  In this paper we use optimal control to design minimum-time adiabatic-like
paths for the expansion of a quantum piston. Under realistic experimental
constraints, we calculate the minimum expansion time and compare it with that
obtained from a state of the art inverse engineering method. We use this result
to rederive the known upper bound for the cooling rate of a refrigerator, which
provides a quantitative description for the unattainability of absolute zero,
the third law of thermodynamics. We finally point out the relation of the
present work to the fast adiabatic-like expansion of an accordion optical
lattice, a system which can be used to magnify the initial quantum state
(quantum microscope).

Optimization and Control;Quantum Physics