Jordan-Schwinger map, 3D harmonic oscillator constants of motion, and classical and quantum parameters characterizing electromagnetic wave polarization

TL;DR

This paper generalizes quantum Stokes operators using SU(3) symmetry, linking them to harmonic oscillator constants of motion to characterize electromagnetic wave polarization in both classical and quantum frameworks.

Contribution

It introduces a novel SU(3)-based generalization of Stokes operators and connects them to harmonic oscillator constants, expanding polarization analysis methods.

Findings

Generalized Stokes operators are constants of motion for 3D harmonic oscillators.

The approach allows expansion of polarization density matrices using Gell-Mann matrices.

Classical Stokes parameters are derived from quantum expectation values.

Abstract

In this work we introduce a generalization of the Jauch and Rohrlich quantum Stokes operators when the arrival direction from the source is unknown {\it a priori}. We define the generalized Stokes operators as the Jordan-Schwinger map of a triplet of harmonic oscillators with the Gell-Mann and Ne'eman SU(3) symmetry group matrices. We show that the elements of the Jordan-Schwinger map are the constants of motion of the three-dimensional isotropic harmonic oscillator. Also, we show that generalized Stokes Operators together with the Gell-Mann and Ne'eman matrices may be used to expand the polarization density matrix. By taking the expectation value of the Stokes operators in a three-mode coherent state of the electromagnetic field, we obtain the corresponding generalized classical Stokes parameters. Finally, by means of the constants of motion of the classical three-dimensional isotropic harmonic oscillator we describe the geometric properties of the polarization ellipse