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As Wikipedia explains, one photon passing through a crystal sometimes down-converts to two photons. Wikipedia says total energy and momentum are conserved by just considering the three photon states; is Wikipedia wrong here?

It seems a phonon (or something else) is needed too. If Wikipedia is right, can you provide 3 example (non-parallel) momenta vectors so that I can see my logic mistake?

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The crystal itself breaks translation symmetry and absorbs a tiny amount of momentum as a whole during a seeming momentum violating process. This is because the phenomenon is coherent along all the crystal atoms, it's not paradoxical. – Ron Maimon Sep 7 '12 at 15:28

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I'm not sure why you don't want the momenta vectors to be parallel. Normally they are parallel in parametric downconversion.

That doesn't solve the problem, because the parametric downconversion happens in a material, and materials always have dispersion (different refractive index at different wavelengths). The nature of dispersion makes it difficult in normal circumstances to simultaneously have $\omega = \omega_1 + \omega_2$ and $k = k_1 + k_2$, even when the wavevectors are parallel. But with a bit of cleverness and effort it is possible.

This field of knowledge is called PHASE MATCHING. It is a basic and important topic in nonlinear optics. In a nonlinear optics textbook, it would normally be discussed in the first chapter. I doubt I would do it justice in a few sentences.

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Wikipedia says they're usually off-parallel for this interesting case: en.wikipedia.org/wiki/Spontaneous_parametric_down_conversion Whether we talk about conservation of crystal momentum (for a bulk infinite crystal) or mechanical momentum (for finite crystal comparing global state before and after crystal passage), I don't get how that first Wikipedia figure can be true. – bobuhito Sep 7 '12 at 2:42
Or, maybe crystal momentum is what is being drawn and dispersion is the reason (the two emitted photons are moving slower than the original photon, so their energies are less than I had thought)? – bobuhito Sep 7 '12 at 2:45
You, the experimenter, are free to decide whether they are parallel or not. You decide based on whatever makes the experiment work best. If you want it to be parallel, you make it phase-matched for parallel waves. If you want it to be non-parallel, you make it phase-matched for non-parallel waves. – Steve B Sep 7 '12 at 19:33
Phase matching is NOT really related to conservation of momentum. With phase-matching, the waves add in phase to become very strong. Without phase-matching, the waves add with random phase offsets to give a small (but nonzero) total. The fact that nonlinearly-generated light goes primarily in phase-matched directions is analogous to the fact that light bouncing off a diffraction grating goes primarily in specific directions. – Steve B Sep 7 '12 at 19:39
Sorry, I think you have taken my question differently than intended. The 3 momenta are not inputs. I only get to input one momentum. The other 2 momenta are outputs. – bobuhito Sep 7 '12 at 21:36
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