Physical meaning of parametric down conversion of photon

I would like to understand what is the physical processes occurring in the particle picture when parametric down conversion of photon happens.

The description in Wikipedia states:

"A nonlinear crystal is used to split photon beams into pairs of photons that, in accordance with the law of conservation of energy and law of conservation of momentum, have combined energies and momenta equal to the energy and momentum of the original photon and crystal lattice, are phase-matched in the frequency domain"

What if I consider a single highly energetic photon entering the crystal? What mechanism is doing the splitting? Is simulated emission occurring i.e the atoms of the crystal become excited by this highly energetic photon and 2 lower energy photons are emitted (while respecting energy momentum and spin conservation and thereby producing 2 correlated photons). Or does the highly photon interact via scattering in the crystal, with an electron for example, and the electron emits two lower energy photons?

• The photon is interacting with the band structure of the crystal, including the electron, phonon, and plasmon/polariton energy vs momentum states. Considering this interaction as with an atom or an electron is doomed to failure. – Jon Custer Jul 30 '18 at 22:34
• @Jon Custer, is it true that when you calculate the nonlinear polarizability of the atoms inside a nonlinear crystal, you also have to take phonon into account? I thought its effect would be rather weak. – wcc Jul 30 '18 at 23:14
• @JonCuster can you direct me to a good reference for this? I am coming from the background of a particle physicist and not quantum optics. – SAMCRO Jul 31 '18 at 0:45
• @SAMCRO, when it comes to nonlinear optics, Robert Boyd's "Nonlinear Optics" is a standard reference. It shows you how to calculate nonlinear polarizability using perturbation theory, and I think it does treat fully quantized EM field at some point, although I'm not 100% sure. Also check ou "Introduction to Quantum Optics" by Alain Aspect. – wcc Jul 31 '18 at 6:06
• To get a strong second order polarizibility you usually need some of the following: a crystal with strong inversion symmetry breaking, fine tuning of the incident beams (to maximize $E^2$) through phase matching etc., and (most simply) intense laser beams. The frequency dependence of the non linear susceptibility is heavily natural dependent as Jon Custer alludes to. Nonetheless, you can still model the process with a single function $\chi^{(2)}$. – KF Gauss Aug 1 '18 at 7:31

Parametric down-conversion is a nonlinear optical effect. What that means is that the electric flux density ${\bf D}$ is a nonlinear function of the electric field ${\bf E}$. In vacuum, the relationship between these two is given by the constituent relation ${\bf D}=\epsilon_0 {\bf E}$, but in a dielectric material one also gets the polarization ${\bf P}$ (not to be confused with the state of polarization of light), so that ${\bf D}=\epsilon_0 {\bf E}+{\bf P}$. In a linear dielectric medium ${\bf P}$ is still proportional to ${\bf E}$, so that it can be combined with the vacuum term to give ${\bf D}=\epsilon {\bf E}$, where $\epsilon=n^2 \epsilon_0$ (assuming an isotropic medium), with $n$ being the refracive index.
In a nonlinear medium, ${\bf P}$ (and therefore also ${\bf D}$) is a nonlinear function of ${\bf E}$. Physically, it means that the polarization (separation of positive and negative charges) that is caused by an electric field when it is applied to that nonlinear medium, is not a linear function of the amplitude of that electric field. One can expand ${\bf D}$ in increasing powers of ${\bf E}$, but since we are dealing with vectorial quantities, the coefficients in this expansion would be tensors. Therefore, it is more convenient to represent such an expression in terms of the components denoted by indices, such that repeated indices imply a summation over all the values of that index. So ${\bf D}$ can be expressed as $$D_a = \epsilon_0 \left( \chi^{(1)}_{ab} E_b + \chi^{(2)}_{abc} E_b E_c + \chi^{(3)}_{abcd} E_b E_c E_d + ... \right) .$$ The tensors $\chi^{(2)}$ and $\chi^{(3)}$ are referred to as the nonlinear susceptabilities of the medium, associated with Pockels effect and Kerr effect, respectively. Parametric down-conversion, where one photon is converted into two down-converted photons, is represented by $\chi^{(2)}$. (This should make sense from a particle physics prespective.)
To see this as an interaction term in a Hamiltonian or a Lagrangian, one needs to contract ${\bf D}$ with another ${\bf E}$ to get $E_a D_a$. The interaction term associated with the parametric down-conversion is then $\epsilon_0 \chi^{(2)}_{abc} E_a E_b E_c$. If these are represented as quantized field operators, one can see how it would allow one photon to be converted into two photons.