Quantum theory of photonic vortices and quantum statistics of twisted photons

TL;DR

This paper develops a quantum formalism for photonic vortices, defining topological charge via circulation, and explores the quantum statistics of twisted photon pairs based on their spin states.

Contribution

It introduces a comprehensive quantum framework for photonic vortices, including operators for current density and circulation, and links quantum statistics to photon spin states.

Findings

Quantized circulation defines photonic topological charge as a winding number.

Helicity current vortices can exist with zero photon current.

Quantum statistics of twisted photons depend on their spin states.

Abstract

The topological charge of a photonic vortex is an essential quantity in singular optics and the critical parameter to characterize the vorticity of twisted light. However, the definition of the photonic topological charge remains elusive. Here we put forth a theoretical formalism to provide a comprehensive treatment of photonic vortices. We introduce quantum operators for the photon current density and helicity current density based on the continuity equations from the paraxial Helmholtz equation. Our formalism allows us to introduce flow velocity and circulation for photonic currents in parallel to their counterparts in superfluids. The quantized circulation of the photonic currents is conserved during propagation and it gives an explicit definition of the photonic topological charge as the winding number of a photonic vortex. In particular, we predict helicity current generated pure helicity vortices, in which the photon current vanishes. Finally, we show an interesting effect that the quantum statistics of twisted photon pairs are essentially determined by their spin states.