  Stochastic methods with quantum jumps are often used to solve open quantum
system dynamics. Moreover, they provide insight into fundamental topics, as the
role of measurements in quantum mechanics and the description of non-Markovian
memory effects. However, there is no unified framework to use quantum jumps to
describe open system dynamics in any regime. We solve this issue by developing
the Rate Operator Quantum Jump (ROQJ) approach. The method not only applies to
both Markovian and non-Markovian evolutions, but also allows us to unravel
master equations for which previous methods do not work. In addition, ROQJ
yields a rigorous measurement-scheme interpretation for a wide class of
dynamics, including a set of master equations with negative decay rates, and
sheds light on different types of memory effects which arise when using
stochastic quantum jump methods.
