  We present measurements of the polarization dependencies of the x-ray signal
photons generated by the effect of parametric down-conversion of x rays into
ultraviolet radiation. The results exhibit pronounced discrepancies with the
classical model for the nonlinearity but qualitatively agree with a newly
developed quantum mechanical theory for the nonlinear interaction. Our work
shows that the reconstruction of the atomic scale charge distribution of
valence electrons in crystals by using nonlinear interaction between x rays and
longer wavelength radiation, as was suggested in previous works, requires the
knowledge of polarization of the generated x-ray signal beam. The results
presented in this work indicate a new methodology for the study of properties
of the Wannier functions in crystals.
