Photonic time crystals (PTCs) are characterized by the rapid modulation of the material properties in time, causing a momentum bandgap for light. However, the observation of these bandgaps at optical frequencies remains elusive as the necessary temporal modulation amplitudes to show notable momentum bandgaps are relatively high, inaccessible with available materials. While it has been known that structuring PTCs at the subwavelength scale can improve the bandgap size, we push this concept to the extreme by leveraging the nanophotonic toolbox. Specifically, we demonstrate that structures composed of scatterers supporting quasi-bound states in the continuum can substantially reduce the required modulation amplitudes by enhancing the interaction time between light and time-varying matter. This allows us to observe noticeable momentum bandgaps despite the weak temporal modulation. Our approach bridges the concepts of bound states in the continuum and time-varying metamaterials, paving the way toward realizable PTCs at optical frequencies.

We present a theoretical framework for electromagnetic scattering by particles with a permittivity that is periodically varying in time, based on a perturbative approach. Within this framework, we derive explicit expressions for the scattering matrix of the dynamic system in a first-order Born approximation, relating it directly to the corresponding static problem. We show that inelastic scattering amplitudes are governed by overlap integrals between static modes at the input and output frequencies. Using this insight, we analyze scattering from a time-modulated, isotropic, dielectric sphere and a high-permittivity dielectric cylinder, and demonstrate how modal orthogonality can suppress inelastic channels, while appropriate tuning of geometric parameters can significantly enhance them. In particular, we show that cylindrical resonators support strong inelastic scattering when resonance-to-resonance optical transitions, induced by the temporal variation, involve a high-Q supercavity mode. Comparison with full time-Floquet calculations confirms that the first-order Born approximation remains quantitatively accurate for modest modulation amplitudes and provides clear physical intuition for frequency conversion and resonance-mediated scattering processes in time-modulated photonic resonators.