Phase Synchronization of Two Anharmonic Nanomechanical Oscillators

We investigate the synchronization of oscillators based on anharmonic nanoelectromechanical resonators. Our experimental implementation allows unprecedented observation and control of parameters governing the dynamics of synchronization. We find close quantitative agreement between experimental data and theory describing reactively coupled Duffing resonators with fully saturated feedback gain. In the synchronized state we demonstrate a significant reduction in the phase noise of the oscillators, which is key for sensor and clock applications. Our work establishes that oscillator networks constructed from nanomechanical resonators form an ideal laboratory to study synchronization— given their high-quality factors, small footprint, and ease of cointegration with modern electronic signal processing technologies.

Figure 1

Simplified circuit schematic for experiment. Each NEMS resonator (colored SEM micrograph) is embedded in two feedback loops: one is used for creating oscillations in each resonator, and the other creates coupling between the oscillators. The attenuators after each limiter (single heavy line boxes) sets the level of oscillation, and constitutes a means to control the frequency pulling. In the coupling loop the signal is amplified so that an attenuator (double heavy line boxes) adjusts the signal level in the common loop, thereby setting the coupling strength. The frequency difference is controlled by adjusting the stress induced in the left resonator by the piezovoltage.

Figure 2

Synchronization in the limit of small coupling described by Eqs. (5) and (6) with a frequency pulling $\alpha =1.25$. (a) Experimental data (points) are compared against theoretical predictions (lines) for the amplitudes of the two oscillators as the system moves through synchronization; the dependence upon detuning $\mathrm{\Delta }\omega$ for a coupling of $\beta =0.068$ is shown. The synchronization regime is shown by orange shading. (b) Data and predictions for the frequency difference ${\phi }^{\prime }$ for three different values of coupling. The set of data with the largest value of coupling $\beta =0.068$ corresponds to the amplitude data from the upper plot. Frequency locking (synchronization regime) is shown where values ${\phi }^{\prime }=0$ occur. SR 0.012, SR 0.044, SR 0.068 denote the synchronization regimes (shaded regions) for the three couplings.

Figure 3

Experimentally measured synchronization space as a $\beta$ and $\alpha$ for $\mathrm{\Delta }\omega =0.6$, 1, 2. The basins of attraction, found from the time domain simulation of Eqs. (1) and (2), are shown by the gradient between white and blue, and correspond to the average number of times the simulation synchronized under 100 random initial phases. All lines show boundaries with respective regions to the right of the line. The green solid line (REP) is the experimental boundary for the transition from unsynchronized to the synchronized state for repulsive coupling. The red solid line (ATT) is the experimental boundary for the transition from the unsynchronized to the synchronized state for attractive coupling. The experimental synchronized state is defined as ${\phi }^{\prime }<0.05$ (in units of the resonator width). The orange dashed line (LSA-1) depicts the predicted (linear stability analysis) boundary for which at least one synchronized state is stable. Similarly, the purple dashed line (LSA-2) bounds the space for which both synchronized states are stable.

Figure 4

Oscillator phase noise at 1 kHz offset from carrier frequency (blue and green spheres, left axis) and oscillator frequency difference (red diamonds, right axis) as coupling is increased. At the value of coupling $\beta =0.086$ the oscillator frequency difference goes to zero and the phase noise for both oscillators decreases by 3 dB, i.e., corresponding to reducing the phase noise by half.