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The coherent state from a laser source has spin and orbital degrees of freedom, which allow an arbitrary superposition state among orthogonal states with varying amplitudes and phases. Here, we theoretically show coherent photons with SU(N) symmetry are characterised by expectation values of angular momentum shown on a hypersphere in SO(N2−1) space. To demonstrate expected unitary transformations in experiments, we have constructed generators of transformations in the Lie group simply by combining widely available optical components such as waveplates and vortex lenses. We show a superposition state between twisted and Gaussian states is characterised by the dynamics of the topological charge upon the transformation in SU(3) states. We also realised photonic singlet and triplet states corresponding to SU(4) states, which were projected to SU(2)×SU(2) states upon passing through a rotated polariser.

The coherent state from a laser source has spin and orbital degrees of freedom, which allow an arbitrary superposition state among orthogonal states with varying amplitudes and phases. Here, we theoretically show coherent photons with SU( N ) symmetry are characterised by expectation values of angular momentum shown on a hypersphere in SO( N 2 - 1 ) space. To demonstrate expected unitary transformations in experiments, we have constructed generators of transformations in the Lie group simply by combining widely available optical components such as waveplates and vortex lenses. We show a superposition state between twisted and Gaussian states is characterised by the dynamics of the topological charge upon the transformation in SU(3) states. We also realised photonic singlet and triplet states corresponding to SU(4) states, which were projected to SU(2)×SU(2) states upon passing through a rotated polariser., Competing Interests: The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Shinichi Saito reports financial support was provided by 10.13039/501100001691Japan Society for the Promotion of Science. Shinichi Saito reports financial support was provided by Hitachi Ltd. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Author(s).)