Activation Barrier Scaling and Crossover for Noise-Induced Switching in Micromechanical Parametric Oscillators

We explore fluctuation-induced switching in parametrically driven micromechanical torsional oscillators. The oscillators possess one, two, or three stable attractors depending on the modulation frequency. Noise induces transitions between the coexisting attractors. Near the bifurcation points, the activation barriers are found to have a power law dependence on frequency detuning with critical exponents that are in agreement with predicted universal scaling relationships. At large detuning, we observe a crossover to a different power law dependence with an exponent that is device specific.

Figure 1

Response of sample $A$ at $\omega /2$ vs the frequency of parametric driving $\omega$. The solid and dashed lines represent the stable attractors and the unstable oscillation states, respectively. Inset: cross-sectional schematic of the device (not to scale).

Figure 2

Oscillation amplitude (a) and phase (c) for $\omega =41163.0\mathrm{rad}{\mathrm{s}}^{-1}$. For ${\omega }_{b2}<\omega <{\omega }_{b1}$, transitions occur when the phase slips by $\pi$. (b) When $\omega (=41124.8\mathrm{rad}{\mathrm{s}}^{-1})$ is lower than ${\omega }_{b2}$, transitions involve jumps in the amplitude. (d) In the high amplitude state, the oscillation phase takes on either one of two values that differ by $\pi$. The phase fluctuates when the oscillator is in the zero-amplitude state.

Figure 3

The occupation in phase space at four different $\omega$’s. $X$ and $Y$ denote the two quadratures of oscillation. The gray scale represents the number of times that the oscillator is measured to lie within a certain location in phase space. (a) $\omega =41171.6\mathrm{rad}{\mathrm{s}}^{-1}$. A pair of oscillation states emerges near ${\omega }_{b1}$. (b) $\omega =41153.0\mathrm{rad}{\mathrm{s}}^{-1}$. As $\omega$ decreases, the two states move further apart in phase space. (c) $\omega =41139.8\mathrm{rad}{\mathrm{s}}^{-1}$. When $\omega <{\omega }_{b2}$, an addition attractor at the origin appears. (d) $\omega =41124.8\mathrm{rad}{\mathrm{s}}^{-1}$. With further decrease in $\omega$, the occupation of the zero-amplitude state increases.

Figure 4

(a) Histogram of the residence time in one of the oscillation states ($\omega =41163.0\mathrm{rad}{\mathrm{s}}^{-1}$) before switching occurs. The solid line is an exponential fit. (b) Logarithm of the transition rate as a function of inverse noise intensity.

Figure 5

(a) Dependence of the activation barriers $R_1$ (solid squares) and $R_2$ (open circles) on the frequency of parametric modulation for sample $A$. (b) $\log R_2$ vs $\log \Delta {\omega }_2$ for sample $A$. The lines are power law fits to different ranges of $\Delta {\omega }_2$. (c), (d) $\log R_1$ vs $\log \Delta {\omega }_1$ for samples $A$ and $B$, respectively.