$SU(1,1)$ Approach to Stokes Parameters and the Theory of Light Polarization

TL;DR

This paper presents a novel quantum algebra-based method to describe light polarization, introducing Stokes-like parameters derived from $su(1,1)$ Lie algebra operators, offering a new geometric interpretation of polarization states.

Contribution

The authors develop a new polarization framework using $su(1,1)$ algebra operators, linking quantum operators to classical polarization parameters and geometric structures.

Findings

Defined Stokes-like parameters from $su(1,1)$ operators

Expressed polarization ellipse in terms of new parameters

Revealed hyperboloid geometry of polarization states

Abstract

We introduce an alternative approach to the polarization theory of light. This is based on a set of quantum operators, constructed from two independent bosons, being three of them the $su(1,1)$ Lie algebra generators, and the other one, the Casimir operator of this algebra. By taking the expectation value of these generators in a two-mode coherent state, their classical limit is obtained. We use these classical quantities to define the new Stokes-like parameters. We show that the light polarization ellipse can be written in terms of the Stokes-like parameters. Also, we write these parameters in terms of other two quantities, and show that they define a one-sheet (Poincar\'e hyperboloid) of a two-sheet hyperboloid. Our study is restricted to the case of a monochromatic plane electromagnetic wave which propagates along the $z$ axis.