Topologically-protected superluminal pair annihilation in photonic time crystals

Photonic time crystals (PTCs) - dielectric media whose permittivity is periodically modulated in time - map to a Dirac equation with an imaginary mass, opening a momentum gap (k-gap) where modes grow or decay exponentially. Here, we introduce a sequence of temporal Jackiw-Rebbi kinks that act as a programmable flip of the Dirac mass, exchanging the amplifying and decaying in-gap modes. By launching two seeded pulses with a controlled relative phase, we demonstrate topological pair annihilation in spacetime domain, the phase-selective cancellation of counter-propagating, k-gap-amplified modes. The resulting spatiotemporal cascade appears superluminal, yet causality is preserved because the cascaded pattern carries no net energy flux. To facilitate implementation, we construct a minimal time-varying non-Hermitian lattice model and reproduce the phase-selective pair annihilation behavior, establishing a direct continuum-lattice correspondence. Our results identify topological kinks as temporal gating to manipulate the growth and wave propagation of time-varying media.