  To characterize the statistics and indistinguishability of a source, it is
common to measure the correlation functions of the emitted field using various
interferometers. Here, we present a theoretical framework for the computation
of the correlation functions of a two-level system that is resonantly driven by
a realistic noisy cw excitation laser. Analytic expressions of the first and
second-order auto-correlation functions are obtained where the various
contributions of the noisy excitation source are correctly taken into account.
We predict that, even in the low power regime, the noise source has a strong
influence on the two-level system dynamics, which is not anticipated by simpler
models. The characterization of photon indistinguishability in the pulsed
excitation regime is usually done by measuring the value of the zero-delay
intensity correlation obtained with a Hong-Ou-Mandel interferometer. We show
that this figure is irrelevant in the cw excitation regime and we introduce the
coalescence time window, a figure of merit based on a probabilistic
interpretation of the notion of photon indistinguishability. We finally use the
coalescence time window to quantify how noisy cw excitation influences photon
indistinguishability.
