  We report on the design of a Hamiltonian ratchet exploiting periodically at
rest integrable trajectories in the phase space of a modulated periodic
potential, leading to the linear non-diffusive transport of particles. Using
Bose-Einstein condensates in a modulated one-dimensional optical lattice, we
make the first observations of this new spatial ratchet transport. In the
semiclassical regime, the quantum transport strongly depends on the effective
Planck constant due to Floquet state mixing. We also demonstrate the interest
of quantum optimal control for efficient initial state preparation into the
transporting Floquet states to enhance the transport periodicity.
