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Nonlinear optical process called the 'Spontaneous Parametric Down Conversion' (SPDC) divides a photon of high frequency into two photons of lower frequencies with energy conservation. But I see no term there corresponding to the decay of photon into two photons in the QED Lagrangian where only available interaction term reads $\mathcal{L}_{int}=-ie\overline{\psi}\gamma^{\mu}A_{\mu}\psi$.

I know that SPDC occurs when the light is incident on special crystals which facilitate division of a photon into two, but how exactly is it possible in terms of perturbation theory in QED? What diagram corresponds to this SPDC process? Is it like a complex process where multiple Fermions (particles in the crystal) interacting with multiple photons?

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    $\begingroup$ If you skip over all the details, the fact that the process only happens in very few materials under very special conditions tells you that there can be no microscopic QED description that does not also model the specific atomic structure of these materials, which is unlikely to be easy. Schematically you have to absorb the energy, momentum and spin of the incoming photon in a blob and emit two photons of opposite momentum and spin from the blob. The schematic process may not even obey relativity because of the preferred rest system of the crystal. Maybe I am overly pessimistic here. $\endgroup$ Commented Jun 16, 2023 at 13:33

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QED is a relativistic Lagrangian formulation of the fields, and (mostly) describes scattering amplitudes.

SPDC is describe by quantum optics, which is a non-relativistic Hamiltonian formulation of photons.

They are very different.

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