  Two strongly-pumped parametric interactions are simultaneously realized in a
single nonlinear crystal in order to generate three strongly correlated optical
fields. By combining together the outputs of two of the three detectors
measuring intensities of the generated fields, we obtain the joint photocount
statistics between the single field and the sum of the other two. Moreover, we
develop a microscopic quantum theory to determine the joint photon-number
distribution and the joint quasi-distributions of integrated intensities and
prove nonclassical nature of the three-mode state. Finally, by performing a
conditional measurement on the single field, we obtain a state endowed with a
sub-Poissonian statistics, as testified by the analysis of the conditional Fano
factor. The role of quantum detection efficiencies in this conditional
state-preparation method is discussed in detail.
