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.

Figures (2)

• Figure 1: Energy transport versus band-slope in a PTC. (a) Mode-resolved velocities on a passband: geometrical slope $v_g=d\omega/dk$ (black) and energy-transport velocity $v_E=\langle S_z \rangle_T/\langle u \rangle_T$ (red), both normalized by $v_{\mathrm{ph,max}}$. The wavevector is shown in units of the momentum-gap edge $k_c$ (vertical dashed line at $k=1$); the blue shading indicates the gap. Inset: Floquet band structure near the edge. (b) Velocity-product law: $v_g v_E$ is constant throughout the passband; thus, as $k\to k_c$, the divergence of $v_g$ is exactly compensated by $v_E\to 0$.
• Figure 2: Cycle-averaged centroid-velocity decomposition. In the narrowband limit $\sigma_k\!\to\!0$, $\langle \dot Z \rangle_T=\langle v_{\rm flux} \rangle_T+\langle v_{\rm mod} \rangle_T$. Curves show $\langle v_{\rm flux} \rangle_T$ (red), $\langle v_{\rm mod} \rangle_T$ (blue), and their sum (green), compared with $v_g=d\omega/dk$ (black dashed); all velocities are normalized by $v_{\mathrm{ph,max}}$. The wavevector is normalized by the momentum-gap edge $k_c$ (vertical dashed line at $k=1$); the blue shading indicates the gap.