Spatiotemporal Metamaterials and their Symmetries

Spatiotemporal metamaterials harness ultrafast modulations of the refractive index to implement synthetic motion, resulting in light-matter interactions where neither the energy nor the momentum of the electromagnetic field are conserved. Despite this fact, here we demonstrate that spatiotemporal metamaterials exhibit reduced spatiotemporal symmetries. Moreover, using Noether's theorem we demonstrate that such spatiotemporal symmetry results in the conservation of the energy-momentum. We present a numerical examples of the scattering of an electromagnetic pulse by a modulation pulse to validate our theoretical results.