  We present an investigation of the fast decompression of a three-dimensional
(3D) Bose-Einstein condensate (BEC) at finite temperature using an engineered
trajectory for the harmonic trapping potential. Taking advantage of the scaling
invariance properties of the time-dependent Gross-Pitaevskii equation, we
exhibit a solution yielding a final state identical to that obtained through a
perfectly adiabatic transformation, in a much shorter time. Experimentally, we
perform a large trap decompression and displacement within a time comparable to
the final radial trapping period. By simultaneously monitoring the BEC and the
non-condensed fraction, we demonstrate that our specific trap trajectory is
valid both for a quantum interacting many-body system and a classical ensemble
of non-interacting particles.
