Energy Transport Velocity in Photonic Time Crystals

TL;DR

This paper clarifies that the apparent superluminal energy transport in photonic time crystals is a geometric effect caused by temporal modulation, not actual energy flow exceeding physical limits, and establishes a universal velocity-product law.

Contribution

It introduces a Maxwell-flux Hellmann-Feynman relation and a universal velocity-product law, revealing the true bounds of energy velocity in photonic time crystals.

Findings

Energy velocity remains bounded despite divergent group velocity.

Superluminal effects are due to geometric phase mismatches, not energy flow.

A universal velocity-product law links energy velocity and group velocity.

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.