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 co-integration with modern electronic signal processing technologies. 2 Synchronization is a ubiquitous phenomenon both in the physical and biological sciences. It has been observed to occur over a wide range of scales-from the ecological[1], with oscillation periods of years, to the microscale[2], with oscillation periods of milliseconds. Although synchronization has been extensively studied theoretically[3-5], relatively few experimental systems have been realized that provide detailed insight into the underlying dynamics. Here we show that oscillators based on nanoelectromechanical systems (NEMS) can readily enable the resolution of such details, while providing many unique advantages for experimental studies of nonlinear dynamics[6-8]. Nanomechanical oscillators also have been exploited for a variety of applications. In particular, nanoscale mechanics exhibits enhanced nonlinearity[9] and tunability[10], which has been used to suppress feedback noise[11, 12] and create new types of electromechanical oscillators[13, 14]. These oscillators may find application as mass[15], gas[16, 17], or force[18] sensors, without the need of an external frequency source. In addition to their extreme sensitivity, they dissipate very little power due to their high quality factors, reducing the sustaining power needed for sensor arrays. Although NEMS arrays can provide exceptional performance as frequency-shift sensors or frequency sources, their implementation can be challenging. For example, statistical deviations in batch fabrication inevitably lead to undesirable array dispersion[16]. If a sensor array has appreciable frequency dispersion, global sensor responsivity gets reduced due to an overall increase in signal phase noise. However, upon synchronization, dispersive elements lock to a single frequency. If the oscillators are not only frequency locked, but phase locked, the phase noise of this array may be reduced[3]. Attainment of this can mitigate the deleterious effects from an array's frequency dispersion. Since NEMS have numerous applications, and are useful in studying nonlinear dynamics, we set an important milestone by demonstrating synchronization in nanomechanical systems. There are previous reports[19, …