  We study the dynamics of phase transitions in the one dimensional
Bose-Hubbard model. To drive the system from Mott insulator to superfluid
phase, we change the tunneling frequency at a finite rate. We investigate the
build up of correlations during fast and slow transitions using variational
wave functions, dynamical Bogoliubov theory, Kibble-Zurek mechanism, and
numerical simulations. We show that time-dependent correlations satisfy
characteristic scaling relations that can be measured in optical lattices
filled with cold atoms.
