Time-Domain Mode-Sum Method

• Time-Domain Mode-Sum Method is defined by expressing complex wave phenomena as time-dependent expansions in complete orthogonal modes tailored to specific system symmetries.
• It enables precise analysis of scattering, transient dynamics, and parameter estimation in quantum, elastic, and electromagnetic systems through systematic modal decompositions and pole expansions.
• Practical implementations rely on controlled truncation, quantitative error estimates, and experimental techniques such as mode-selective sum-frequency generation for quantum-limited measurements.

The time-domain mode-sum method encompasses a class of analytical and computational frameworks for representing and solving wave phenomena—acoustic, elastic, electromagnetic, or quantum—in the time domain by expansion in suitable sets of orthogonal modes. By leveraging modal decompositions, such methods enable rigorous treatment of scattering, transient dynamics, and parameter estimation, with particular relevance in systems where dissipative, dispersive, and coupling effects are prominent. Major applications include quantum-limited time-frequency estimation, wide-band elastic scattering from metamaterial structures, and electromagnetic transients in dispersive nano-objects.

1. Foundational Principles of Time-Domain Mode-Sum Expansion

The mode-sum methodology hinges on expressing the physical fields (e.g., electric, displacement, or polarization fields) as expansions in complete orthogonal mode sets. These modes are tailored to the system’s geometry or symmetries—such as Hermite–Gaussian functions, vector spherical harmonics, or static eigenmodes—and may be defined in temporal, spectral, or spatial domains.

A general expansion takes the form: $$F(\mathbf{r},t) = \sum_{n} a_n(t) \, u_n(\mathbf{r})$$ where $u_n$ are the orthogonal modes and $a_n(t)$ are time-dependent amplitudes. Such decomposition is central to diverse fields:

• Quantum optics: Single-photon wave packets expanded in temporal or spectral orthonormal modes $u_n(t), \tilde{u}_n(\nu)$, with photon annihilation operators $\hat{a}_n$, enabling projection-based measurement and quantum estimation (Donohue et al., 2018).
• Elastodynamics: Boundary fields expanded in eigenfunctions $\phi_{i,n,m}$ of the Neumann–Poincaré (NP) operator, yielding explicit modal expansions for scattered fields (Chen et al., 2022).
• Electromagnetism: Expansion of polarization induced fields and radiation fields in static (longitudinal/transverse) eigenmodes and free-space propagation modes, with time-dependent amplitudes governed by coupled ODEs/IDEs (Forestiere et al., 2021).

The completeness and orthogonality properties enable systematic truncation for numerical efficiency while maintaining controlled approximation errors.

2. Modal Analysis and Model Formulation in Specific Physical Systems

2.1 Quantum-Limited Time-Frequency Estimation

The time-domain mode-sum estimation technique in (Donohue et al., 2018) projects single-photon signals onto Hermite–Gaussian temporal or spectral modes: $$\hat{E}^{(+)}(t) = \sum_n u_n(t) \hat{a}_n;\quad \hat{a}_n = \int dt\,u_n^*(t)\hat{E}^{(+)}(t)$$ Projective measurements $\hat{P}_n = \hat{a}_n^\dagger \hat{a}_n$ are realized experimentally by mode-selective sum-frequency generation (quantum pulse gating).

2.2 Elastic Scattering from Metamaterial Quasiparticles

In (Chen et al., 2022), elastic wave scattering from small inclusions is analyzed via expansion of the boundary data in the NP operator’s vector spherical harmonic eigenfunctions. The modal coefficients $a_{i,n,m}(\omega)$ encapsulate inclusion and host properties, with the field outside represented as: $$u_{\rm sc}^\omega(x) \approx \sum_{i,n,m} a_{i,n,m}(\omega)\,\Phi_{i,n,m}(x)$$ and the time-domain response $\hat{u}_{\rm sc}(x,t)$ recovered by inverse Fourier transform.

2.3 Electromagnetic Transients in Dispersive Particles

The formalism of (Forestiere et al., 2021) employs a Hopfield model for material polarization, expanding in static longitudinal $\{U_p^\parallel\}$ and transverse $\{U_p^\perp\}$ eigenmodes, with associated time-dependent coordinates. Coupled ODEs, including convolution (memory) terms from material and radiative damping, govern the system: $$\left[\frac{d^2}{dt^2} + (\gamma_{\text{bath}}*\,)(t) + \omega_0^2 \right] x_p^a(t) + \omega_P^2 \sum_{p'}(s_{pp'}^{ab}*\,\dot{x}_{p'}^b)(t) = f_p^a(t)$$ yielding a rigorous time-domain dynamical description.

3. Resonant Modal Structures and Pole Expansions

A distinguishing feature is the identification of physical resonances—polaritons or plasmonic poles—as simple poles in the modal coefficients’ frequency dependence. In elastodynamics (Chen et al., 2022), for each mode $(i,n)$ the resonance $\omega_{i,n}$ arises as the simple zero of the denominator in the modal amplitude: $\tau_{i,n}(\omega\delta)=0$ Mode residues at these poles determine the late-time causal dynamics (ring-downs) of the field: $$\hat u_{\rm sc}(x,t) = \sum_{i,n,m} R_{i,n,m}\,\Phi_{i,n,m}(x)\,H(t-t_0) e^{-i\omega_{i,n}(t-t_0)}$$ In the electromagnetic context (Forestiere et al., 2021), a vector-fitted rational approximation yields the dominant complex resonance poles $\zeta_j$ and residues, summarizing the impulse response as: $h(t) = \sum_j r_j e^{\zeta_j t} (t>0)$ A plausible implication is that a handful of low-lying resonances typically suffice for accurate descriptions of ultrafast and transient phenomena.

4. Practical Implementation and Quantitative Error Estimates

Sharp error bounds and practical truncation guidelines are essential for computational efficiency:

• For elastic scattering, the error in approximating the time-domain field by retaining $N$ modes and a frequency cutoff $p$ decays rapidly both in $N$ and $p$, allowing arbitrary precision given sufficient mode and frequency bandwidth (Chen et al., 2022).
• In quantum-limited mode-sum estimation, the variance of the mode-sum estimator sits a factor of ten below the classical (intensity-only) Cramér–Rao bound, achieving a quantum-limited variance $\mathrm{Var}(s) = 4\sigma^2/N$ with resolution ten times finer than the source bandwidth or pulse duration (Donohue et al., 2018).
• In electromagnetic problems, convolution ODE systems are truncated to the most relevant low-frequency (static) modes (by eigenvalue magnitude), resulting in a reduced finite system capturing both transient and steady-state behaviors (Forestiere et al., 2021).

5. Experimental Realization and Applications

Mode-sum methods have enabled significant experimental advances:

• Time-frequency quantum estimation employs mode-selective sum-frequency generation with shaped ultrafast pulses (4$f$ pulse shaper, spatial light modulator) for precise temporal or spectral decomposition of single-photon pulses, resolving separations much smaller than bandwidth limitations and directly measuring projections on individual Hermite–Gaussian modes (Donohue et al., 2018).
• Elastic metamaterials and electromagnetic nano-particles: Modal decompositions provide the foundation for analyzing transient scattering, extraction of impulse responses, and systematic investigation of ultrafast dynamics, applicable up to particle sizes near the wavelength (Forestiere et al., 2021, Chen et al., 2022).

A summary of selected applications is given below:

Domain	Mode Basis	Application Area
Quantum optics	Hermite–Gaussian (temporal/spectral)	Time-frequency superresolution (Donohue et al., 2018)
Elastodynamics	NP operator spherical harmonics	Wide-band scattering by inclusions (Chen et al., 2022)
Electromagnetism	Static longitudinal/transverse modes	Plasmonic/dielectric transients (Forestiere et al., 2021)

6. Extensions, Limitations, and Future Directions

Several extensions and practical limitations have been identified:

• Higher-order corrections: Mode selectivity is limited in practice by non-ideal phasematching, finite bandwidth, and group velocity mismatch; analysis is typically restricted to first-order perturbative regimes, low upconversion efficiency, and Gaussian point-spread functions (Donohue et al., 2018).
• Alternative platforms: Extensions to platforms such as Raman memories or homodyne-based mode selection have been proposed for enhanced sensitivity in time-frequency metrology (Donohue et al., 2018).
• Geometric generality: Modal expansions generalize to arbitrary geometries through numerical solution of electrostatic and magnetostatic problems to extract relevant static modes and eigenvalues, with only the radiation Green’s function inducing mode coupling (Forestiere et al., 2021).

A plausible implication is that future developments in modal analysis, pole fitting, and pulse-shaping strategies will continue to expand the reach of time-domain mode-sum methods across nonlinear optics, acoustics, and ultrafast phenomena.

7. Comparative Overview and Cross-Disciplinary Significance

The time-domain mode-sum method establishes a versatile formalism bridging quantum optics, classical elastic and electromagnetic wave physics. By reducing complex systems to modal (oscillator) representations paired with analytical and numerical pole expansions, the approach affords both physical insight and technical power in analyzing quantum limits, resonance behaviors, and ultrafast transients. Its cross-disciplinary adoption demonstrates the centrality of modal decomposition in the quantitative analysis of wave phenomena in complex, structured, or dispersive media (Donohue et al., 2018, Forestiere et al., 2021, Chen et al., 2022).