Local versus global equilibration near the bosonic Mott-superfluid transition

We study the response of trapped two dimensional cold bosons to time dependent lattices. We find that in lattice ramps from 11 (superfluid, $\mathrm{\hslash }\mathrm{/}{U}_{\text{i}}=3$ms, $\mathrm{\hslash }\mathrm{/}{J}_{\text{i}}=45$ms) to 16 recoils (Mott, $\mathrm{\hslash }\mathrm{/}{U}_{\text{f}}=2$ms, $\mathrm{\hslash }\mathrm{/}{J}_{\text{f}}=130$ms) the local number fluctuations remains at their equilibrium values if ramps are slower than 3 ms. Global transport, however, is much slower (1s), especially in the presence of Mott shells. This separation of timescales has practical implications for cold atom experiments and cooling protocols.