On-Chip Frequency Noise Cancellation in Nanomechanical Resonators using Cavity Optomechanics

Abstract

Understanding and minimizing the sources of frequency noise in nanomechanical resonators is crucial for many sensing applications. In this work, we report an ultracoherent perimeter-mode nanomechanical resonator co-integrated with an on-chip optical cavity. This device combines low thermomechanical force noise and low detector noise, allowing us to study its intrinsic frequency fluctuations in detail. We find that the fluctuations of two mechanical modes are strongly correlated. Moreover, we demonstrate the generation of a signal at the frequency difference between the two modes directly on chip via nonlinear optomechanical transduction. This `difference signal' has vastly reduced intrinsic frequency fluctuations and can be used for frequency tracking with high precision, as we establish in a proof-of-principle experiment.

Figures (22)

• Figure 1: Measurement setup and calibration. (a) False-color scanning electron microscope micrograph of the Si$_3$N$_4$ optomechanical system. Red: Polygon resonator. Blue: Fabry Pérot cavity (scalebar $50µm$). (b)Detail showing the photonic crystal mirrors (scalebar $5µm$). Inset: Polygon resonator suspended near the cavity (scalebar $200nm$). Nanobeam-waveguide gap: 200nm. (c) Schematic of the optical detection scheme. $\Delta\omega$ = measured frequency shift, see Fig. \ref{['fig:3']}. (d) Normalized reflection trace of the optical mode used in the experiment. Yellow circles: measured reflection as a function of laser wavelength. Yellow dashed line: fit of the optical reflection. Vertical dashed line: laser operating point at $\Delta =$-7GHz, used in experiments. (e) Displacement PSD of the calibrated optical frequency noise. The peaks correspond to the thermomechanical noise stemming from the perimeter modes of the polygon resonator. Modes marked with red and blue are used in the experiment. (f) Mechanical frequency noise PSD of the high-$Q$ mode at different optical powers. Inset: Allan variance corresponding to the $A/\Omega$ component of the PSDs as a function of the input optical power. The gray shading shows the standard deviation of all values. PZT = piezoelectric actuator, PID = feedback loop, PLL = phase-locked loop
• Figure 2: Properties of signal at $\omega_\mathrm{d} = \omega_2 - \omega_1$. Displacement PSD of demodulated signal around (a) $\omega_\mathrm{d} = \omega_2 - \omega_1$ (b) $\omega_1$, (c) $\omega_2$, and (d) $\omega_s = \omega_1 + \omega_2$. The mechanical modes at $\omega_1$ (red) and $\omega_2$ (blue) are driven with separate PLLs. (e)-(g) Spectograms of the difference signal and the displacements of the OOP and the IP modes. The colour bar corresponds to the normalized PSD. (h) Double-sided frequency-PSD for $\omega_1$, $\omega_2$, and $\omega_\mathrm{d}$ spanning 150s with a data rate of 14.3kHz. We use the PLLs for estimating $\omega_1$ and $\omega_2$, while $\omega_\mathrm{d}$ is calculated using phase-to-frequency conversion Besic2023ResonanceResonators. The dashed-gray line is the fit for the $1/\Omega$-component of the frequency noise PSD. (i) Absolute Allan deviation $\sigma$ in Hz calculated using the same traces as in (h).
• Figure 3: Wide cavity scan and narrow scan with calibration. (a) Wide-range reflection spectrum of the cavity showing multiple resonances with a FSR of 4.25nm. (b) Reflected signal of a fibre loop cavity (FLC) used for calibrating the x-axis (c) Reflected power from the rightmost cavity mode in (a) as a function of laser detuning $\Delta/2\pi$. $\Delta = 0$ indicates the cavity resonance. The fit in orange dashed yields a linewidth of $\kappa/2\pi = 3.33$ GHz.
• Figure 4: Optomechanical coupling. (a) A broadband spectrum obtained in direct detection from the device. The red line corresponds to the thermomechanical spectrum acquired when the laser is red detuned from the cavity. The peaks correspond to the thermal motion of different modes of the polygon resonator, a few of which are marked alongside their mode shapes. (b) FEM simulation of the electric field intensity
• Figure 5: Ringdown measurements of the device's perimeter modes $\omega_1$ in red and $\omega_2$ in blue with their respective fits (dashed).