Kinetics of many-body reservoir engineering

Quantum simulators based on superconducting circuits can be used to study
many-body physics with microwave photons. In particular, lattice systems can be
used to study the thermalization of an engineered condensed matter system to a
cold reservoir. However, in stark contrast to naturally occurring condensed
matter systems, most engineered quantum lattice systems do not have a
well-defined chemical potential. Here, we show that it is possible to cool a
reservoir engineered many-body system with a well-defined chemical potential if
the coupling with the reservoir is a density-density type coupling.
Specifically, we consider an array of single-mode microwave cavities with each
site of the array coupled to a driven, lossy cavity. The lossy cavities will
act as an engineered non-equilibrium bath that mediates transitions between the
eigenstates of the array while simultaneously conserving the number of
excitations. We show that the steady state of the system can be described via a
modified Bose-Einstein distribution with a momentum-dependent temperature.