Quantum field theory of photons with orbital angular momentum

A quantum-field-theory approach is put forward to generalize the concept of classical spatial light beams carrying orbital angular momentum to the single-photon level. This quantization framework is carried out both in the paraxial and nonparaxial regimes. Upon extension to the optical phase space, closed-form expressions are found for a photon Wigner representation describing transformations on the orbital Poincaré sphere of unitarily related families of paraxial spatial modes.

Figure 1

Orbital Poincaré sphere of second-order modes. The poles of the sphere correspond to Laguerre-Gaussian modes with $l=\pm 2$ and $p=0$ (positive and negative signs for north and south poles, respectively). The states in the equatorial plane $(\theta =\pi ∕2)$ with $\phi =0$, and $\phi =\pi$ yield the Hermite-Gaussian modes having indexes $n_x=2$, $n_y=0$, and $n_x=0$, $n_y=2$ (not shown), respectively.