= Solution
An <atom-field dressed state> is an eigenstate of the coupled atom and drive, rather than an eigenstate of the isolated atom. A strong near-resonant field mixes the bare two-level states and changes the transition spectrum. In a frame rotating at the drive frequency, the <rotating-wave approximation> gives, for one convenient convention,
$$
H_{\rm rot}=\frac\hbar2(-\Delta\sigma_z+\Omega\sigma_x),\qquad \Delta=\omega_L-\omega_0.
$$
Here $\Omega$ is the resonant <Rabi frequency>. The two eigenvalues are
$$
\boxed{E_\pm=\pm\frac\hbar2\Omega_R,\qquad \Omega_R=\sqrt{\Omega^2+\Delta^2}.}
$$
At resonance, the dressed states are equal-weight symmetric and antisymmetric combinations of the bare atom–field configurations. For a quantized drive, the appropriate paired configurations are $|e,n\rangle$ and $|g,n+1\rangle$; their coupling scales as $\sqrt{n+1}$. A strong coherent drive gives the familiar approximately classical Rabi frequency over its occupied photon-number range.
Radiative transitions connect neighboring dressed manifolds. Same-branch transitions have frequency near $\omega_L$, while opposite-branch transitions give sidebands near $\omega_L\pm\Omega_R$. Under sufficiently strong resonant driving this produces the <Mollow triplet> in resonance fluorescence, with sidebands separated from the center by $\Omega$. Radiation damping determines widths and weights; detuning changes the relative intensities. Sidebands must be sufficiently separated compared with the decay scale to be resolved.<Power Spectrum of Light Scattered by Two-Level Systems by B. R. Mollow>[1969 primary calculation of near-resonant driven two-level fluorescence with radiative damping]{ref}
A fluorescence experiment drives the transition with a narrow-band laser, collects scattered light away from the intense drive beam, and resolves its spectrum. Changing drive intensity changes $\Omega$ and hence the splitting. This measures transitions of the interacting system, not merely a population oscillation in time. The emitted spectrum can include an elastic coherent contribution as well as the inelastic fluorescence peaks, so the observed line shape needs the specified drive, damping and detection model.
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