Extractable work in quantum electromechanics.

We investigate quantum electromechanics from the perspective of work extraction. Our work draws inspiration from recent experiments which have coupled a vibrational mode of a suspended carbon-nanotube to a single electron transistor. In one such experiment a threshold voltage has been observed for the onset of self-sustained oscillations in analogy to the theory of the laser. This is suggestive that the vibrational degree of freedom can serve as a fly-wheel or quantum battery for work deposition. In this paper we derive a microscopic model that qualitatively matches the experimental finding. We use our model to calculate the Wigner function of the quantum vibrational mode in the non equilibrium steady state and study the phonon lasing threshold. We characterise the threshold by means of a concept routinely used in the thermodynamics of quantum systems called the ergotropy. This set an upperbound for extractable work from the quantum state by means of a cyclical unitary operation. We find ergotropy serves as an order parameter for the phonon lasing threshold.