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 Möbius 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 Poincaré 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.
Instituto Nacional de Astrofísica, Óptica y Electrónica（墨西哥国家天体物理学、光学和电子学研究所）；Tecnologico de Monterrey（蒙特雷科技大学）；Universidad Politécnica Metropolitana de Hidalgo（伊达尔戈大都会理工大学）
A unified matrix representation for spin and orbital angular momentum in partially coherent beams
一个矩阵表示引入静止的光束状光场，允许同时治疗的轨道角动量（OAM）和空间相干性的二阶状态。该相干-OAM矩阵是笛卡尔偏振基中的2 X 2光束相干-偏振矩阵或圆偏振基中的相干-自旋角动量（SAM）矩阵的对应物。讨论了相干-OAM矩阵的一般性质，并引入相干-SAM-OAM矩阵作为处理具有任意相干、自旋和轨道角动量态光束的统一实体。
Quantum Skyrmions in Mixed States of Light and Their Nested Topology
Technion - Israel Institute of Technology（以色列理工学院）；Stanford University（斯坦福大学）；Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）；Massachusetts Institute of Technology（麻省理工学院）
X-ray Fractional Orbital Angular Momentum from Coherent Magnetic Scattering
University of Kentucky（肯塔基大学）；Argonne National Laboratory（阿贡国家实验室）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Brookhaven National Laboratory（布鲁海文国家实验室）
