Energy Transport Velocity in Photonic Time Crystals

Abstract

Steep or near-vertical Floquet dispersion in photonic time crystals (PTCs) is often read as fast, even apparently superluminal, transport. Here, we demonstrate that this anomaly arises from modulation-driven geometric drift, not energy flow. By deriving a Maxwell-flux Hellmann-Feynman relation, we prove that the cycle-averaged energy velocity remains strictly bounded. We further establish a universal velocity-product law conserved throughout the passband, $ v_E v_g=\langle v_{\rm ph}^2\rangle_T $, fixing transport solely by the temporal average of the inverse permittivity. The divergent group velocity is then traced to a mismatch between electric and magnetic geometric phase connections, revealing apparent superluminality as a geometric effect of temporal modulation.