Continuously measured quantum systems are characterized by an output current,
in the form of a stochastic and correlated time series which conveys crucial
information about the underlying quantum system. The many tools used to
describe current fluctuations are scattered across different communities:
quantum opticians often use stochastic master equations, while a prevalent
approach in condensed matter physics is provided by full counting statistics.
These, however, are simply different sides of the same coin. Our goal with this
tutorial is to provide a unified toolbox for describing current fluctuations.
This not only provides novel insights, by bringing together different fields in
physics, but also yields various analytical and numerical tools for computing
quantities of interest. We illustrate our results with various pedagogical
examples, and connect them with topical fields of research, such as
waiting-time statistics, quantum metrology, thermodynamic uncertainty
relations, quantum point contacts and Maxwell's demons.Comment: This is a tutorial paper, submitted to PRX Quantu