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In regular SPDC, a photon from a (pump) laser enters a transparent nonlinear crystal at rest, and gets converted into a pair of photons whose total energy and momentum add up to that of the original (pump) photon.

[itex]E_{p}=\hbar \omega_{p} = \hbar \omega_{1} + \hbar \omega_{2}[/itex]

[itex]\vec{p}_{p}=\hbar \vec{k}_{p} = \hbar \vec{k}_{1} + \hbar \vec{k}_{2}[/itex]

In CP-SPDC, the energy of the photon pair adds up to the energy of the original pump photon, but the momenta of the photon pair adds to zero since they are propagating in opposite directions. Since the total energy of the field is the same before and after the interaction, but the momentum is different, it would look like there is a violation of conservation of momentum, since the crystal gets a momentum kick without changing its energy.

This is a process that really exists, and has been demonstrated in the laboratory.

I can work through the math and show that the technique known as quasi-phase matching (a way of periodically switching the crystal structure of the medium) allows one to add a term that offsets the momentum imbalance, but that much is theoretical.

If anyone out there is intimately familiar with this process or has a photon-level understanding of quasi-phase matching, what is physically going on here? I can imagine the kinetic energy of the crystal remaining the same, while internal kinetic energy is converted to center-of-mass kinetic energy, allowing for a momentum kick, but that would be a first for me. Thermodynamically, it always goes the other way.

Thoughts?