If one pump photon "decays" through the process of spontaneous parametric down-conversion into two photons how can the spin be conserved?


It isn't.

SPDC, like all second-order nonlinear optical phenomena, happens in an anisotropic medium, which means that the optical interaction (when seen exclusively as an interaction between optical modes) is not rotationally invariant and therefore does not conserve angular momentum.

As such, it is perfectly possible for the spin of the input and output states to differ; the difference in angular momentum is taken up by the medium.

If you want a fully spin-conserving formalism, then you need to include the rotational state of the generating medium as a full-fledged dynamical degree of freedom into your equations of motion. This will then allow you to explicitly describe the transfer of spin from the optical degrees of freedom to the mechanical ones. However, unless you're actively going to measure the mechanical angular momentum after the generation (which will be an extremely challenging task), then expanding the formalism in this way just isn't worth it.


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