A photon with frequency W0 interacts with the non linear crystal. A photon can be considered as an oscillating electric field with frequency W0. This electricc field interacts with the crystal's dipoles. The electric field will try to induce a polarization in the crystal. However since the crystal is non linear, polarization induced in it will depend on two terms: 1) linearly  on the electric field and 2) Quadratically on the electric field.  

It can be visualized as the following: An electric field oscillating at frequency W0 tries to induce a polarization at frequency W0 in the crystal (Which is consistent with a linear behavior) plus since the crystal is non linear there are dipoles which will be radiating at 2W0 (due it's square dependence on the electric field).

Conservation of energy also has to be looked in this case, since a photon at frequency W0 is generating 2 photons of different frequency. This generation is only possible when a certain condition known as the phase matching condition is fulfilled. this condition can be read about in any non linear optics book. It only states that the velocity of the incoming electric field should be equal to the velocity of the radiated electric field. 

parametric down conversion can be looked in the same manner as above. 

Hi Juan!

I'm considering a similar situation; I don't like the classical perspective of optics, and after reading a couple papers in the history of quantum mechanics and quantum field theories, I've come to realize that the idea of a "photon" is mostly a simplification introduced by quantum electrodynamics.

In reality, we just have an EM field, and due to the quantized nature of excitation and decay, the waves in this field transfer their energy in quantized amounts. In an active optical material, an EM field can be said to excite modes in the electron structure in a classical way; even without quantum mechanics, it's a pretty accurate model. I believe this is likely due to the very fast absorption-emission of photons in the glassy insulator; the duration a "photon" stays within an electron is extremely small, pretty much negligible...we simply handwave away that time and call it a refractive index - that is, the light goes slower because of this cascading phonon-like process.

Active optical materials are pretty tricky though. The crystal structure defines an interlocking electron band structure; when a "photon" comes in, if it has enough energy, it may be absorbed into a lower electron band, exciting that electron above most of the others. Ordinarily it would just want to fall back down, emitting the same energy as it absorbed, but in some situations it may actually be preferable for it to stay in that higher energy band. In such a situation, a lower-energy electron might fall, leading to a hole where it was; in turn, another electron falls into where that one was...until everything is in its happy ground state.

The important bit to get here, going off your second post, is that a photon is kind of...well, it's a pseudoparticle. We consider it a real thing because it's highly convenient. Light is an EM field, and yes it is quantized, but this is due to the quantized energy levels of the electrons in the field, and how they form these quantized EM wavepackets in the field.

Yes yes Juan you are right. A photon cannot be considered as an oscillating electric field or for that case any fixed number of photons. When you take the expectation value for the electric field for the number states of photons you always get zero. But that was not what I meant. It might have come out wrong now that I think about it. 

What I meant to imply was instead of a single photon consider an oscillating field. In that case the argument becomes simplified. 

I am extremely sorry for any misconceptions!!