Virtual Absorption Modes in Resonant Systems

• Virtual absorption modes are defined by wave excitations at complex frequencies that temporarily trap energy in a structure.
• They bridge wave physics, photonics, plasma, and gravitational scattering, enabling coherent control and selective energy management.
• Precise temporal and spatial engineering is essential to achieve transient energy storage, spectral selectivity, and controlled release.

Virtual absorption modes describe the excitation and response of resonant systems—ranging from magnetohydrodynamic waves in plasmas to photonic cavities, metamaterials, and even black hole scattering—at complex frequencies that do not correspond to traditional propagating or bound eigenstates. These modes govern phenomena in which energy is not permanently absorbed by material loss, but is temporarily stored, redistributed, or converted within a structure through precisely engineered temporal, spatial, or parameteric conditions. The virtual absorption paradigm connects wave physics, non-Hermitian operator theory, and advanced scattering control, with significant implications for energy manipulation, spectral selectivity, and the emulation of perfect absorption in lossless or low-loss media.

1. Mathematical and Physical Foundations

The core mathematical underpinning of virtual absorption modes is the concept of excitation at a complex frequency—specifically, the zeros of the scattering matrix (S-matrix) located off the real axis. For a wave equation or a photonic/electromagnetic structure,

$(A - zI) \psi = 0,$

a virtual absorption mode occurs when $z$ (or equivalently the frequency $\omega$) is complex, such that the system supports neither a true bound state ($L^2$ eigenfunction) nor a regular propagating mode. In scattering frameworks, this often manifests through singular or nearly singular features in the resolvent $(A - zI)^{-1}$ as $z \to z_0$ at or near the spectral threshold, as detailed in the limiting absorption principle (Boussaid et al., 2021, Comech et al., 30 Dec 2024).

Physically, the principle is that a specific form of wave excitation—most notably, an incident signal with an exponentially growing (or decaying) temporal envelope matching the imaginary part of a complex resonance—can force all incident energy to be stored or transiently trapped in the system, with no outgoing (reflected or transmitted) component during excitation (Baranov et al., 2017, Delage et al., 2023, Tuncer et al., 23 Sep 2025). This process is entirely determined by destructive interference and the precise matching to the system’s non-Hermitian spectral structure.

2. Mechanisms and Exemplars in Diverse Systems

Solar and Plasma MHD

Resonant absorption and mode conversion in magnetohydrodynamics (MHD) exemplify virtual absorption modes in plasmas. In the classic scenario of a cold, inhomogeneous plasma threaded by a uniform magnetic field, a fast magnetoacoustic wave encountering a resonant Alfvén layer experiences energy transfer. If the geometry is such that the Alfvén wave cannot propagate energy away (due to confinement or boundary conditions), phase mixing pushes energy to ever smaller wavenumbers—an in situ process synonymous with resonant absorption. If energy can escape (i.e., the converted wave is not spatially trapped), the process appears as pure mode conversion (Cally et al., 2010, Hanson et al., 2010). The distinction is determined by whether energy is redistributed in real space (escape) or in k-space (phase mixing, virtual absorption).

Photonics, Resonators, and Metamaterials

Virtual absorption also arises in optical settings where perfect absorption can be dynamically emulated in lossless or low-loss systems. In dielectric slabs, microring resonators, or complex photonic structures, tailored incident waveforms—such as an input field of the form $E(t) \sim \exp(i\omega' t)\exp(\omega'' t)$ with $\omega'' > 0$, over a finite interval—engage a complex S-matrix zero and result in zero scattering during excitation. This is “coherent virtual absorption” (CVA): energy is stored in the resonator and later released on demand once the driving signal is terminated (Baranov et al., 2017, Zhong et al., 2020).

A parallel occurs in critical coupling to metamaterials, where the match between radiative and absorbptive decay rates (e.g., $\gamma_{\text{rad}} = \gamma_{\text{abs}}$) allows for near-unity absorption of, for instance, molecular vibrational resonances without overtly destroying the high quality (Q) factor of the mode. The resulting absorption lines are extremely narrow, indicating that the energy is not lost to broad material absorption, but is “stored” in a sharp resonance—the “virtual” mode of the molecular or material subsystem (Dayal et al., 13 Nov 2024).

Discrete and Non-Hermitian Lattices

In discrete systems such as coupled resonator optical waveguides (CROW) or waveguide arrays, virtual absorption can be accomplished by engineering not only the temporal but also the spatial envelope of the excitation. Both scattering matrix-based and time-reversal-based “self-focusing” wavepacket constructions demonstrate that reflection and transmission can be transiently cancelled, even when complex S-matrix zeros are absent, as in certain regimes of non-Hermitian or PT-symmetric lattices (Longhi, 2018, Huang et al., 2013, Novitsky et al., 2023).

Atomic and Molecular Systems

In atomic physics, virtual absorption modes manifest in the presence of strong-field dressing—intense driving fields that create virtual dressed states of the atom (e.g., in helium subjected to intense IR and XUV fields). These dressed states appear as additional, non-perturbative absorption features in the spectrum—so-called light-induced virtual absorption lines—which are present only for certain polarization and delay configurations, and vanish otherwise. They reflect coherent superpositions of dipole-allowed and normally forbidden transitions that are only “activated” under the virtual absorption scenario (Reduzzi et al., 2019).

Gravitational Scattering and Black Holes

In gravitational physics, virtual absorption modes correspond to complex-frequency resonances (total transmission/zero-reflection or “algebraically special” modes) in the effective potential of compact objects such as black holes or ultracompact stars. When the incoming wave is modulated to match the complex frequency of the virtual absorption mode—i.e., $$\,\psi(0, x) \propto \exp(\Omega_I \eta) \cos(\Omega_R \eta)$$ with $\Omega_R + i\Omega_I$ equal to the mode frequency—total absorption is realized during the excitation phase, with energy later re-emitted according to the system’s ringdown properties. In higher-dimensional spacetimes, a particularly rich structure of VA modes emerges, illustrating the universality of these phenomena (Tuncer et al., 23 Sep 2025).

3. Temporal and Spectral Control: Engineering Virtual Absorption

The excitation of a virtual absorption mode requires precise control of the incident signal's temporal profile, such that it matches the exponential growth dictated by the imaginary part of the mode's complex frequency. In practice, this typically takes the form: $$E_{\text{inc}}(t) = A \exp(-i\omega' t)\exp(\omega'' t)\theta(-t) + \ldots$$ where $\omega' + i\omega''$ is the relevant complex resonance, $A$ is the amplitude, and $\theta(t)$ is the Heaviside function enforcing temporal localization (Baranov et al., 2017, Delage et al., 2023, Tuncer et al., 23 Sep 2025).

For certain applications, counterpropagating excitation or spatially asymmetric inputs can be used (as in PT-symmetric structures or photonic devices) to localize the energy storage or direct the subsequent emission (e.g., quasilasing pulses) along chosen paths (Novitsky et al., 2023). In photonic structures, the modulation of array periodicity or cavity thickness allows tuning of the condition for critical coupling and thus the virtual absorption bandwidth and selectivity (Dayal et al., 13 Nov 2024).

4. Operator-Theoretic and Scattering Perspectives

The abstract mathematical structure relates to the limiting absorption principle (LAP) and the emergence of virtual levels or exceptional points in operator spectra (Boussaid et al., 2021, Comech et al., 30 Dec 2024). When the resolvent $(A - zI)^{-1}$ fails to remain bounded (e.g., as $z\to 0^+$ for low-energy thresholds), but a non-square-integrable solution to $(A-z_0)u = 0$ exists (the “virtual state”), physical scattering processes manifest anomalously strong absorption or reflection features at thresholds. This operator-theoretic view rigorously justifies the physical notion of “virtual absorption modes” as threshold resonances or quasi-bound states that, while not contributing true eigenvalues, dramatically affect spectral and dynamical properties in the vicinity of $z_0$.

Weighted spaces $L^2_s$ and mappings $E \to F$ (with $E, F \subset X$ Banach spaces) are instrumental in extracting uniform resolvent estimates and identifying the presence of virtual absorption modes or the necessity for finite-rank regularization.

5. Applications and Implications

Virtual absorption modes underpin or enable:

• Efficient transient energy storage and release: Used to trap light or other waves in lossless (or low-loss) structures and then controllably release them (CVA in photonics (Baranov et al., 2017, Zhong et al., 2020), VPA in plasma ignition (Delage et al., 2023), gravitational wave trapping (Tuncer et al., 23 Sep 2025)).
• Enhanced sensing and selectivity: Near-unity absorption of specific vibrational or dressed modes—useful for high-sensitivity molecular spectroscopy or quantum state preparation (Dayal et al., 13 Nov 2024, Reduzzi et al., 2019).
• Nontrivial control in non-Hermitian and PT-symmetric systems: Virtual absorption enables asymmetric absorption/amplification, band engineering, and pulse shaping in photonic devices (Huang et al., 2013, Novitsky et al., 2023).
• Wave manipulation at and near spectral thresholds: In quantum mechanics and waveguides, the presence or absence of virtual absorption modes (as virtual levels) determine the emergence of new bound states or anomalous scattering in response to perturbations (Boussaid et al., 2021, Comech et al., 30 Dec 2024).

A table summarizing key manifestations:

System Type	Virtual Absorption Mechanism	Role/Application
MHD/Plasma	Mode conversion at resonance	Solar coronal heating, wave damping
Photonic Resonators & Metasurfaces	Excitation at complex S-matrix zero, critical coupling	Light storage, perfect absorber, sensing
Discrete Lattices	Engineered wavepacket (Gamow-like or self-focusing)	Transient trapping, photonic circuit control
Atomic/Molecular	Field-dressed (Floquet) virtual states	Attosecond pump-probe, quantum state control
Black Holes/Gravitational	Temporal modulation matching complex VA mode frequency	Energy trapping, gravitational collapse
Operators/Scattering Theory	Virtual levels, failure of uniform resolvent limit	Threshold resonances, spectral stability

6. Limitations, Challenges, and Open Questions

The design and observation of virtual absorption modes face several challenges:

• Sensitivity to input waveform: Exact temporal and phase matching are often required—deviations (e.g., in growth rate) degrade the perfect storage/absorption condition (Zhong et al., 2020). Nonlinear effects (Kerr, thermal, gain depletion) may terminate virtual absorption if the resonance shifts (Zhong et al., 2020).
• Material and fabrication constraints: In practical devices, losses, gain saturation, and fabrication tolerances limit the ideality of the virtual absorption process (Delage et al., 2023, Dayal et al., 13 Nov 2024).
• Dimension and topology dependence: The richness of virtual absorption mode spectra depends on system geometry—e.g., higher dimensions in gravitational settings allow for more complex VA mode structure (Tuncer et al., 23 Sep 2025), while in certain discrete lattices the standard S-matrix formalism may be insufficient (Longhi, 2018).
• Dynamic or time-varying environments: In plasma or gravitational systems, the background parameters can change during excitation, potentially detuning the virtual absorption condition (Delage et al., 2023).

A plausible implication is that further development of adaptive or feedback-controlled approaches to maintain or track the complex mode condition during excitation could extend the robustness and applicability of virtual absorption in realistic systems.

7. Outlook and Connections

Virtual absorption modes represent a broad and unifying paradigm in wave physics for perfect or near-perfect transient energy coupling, relying on precision engineering of the system's spectral and temporal characteristics rather than material dissipation. These ideas bridge magnetohydrodynamics, optics, quantum mechanics, and general relativity—demonstrating a fundamental and transferable principle for lossless energy trapping, amplification, and release across scales and physical platforms. Future research will likely expand their applicability to active matter, non-Hermitian topological systems, and dynamic media with feedback, enhancing both theoretical understanding and technological capability.