  In this article we use geometric optimal control to completely solve the
problem of minimum-time transitions between thermal equilibrium and fixed
average energy states of the quantum parametric oscillator, a system which has
been extensively used to model quantum heat engines and refrigerators. We
subsequently use the obtained results to find the minimum driving time for a
quantum refrigerator and the quantum finite-time availability of the parametric
oscillator, i.e. the potential work which can be extracted from this system by
a very short finite-time process.
