Superluminal k -Gap Solitons in Nonlinear Photonic Time Crystals

We propose superluminal solitons residing in the momentum gap ($k$ gap) of nonlinear photonic time crystals. These gap solitons are structured as plane waves in space while being periodically self-reconstructing wave packets in time. The solitons emerge from modes with infinite group velocity causing superluminal evolution, which is the opposite of the stationary nature of the analogous Bragg gap soliton residing at the edge of an energy gap (or a spatial gap) with zero group velocity. We explore the faster-than-light pulsed propagation of these $k$-gap solitons in view of Einstein's causality by introducing a truncated input seed as a precursor of a signal velocity forerunner, and find that the superluminal propagation of $k$-gap solitons does not break causality.