  In this work we experimentally demonstrate a photon-pair source with
correlations in the frequency and polarization degrees of freedom. We base our
source on the spontaneous four-wave mixing (SFWM) process in a photonic crystal
fiber. We show theoretically that the two-photon state is the coherent
superposition of up to six distinct SFWM processes, each corresponding to a
distinct combination of polarizations for the four waves involved and giving
rise to an energy-conserving pair of peaks. Our experimental measurements, both
in terms of single and coincidence counts, confirm the presence of these pairs
of peaks, while we also present related numerical simulations with excellent
experiment-theory agreement. We explicitly show how the pump frequency and
polarization may be used to effectively control the signal-idler photon-pair
properties, defining which of the six processes can participate in the overall
two-photon state and at which optical frequencies. We analyze the signal-idler
correlations in frequency and polarization, and in terms of fiber
characterization, we input the SFWM-peak experimental data into a genetic
algorithm which successfully predicts the values of the parameters that
characterize the fiber cross section, as well as predict the particular SFWM
process associated with a given pair of peaks. We believe our work will help
advance the exploitation of photon-pair correlations in the frequency and
polarization degrees of freedom.
