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In this article we use geometric optimal control to completely solve theproblem of minimum-time transitions between thermal equilibrium and fixedaverage energy states of the quantum parametric oscillator, a system which hasbeen extensively used to model quantum heat engines and refrigerators. Wesubsequently use the obtained results to find the minimum driving time for aquantum refrigerator and the quantum finite-time availability of the parametricoscillator, i.e. the potential work which can be extracted from this system bya very short finite-time process.

Minimum-Time Transitions between Thermal and Fixed Average Energy States of the Quantum Parametric Oscillator