In order to obtain the Jaynes-Cummings Hamiltonian, the RWA is applied to the Rabi Hamiltonian: $$H=\frac{1}{2}\hbar\omega_0 \sigma_z+\hbar\omega \hat{a}^{\dagger}\hat{a}+\hbar g(\sigma_{+}+\sigma_{-})(\hat{a}+\hat{a}^{\dagger}),$$ under two conditions: the near-resonance $\omega_0\approx\omega$, and the relatively weak coupling strength $g\ll \text{min}\{\omega_0,\omega\}$. While the requirement for the first condition is reasonable (the terms $\sigma_{+}\hat{a}^{\dagger}$ and $\sigma_{-}\hat{a}$ become rapidly oscillating for $|\omega_0-\omega|\ll \omega_0+\omega$, as seen in the interaction picture w.r.t. the free Hamiltonian, where they acquire phase factors $e^{\pm i(\omega_0+\omega)t}$), the condition for the weak coupling is not so evident and usually is not explained in the introductory quantum optics textbooks. So, why is it needed?
1 Answers
I will not go into much detail here, but rather give you the link to this answer.
In order to sum it up very quickly let us simply state that the RWA which gives rise to the Jaynes-Cummings Hamiltonian is an on-resonance perturbative theory, where we neglect the fast rotating terms in the Rabi Hamiltonian when written in the interaction picture.
In the answer, a simple model was given where an atom is classically driven by a field. The coupling constant is thus proportional to the driving field, and it is stated that :
It is essential to emphasize that, as the applied field increases, this approximation becomes even less reliable and it is just the leading order of a perturbation series in a near-resonance regime.
This is a direct analogue of the $g \ll min\{ω_0,ω\}$ condition.
Hence one could say that the Rabi and Jaynes-Cummings Hamiltonian describe the same physics as soon as both conditions (near-resonance and weak coupling) are verified. If the coupling becomes strong (as in superconducting qubits for instance), the Jaynes-Cummings Hamiltonian no longer describes completely the physics, since higher order terms start to play a role. (cf. Bloch-Siegert shift and/or AC Stark shift).
An interesting paper on this topic : A modern review of the two-level approximation by Marco Frasca.
Edit : Also, a very elegant way to look at these light-atom interaction problems, is through the dressed-atom formalism (Atom-Photon Interactions - Chapter 6 The Dressed Atom Approach by Claude Cohen-Tannoudji , or any introductory ressource that builds the dressed-atom approach starting from the Rabi Hamiltonian and not the Jaynes-Cummings one).
