 Solution

Let $|g⟩,|e⟩$ be the atomic levels, ${\sigma }_{+}=|e⟩⟨g|$, and $a$ the annihilation operator of the single field mode. Choose the joint ground energy to be zero. The resonant Jaynes-Cummings model isThe 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 $\mathcal{N}=a^{†}a+|e⟩⟨e|$ 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 interaction and the rotating-wave approximation, retaining the energy-conserving terms while dropping $a{\sigma }_{-}$ and $a^{†}{\sigma }_{+}$. We assume a single quantized monochromatic mode and negligible cavity leakage and atomic relaxation over the time considered. With this convention, the resonant population Rabi angular frequency in the one-excitation subspace is $2g$.