Transition rates with incoherent radiation When deriving the transition rates for atoms in radiation, one calculates the rate for a single frequency and then adds the rates corresponding to all frequencies in the spectrum. The reason for this comes down to the fact that the radiation is incoherent, so we add intensities instead of amplitudes.
I was under the impression however that there is no such thing as perfectly coherent/incoherent radiation.  So when the radiation here is described as incoherent, does this mean that it is incoherent relative to some intrinsic atomic timescale which we average over (i.e. we average over a time much longer than the coherence time)?
If this is the case, what is the timescale, and how would one go about finding the transition rates if the radiation were not sufficiently incoherent?
 A: Typically when calculating transition rates one uses the perturbative treatment, where the incident electric field induces a small dipole energy compared to the energy separation $E_2 - E_1$ between the ground state $1$ and excited state $2$. Now about the field itself you may characterize its 'coherence degree' by the coherence time $\tau$, which can be thought as the average amount of time you can approximate your field as a coherent plane wave.
As a rule of thumb, you should always start by comparing the coherence time with the transition frequency $\omega_0 = (E_2 - E_1)/\hbar$. If the coherence time is much larger than a single transition cycle ($\tau \gg 2\pi/\omega_0$), the field drives the atom coherently for many cycles, thus a coherent treatment is appropriate. In the other hand, if the coherence time is much shorter than the transition time ($\tau \ll 2\pi/\omega_0$) the field loses its phase relationship before the atom can make a transition, and thus the incoherent approach is better suited. When the coherence time is close to the transition time ($\tau \sim 2\pi/\omega_0$) a more careful approach is needed.
Anyway, in the coherent case you may not be able to apply the perturbative approach because perturbation theory is well suited for weak broadband incident fields (like incoherent ones), whereas coherent radiation is usually narrowband, and can have very high field amplitudes. There is a more general approach where you treat your field in a more general way, and it involves solving the Optical Bloch Equations. Such an approach can be found in many Quantum Optics textbooks.
