Coherent Two-Mode Dynamics of a Nanowire Force Sensor

Classically coherent dynamics analogous to those of quantum two-level systems are studied in the setting of force sensing. We demonstrate quantitative control over the coupling between two orthogonal mechanical modes of a nanowire cantilever through measurement of avoided crossings as we deterministically position the nanowire inside an electric field. Furthermore, we demonstrate Rabi oscillations between the two mechanical modes in the strong-coupling regime. These results give prospects of implementing coherent two-mode control techniques for force-sensing signal enhancement.

Figure 1

(a) Overview of the experimental setup. (b) SEM of the NW used in this work. (c) Simulated electric field of the gate electrodes with the tip of the NW placed at a height of 70 nm above the sample surface. The arrows indicate the direction of the field, while the magnitude is encoded in color scale. The red rectangle indicates the region of the sample measured in the remainder of this work. (d) Interferometric measurement of the power spectral density of the two fundamental flexural modes of the NW taken with a tip-sample distance of 70 nm.

Figure 2

(a) Mode frequencies as a function of voltage applied over the two gates. Data points represented by light blue circles are taken at the position indicated by the dashed circle drawn in the left panel of (b), while the dark blue squares correspond to data taken at the position of the solid circle drawn in the right panel. The curves are fits of Eq. (2) to the data. Gray points are data points not used in the fits. Both sets of data show avoided crossings, with different splittings due to different coupling strengths. (b) Spatial maps of the measured coupling term $F_{12}$ for $V=-6.5\mathrm{V}$ (left) and $V=-5\mathrm{V}$ (right). The dashed black line indicates the position of the nearest gate electrode.

Figure 3

(a) Spatial maps of the experimentally determined force derivatives $F_{11}$ (left), $F_{22}$ (middle), and $F_{12}$ (right). (b) Simulated derivatives of the electric field $-d{E}_1/d{r}_1$ (left), $-d{E}_2/d{r}_2$ (middle), and $|d{E}_2/d{r}_1|$ (right). For all plots $V=-10\mathrm{V}$.

Figure 4

(a) Schematic of Rabi pulsing scheme. (b) Bloch sphere representation of Rabi oscillations. Light blue (vertical) arrow indicates Rabi oscillations for ${\delta }_d=0$; dark blue (tilted) arrow indicates Rabi oscillations for ${\delta }_d\ne 0$. Boxed arrows on the right side schematically indicate NW in the plane for different population distributions. Right panel illustrates approximate oscillation directions of the $|+⟩$ and $|-⟩$ modes with respect to the optical axis of the detector and the oscillation directions of the original modes. (c) Power spectral densities of both modes as a function of Rabi pulse duration. Points indicate measured data; solid lines are fits to the data. The fit for mode $|-⟩$ returns a Rabi frequency of 1.9 kHz ($\mathrm{\Delta }\mathrm{\Omega }/2\pi =10.9\mathrm{kHz}$). Displacement of mode $|+⟩$ is undetectable due to misalignment of this mode with respect to the optical axis of the detector. (d) Similar to (c) but now for ${\delta }_d\ne 0$. Fits yield Rabi frequencies of 2.8 kHz (mode $|-⟩$) and 3.0 kHz (mode $|+⟩$) ($\mathrm{\Delta }\mathrm{\Omega }/2\pi =12.1\mathrm{kHz}$). Note that the offset in the measured Rabi oscillations is due to thermal excitation of the mode.