Quantum Shuttle in Phase Space

We present a quantum theory of the shuttle instability in electronic transport through a nanostructure with a mechanical degree of freedom. A phase space formulation in terms of the Wigner function allows us to identify a crossover from the tunneling to the shuttling regime, thus extending the previously found classical results to the quantum domain. Further, a new dynamical regime is discovered, where the shuttling is driven exclusively by the quantum noise.

Figure 1

Phase space picture of the tunneling-to-shuttling crossover. The respective rows show the Wigner distribution functions for the discharged ($W_{00}$), charged ($W_{11}$), and both ($W_{\mathrm{t}\mathrm{o}\mathrm{t}}$) states of the oscillator in the phase space (horizontal axis–coordinate in units of $x_0=\sqrt{\hslash /m\omega }$, vertical axis–momentum in $\hslash /x_0$). The values of the parameters are $\lambda =x_0$, $T=0$, $d=0.5x_0$, $\Gamma =0.05\hslash \omega$. The values of $\gamma$ are in units of $\hslash \omega$. The Wigner functions are normalized within each column.

Figure 2

$I\mathrm{\text{−}}\gamma$ curve. The $\gamma$ dependence of the stationary current through the grain for different transfer rates and electric fields. Their values are $d=0.5x_0$, $\Gamma =0.05\hslash \omega$ (pluses; corresponds to Fig. 1); $d=0.5x_0$, $\Gamma =0.01\hslash \omega$ (circles); $d=0.0$, $\Gamma =0.05\hslash \omega$ (asterisks); $d=0.0$, $\Gamma =0.01\hslash$ (crosses). Other parameters are $\lambda =x_0$, $T=0$. The current is in units of $e\omega$ while $\gamma$ is in $\hslash \omega$.