  The development and use of Monte Carlo algorithms plays a visible role in the
study of non-Markovian quantum dynamics due to the provided insight and
powerful numerical methods for solving the system dynamics. In the Markovian
case, the connections between the various types of methods are fairly
well-understood while for non-Markovian case there has so far been only a few
studies. We focus here on two jumplike unravelings of non-Markovian dynamics,
the non-Markovian quantum jump (NMQJ) method and the property state method by
Gambetta, Askerud, and Wiseman (GAW). The results for simple quantum optical
systems illustrate the connections between the realizations of the two methods
and also highlight how the probability currents between the system and
environment, or between the property states of the total system, associate to
the decay rates of time-local master equations, and consequently to the jump
rates of the NMQJ method.
