Normalized correlation functions provide expedient means for determining the photon-number properties of light. These higher-order moments, also called the normalized factorial moments of photon number, can be utilized both in the fast state classification and in-depth state characterization. Further, non-classicality criteria have been derived based on their properties. Luckily, the measurement of the normalized higher-order moments is often loss-independent making their observation with lossy optical setups and imperfect detectors experimentally appealing. The normalized higher-order moments can for example be extracted from the photon-number distribution measured with a true photon-number-resolving detector or accessed directly via manifold coincidence counting in the spirit of the Hanbury Brown and Twiss experiment. Alternatively, they can be inferred via homodyne detection. Here, we provide an overview of different kind of state classification and characterization tasks that take use of normalized higher-order moments and consider different aspects in measuring them with free-traveling light.
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