  We study the nonequilibrium steady state of the Bose Hubbard model coupled to
Lindblad reservoirs, using the density matrix renormalization group in operator
space. We observe a transition from a flat particle density profile in the
noninteracting limit to the linear profile with onset of the interparticle
interaction. Analyzing the effect of coherent pumping on the nonequilibrium
steady state we find a subspace which remains unaffected by the pumping in the
noninteracting limit with the protection gradually diminishing due to
interparticle interaction. In the equilibrium situation with one or more
symmetric reservoirs we show analytically that the steady state of the system
is a product state for any interaction strength. We also provide analytical
results in the noninteracting limits, using the method of the third
quantization in operator space.
