Photon statistics in the fluorescence of single molecules and nanocrystals: Correlation functions versus distributions of on- and off-times

The stream of photons emitted by a single quantum system such as a molecule or a nanocrystal is often statistically characterized by the distribution of delays between consecutive photons, or by the autocorrelation function of the intensity, or by the distributions of on- and off-times. We derive and discuss general relations between their Laplace transforms, addressing the influence of detection yield and background. Our analytical treatment applies to any distribution of delays and to random telegraph signals, including nonexponential distributions. We examine the special case of systems switching between two states characterized by different distributions of delays, where the switching can obey various statistics. We show that the second-order autocorrelation function keeps track of long-time fluctuations which are obviously lost in averaging the distributions of delays. We apply our formalism to random telegraphs, in particular to those with power-law distributions of on- and/or of off-times, which ar...