Symmetries in time-varying media: on the conservation of spin angular momentum, helicity, and chirality

The introduction of time as an extra degree of freedom in optical systems, known as temporal metamaterials, significantly expands the realms of optics and nanophotonics by allowing dynamic manipulation of matter at ultrafast time scales [1]. Time-varying media break temporal symmetries and thus also have implications on fundamental physical principles potentially affecting conservation laws and symmetry properties. Regarding symmetries and conservation laws, Noether’s theorem is a crucial statement. In particular, it has been used to address the conservation of spin-dependent optical properties, like electromagnetic helicity and optical chirality, with investigations revealing insights into their conservation conditions and relationships between these dynamical properties and their associated [2–4]. In this respect, our work extends previous formalisms by using the Lagrangian formalism of time-varying media, shedding light on the interplay between time-modulation, the conservation and symmetries of spin angular momentum (SAM), helicity, and chirality. To this end, first, we define Maxwell’s equations and the Lagrangian of a time-varying homogeneous system. According to Noether’s theorem, any transformation of any dynamical variable that keeps the action integral unchanged is known as a classical system’s symmetry. So, there is a time-invariant quantity for each global continuous symmetry. Upon this ground, we investigate SAM and other spin-dependent quantities such as helicity and chirality. To do that, we introduce the symmetry for each of them and then concerning those symmetries find which one is conserved or not [5]. The symmetries for SAM, helicity, and chirality in our work are: