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I am reading some papers on spontaneous parametric downconversion, but I don't grasp the entire concept of the spontaneous emission. I read a book on classical nonlinear optics and I understood most of the concepts, but I don't get the difference between, e. g. the classical process difference frequency generation, and the quantum mechanical spontaneous parametric downconversion.

What is the difference?

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Actually the quantum-mechanical description of spontaneous parametric down-conversion (SPDC) is more rigorous and accurate. It has wider applicability than the simple toy-description from classical non-linear optics. For instance, the classical description fails to explain the photon statistics of the generated signal-idler field, non-local correlations of the signal and idler photons,etc. For a rigorous treatment of PDC, see:

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.31.2409

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