Quantum Commutation Relationship for Photonic Orbital Angular Momentum

Orbital Angular Momentum (OAM) of photons are already ubiquitously being used for numerous applications. However, there is a fundamental question whether photonic OAM operators satisfy standard quantum mechanical commutation relationship or not; this also poses a serious concern on the interpretation of an optical vortex as a fundamental quantum degree of freedom. Here, we examined canonical angular momentum operators defined in a cylindrical coordinate, and applied them to Laguerre-Gauss (LG) modes in a graded index (GRIN) fibre. We confirmed the validity of commutation relationship for the LG modes and found that ladder operators also work properly with the increment or decrement in units of the Dirac constant ($\hbar$). With those operators, we calculated the quantum-mechanical expectation value of the magnitude of angular momentum, which includes contributions from both intrinsic and extrinsic OAM. The obtained results suggest that OAM characterised by the LG modes exhibits a well-defined quantum degree of freedom.