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Polarization-squeezed triphoton states and their Wigner representation on the Poincaré sphere

Lynden K. Shalm, Robert B. A. Adamson, Aephraim M. Steinberg

Published May 4, 2008• 0 citations
OpenAlex

Abstract

We present reconstructions of Wigner distributions for the polarization state of three indistinguishable photons on the generalized Poincare sphere. We study a variety of states including coherent, phase and NOON states.

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We reconstruct the polarization sector of a bright polarization squeezed beam starting from a complete set of Stokes measurements. Given the symmetry that underlies the polarization structure of quantum fields, we use the unique SU(2) Wigner distribution to represent states. In the limit of localized bright states, the Wigner function can be approximated by an inverse three-dimensional Radon transform. We compare this direct reconstruction with the results of a maximum likelihood estimation, thus finding excellent agreement.
We derive quantum polarization distributions on the Poincar\'e sphere for two-mode fields as suitable marginals of the standard sss-ordered quadrature distributions that include the Wigner function. We compare them with previously introduced distributions directly defined on the sphere from first principles. We apply this approach to some relevant field states such as coherent and number states.
We comprehensively review the quantum theory of the polarization properties of light. In classical optics, these traits are characterized by the Stokes parameters, which can be geometrically interpreted using the Poincaré sphere. Remarkably, these Stokes parameters can also be applied to the quantum world, but then important differences emerge: now, because fluctuations in the number of photons are unavoidable, one is forced to work in the three-dimensional Poincaré space that can be regarded as a set of nested spheres. Additionally, higher-order moments of the Stokes variables might play a substantial role for quantum states, which is not the case for most classical Gaussian states. This brings about important differences between these two worlds that we review in detail. In particular, the classical degree of polarization produces unsatisfactory results in the quantum domain. We compare alternative quantum degrees and put forth that they order various states differently. Finally, intrinsically nonclassical states are explored, and their potential applications in quantum technologies are discussed.