  We calculate how correlations in a Bose lattice gas grow during a finite
speed ramp from the Mott to the Superfluid regime. We use an interacting
doublon-holon model, applying a mean-field approach for implementing hard-core
constraints between these degrees of freedom. Our solutions are valid in any
dimension, and agree with experimental results and with DMRG calculations in
one dimension. We find that the final energy density of the system drops
quickly with increased ramp time for ramps shorter than one hopping time,
$J\tau_{ramp}\lesssim 1$. For longer ramps, the final energy density depends
only weakly on ramp speed. We calculate the effects of inelastic light
scattering during such ramps.
