Description of AMO interactions in terms of QED Lagrangian

Question

I consistently had this question of how could the light-matter interaction be described in terms of the fundamental language of QED. To be more specific, is there a way to 'derive' the interaction Hamiltonian such as Jaynnes-Cummings Hamiltonian from the fundamental interaction Lagrangian such as

$$\mathcal{L}=i\overline{\psi}\gamma^{\mu}(\partial_{\mu}-ieA_{\mu})\psi-m\overline{\psi}\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$$

used nearly exclusively in the particle physics context to describe electron photon interaction?

I found a similar question raised in

What is the difference between QED and quantum optics?

but the answer to the question only stated that full quantum field theoretic description is rarely used and semiclassical description suffices and is convenient in most cases.

Answer 1

I'm not an expert in this field but I comment briefly: The answer to your question is yes. The first term in your expression essentially couples the electron momentum $p$ to the field amplitude $A$ so that you have $H = A \cdot p$. From this you can go through a few steps (including the Power Zienau transform and dipole approximation) to eventually get the $E\cdot d$ Hamiltonian from which usual "AMO" type interactions are derived.