According to Clausius formulation of the second law of thermodynamics, for any thermal machine withdrawing heats $Q_{1,2}$ from two heat reservoirs at temperatures $T_{1,2}$, it holds that $Q_1/T_1+Q_2/T_2\le 0$. Combined with the observation that the quantity $Q_1+Q_2$ is the work $W$ done by the system, that inequality tells us that only four operation modes are possible for the thermal machine, namely, heat engine, refrigerator, thermal accelerator, and heater. We illustrate their emergence in the finite time operation of a quantum Otto engine realized with a single qubit. We first focus on the ideal case when isochoric and thermally insulated strokes are well separated and give general results as well as results pertaining to the specific finite-time Landau-Zener dynamics. We then present realistic results pertaining to the solid-state experimental implementation proposed by Karimi and Pekola [Phys. Rev. B 94, 184503 (2016)]. That device is nonadiabatic both in the quantum mechanical sense and in the thermodynamical sense. Oscillations in the power extracted from the baths due to coherent LZ tunneling at too low of temperatures are observed that might hinder the robustness of the operation of the device against experimental noise on the control parameters.

• Received 2 December 2019
• Revised 3 February 2020
• Accepted 4 February 2020

DOI:https://doi.org/10.1103/PhysRevB.101.054513