Fermi's Golden rule: Accounting for Decoherence On the Wikipedia page for Fermi's golden rule, there is a vague statement that is given in passing:

... if there is some decoherence in the process, like relaxation or collision of the atoms, or like noise in the perturbation, in which case the density of states is replaced by the reciprocal of the decoherence bandwidth.

I couldn't find any information about this in any of my textbooks/references. I was wondering if someone could point me to a resource that discusses this in depth, or at least provide a more precise statement of it. What is the exact definition of a "decoherence bandwidth"? How is this result derived? I appreciate any help!
 A: I could point you to my two answers on the subject: here and here. Standard derivations of Fermi Golden rule make some assumptions, which allow for really simple derivation, but hide some important physics.
Finite density-of-states
One important point is that delta-function in the FGR is impractical for any practical purposes, unless the transitions are happening from/to continuous spectrum, so that the integral over the initial/final states takes care of the delta function. In solid state physics this is not a problem, but in atomic physics it is hard to have continuous spectrum without considering explicitly coupling to quantized photon field, which is beyond the basic QM. This is why one uses hand-waving to justify replacing the delta function by the density of final states or to insert Lorentz broadening in it, or something of the kind - which is just a way to introduce in a simple way the elements that could be obtained rather rigorously from more complex/details models, e.g., calculating cross-section for an atom coupled to a quantized EM field or using Keldysh formalism.
This is probably what is meant by "the density of states is replaced by the reciprocal of the decoherence bandwidth". We account for the natural broading of the linewhidth by
$$
\delta (E_f-E_i-\hbar\omega) \longrightarrow \frac{1}{2\pi}\frac{\hbar\gamma}{(E_f-E_i-\hbar\omega)^2+\hbar^2\gamma^2},
$$
where $\gamma$ is the natural linewidth, and $\tau=1/\gamma$ is the lifetime of the excited state. Then at the resonance the delta function is replaced simply by
$$
\frac{1}{2\pi\hbar\gamma}
$$
Long times limit
Another rather obvious elephant in the Fermi Golden Room is taking the limits like
$$
\lim_{t\rightarrow+\infty}\frac{\sin^2(\omega t)}{\omega^2}
$$
Obviously, the duration of an experiment is not infinite. Moreover, saying "that $t$ is big" is meaningless, unless we compare it with something. This is where one needs to compare it with typical time (or inverse frequency) scales, such as decoherence time or spacing between the energy levels (which usually implies that that the levels are much closer than the level broadening, that is inverse decoherence time).
Ironically, the introduction of the finite density-of-states is mostly undoing of this earlier step in the derivation.
