Delayed-response quantum back action in nanoelectromechanical systems

We present a semiclassical theory for the delayed response of a quantum dot (QD) to oscillations of a coupled nanomechanical resonator (NR). We prove that the back action of the QD changes both the resonant frequency and the quality factor of the NR. An increase or decrease in the quality factor of the NR corresponds to either an enhancement or damping of the oscillations, which can also be interpreted as Sisyphus amplification or cooling of the NR by the QD.

Figure 1

Schematic diagram of a system composed of a nanomechanical resonator (green) electrostatically coupled to a quantum dot (red). The source (left) and drain (right) electrodes of the QD are biased by the voltage $V_{\mathrm{SD}}$; the QD state is controlled by the gate voltage $V_{\mathrm{g}}$. The NR is actuated by the voltage $V_{\mathrm{NR}}\left(t\right)=V_{\mathrm{NR}}+V_{\mathrm{A}}sin{\omega }_{0}t$. The coupling between the NR and the QD is characterized by the displacement-dependent capacitance $C_{\mathrm{NR}}\left(u\right)$.

Figure 2

The gate-voltage offset ${\varepsilon }_0$ dependence of (a) the energy levels $E_{\pm }/\Delta$; (b) average electron number $〈n〉$; (c) QD current $I$ and the force $F_{\mathrm{q}}$ which influences the NR, where $I/I_0=F_{\mathrm{q}}/F_0$; and (d) the NR quality-factor changes $\Delta Q$ and the work $W$ on the NR, where $\Delta Q/Q_{\mathrm{S}}=W/W_0$. Closed trajectories in (c) describe the delayed value of the force, and the hatched areas give the work $W=\oint du{F}_{\mathrm{q}}$. If there is no delay ($\tau \to 0$), then the red and blue hatched ovals in (c) merge with the solid green curve, which describes the adiabatic evolution. The source-drain current $I$ in (c) is given by the Lorentzian (green) curve and the quality factor changes $\Delta Q$ in (d) are defined by its derivative, of which the maximum and the minimum are indicated by a square and a rhombus.

Figure 3

The bias ${\varepsilon }_0$ dependence of the response function $\Xi$ for several values of the NR voltage, $n_{\mathrm{NR}}=-C_{\mathrm{NR}}{V}_{\mathrm{NR}}/e$. For $n_{\mathrm{NR}}\sim 1$ the response is described by a peak at ${\varepsilon }_0=0$, while for $\left|n_{\mathrm{NR}}\right|\gg 1$ there are both an increase and a decrease of the function, which relates to the quality factor changes $\Delta Q$, as demonstrated in Fig. 2.

Figure 4

Normalized quality factor shift $\Delta Q$ as the function of the energy bias ${\varepsilon }_0$ calculated with Eq. (A14).

Figure 5

(a) Average excessive electron number on the QD $〈n〉$ as a function of the bias ${\varepsilon }_0$ and (b) changes of the bias due to the periodic evolution of the NR. The right and left halves of the graph illustrate the cycles in which resonator changes the average number of electrons from 0 to 1 and vice versa. See text for a detailed description of these cycles.