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For example, I want to consider the following situation: photon transit from $m$ energy level to $m+2$ after absorption of two phonons with frequency $\Omega$. I want to calculate a transition rate for this process and now I'm little stuck in the choosing the way to do it. Should I use a second-order perturbation theory to find this rate, or can I just multiply the probabilities as a sequential processes ($m \to m+1 \to m+2$ states) to find total probability? Which of these options is more correct?

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  • $\begingroup$ A sequential process would emit photons with very sharply defined energies, whereas in the 2nd order process, the photons' energies could divide up more flexibly. $\endgroup$ – Bert Barrois Jun 15 at 11:22
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To be honest, it depends on the situation, and particularly on how long the pump lasts and what the lifetime of the states is. If you have long-lived states, then sequential processes become possible and they can dominate, but at shorter timescales you expect second-order perturbation theory to be more crucial.

It also depends on how close you are to the resonance on the middle transition. If you're exactly on resonance, then SOPT in its naive form will diverge, until you include the lifetime of the state, and you might be better off with a sequential process. On the other hand, it's quite common to have the full two-photon process be exactly on resonance, but not the middle step; in these cases SOPT is essential.

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  • $\begingroup$ To be more specific, photon initially have frequency $\omega$, my beam passing through the crystal, so the time of perturbation is, for example, $\tau = \frac{Ln}{c}$, where $L$ - length of the crystall in the direction of laser propagation. If we consider two phonon process, then photon can transit on levels $m \pm 2$ (besides other levels) with defined energy through $m\pm1$ level. So what are the difference in using SOPT and sequential processes in this case? $\endgroup$ – Kirill Jun 15 at 11:28

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