Background:

Spontaneous parametric down-conversion (SPDC) is the process when one photon spontaneously splits into two entangled photons. In the classical regime, this is a reverse process of sum frequency generation (SFG) where two photons add up to be a single photon. The classical SFG depends on the second-order susceptibility tensor, which dictates what polarization of each interacting light should be. For example, chi(2)_zxy will generate polarization density in the z direction by using electric fields polarized in x and y directions. By this description, 'two fields' are used to determine the polarization of the generated field. As such, if only chi(2)_zxy and its permutations, such as chi(2)_zyx, are non-zero, it is impossible to generate z-polarized f3 (frequency-3) field with z-polarized f1 field and x-polarized f2 field because there is no y-polarized ingredient (where f1+f2=f3).

Problem:

However, in SPDC, it starts with one photon. I have difficulty picturing the polarization selection rule. What I found in the literature in the case of non-zero chi(2)_zxy (and its permutation) is only when z-polarized f3 photon spontaneously splits into one xy-polarized f1 photon and one xy-polarized f2 photon where f3=f1+f2. (xy-polarized light corresponds to a TE mode in the waveguide the paper discusses)

I believe there should be a relationship between classical susceptibility tensor and quantum operator that dictates spontaneous frequency conversion process, but I couldn't find a good reference so far and I'm not skilled at quantum optics.