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since this expansion follows directly from maxwell's equations in Lorentz gauge, it should still hold identically for the cavity because the former also holds. However what changes are the boundary conditions so that the sum over momentum goes from being over $\mathbb{R}$ to being over $\mathbb{N}$ for $p = 2 \pi n\hbar/L$


The OP's characterization of these regimes is not quite universal, and there is some variation. This is partly because there are five important timescales for a Rabi-like system: the resonance and driving frequencies, $\omega_0$ and $\omega$, and the coupling strength $g$, but also the cavity decay rate $\kappa$ and the atomic decay rate $\gamma$. The ...


It is really quite difficult. This gets you to the original paper by Cummings in Phys. Rev. Lett. 140 in 1965, page A1051: http://journals.aps.org/pr/abstract/10.1103/PhysRev.140.A1051 . You must notice first that if $P_e(t)$ is given as you wrote then the coefficients $\frac{e^{-\alpha^2}|\alpha|^{2n}}{{n!}}$ multiplying $\cos()^2$ terms are localized ...

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