Measuring higher-order photon correlations of faint quantum light: A short review

•As a tutorial review provides an overview on the experimental aspects in measuring photon correlations.•Highlights the practicality of the higher-order moments in quantum state classification and characterization.•Sums up the usage of the normalized moments with orders higher than two in quantum optics and brings together crucial experimental considerations.•Anchors the normalized higher-order moments as quantum optical characterization tools and emphasizes their importance in the state tomography. 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.