Quantum polarization distributions via marginals of quadrature distributions

Polarization is a crucial ingredient of light in both the classical and quantum domains. However, quantum polarization is mainly addressed in terms of the abstract Hilbertspace logic in a finite-dimensional space. This is rather divorced from the usual language used in the classical domain in terms of the Stokes parameters and the Poincaré sphere. Because of this, we think that it is worth developing and investigating formulations of quantum polarization as distributions on the Poincaré sphere. Such approaches are closer to our common intuition about polarization phenomena, so they can be very helpful in understanding quantum polarization properties. In this regard we can follow two different routes. On the one hand, we have a direct approach by translating to polarization previously introduced phase-space formalisms for angular momentum and spin variables f1–7g. This is possible because the Stokes parameters sthe basic variables describing polarizationd are formally equivalent to an angular momentum. For definiteness we focus on the proposals in Refs. f1–5g referring to them as SUs2d distributions. On the other hand, we can derive polarization distributions via suitable marginals of distributions for the complex amplitudes sor field quadraturesd by removing the degrees of freedom irrelevant for the specification of polarization. This is the approach investigated in this work, and we refer to them as marginal distributions. More specifically, we consider proper marginals of s-ordered quadrature distributions that include distinguished particular examples such as the Q, P, and Wigner functions f8g. There are several reasons supporting the expediency of this approach to quantum polarization. From a practical perspective, we have that polarization distributions provide a feasible approach to examine and measure diverse polarization properties. This is the case of the degree of polarization recently introduced as the distance between the polarization distribution and the uniform distribution associated with unpolarized light f9g. This can be also the case of the proper assessment of polarization correlations derived from the properties of the joint polarization distribution f10g. Polarization correlations are crucial for recently developed applications of quantum theory f11g. Marginals of suitable quadrature distributions have been used also to study phase properties of oneand two-mode quantum field states f12g. In this regard we may say that marginal distributions provide the most down-to-earth approach to quantum polarization distributions, since, by definition, they are to be obtained exactly in the same way as polarization distributions are derived in classical optics f13g. This is in sharp contrast to SUs2d distributions whose definitions bear no definite relation with optics, being introduced specifically for the description of abstract angular momenta. Among the large family of quantum phase-space distributions known in physics, the s-ordered distributions are distinguished by their good theoretical and experimental properties. For example, in the context of polarization we have that s-ordered quadrature distributions transform properly under the transformations that represent action of standard polarization changing devices f14g. From the experimental perspective, s-ordered distributions can be determined in practice by using diverse experimental procedures, such as homodyne and heterodyne detection, tomography, and atom-field interactions, to mention just the most popular and repeatedly carried out experimentally f15g. As a matter of fact, most of these practical schemes are very simple so that their operation can be understood even within a purely classical framework. Moreover, they are robust against experimental imperfections, such as detection inefficiencies, that imply just a change of the value of the s parameter. This widespread measurability is not matched by any other family of phase-space distributions. This is also in sharp contrast to the case of SUs2d distributions which, to the best of our knowledge, have not been determined experimentally yet. In any case, the theoretical proposals for their practical determination are rather cumbersome and lack the simple and intuitive picture provided by schemes measuring the s-ordered distributions f3,16g. Finally, s-ordered distributions provide a simple measure of the degree of nonclassical behavior of quantum states f17g. In Sec. II we derive the main formulas establishing the operator-function correspondence obtained after removing from the quadrature distributions the variables not related to the polarization. In Secs. III and IV we examine their main properties, comparing the marginal distributions with the *Electronic address: [email protected]; URL: http://www.ucm.es/ info/gioq PHYSICAL REVIEW A 71, 053801 s2005d