Momentum flatband and superluminal propagation in a photonic time Moir\'e superlattice

Flat bands typically describe energy bands whose energy dispersion is entirely or almost entirely degenerate. One effective method to form flat bands is by constructing Moir\'e superlattices. Recently, there has been a shift in perspective regarding the roles of space (momentum) and time (energy) in a lattice, with the concept of photonic time crystals that has sparked discussions on momentum dispersion such as the presence of a bandgap in momentum. Here we propose a photonic time moir\'e superlattice achieved by overlaying two photonic time crystals with different periods. The resulting momentum bandgap of this superlattice supports isolated momentum bands that are nearly independent of energy, which we refer to as momentum flat bands. Unlike energy flat bands, which have zero group velocity, momentum flat bands exhibit infinitely large group velocity across a broad frequency range. Unlike previous optical media supporting broadband superluminal propagation based on gain, the effective refractive index of the momentum flat bands is real-valued, leading to more stabilized superluminal pulse propagation.