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If we look at a system of two two-level atoms interacting with light, most papers start with a Hamiltonian

[tex]

H_{int}=(\sigma_{1}^{+}+\sigma_{2}^{+})a_{\textbf{k},\lambda} + h.c.

[/tex]

That is, we absorb a photon and lost one excitation in the atoms or vice versa. Why do we never consider terms like

[tex]

\sigma_{1}^{+}\sigma_{2}^{+}a_{\textbf{k},\lambda}a_{\textbf{k},\lambda}

[/tex]

Here, the two photons are absorbed simultaneously and we transition directly from the ground state of both to the excited state of both atoms. I suspect that it is because this process is much less likely but how do I prove it?

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# Two two-level atoms and form of the Hamiltonian

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