytic expectations.
• For mixed channels (e.g., $2 \times \ell \rightarrow \ell+2$), linear growth with $\ell$ is observed:

$$\mathcal{R}^{\ell+2}_{2 2 \times \ell \ell} \sim 0.11 \ell$$

This growth does not invalidate perturbation theory because the high-$\ell$ modes are not strongly excited in physically realistic initial data (Bucciotti et al., 29 Jan 2025).

6. Implications for Black Hole Spectroscopy and Observational GW Astronomy

The detection and interpretation of quadratic QNM ratios are pivotal for several reasons:

• They provide an explicit observable encoding both the intrinsic black-hole parameters and the symmetry properties of the merger process, hence serving as a nonlinear "fingerprint" for black hole formation (Bourg et al., 16 May 2024, Bourg et al., 10 Mar 2025).
• In gravitational wave data analysis, inclusion of quadratic QNMs in ringdown templates can reduce systematic parameter bias, enable nonlinear tests of general relativity, and probe the presence of up/down symmetry breaking or environmental couplings (Yi et al., 14 Mar 2024, Khera et al., 18 Oct 2024).
• The amplitude ratios for dominant couplings (e.g., the $\ell=2\times 2 \rightarrow 4$ channel with $|\mathcal{R}|\approx 0.154$ and phase shift $e^{-0.068i}$) have been found to agree with ratios extracted from numerical relativity, strengthening the case for their physical measurability (Bucciotti et al., 9 May 2024, Bucciotti et al., 20 Jun 2024).

7. Connection to Broader Resonance Physics and Outlook

While the primary focus is gravitational ringdown, similar quadratic QNM ratios appear wherever open systems are modeled by dissipative boundary value problems: in nanophotonic cavities, parametric amplifiers, or quantum scattering with nontrivial topologies. The methodology for extracting and regularizing quadratic QNM ratios—especially via residue calculus and spectral decomposition—is mathematically universal, as are selection rules emerging from underlying symmetry groups (e.g., parity, angular momentum). Ongoing advances aim to generalize these results to rotating/Kerr backgrounds, multimode GW detectors, and high-precision nanophotonics, with continued emphasis on the interplay between exact analytic results and robust numerical computation.

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Aspect	Gravitational Case	Optical/Resonator Case
Frequency structure	$\omega^{(2)} = \omega^{(1)}_1 + \omega^{(1)}_2$	Quadratic mixing of complex eigenfrequencies
Amplitude ratio	$$\mathcal{R} = \mathcal{A}^{(2)} / (\mathcal{A}^{(1)}_1 \mathcal{A}^{(1)}_2)$$	Modal quadratic expansions for power, absorption
Regularization	Contour integration/hyperboloidal slicing	Volume/surface integral regularized QNMs
Selection rules	Wigner 3j, parity, symmetry	Mode matching, spatial overlap constraints

Quadratic QNM ratios, therefore, provide critical quantitative links between nonlinear theory and physically measurable resonance features in both gravitational and non-gravitational dissipative systems. Their computation, constraints, and numerical universality are central to contemporary and future precision resonance spectroscopy.