Quantum Photonic Time Crystals: From Temporal Boundaries to Floquet Light-Matter Interactions

Photonic time crystals (PTCs) are temporally periodic media whose Floquet spectra can exhibit momentum gaps, parametric amplification, and effective non-Hermitian descriptions, making them an idealized setting for vacuum amplification and nonequilibrium light-matter dynamics. Their classical electrodynamics is now well developed; the quantum side is less so, and this focused review is an attempt to organize what exists. We trace that account from temporal boundaries to homogeneous Floquet media and light-matter dynamics. A single temporal boundary induces Bogoliubov mode mixing and photon-pair creation; in homogeneous bulk media, momentum conservation isolates counter-propagating $(k,-k)$ sectors and yields a two-mode $SU(1,1)$ squeezing structure. Temporal periodicity promotes this to a Floquet problem with band and momentum-gap regimes, compactly described in a fixed Nambu basis. We then relate PTCs to the dynamical Casimir effect and parametric amplification, which share the same pair-creation mechanism but organize it through discrete resonances rather than a momentum-resolved bulk spectrum. We close with light-matter settings: spontaneous-emission decay and modulation-assisted excitation, atom-PTC dynamics, LDOS-based observables and their limits, and finite, dispersive, and experimentally accessible platforms.