= Solution
Let $|g\rangle,|e\rangle$ be the atomic levels, $\sigma_+=|e\rangle\langle g|$, and $a$ the <annihilation operator> of the single field mode. Choose the joint ground <energy> to be zero. The resonant <Jaynes-Cummings model> is
$$
\boxed{H=\hbar\omega a^\dagger a+\hbar\omega|e\rangle\langle e|
+\hbar g(a\sigma_++a^\dagger\sigma_-).}
$$
The first term counts field <photons>, the second the atomic excitation, and the interaction exchanges one <photon> for one atomic excitation. A phase redefinition makes $g$ real and positive. Resonance means $\omega=\omega_{eg}$; the total excitation $\mathcal N=a^\dagger a+|e\rangle\langle e|$ is conserved. Adding an overall constant gives the equivalent convention using $\sigma_z/2$ and a zero-point field term.
The model follows from the <electric-dipole approximation>[electric-dipole interaction] and the <rotating-wave approximation>, retaining the energy-conserving terms while dropping $a\sigma_-$ and $a^\dagger\sigma_+$. We assume a single quantized monochromatic mode and negligible <optical cavity>[cavity] leakage and atomic relaxation over the time considered. With this convention, the resonant population <Rabi frequency>[Rabi angular frequency] in the one-excitation subspace is $2g$.
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