  We study, experimentally and theoretically, the controlled transfer of
harmonically trapped ultracold gases between different quantum states. In
particular we experimentally demonstrate a fast decompression and displacement
of both a non-interacting gas and an interacting Bose-Einstein condensate which
are initially at equilibrium. The decompression parameters are engineered such
that the final state is identical to that obtained after a perfectly adiabatic
transformation despite the fact that the fast decompression is performed in the
strongly non-adiabatic regime. During the transfer the atomic sample goes
through strongly out-of-equilibrium states while the external confinement is
modified until the system reaches the desired stationary state. The scheme is
theoretically based on the invariants of motion and scaling equations
techniques and can be generalized to decompression trajectories including an
arbitrary deformation of the trap. It is also directly applicable to arbitrary
initial non-equilibrium states.
