  We suggest a simple approach to populate photonic quantum materials at
non-zero chemical potential and near-zero temperature. Taking inspiration from
forced evaporation in cold-atom experiments, the essential ingredients for our
low-entropy thermal reservoir are (a) inter-particle interactions, and (b)
energy-dependent loss. The resulting thermal reservoir may then be coupled to a
broad class of Hamiltonian systems to produce low-entropy quantum phases. We
present an idealized picture of such a reservoir, deriving the scaling of
reservoir entropy with system parameters, and then propose several practical
implementations using only standard circuit quantum electrodynamics tools, and
extract the fundamental performance limits. Finally, we explore, both
analytically and numerically, the coupling of such a thermalizer to the
paradigmatic Bose-Hubbard chain, where we employ it to stabilize an $n=1$ Mott
phase. In this case, the performance is limited by the interplay of dynamically
arrested thermalization of the Mott insulator and finite heat capacity of the
thermalizer, characterized by its repumping rate. This work explores a new
approach to preparation of quantum phases of strongly interacting photons, and
provides a potential route to topologically protected phases that are difficult
to reach through adiabatic evolution.
