  We investigate the resonantly enhanced tunneling dynamics of ultracold bosons
loaded on a tilted 1-D optical lattice, which can be used to simulate a chain
of Ising spins and associated quantum phase transitions. The center of mass
motion after a sudden tilt both at commensurate and incommensurate fillings is
obtained via analytic, time-dependent exact diagonalization and density matrix
renormalization group methods (adaptive t-DMRG). We identify a maximum in the
amplitude of the center of mass oscillations at the quantum critical point of
the effective spin system. For the dynamics of incommensurate systems, which
cannot be mapped to a spin model, we develop an analytical approach in which
the time evolution is obtained by projecting onto resonant families of small
clusters. We compare the results of this approach at low fillings to the exact
time evolution and find good agreement even at filling factors as large as 2/3.
Using this projection onto small clusters, we propose a controllable transport
scheme applicable in the context of Atomtronic devices on optical lattices
(`slinky scheme').
