Various Monte Carlo methods allow the simulation of the dynamics of open quantum systems. We highlight the
central features a commonly used method for Markovian systems, i.e., Monte Carlo Wave Function method and
another one used for non-Markovian systems, i.e., the doubled Hilbert space method. One of the open problems
in the field has been the lack of the existence of Monte Carlo method with quantum jumps for non-Markovian
systems. We characterize the key problems that should be solved in this context and give indications for the way
towards the development of quantum jump method without auxiliary states for non-Markovian systems.
The quantum Zeno effect predicts that the decay of an unstable system can be slowed down by measuring the
system frequently enough. In some systems, however, an enhancement of the decay due to frequent measurements,
namely the anti-Zeno effect, may occur. In this paper we study the Zeno-anti-Zeno crossover for quantum
Brownian motion. Moreover, we propose a way to manipulate the decay of the system and to observe a controlled
continuous passage from decay suppression to decay acceleration by using reservoir engineering techniques.
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