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 modeling a nanoelectromechanical 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 we compute the Wigner function of the quantum vibrational mode in its nonequilibrium steady state. We characterize the threshold for self-sustained oscillations using two approaches to quantifying work deposition in nonequilibrium quantum thermodynamics: the ergotropy and the nonequilibrium 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.

Figure 1

Schematic depiction of the electromechanical system. A central quantum dot is coupled to two electrodes as well as a quantum harmonic oscillator representing the mechanical degree of freedom. The electrodes are described by Fermi-Dirac distributions $f_{\alpha }\left(\omega \right)$ at the same temperature but different chemical potentials. Electrons tunneling through the quantum dot under this voltage bias excite self-sustained oscillations of the mechanical motion.

Figure 2

Wigner functions of the oscillator at various biases. The left panel shows the Wigner function at zero applied bias showing the thermal blob, the central panel shows the Wigner function at 6 applied bias just after the threshold voltage, the right panel shows the Wigner function at 16 bias where the voltage has reached its saturation point. Parameters: ${\omega }_0=0.2$, $m=1$, $\lambda =0.1$, ${\omega }_L=0.5$, ${\omega }_R=-0.5$, ${\delta }_L={\delta }_R=1$, ${\mathrm{\Gamma }}_L={\mathrm{\Gamma }}_R=2$, ${\beta }_L={\beta }_R=0.5$, $x_0={(2/m{\omega }_0)}^{1/2}$, and $p_0={\left(2m{\omega }_0\right)}^{1/2}$.

Figure 3

Second-order coherence function of the oscillator as a function of voltage, for the same parameters as in Fig. 2.

Figure 4

Total work extractable from the steady state of the vibrational mode via a unitary transformation (${\mathcal{W}}_E$, red, dotted) and a thermal operation (${\mathcal{W}}_F$, red, dashed). The threshold voltage is indicated by the vertical dot-dashed line and the von Neumann entropy of the oscillator is shown in blue. The same parameters as Fig. 2 are used.