Rotating wave approximation (RWA) in Dicke model The interaction part of Rabi Hamiltonian is defined as
$$H_I = g (a + a^{\dagger})(\sigma_{+}+\sigma_{-})$$
where $g$ is the coupling between atom and the field, the time dependencies of field and atomic operators are
$$a(t) = a(0) e^{-i\omega t} $$
$$a^{\dagger}(t) = a^{\dagger}(0) e^{i\omega t} $$
$$\sigma_{+}(t) = \sigma_{+}(0) e^{-i\omega_{mn} t} $$
$$\sigma_{-}(t) = \sigma_{-}(0) e^{i\omega_{mn} t} $$
The product of atomic and field operators has four terms with two detunings  $\delta_{-} = \omega_{mn}-\omega$ (corresponds to co-rotating terms) and  $\delta_{+} = \omega_{mn} + \omega$ (corresponds to counter-rotating terms). Under the condition $\color{red} {g << \delta_{-} <<\delta_{+}}$ (condition for RWA), we can neglect the counter-rotating terms and the resulting Hamiltonian is called Jaynes-Cummings Hamiltonian
$$H_{I} = g (\sigma_{+}a + \sigma_{-}a^{\dagger}).$$
My question: When it comes to Dicke Hamiltonian, the coupling between atom and field have explicit dependence on the size of the system.  Suppose there are $j$ atoms, the coupling is scaled as $jg$, does this scaling affect the condition for RWA? I mean instead of ${g << \delta_{-} <<\delta_{+}}$, do we need $jg << \delta_{-} <<\delta_{+}$ for RWA to be valid in Dicke Hamiltonian?
 A: First of all the RWA condition is not $g\ll\delta$ but $g\ll \omega$ since the counter-rotating term gives $e^{\pm i(\omega+\omega_{mn})}$ oscillatory term.
Also, you have the scaling of the (collective) coupling strength wrong since it scales as $\sqrt{j}g$. This is because the interaction is ''coherent'', or ''superradiant'', or whatever other names you prefer. To be more specific, the matrix element between $(n+1)$-photon and $0$-atomic excitation and $n$-photon and $1$-atomic excitation is (here I use $\chi$ to represent the coupling strength and $g$ for ground state, to avoid confusions)
$$H_{n,01}=\langle g\cdots g|\otimes\langle n+1|H|n\rangle\otimes|\text{1 atomic excitation}\rangle$$
where due to the symmetry, the 1 atomic excitation state should be
$$|\text{1 atomic excitation}\rangle=\frac{1}{\sqrt{j}}\sum_{i=1}^j |g\cdots ge_ig\cdots g\rangle,$$
and using $H=\sum_{i=1}^j\chi(a\sigma_+^i+a^\dagger\sigma_-^i)$, one can easily evaluate that
$$\begin{split}H_{n,01}&=\left\langle g\cdots g\middle|\otimes\middle\langle n+1\middle|\sum_{i=1}^j\chi(a\sigma_+^i+a^\dagger\sigma_-^i)\middle|n\middle\rangle\otimes\frac{1}{\sqrt{j}}\sum_{i'=1}^j \middle|g\cdots ge_{i'}g\cdots g\right\rangle\\
&=\frac{\chi}{\sqrt{j}}\sum_{i=1}^j\langle n+1|a^\dagger|n\rangle\cdot\langle g\cdots g|\sigma_-^i|ge_{i}g\cdots g\rangle\\
&=\frac{\chi\sqrt{n+1}}{\sqrt{j}}\sum_{i=1}^j1=\chi\sqrt{n+1}\sqrt{j}.\end{split}$$
With these corrections, let me give you some intuitions about real numbers. The $g$ for optical cavity QED experiments can vary from 100kHz to MHz, or, let's put some extreme values, can go to GHz. The atom number is usually no more than $10^5$. Together this gives $\sqrt{j}g$ less than THz, while the atomic transition is usually hundreds of THz. The RWA holds.
However, in other cases like the superconducting circuit's artificial atom-cavity mode interaction, the $g$ can indeed go higher than $\omega$, a regime called the ultrastrong coupling regime. I recommend the following review paper so that you can make a bit more sense.
"Ultrastrong coupling regimes of light-matter interaction" by P. Forn-Díaz, L. Lamata, E. Rico, J. Kono, and E. Solano, Rev. Mod. Phys. 91, 025005 (2019)
