Thermal frequency noise in micromechanical resonators due to nonlinear mode coupling

Nonlinear coupling between different normal modes of a mechanical resonator is a relevant issue in nanomechanical resonators with high aspect ratio, as well as in very low mass resonators based on graphene and nanotube resonators or in trapped nanoparticles. Here we demonstrate that nonlinear coupling between the two orthogonal flexural modes of a high aspect ratio microcantilever results in measurable effects down to the thermal noise level at liquid helium temperature. In particular, thermal amplitude fluctuations of the first mode are mapped into frequency fluctuations of the second mode. Furthermore, we point out non-Gaussian features in the frequency noise due to a single individual mode, an effect which is a direct consequence of the nonlinear coupling. Finally, we discuss possible implications of nonlinear thermal frequency noise in ultrasensitive force microscopy technologies.

Figure 1

(a) Sketch of the experimental setup. The motion $x$ of the cantilever causes a change of the magnetic flux $\Phi$ coupled by the magnetic particle $\vec{\mu }$ on the cantilever into the SQUID. A current $I$ in the external loop (field coil) allows us to drive the cantilever via a dipole-gradient force $F$. In the real system SQUID and field coil are composed of two pairs of concentric loops in a gradiometric microsusceptometer configuration, which results in nominally zero flux crosstalk between field coil and SQUID. (b) Measurement scheme for the correlation measurements. Mode 2 is driven on resonance by a fixed sinusoidal excitation at frequency $f_2$. Mode 1 is either undriven or driven by a white noise source $V_n$ bandpass filtered around $f_1$. Amplitude $R_i$ and phase ${\varphi }_i$ of mode 1 and 2 are measured respectively by lock-in amplifiers 1 and 2. Cross correlation is performed between the phase $\varphi ={\varphi }_2$ of mode 2 and the energy $E=x_1^2$ of mode 1, obtained by $R_1^2$ after calibrating the signal in terms of cantilever displacement.

Figure 2

(a) Resonant frequency of the first mode $f_1$ as a function of the amplitude of the second mode $x_2$. (b) Resonant frequency of the second mode $f_2$ as a function of the amplitude of the first mode $x_1$.

Figure 3

(a)–(c) Experimental histograms of the distributions of the energy of the first undriven mode $E=x_1^2$, corresponding to three representative effective temperatures of mode 1: (a) $4.2$ K, (b) 91 K, and (c) 690 K. Temperatures above $4.2$ K are obtained by driving only mode 1 with white noise, and do not correspond to a true thermodynamic temperature of the cantilever. Temperatures are obtained by the slope of the exponential fit. (d)–(f) Experimental histograms of the distributions of the phase $\varphi$ of the second driven mode, corresponding to three representative effective temperatures of mode 1: (d) $4.2$ K, (e) 91 K, and (f) 690 K. A Gaussian fit and exponential fit of the left tail are respectively shown in (d) and in (f). (g)–(i) Thick (red) line: measured cross correlation $R_{E\varphi }$ as a function of time lag for three representative effective temperatures of mode 1: (g) $4.2$ K, (h) 91 K, and (i) 690 K. The data are obtained by averaging many repetitions of the experiment. The thin (blue) lines represent the experimental standard deviation of the averaged data.

Figure 4

Dependence of the zero-lag cross correlation $R_{E\varphi }^{\left(0\right)}$ on the zero-lag autocorrelation of mode 1 energy $R_{EE}^{\left(0\right)}$. The data points are labeled by the effective temperature of mode 1. The straight line represents the best linear fit with zero intercept. The linear dependence is in agreement with Eq. (3), demonstrating quadratic nonlinear coupling down to the true thermal noise level $(4.2$ K data point).