Extractable work in quantum electromechanics

Recent experiments have demonstrated the generation of coherent mechanical oscillations in a suspended carbon nanotube, which are driven by an electric current through the device above a certain voltage threshold, in close analogy with a lasing transition. We investigate this phenomenon from the perspective of work extraction, by modelling a nano-electromechanical device as a quantum flywheel or battery that converts electrical power into stored mechanical energy. We introduce a microscopic model that qualitatively matches the experimental finding, and compute the Wigner function of the quantum vibrational mode in its non-equilibrium steady-state. We characterise the threshold for self-sustained oscillations using two approaches to quantifying work deposition in non-equilibrium quantum thermodynamics: the ergotropy and the non-equilibrium free energy. We find that ergotropy serves as an order parameter for the phonon lasing transition. The framework we employ to describe work extraction is general and widely transferable to other mesoscopic quantum devices.