Optimal Sensing of Momentum Kicks with a Feedback-Controlled Nanomechanical Resonator

Abstract

External disturbances exciting a mechanical resonator can be exploited to gain information on the environment. Many of these interactions manifest as momentum kicks, such as the recoil of residual gas, radioactive decay, or even hypothetical interactions with dark matter. These disturbances are often rare enough that they can be resolved as singular events rather than cumulated as force noise. While high-Q resonators with low masses are particularly sensitive to such momentum kicks, they will strongly excite the resonator, leading to nonlinear effects that deteriorate the sensing performance. Hence, this paper utilizes optimal estimation methods to extract individual momentum kicks from measured stochastic trajectories of a mechanical resonator kept in the linear regime through feedback control. The developed scheme is illustrated and tested experimentally using a pre-stressed SiN trampoline resonator. Apart from enhancing a wide range of sensing scenarios mentioned above, our results indicate the feasibility of novel single-molecule mass spectrometry approaches.

Figures (3)

• Figure 1: (\ref{['fig:resonator_picture']}) Microscope picture of the used resonator. (\ref{['fig:expSetup1']}) Experimental setup used for interferometric velocity measurement and feedback control of the resonator: Vacuum chamber with pressure below 1e-5mbar. Lorentz force feedback and momentum kicks synthesized by the FPGA board. Computer for data post-processing. (\ref{['fig:ResSpec']}) Power spectral density of the resonator. The measured spectrum and the spectrum resulting from theory for the first three modes are depicted in blue and green, respectively. The dotted black lines represent the expected modal frequencies with their modal shapes next to them, calculated from FEM simulations.
• Figure 2: Schematic overview of the complete estimation process: Figure (\ref{['fig:impSine']}) shows the estimated time traces before and after the kick at $t=t_p$, which results in a strong discontinuity in velocity that can be retrieved by applying Kalman-Bucy filtering to the data before the kick and RTS-smoothing to the data after the kick. Figure (\ref{['fig:impCov']}) shows the corresponding time evolution of the velocity entries of the covariance matrices.
• Figure 3: (\ref{['fig:SpecCon']}) Measurement spectra with (orange) and without (blue) feedback and the innovation (green).(\ref{['fig:time_traces']}) Example time trace of a kick applied to the resonator. Top: Measured output signal in blue compared to the estimated output signal in red, buried in noise. Below: Estimated velocities in red of the first three modes. The vertical black line indicates the time of the kick. Uncertainty of the estimated signals (one standard deviation) is shaded in blue. (\ref{['fig:ImpReCon']}) Statistical relation between applied and estimated kicks. The blue shaded area marks expected uncertainty from theory. The blue dots show individual data points. Red dots and bars show mean and standard deviation of the recorded data.