We present a simple treatment of the phenomenon of spontaneous parametric downconversion consisting of the coherent scattering of a single pump photon into an entangled photon pair inside a nonlinear crystal. The energy and momentum entanglement of the quantum state of the generated twin photons are seen as a consequence of the fundamental indistinguishability of the time and the position in which the photon pair is created inside the crystal. We also discuss some consequences of photon entanglement.
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We could use spherical-wave modes to describe the propagation of photons 1 and 2 from (r′,t′) to (r1,t) and (r2,t) in Eq. (1). This would be more precise, but the plane-wave decomposition that we use here, in which all wavevectors appear with the same probability amplitude, leads to simpler calculations with essentially the same results.
If the photon is not monochromatic, the change of representation is not so simple, as discussed in Refs. 9 and 10.
It is important to stress that the form ℏk is not the only possibility for the momentum of a photon with wavevector k in a medium, as discussed in Refs. 30 and 31. There are many different ways of dividing the total momentum of an electromagnetic wave in a linear medium into electromagnetic and material parts, all of which are compatible with momentum conservation, and for each chosen division the photon momentum will have a different expression. The form ℏk is the simplest one for the present case, which is why we have used it.
