  We review recent theoretical developments on the stabilization of strongly
correlated quantum fluids of light in driven-dissipative photonic devices
through novel non-Markovian reservoir engineering techniques. This approach
allows to compensate losses and refill selectively the photonic population so
to sustain a desired steady-state. It relies in particular on the use of a
frequency-dependent incoherent pump which can be implemented, e.g., via
embedded two-level systems maintained at a strong inversion of population. As
specific applications of these methods, we discuss the generation of Mott
Insulator (MI) and Fractional Quantum Hall (FQH) states of light. As a first
step, we present the case of a narrowband emission spectrum and show how this
allows for the stabilization of MI and FQH states under the condition that the
photonic states are relatively flat in energy. As soon as the photonic
bandbwidth becomes comparable to the emission linewidth, important
non-equilibrium signatures and entropy generation appear. As a second step, we
review a more advanced configuration based on reservoirs with a broadband
frequency distribution, and we highlight the potential of this configuration
for the quantum simulation of equilibrium quantum phases at zero temperature
with tunable chemical potential. As a proof of principle we establish the
applicability of our scheme to the Bose-Hubbard model by confirming the
presence of a perfect agreement with the ground-state predictions both in the
Mott Insulating and superfluid regions, and more generally in all parts of the
parameter space. Future prospects towards the quantum simulation of more
complex configurations are finally outlined, along with a discussion of our
scheme as a concrete realization of quantum annealing.
