Quasi-Bound States in the Continuum (QBIC) Modes

• QBIC modes are resonant states with eigenenergies inside the continuum that achieve high quality factors through suppressed radiative coupling.
• They are realized by mechanisms such as symmetry protection, destructive interference, and parameter tuning across quantum, photonic, acoustic, and many-body platforms.
• A unified operator framework and rigorous spectral selection criteria enable precise engineering of QBIC modes, guiding applications like ultra-high-Q lasers and sensitive modulators.

Quasi-bound states in the continuum (QBIC) modes are a broad and rigorously characterized class of spatially localized or near-localized resonant states whose eigenenergies are embedded within the continuum spectrum of an open system, yet which exhibit extraordinarily suppressed coupling to the continuum. While true bound states in the continuum (BICs) possess infinite radiative lifetimes and are perfectly decoupled from the embedding continuum, QBICs represent analogous states with finite but typically very high quality factors (Q)—often arising from symmetry-breaking, interference, or parameter detuning—which leads to partial leakage and observable resonant features. Research on QBICs spans quantum, photonic, acoustic, metamaterial, and many-body platforms, providing both a unifying theoretical framework and a set of concrete design principles for realizing ultra-high Q resonances, tunable light–matter and wave–matter interactions, and nontrivial topological and correlated effects.

1. Unified Theoretical Framework for Quasibound States

QBICs are formally captured within a unified operator framework that generalizes the standard stationary eigenvalue problem for open quantum and wave systems. By constructing a non-orthogonal but complete quantum history (timeline) basis $\{|\tau\rangle\}$ satisfying a translation property $$|\tau+\tau'\rangle = \exp(-i\hat{H}_0\tau)|\tau'\rangle$$, the general solution to $(\hat{H}_0 + V)|\psi_E\rangle = E|\psi_E\rangle$ can be recast as

$$|\psi_E\rangle = \frac{\langle \tau|\psi_E\rangle}{\langle \tau|E\rangle}|E\rangle + \hat{G}_E(\tau)V|\psi_E\rangle,$$

where $\hat{G}_E(\tau)$ is a Green operator defined to ensure the independence from continuum background states. Quasibound states are precisely those for which the "free" continuum part vanishes, that is, they are sourced by $V$ and must satisfy the homogeneous criterion $\langle \tau|\psi_E\rangle = 0$, yielding

$|\psi_E\rangle = \hat{G}_E(\tau)V|\psi_E\rangle.$

This structure separates QBICs from true continuum eigenstates and governs spectral selection rules: in the coordinate representation, stationary QBICs with real energies and everywhere bounded wavefunctions satisfy

$\int_{-\infty}^{\infty} \psi_E(x)\, dx = 0,$

or analogous projections for degenerate continua. This framework encompasses both stationary QBICs (real energy, physically acceptable wavefunctions) and complex-energy resonances (resonant poles of the analytic continuation of the resolvent) (Moyer, 2013).

2. Physical Mechanisms: Symmetry Protection, Interference, and Parameter Tuning

Most QBICs originate from one of several universal mechanisms suppressing continuum leakage:

• Symmetry-protected BICs/QBICs: When a system's symmetry forbids coupling of a mode to the continuum (e.g., inversion, mirror, or polarization selection rules), a true BIC exists. Breaking symmetry (via structural perturbations, geometric asymmetry, chiral deformations) opens a finite leakage channel, converting the BIC to a QBIC with $Q \propto \alpha^{-2}$ or $Q \propto \sin^{-2}(\theta)$ scaling, where $\alpha$ is an asymmetry parameter and $\theta$ an angular parameter governing radiative mixing (Mangach et al., 17 Apr 2024, Watanabe et al., 2023, Nan et al., 27 Mar 2024).
• Friedrich–Wintgen (accidental) mechanism: Destructive interference between two or more radiative decay channels leads to a parameter-tuned dark state. Adjustment of geometry, frequency, or coupling strength yields destructive cancellation, producing a BIC at critical points; detuning turns such a mode into a QBIC with finite but large $Q$ (Liu et al., 30 Aug 2024, Bulgakov et al., 2017, Zhang et al., 2022).
• Multipolar interference: In high-index dielectric metasurfaces and photonic crystals, out-of-plane dipoles and in-plane quadrupoles can constructively or destructively interfere in their radiation, so careful phase engineering of multipolar contributions (as determined by the system's irreducible representations) produces high-Q QBIC modes (Mangach et al., 17 Apr 2024, Ghahremani et al., 18 Apr 2024).
• Polarization-mismatch and material contrast: For acoustics, a QBIC can be realized when the mode profile (e.g., torsional/shear) lacks a component compatible with leakage into the surrounding continuum (e.g., pressure-only fluid), and symmetry reduction introduces a tunable radiative leakage (Deriy et al., 2021, Farhat et al., 2023).
• Many-body hybridization: In modulated Bose–Hubbard models, interactions localize multiparticle wavefunctions even as the total energy remains inside a band of extended states. Under periodic boundary conditions, the resulting hybrid state is a genuine quasi-BIC whose decay is slow and governed by the flatness of the multiparticle band (Huang et al., 2023).

3. Classification and Scaling Laws

The QBICs embodied in diverse systems can be classified by their dependence on symmetry, parameter tuning, and multipolar content, each resulting in distinct Q-factor scaling:

Type	Protection Mechanism	$Q$-factor Scaling
Symmetry-protected	Exact symmetry (broken weakly)	$Q \propto \alpha^{-2},\ \sin^{-2}(\theta)$
Accidental (FW)	Destructive interference of decay channels	$Q \propto \delta^{-2}$ (perturbation)
Super QBIC	Avoided-crossing + momentum merging	$Q \propto N^3$ (number of elements)
Fundamental BIC	Full symmetry, no leakage	$Q \to \infty$

Here, $\alpha$ and $\delta$ denote small symmetry-breaking parameters or perturbations, $\theta$ an angular deviation, and $N$ the number of unit cells/resonators (Zhang et al., 2022, Mangach et al., 17 Apr 2024, Bulgakov et al., 2017, Liang et al., 19 Nov 2024).

4. Experimental Realizations and Material Platforms

QBIC modes have been realized in a wide range of settings:

• Photonic Structures: Dielectric metasurfaces, photonic crystal slabs, and periodic arrays of rods/spheres routinely employ symmetry-protected and FW QBICs to achieve narrow Fano resonances for high-Q lasing or nonlinear optics (Taghizadeh et al., 2017, Bulgakov et al., 2017, Mangach et al., 17 Apr 2024, Nan et al., 27 Mar 2024, Sun et al., 2022). In the mid-infrared, surface phonon polariton (SPhP) metasurfaces based on SiC membranes exhibit deeply subwavelength QBIC modes robust to incident angle, enabling mode volumes up to $10^4$ times smaller than the diffraction limit (Nan et al., 27 Mar 2024).
• Metamaterial Devices: Hybrid metal–dielectric metasurfaces leveraging the electro-optic Pockels effect in lithium niobate exploit the extreme sensitivity of qBIC resonance wavelength and amplitude to refractive index perturbations under bias voltage, yielding ultrafast, high-contrast, and angle-tunable optical modulation at telecom wavelengths (Damgaard-Carstensen et al., 2 Dec 2024).
• Acoustic Systems: Elastic metasurfaces in underwater ultrasound achieve Q-factors of several hundred by tuning slit geometries and gap sizes such that radiative channels destructively interfere, as predicted by a TCMT formalism. This represents at least an order-of-magnitude enhancement over traditional acoustic resonators (Farhat et al., 2023).
• Many-body Quantum Lattices: Interaction-modulated Bose–Hubbard chains support multiparticle BICs/quasi-BICs under open and periodic boundary conditions. Quasi-BICs constructed via maximally localized Wannier states display robust topological pumping and correlated internal dynamics (Huang et al., 2023).
• Finite and Infinite Arrays: The formation and scaling of QBICs are similar in both finite and infinite periodic gratings, provided strong Bragg reflection and appropriate boundary conditions are present. When perfect reflection is realized, dark BICs with infinite Q-factor occur; geometrical or phase perturbations reduce $Q$ and render the states bright (quasi-BICs) (Liang et al., 19 Nov 2024, Yezekyan et al., 10 Feb 2025). Stringent scaling, $Q \propto N^2$ or $Q \propto N^3$, is achieved in finite arrays subject to conditions for super QBICs (Zhang et al., 2022, Bulgakov et al., 2017).

5. Mathematical and Topological Structure

The analysis of QBICs relies on the interplay of spectral properties of open systems, pole structure in the complex frequency/momentum domain, and interference criteria:

• Selection Rules: Stationary QBICs are selected by vanishing of certain Hilbert-space projections, e.g., $\int \psi_E(x)dx = 0$ or mode-specific overlap integrals, ensuring the state is "sourced" only by the interaction $V$ and not by background continuum (Moyer, 2013).
• Pole Structure: In multi-channel, multi-band systems, QBICs correspond to real or near-real spectral poles of the resolvent or scattering matrix. The transition to a true BIC requires eliminating the contribution from real (oscillatory) momentum poles in the Fourier domain by enforcing cancellation of associated residues. This can be achieved without any symmetry, via parametric tuning such that Fourier transforms of the potential–state product vanish at continuum momenta (Rao et al., 24 Jun 2025).
• Topological Conservation: In periodic arrays supporting multiple BICs (e.g., degenerate modes in high-symmetry lattices), total vorticity associated with the winding of field vectors in momentum space is conserved when parameters are tuned through BIC–quasi-BIC–BIC transitions. For example, moving sphere radius through the accidental BIC point in $C_{6v}$ symmetry lattices results in preservation of topological charge (vorticity) among bands, manifest in the pinning of high-Q channels (Ochiai, 25 Apr 2024).

6. Applications and Functional Implications

The engineering and utilization of QBICs enable a diverse set of advanced functionalities:

• Light and Field Enhancement: Near-field amplitude enhancements by factors of 10–25 have been reported in lossy Si arrays at red-to-near-IR wavelengths (Bulgakov et al., 2017, Taghizadeh et al., 2017). Deeply subwavelength confinement in SPhP metasurfaces provides high local field intensities for vibrational strong coupling and ultrasensitive molecular detection (Nan et al., 27 Mar 2024).
• Switching and Modulation: Ultrafast, high-contrast optical modulation in electro-optic metasurfaces is achieved by exploiting the high susceptibility of qBIC modes to refractive index changes (Damgaard-Carstensen et al., 2 Dec 2024).
• High-Q Lasers and Filters: QBICs serve as building blocks for low-threshold lasers, mirrorless microcavities, and high-Q filtering elements (Zhang et al., 2022, Taghizadeh et al., 2017, Bulgakov et al., 2017).
• Sensing and SEIRA: Surface-enhanced infrared absorption (SEIRA) platforms optimized via tailored radiative loss engineering push the sensitivity of biosensors and refractive index sensors (Watanabe et al., 2023, Mangach et al., 17 Apr 2024).
• Topological Control: In many-body platforms, QBICs allow quantized Thouless pumping with internal correlated motion, providing a resource for robust state transfer and state engineering (Huang et al., 2023).
• Nonlinear and Quantum Optics: High-Q QBICs enhance frequency conversion, strong coupling, and nonclassical photon generation in microcavities and integrated circuits (Liu et al., 30 Aug 2024).

7. Frontiers, Generalizations, and Theoretical Advances

Recent research generalizes the QBIC concept beyond traditional symmetry-protected/topologically-protected paradigms:

• General conversion criteria: A broad two-band model demonstrates that by careful tuning of interaction/coupling matrices and potentials, one can convert a quasi-BIC mode (with an oscillatory radiative tail) into a true BIC even in the absence of any symmetry or topology, provided the Fourier components of the coupling at continuum momenta vanish (Rao et al., 24 Jun 2025). This provides a comprehensive criterion for exact BIC formation in arbitrary discrete–continuum coupled systems, accessible to both linear and nonlinear, single-particle and many-body regimes.
• Multi-band and Many-body Extensions: The formalism is systematically extensible to multiband and many-body contexts, where interference and hybridization of multiple channels or correlated states allow the design of embedded localization with strong interaction or in topologically nontrivial bands (Huang et al., 2023, Rao et al., 24 Jun 2025).
• Practical Engineering: Super-QBICs leveraging hybrid parametric-momentum space engineering enable cubic scaling of $Q$ with the number of resonators even within compact and lossy photonic crystals, establishing a path to miniaturized, efficient photonic devices (Zhang et al., 2022).

---

In summary, quasi-bound states in the continuum represent a unifying physical and mathematical framework encompassing a wide variety of high-Q, localized (or nearly localized) resonances embedded in the radiation continuum. They are governed by universal interference, symmetry, and coupling principles, are accessible to rigorous spectral selection criteria, and serve as an enabling engine for next-generation photonic, acoustic, and quantum devices in both fundamental and application-driven research.