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Continuously measured quantum systems are characterized by an output current,in the form of a stochastic and correlated time series which conveys crucialinformation about the underlying quantum system. The many tools used todescribe current fluctuations are scattered across different communities:quantum opticians often use stochastic master equations, while a prevalentapproach in condensed matter physics is provided by full counting statistics.These, however, are simply different sides of the same coin. Our goal with thistutorial is to provide a unified toolbox for describing current fluctuations.This not only provides novel insights, by bringing together different fields inphysics, but also yields various analytical and numerical tools for computingquantities of interest. We illustrate our results with various pedagogicalexamples, and connect them with topical fields of research, such aswaiting-time statistics, quantum metrology, thermodynamic uncertaintyrelations, quantum point contacts and Maxwell's demons.

Current fluctuations in open quantum systems: Bridging the gap between quantum continuous measurements and full counting statistics