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The question itself is pretty self explanatory. I asked this to my chemistry teacher when he was doing the photoelectric effect while teaching atomic structure, and he just shrugged it off.

One possible mechanism that I thought up, is an atom with a unpaired electron (say hydrogen) to absorb one photon, and jump to a higher energy level, and then just immediately be hit by another photon with a different energy which raises it to an even higher level. Then finally the electron falls back into its proper non exited state releasing not two but only one photon which has the energy of both the initial photons, ideally combining photons.

Is this mechanism correct? Does it occur in nature? Do photons have other ways in which they merge? or do they not merge at all?

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    $\begingroup$ Yes: two-photon absorption $\endgroup$ – lemon Oct 2 '17 at 15:09
  • $\begingroup$ As @lemon pointed out, this has many applications. $\endgroup$ – Sayan Mandal Oct 2 '17 at 15:11
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To answer the question in the title, yes, see this link. It is also possible to combine more photons in order to get superior harmonics. For this you have a list that you can use to get some fast info on Wikipedia on this link.

Now, related to the mechanism that you proposed, in principle it could work, but you have to take into account the selection rules, which allows only for some transition between the energy levels on a system, in your case the hydrogen atom. What this means, is the following. Say the hydrogen atom is on its ground state $0$. First photon changes this state $n$ and the second from state $n$ to state $m$. Now, if the transition from $m$ to $0$ is not allowed by the selection rules, then your mechanism won't work the way you intended. You might however get 2 photons with different energies from the initial ones.

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    $\begingroup$ This (and every) answer should make it clear that it is not possible to do this in free space, but occurs in interaction with a material medium. $\endgroup$ – dmckee Oct 2 '17 at 15:42
  • $\begingroup$ Firstly, the Wikipedia page on Second-harmonic generation mentions that it is a process that can combine photons of the same frequency (hence same energy). Is there no way to combine photons of different energies? Secondly, I did not understand the part where you mention that I might end up getting two photons of different energies. Is it such that the electron selectively absorbs a part of the energy of the photon which gives it sufficient energy to jump to the higher level? $\endgroup$ – Debaditya Oct 2 '17 at 16:46
  • $\begingroup$ Third, as @dmckee mentioned, as second-harmonic generation requires interactions with material mediums, aren't these two phenomenon two sides of the same coin where we the electrons of the material's atoms absorb and re-emit these photons? $\endgroup$ – Debaditya Oct 2 '17 at 16:50
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    $\begingroup$ About adding 2 photons of different energies. On the second link (the one with the list of nonlinear processes) you wave Sum Frequency Generation. $\endgroup$ – Victor Palea Oct 2 '17 at 16:59
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    $\begingroup$ And the third question, I'm won't say that harmonic generation and the hydrogen transition cases are similar. In the case of the H atom, the transition takes place only for some wavelengths so your wavelength mixing can be made only using some specific values. This is only due to the fact that the photons have to be absorbed by the atom. On the harmonics case, the phenomenon is not dependent to wavelength, in the sense that it is not restricted to one some distinct values of it. I might be wrong here, but based on the materials I've seen, the thing is presented for "any" wavelength. $\endgroup$ – Victor Palea Oct 2 '17 at 17:12

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