Telegraph frequency noise in electromechanical resonators

We demonstrate experimentally the possibility of revealing fluctuations in the eigenfrequency of a resonator when the frequency noise is of the telegraph type. Using a resonantly driven micromechanical resonator, we show that the time-averaged vibration amplitude spectrum exhibits two peaks. They merge with an increasing rate of frequency switching and the spectrum displays an analog of motional narrowing. We also show that the moments of the complex amplitude depend strongly on the frequency noise characteristics. This dependence remains valid even when strong thermal noise or detector noise is present.

Figure 1

(a) Scanning electron micrograph of the micromechanical torsional oscillator. (b) Cross-sectional schematic of the device and the measurement circuitry. (c) Typical telegraph noise voltage that is applied to the left electrode. (d) The amplitude spectra of the resonator $|\langle u\rangle |$ with eigenfrequency shifts $\xi =\pm \Delta /2$ (right and left peak, respectively). $\Delta =1.665\text{rad}{\text{s}}^{-1}$ is constant in time. The dashed-dotted curve is an average of the two solid curves.

Figure 2

(a) Measured spectra of the time-averaged complex amplitude $|\langle u\rangle |$ of the resonator under RTFN with different $W/\Delta$. $\Delta$ is kept constant at 1.665 rad ${\text{s}}^{-1}$. For clarity, successive curves are shifted by $10\mu \text{rad}$. (The symbols $▹, *, \circ , \blacksquare , \diamond$, and $•$ correspond to $W/\Delta =0.05$, 0.15, 0.45, 2.36, 43.26 ${\text{rad}}^{-1}$, and no frequency noise applied, respectively.) (b) The corresponding ratios $|\langle u^2\rangle /{\langle u\rangle }^2|$ and (d) $|\langle u^3\rangle /{\langle u\rangle }^3|$. (c) The expanded views of the ratios $|\langle u^2\rangle /{\langle u\rangle }^2|$ and (e) $|\langle u^3\rangle /{\langle u\rangle }^3|$ around 1. In all panels, the lines are the calculated spectra.

Figure 3

(a) Measured amplitude spectrum under weak frequency noise ($\Delta =0.5\Gamma , W/\Delta =1.76{\text{rad}}^{-1}$). The solid line is a fit of the square root of a Lorentzian, yielding a width that is $4.6\%$ larger than $\Gamma$. (b) The distribution of the measured complex amplitudes in the $X-Y$ phase diagram. With the external frequency noise removed, thermal motion of the resonator leads to isotropic distributions centered at the origin (no periodic drive, red) and at the bottom (periodic drive at ${\omega }_F={\omega }_0$, orange). The dashed line represents the calculated $X$ and $Y$ when the driving frequency is swept through ${\omega }_0$. When the resonator is subjected to frequency noise as in (a), the distribution (blue) elongates along the dashed line. (c) The bandwidth of the lock-in is increased by about 1000 so that detector noise makes it more difficult to identify frequency fluctuations.

Figure 4

Spectra of the ratios (a) $|\langle u^2\rangle /{\langle u\rangle }^2|$ and (b) $|\langle u^3\rangle /{\langle u\rangle }^3|$ for the same experimental conditions and under the same frequency noise as in Fig. 3. The open and solid circles correspond to lock-in time constants of 300 ms and $300\mu \text{s}$, respectively. They correspond to (b) and (c) of Fig. 3, respectively. The solid line is the theoretical prediction using Eq. (4). The dashed line shows the calculated values when the telegraph frequency noise is replaced by white Gaussian frequency noise with the intensity $D=\Delta /2\sqrt{2}$.