  We develop a method to study quantum impurity models, small interacting
quantum systems linearly coupled to an environment, in presence of an
additional Markovian quantum bath, with a generic non-linear coupling to the
impurity. We aim at computing the evolution operator of the reduced density
matrix of the impurity, obtained after tracing out all the environmental
degrees of freedom. First, we derive an exact real-time hybridization expansion
for this quantity, which generalizes the result obtained in absence of the
additional Markovian dissipation, and which could be amenable to stochastic
sampling through diagrammatic Monte Carlo. Then, we obtain a Dyson equation for
this quantity and we evaluate its self-energy with a resummation technique
known as the Non-Crossing-Approximation. We apply this novel approach to a
simple fermionic impurity coupled to a zero temperature fermionic bath and in
presence of Markovian pump, losses and dephasing.
