Scaling laws for the bifurcation escape rate in a nanomechanical resonator

We report on experimental and theoretical studies of the fluctuation-induced escape time from a metastable state of a nanomechanical Duffing resonator in a cryogenic environment. By tuning in situ the nonlinear coefficient $\gamma$ we could explore a wide range of the parameter space around the bifurcation point, where the metastable state becomes unstable. We measured in a relaxation process the distribution of the escape times. We have been able to verify its exponential distribution and extract the escape rate $\mathrm{\Gamma }$. We investigated the scaling of $\mathrm{\Gamma }$ with respect to the distance to the bifurcation point and $\gamma$, finding an unprecedented quantitative agreement with the theoretical description of the stochastic problem. Simple power scaling laws turn out to hold in a large region of the parameter space, as anticipated by recent theoretical predictions. These unique findings, implemented in a model dynamical system, are relevant to all systems experiencing underdamped saddle-node bifurcation.

Figure 1

Top panel: Schematic of the experimental setup with the nanoresonator structure. Bottom panel: Linear and Duffing resonances (respectively gray and black points, with top-right and bottom-left axes). The lines show the fit. The nonlinear resonance is for $V_g=9.4\mathrm{V}$, which shifts the resonance frequency and opens a hysteresis (thin green arrows highlight upward and downward sweeps). The relaxations occur at a detuning $\omega -{\omega }_b$ from the bifurcation frequency (red point and vertical arrow). Inset: Gaussian distribution histogram of the measured intrinsic frequency fluctuations.

Figure 2

Bifurcation parameter space (normalized driving force versus $\mathrm{\Omega }=2|\omega -{\omega }_0|/\mathrm{\Delta }\omega$). The gray area is the NEMS bistability regime where the right edge is the transition from a high-amplitude oscillation to a low one (the left edge is the opposite) and K is the spinode point where hysteresis starts to open. We show within the bistability the data points at different voltages $V_g$. Inset: Typical low-$V_g$ relaxation curve obtained with about 1000 relaxations, and fit with and without fluctuations on ${\omega }_b$.

Figure 3

Escape time as a function of $D^{-1}$ for $V_g=9.4\mathrm{V}$ at different detunings $\omega -{\omega }_b$ from the bifurcation point.

Figure 4

Scaling plots for $E_a$ (left) and ${\mathrm{\Gamma }}_0$ (right) with respect to detuning. The full circles indicate the experimental points, the open (blue) triangles the prediction of the full numerical simulation, the (red) full lines the linear fit to the data, and the dashed (blue) lines the prediction of Eq. (4). Insets: Scaling with the nonlinear parameter ${\mathrm{\Omega }}_b$.