Optimal trajectories for efficient atomic transport without final excitation

We design optimal harmonic-trap trajectories to transport cold atoms without final excitation, combining an inverse engineering technique based on Lewis-Riesenfeld invariants with optimal control theory. Since actual traps are not really harmonic, we keep the relative displacement between the center of mass of the transport modes and the trap center bounded. Under this constraint, optimal protocols are found according to different physical criteria. The minimum time solution has a bang-bang form, and the minimum displacement solution is of bang-off-bang form. The optimal trajectories for minimizing the transient energy are also discussed.