Longitudinal optical phonons in photonic time crystals containing a stationary charge

Lorentzian-type media support optical phonons that oscillate with longitudinal polarization parallel to the wave direction, at a wave vector-independent frequency at which the permittivity becomes zero. Here, we study the interactions between the longitudinal optical phonons and Lorentzian medium-based dispersive photonic time crystals (PTCs). We demonstrate that a stationary charge embedded in the PTCs can excite these longitudinal modes through the conversion of the static polarization field induced by the charge. Furthermore, the PTCs can develop a momentum bandgap across the entire wave vector space to amplify the longitudinal modes. Remarkably, this infinite momentum bandgap can be established with minimal temporal modulation of the refractive index when creating the PTCs. Our approach expands the range of waves that can be manipulated in PTCs and shows potential for observing momentum bandgap phenomenon in realistic optical experiments, where the modulation depth of the refractive index is severely constrained.