Representation theory of coherent photons and application to CNOT operation for spin and orbital angular momentum

Coherent photons have both spin and orbital angular momentum, and the superposition state among orthogonal states can describe a qubit. Here, we show the internal structure of coherent photons with spin and orbital angular momentum can be described by a representation theory of Lie algebra and Lie group. As an application of this theory, we have experimentally demonstrated the controlled-NOT (CNOT) operation, using a standard green laser diode. We have used beam splitters and waveplates to generate the macroscopic entangled light source with amplitudes and phases, controlled by rotating the waveplates. By applying the Bell projection to the entangled light, the far-field image of the light shows an expected dipole shape in the superposition state of left and right vortices, allowing to identify the state from the image. Then, we have used a polarisation dependent beam splitter to split the spin state, and we applied a NOT operation, made of a pair of cylindrical lenses, only for a vortex with a chosen polarisation, and recombined the beams after the operation. The proposed architecture could be used for a platform to manipulate qubits, made of macroscopically coherent photons.