Representation Theory and Topology of Coherent Photons with Angular Momentum

Photons are elementary particles of lights, which have both spin and orbital angular momentum as internal degrees of freedom. Nature of spin is known as polarisation, which is widely used for sunglasses, liquid-crystal displays, digital-coherent communications, while orbital angular momentum is useful for optical tweezers, laser-patterning, and quantum optics. However, spin and orbital angular momentum of photons are considered to be impossible for splitting into two independent degrees of freedom in a proper gauge invariant way, proved by plane wave expansions in a free space. Here, we show these degrees of freedom are well-defined quantum observables in a waveguide and a free space as far as the propagation mode is sufficiently confined in the core. We found Stokes parameters are spin expectation values of coherent photons, which exhibit non-trivial topological features like a torus, a M\"obius strip, and a bosonic Dirac cone. We have applied an SU(N) representation theory to describe both spin and orbital angular momentum of photons, and experimentally demonstrated their controls over a full Poincar\'e sphere to show a fullerene C$_{60}$ and the earth by qubits. We have also ascribed topological colour charge to photonic orbital angular momentum, whose SU(3) states are shown on a proposed Gell-Mann hypersphere in SO(8), whose parameters could be embedded in SO(5). We have also realised photonic SU(4) states of singlet and triplet states, which were successfully projected into SU(2)$\times$SU(2) states by a rotated polariser. Our results indicate that our platform of manipulating spin and orbital angular momentum is useful for exploring a photonic quantum chromodynamics and a higher order macroscopic quantum state.