Nonlinear optics is usually looked at through a classical lens for most purposes. I do a Taylor expansion of the material polarization in terms of light-field strengths and we can derive all the essential mathematics for second-harmonic generation, sub-harmonic generation etc. Another way to look at high-harmonic generation is through multi-photon physics. For instance, I can look at second-harmonic generation as two photon emission. Is this the correct way to understand this? Is the classical description of nonlinear optics analogous to multi-photon physics? Are they just two ways of describing the same thing? Or am I missing a subtle difference in the terminology here? This is more of a terminology than a physics question.
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$\begingroup$ If you think about it, the if you let the Hamiltonian of the system have a nonlinear term, that leads to coupling between modes and by using ladder operators you can see it leads to terms like $\hat{a_2}^{\dagger}\hat{a_1}^2$ which is the quantum way to describe SHG. Meaning, quantum non-linear optics (NLO) describes well classical NLO phenomena and can only add things that couldn't be described earlier, like generation of spontaneous parametric down conversion (which has non-classical features) $\endgroup$– Ofek GillonCommented Nov 1, 2019 at 17:41
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The difference is that multi-photon physics incorporates quantum phenomena whereas classical nonlinear optics does not. For example, multi-photon phenomena can yield entangled pairs of photons which have no classical analog.
answered Oct 30, 2019 at 23:02