Two two-level atoms and form of the Hamiltonian

by McLaren Rulez
Tags: atoms, form, hamiltonian, twolevel
 P: 262 Hello, If we look at a system of two two-level atoms interacting with light, most papers start with a Hamiltonian $$H_{int}=(\sigma_{1}^{+}+\sigma_{2}^{+})a_{\textbf{k},\lambda} + h.c.$$ That is, we absorb a photon and lost one excitation in the atoms or vice versa. Why do we never consider terms like $$\sigma_{1}^{+}\sigma_{2}^{+}a_{\textbf{k},\lambda}a_{\textbf{k},\lambda }$$ 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?
 Sci Advisor HW Helper PF Gold P: 2,602 The interaction Hamiltonian appears in the full Hamiltonian with some coefficient which we can call ##\epsilon##. So ##H = H_0 + \epsilon H_{int}##. We can determine roughly what the value of ##\epsilon## is by considering the fundamental process that leads to ##H_{int}##. This is the quantum electrodynamics interaction term $$L_{int} = e A_\mu\bar{\psi} \gamma_\mu \psi$$ that describes the interaction of a photon, described by the quantum field ##A_\mu##, with the electron, described by the field ##\psi##. Of course ##e## is the electric charge, which in natural units is ##e\sim 1/\sqrt{137} \sim 0.09##. Clearly ##\epsilon## is related to this value, but in order to more accurately estimate it, we would need to add details about the atomic wavefunction, etc. Now, the 2nd order interaction that you consider is an additional correction. It is generated at 2nd order in pertubation theory in QED using the interaction ##L_{int}##. It would also have some coefficient ##\epsilon_2## that is now related to ##e^2##, so it is about 10% as large as the 1st order term. On the other hand, we can also generate 2-photon processes in the 2nd-order perturbation theory of the effective theory using ##H_{int}##. Including this order of correction clearly accounts for some of the lack of precision that we lost by not including the 2nd order term that you wrote down. I don't think it's easy to estimate the difference between the two methods for describing 2-photon processes.