  The non-Markovian nature of quantum systems recently turned to be a key
subject for investigations on open quantum system dynamics. Many studies, from
its theoretical grounding to its usefulness as a resource for quantum
information processing and experimental demonstrations, have been reported in
the literature. Typically, in these studies, a structured reservoir is required
to make non-Markovian dynamics to emerge. Here, we investigate the dynamics of
a qubit interacting with a bosonic bath and under the injection of a classical
stochastic colored noise. A canonical Lindblad-like master equation for the
system is derived, using the stochastic wavefunction formalism. Then, the
non-Markovianity of the evolution is witnessed using the Andersson, Cresser,
Hall and Li measure. We evaluate the measure for three different noises and
study the interplay between environment and noise pump necessary to generate
quantum non-Markovianity, as well as the energy balance of the system. Finally,
we discuss the possibility to experimentally implement the proposed model.
