Unified treatment of the total angular momentum of single photons via generalized quantum observables

TL;DR

This paper develops a unified quantum framework for understanding the total angular momentum of single photons, clarifying the roles of different decompositions into spin and orbital components using POVM and PVM formalisms.

Contribution

It introduces a consistent approach to decompose single-photon angular momentum into compatible and incompatible parts using generalized quantum observables.

Findings

Canonical SAM and OAM are mutually compatible unsharp observables.

Non-canonical SAM and OAM are incompatible but sharp observables.

Explicit calculations for circularly polarized Gaussian states demonstrate the framework.

Abstract

In this paper we provide a consistent framework to address the notorious decomposition of the single-photon total angular momentum (TAM) into a spin (SAM) and an orbital (OAM) component. In particular, we find that the canonical SAM and OAM, generators of internal and spatial rotations respectively in the space of physical states, are mutually compatible but unsharp quantum observables, therefore POVM (Positive Operator-Valued Measures) describe their joint measurements. On the other hand, a non-canonical SAM and OAM can be constructed that is mutually incompatible but represent sharp quantum observables, thus PVM (Projector-Valued Measurements), reflecting their consistency with the transversality condition characterizing singlephoton wavefunctions. We discuss the implementations on joint measurements for both decompositions and provide an explicit calculation of all these quantities for circularly polarized Gaussian single-photon states.