We consider a boundary between a Mott insulator and a superfluid region of a
Bose-Hubbard model at unit filling. Initially both regions are decoupled and
cooled to their respective ground states. We show that, after switching on a
small tunneling rate between both regions, all particles of the Mott region
migrate to the superfluid area. This migration takes place whenever the
difference between the chemical potentials of both regions is less than the
maximal energy of any eigenmode of the superfluid. We verify our results
numerically with DMRG simulations and explain them analytically with a master
equation approximation, finding good agreement between both approaches. Finally
we carry out a feasibility study for the observation of the effect in coupled
arrays of micro-cavities and optical lattices.Comment: 5 pages, 6 figures, to appear in Phys. Rev. Let
