Photon-number statistics in resonance fluorescence

The theory of photon-number statistics in resonance fluorescence is treated, starting with the general formula for the emission probability of $n$ photons during a given time interval $T$. The results fully confirm formerly obtained results by Cook that were based on the theory of atomic motion in a traveling wave. General expressions for the factorial moments are derived and explicit results for the mean and the variance are given. It is explicitly shown that the distribution function tends to a Gaussian when $T$ becomes much larger than the natural lifetime of the excited atom. The speed of convergence towards the Gaussian is found to be typically slow, that is, the third normalized central moment (or the skewness) is proportional to $T^{-\frac{1}{2}}$. However, numerical results illustrate that the overall features of the distribution function are already well represented by a Gaussian when $T$ is larger than a few natural lifetimes only, at least if the intensity of the exciting field is not too small and its detuning is not too large.