# The requirement of a relatively weak coupling for the application of the rotating wave approximation to obtain the Jaynes-Cummings model

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?

• This is a good question. I don't think I've ever seen a really quantitative analysis of the conditions under which the RWA work, much less an analysis of the error in various limits. This issue is of particular interest to me because of this experiment. – DanielSank Mar 4 '17 at 20:49
• @DanielSank possibly linked to the non-inertial nature of the rotating frame? – ZeroTheHero Mar 4 '17 at 21:31
• @ZeroTheHero The rotating wave approximation appears in first order differential equations, such as Hamilton's equations of motion in classical physics or the analogous Heisenberg equations in quantum mechanics. Neither of these cases have inertia, so I'm not sure what you mean by non-inertial nature. – DanielSank Mar 4 '17 at 22:59
• Could you give a source where the condition $g \ll min\{\omega_0,\omega\}$ is clearly stated ? Indeed to my knowledge, which might be wrong, in order to move from the Rabi Hamiltonian to Jaynes-Cummings Hamiltonian, you first go to Interaction Picture with $\hat{H}_{int} = \frac{1}{2}\hbar\omega_0\sigma_z + \hbar\omega\hat{a}^{\dagger}\hat{a}$, then neglect the terms $\propto e^{\pm i (\omega_0 + \omega)t}$, then finally move back to the Schödinger Picture. I seems to my that the weak coupling-strength does not play any role here, or maybe I missed something ... – mhham Mar 13 '17 at 13:16
• Okay it seems that my comment was wrong, look at my answer below ! – mhham Mar 13 '17 at 14:04

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).

• So, briefly speaking, the need for small $g$ can be seen when expanding the time evolution operator in interaction picture (Dyson series). For JC model the higher order terms are negelcted when $g$ is much smaller than $\omega_0$ and $\omega$ (i.e. $\mathcal{O}[g^2/(\omega_0+\omega)^2]$). And in combination with $\omega_0\approx \omega$, the counterrotating terms can be negelcted . – Andreas K. Mar 14 '17 at 8:55
• Exactly, you got it ! – mhham Mar 14 '17 at 10:15
• I also insist on the fact that when $g$ is no longer small with respect to $\omega_0 , \omega$, you do not describe completely the physics (i.e Bloch-Siegert shift and/or AC Stark shift) with the JC model. – mhham Mar 14 '17 at 13:05
• Experimental example : THz driven quantum wells: Coulomb interactions and Stark shifts in the ultrastrong coupling regime, Zaks et al. – mhham Mar 14 '17 at 13:17
• Don't forget to accept my answer, if it answers your questions completely of course ! :) – mhham Mar 16 '17 at 16:18