  Shortcuts to adiabatic expansion of the effectively one-dimensional
Bose-Einstein condensate (BEC) loaded in the harmonic-oscillator (HO) trap is
investigated by combining techniques of the variational approximation and
inverse engineering. Piecewise-constant (discontinuous) intermediate trap
frequencies, similar to the known bang-bang forms in the optimal-control
theory, are derived from an exact solution of a generalized Ermakov equation.
Control schemes considered in the paper include imaginary trap frequencies at
short time scales, i.e., the HO potential replaced by the quadratic repulsive
one. Taking into regard the BEC's intrinsic nonlinearity, results are reported
for the minimal transfer time, excitation energy (which measures deviation from
the effective adiabaticity), and stability for the shortcut-to-adiabaticity
protocols. These results are not only useful for the realization of fast
frictionless cooling, but also help to address fundamental problems of the
quantum speed limit and thermodynamics.
