We present a moment expansion method for the systematic characterization of the polarization properties of quantum states of light. Specifically, we link the method to the measurements of the Stokes operator in different directions on the Poincar{e}
sphere, and provide a method of polarization tomography without resorting to full state tomography. We apply these ideas to the experimental first- and second-order polarization characterization of some two-photon quantum states. In addition, we show that there are classes of states whose polarization characteristics are dominated not by their first-order moments (i.e., the Stokes vector) but by higher-order polarization moments.
We discuss different proposals for the degree of polarization of quantum fields. The simplest approach, namely making a direct analogy with the classical description via the Stokes operators, is known to produce unsatisfactory results. Still, we argu
e that these operators and their properties should be basic for any measure of polarization. We compare alternative quantum degrees and put forth that they order various states differently. This is to be expected, since, despite being rooted in the Stokes operators, each of these measures only captures certain characteristics. Therefore, it is likely that several quantum degrees of polarization will coexist, each one having its specific domain of usefulness.
We propose an operational degree of polarization in terms of the variance of the projected Stokes vector minimized over all the directions of the Poincare sphere. We examine the properties of this degree and show that some problems associated with th
e standard definition are avoided. The new degree of polarization is experimentally determined using two examples: a bright squeezed state and a quadrature squeezed vacuum.
