Noise-activated switching in a driven nonlinear micromechanical oscillator

We study noise induced switching in systems far from equilibrium by using an underdamped micromechanical torsional oscillator driven into the nonlinear regime. Within a certain range of driving frequencies, the oscillator possesses two stable dynamical states with different oscillation amplitudes. We induce the oscillator to escape from one dynamical state into the other by introducing noise in the excitation. By measuring the rate of random transitions as a function of noise intensity, we deduce the activation energy as a function of frequency detuning. Close to the critical point, the activation energy is expected to display system-independent scaling. The measured critical exponent is in good agreement with variational calculations and asymptotic scaling theory.

Figure 1

(a) A cross-sectional schematic of the micromechanical torsional oscillator with electrical connections and measurement circuitry (not to scale). (b) Frequency response of sample A scaled by the excitation voltage amplitudes of $47\mathrm{\mu }\mathrm{V}$ (squares) and $450\mathrm{\mu }\mathrm{V}$ (circles). The dotted line represents a fit to the data at smaller excitation using the response of a damped harmonic oscillator. For the large excitation, two dynamical states coexist from $3278.76\mathrm{Hz}$ to $3280.8\mathrm{Hz}$. The dashed line fits the data to a damped oscillator with cubic nonlinearity (Ref. 18), yielding $\kappa =-1.25\times {10}^7\mathrm{rad}{\mathrm{s}}^{-1}$.

Figure 2

(a) In the presence of noise in the excitation, the oscillator switches from the high-amplitude state to the low-amplitude state at different time intervals. The system is reset to the upper amplitude state between switching events. The detuning frequency is $\mathrm{\Delta }\nu =0.05\mathrm{Hz}$. (b) Histogram of the residence time in the upper state before switching occurs, at a detuning frequency of $0.05\mathrm{Hz}$ for sample A. The dotted line is an exponential fit.

Figure 3

Logarithm of the transition rate from the high-amplitude state as a function of inverse noise intensity at a detuning frequency $\mathrm{\Delta }\nu$ of $0.25\mathrm{Hz}$ for sample A. The slope of the linear fit yields the activation energy.

Figure 4

Dependence of the activation energy on detuning frequency for (a) sample A and (b) sample B. The solid lines are power law fits, yielding critical exponents of $1.38\pm 0.15$ and $1.4\pm 0.15$, respectively.