Frequency stabilization of self-sustained oscillations in a sideband-driven electromechanical resonator

TL;DR

This work addresses phase diffusion in self-sustained oscillations of two nonlinearly coupled micromechanical modes by exploiting dispersive two-mode coupling and a blue-sideband pump. The authors show that the phases of the two modes diffuse anti-correlatively, allowing the sum of the phases to be controlled via the pump phase, which enables stabilization of either mode without a nearby frequency reference. They derive and experimentally verify a stabilization algorithm in which measuring one mode’s phase and adjusting the pump phase reduces the other mode’s phase noise, achieving substantial linewidth narrowing. The approach opens routes to stable, broadband frequency generation through parametric downconversion in nonlinear resonators and can be extended to cavity-optomechanical platforms.

Abstract

We present a method to stabilize the frequency of self-sustained vibrations in micro- and nanomechanical resonators. The method refers to a two-mode system with the vibrations at significantly different frequencies. The signal from one mode is used to control the other mode. In the experiment, self-sustained oscillations of micromechanical modes are excited by pumping at the blue-detuned sideband of the higher-frequency mode. Phase fluctuations of the two modes show near perfect anti-correlation. They can be compensated in either one of the modes by a stepwise change of the pump phase. The phase change of the controlled mode is proportional to the pump phase change, with the proportionality constant independent of the pump amplitude and frequency. This finding allows us to stabilize the phase of one mode against phase diffusion using the measured phase of the other mode. We demonstrate that phase fluctuations of either the high or low frequency mode can be significantly reduced. The results open new opportunities in generating stable vibrations in a broad frequency range via parametric downconversion in nonlinear resonators.

Figures (7)

• Figure 1: (a) A sketch of the spectra of the mode 1 (mechanical mode) and mode 2 (cavity mode). Energy of the pump that leads to self-sustained vibrations is supplied to the upper sideband of mode 2. (b) Scanning electron micrograph of electromechanical resonator including the measurement scheme. The white scale bar at the lower left corner measures 30 $\mu$m. (c) Vibration profiles of mode 1 (top) and mode 2 (bottom). (d) Dependence of the square of vibration amplitude of mode 1 on the detuning of the pump frequency from the upper sideband for mode 1 at pump current amplitude of 189 $\mu$A. (e) Similar plot for mode 2.
• Figure 2: (a) Change of the vibration phase ${\delta\phi}_1$ of mode 1 (green) and ${\delta\phi}_2$ of mode 2 (yellow), measured as a function of time. The black line represents the sum ${\delta\phi}_1+{\delta\phi}_2$. The pump current amplitude is 189 $\mu$A corresponding to $F$ = 2.3 mN$m^{-1}$ and frequency detuning $\Delta_F$ is 300 Hz. The same results are plotted in panels (b) and (c) in the frames rotating at the frequencies $\omega_\mathrm{self1}$ and $\omega_\mathrm{self2}$ of the self-sustained vibrations of the two modes. The variables are $X_i = r_i \cos(\phi_i)$ and $Y_i = r_i \sin(\phi_i)$, $i = 1, 2$.
• Figure 3: (a) Numerical calculation of the dynamics of mode 1 in the frame rotating at frequency $\omega_\mathrm{self1}$ in response to a sudden change of $\theta_F$ by $1^{\circ}$; $X_1 = r_1 \cos \phi_1$ and $Y_1 = r_1 \sin\phi_1$. The pump current amplitude is 379 $\mu$A and frequency detuning $\Delta_F$ is 1000 Hz. Initially, $X_1$ is set to zero. The black line is part of the circles $X_1^2 + Y_1^2 = r_1^{(0)}{}^2$.(b) Corresponding plot for mode 2. (c) Measured dependence of $\phi_1$ on time (green circles). The solid line represents numerical simulations with no fitting parameters. Inset: zoom in at shorter times. (d) Corresponding plot for mode 2. (e) Real parts of the eigenvalues for matrix $\mathbf{\Lambda}$ in the linearized equation of motion plotted as a function of the pump detuned frequency. $\lambda_{1,2,3}$ are plotted as red, blue and green dashed lines respectively. Near the supercritical Hopf bifurcation point, analytical expressions (Appendix \ref{['section:B']}) give the green solid line for $\lambda_3$ and the red square for $\lambda_1$. (f) Similar plot for the imaginary parts of $\lambda_{1,2,3}$.
• Figure 4: (a) Algorithm for stabilizing $\phi_2$ by measuring $\phi_1$ and adjusting $\theta_F$. (b) The duration of one cycle equals $t_\mathrm{wait} + t_\mathrm{measure}$. The $p^{th}$ cycle lasts from time $t_p$ to $t_{p+1}$. $\theta_F$ is changed only at the beginning of each cycle. (c) $\phi_2$ is measured as a function of time when the stabilization routine is turned off for 20 min (yellow) and on for 20 min (brown). The pump current amplitude is 189 $\mu$A and frequency detuning is 300 Hz. Inset: The spectra of the self-sustained vibrations of mode 2 when the stabilization algorithm in (b) is implemented. The spectral width is significantly reduced with stabilization turned on. (d) Dependence of the standard deviation of $\phi_2$ on $t_\mathrm{measure}$ at a fixed $t_\mathrm{wait}$ of 300 ms.
• Figure 5: (a) Algorithm for stabilizing $\phi_1$ by measuring $\phi_2$ and adjusting $\theta_F$. (b) $\phi_1$ is measured as a function of time when the stabilization routine is turned off (green) and on (dark green). The pump current amplitude is 189 $\mu$A and frequency detune is 300 Hz. (c) The spectra of the self-sustained vibrations of mode 1 with and without stabilization. (d) Dependence of the standard deviation of $\phi_1$ on $t_\mathrm{measure}$ at a fixed $t_\mathrm{wait}$ of 300 ms.