  We study a dissipative Bose-Hubbard chain subject to an engineered bath using
a superoperator approach based on matrix product operators. The dissipation is
engineered to stabilize a BEC condensate wave function in its steady state. We
then characterize the steady state emerging from the interplay between
incompatible Hamiltonian and dissipative dynamics. While it is expected that
interactions lead to this competition, even the kinetic energy in an open
boundary condition setup competes with the dissipation, leading to a
non-trivial steady state. We also present results for the transient dynamics
and probe the relaxation time revealing the closing of the dissipative gap in
the thermodynamic limit.
