Noise-Enabled Precision Measurements of a Duffing Nanomechanical Resonator

We report quantitative measurements of the nonlinear response of a radio frequency mechanical resonator with a very high quality factor. We measure the noise-free transitions between the two basins of attraction that appear in the nonlinear regime, and find good agreement with theory. We measure the transition rate response to controlled levels of white noise, and extract the basin activation energy. This allows us to obtain precise values for the relevant frequencies and the cubic nonlinearity in the Duffing oscillator, with applications to parametric sensing.

Figure 1

(a) Circuit and active and reference beams; dotted outline encloses cryogenic part of experiment. Box labeled $0,180$ is a $180°$ phase splitter, and that labeled $A,\varphi$ allows adjustment of amplitude and phase. Arrow indicates $B$ field orientation. (b) Amplitude versus frequency for drive from $-68$ to $-53\mathrm{dBm}$, in 1 dBm steps. (c) Phase for the same drive amplitudes as (b). (d) Hysteresis in $u_1-u_2$ plane, plotted as $i$ vs $q$ in dimensionless units.

Figure 2

(a) Numerically generated phase-space flow for a drive force 9 dB above the critical point ${\mathcal{B}}_c$, and drive frequency 40 kHz above ${\Omega }_0/2\pi$. Flow begins near the saddle point and evolves toward either focus. (b) Experimental phase-space mean trajectory from focus 1 to focus 2 (8000 averages). (c) Data for phase-space mean trajectory from focus 2 to focus 1 (8000 averages). (d),(e) Experimental time traces for the two switching transitions (8000 averages).

Figure 3

(a) Switching histograms $h(\nu )$ for different noise powers, with $\mathcal{B}=-56\mathrm{dBm}$, for $1\to 2$ transitions. Noise intensity is increased from bottom to top. (b) Transition rates $\Gamma (\nu )$ extracted from switching histograms. (c) Calculated activation energy $E_A(\nu )$ extracted from transition rates and variation in noise power. The noise power was varied from $-127$ to $-113\mathrm{dBm}/\mathrm{Hz}$. Solid curve is fit to expected dependence.

Figure 4

(a) Measured activation energy for drive force ranging from 1 to 7 dB above ${\mathcal{B}}_c$. (b) Numerically calculated activation energies between foci 1 and 2. (c) Log-log plot showing $E_A\propto (\nu -{\nu }_U{)}^2$ dependence near the critical point. (d) ${\nu }_U$ versus drive amplitude $\mathcal{B}/{\mathcal{B}}_c$. At large amplitudes the data diverge from the analytic form.

Figure 5

Amplitude hysteresis plots, for no noise power (bottom), with the drive amplitude set at $-59\mathrm{dBm}$, 2 dB above the critical point. Noise power was increased by 2 dB for each succeeding frame.