= Solution
{c}
<Rabi oscillations> require a coherent drive with a stable phase over several population cycles and an isolated, appreciably coupled pair of <quantum states>. For large-amplitude oscillations, the <Rabi detuning> must be comparable to or smaller than the <Rabi angular frequency>. The population lifetime, <optical coherence time>, transit time and drive coherence time should all exceed $2\pi/\Omega_R$ by enough to resolve the cycles. An ensemble also needs sufficiently uniform <Rabi frequencies> and <Rabi detunings> that phase averaging does not wash the oscillations out. The usual sinusoidal formula additionally assumes the <rotating-wave approximation>.
The <Einstein coefficients> describe incoherent transition \i[rates], rather than the evolving optical <probability amplitudes>. Their simplest use is appropriate for broadband, randomly phased <radiation>, with field correlation time short relative to population evolution, or for rapidly dephased <chemical atoms> whose optical coherences can be eliminated. The <radiation energy density> per unit <frequency> must be smooth over the relevant <spectral line>. Stimulated upward and downward transitions and <spontaneous emission> then enter <rate equations>, without predicting coherent population return. A short observation time can give a small excitation <probability> even for a coherent field; that condition alone does not justify an <Einstein coefficients> rate description. Coherent <laser> irradiation can also admit steady <rate equations> after optical coherence has relaxed, with the appropriate <saturation> response.
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