Quantised helicity in optical media

We present a new approach to the definition of optical helicity in a medium. Our approach resolves the problem that duality transformations which simultaneously combine $\mathbf{E}$ with $\mathbf{H}$ and $\mathbf{D}$ with $\mathbf{B}$ are incompatible with linear constitutive relations. We find that the helicity density in a medium, as the conserved quantity associated with duality transforms, must contain an explicit contribution associated with the polarisation and magnetisation of the matter, and that it can be expressed naturally in terms of the elementary polarised excitations of the system. In media for which the helicity is conserved, each circular excitation carries a well-defined helicity. However, in a medium for which the helicity is not conserved, we find that the time-varying helicity can be viewed in terms of oscillations between different helicity eigenstates, analogous to neutrino oscillations. Here we explicitly study the helicity in homogeneous and lossless media but we believe that, differently to other choices, this helicity is readily generalisable to media that may be inhomogeneous, lossy, chiral or nonreciprocal.