Phase Synchronization of Two Anharmonic Nanomechanical Oscillators

Abstract

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. 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][4][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][7][8]. In addition, nanomechanical systems might prove useful for exploring quantum synchronization [9,10].Nanomechanical oscillators also have been exploited for a variety of applications [11][12][13]. In particular, nanoscale mechanics exhibits enhanced nonlinearity [14,15] and tunability [16,17], which has been used to suppress feedback noise [18,19] and create new types of electromechanical oscillators [20][21][22]. These oscillators may find application as mass [23], gas [24,25], or force sensors [26], without the need of an external frequency source.Building frequency sources from arrays of NEMS may yield enhanced applicability, but is challenging. For example, statistical deviations in batch fabrication inevitably lead to undesirable array dispersion [24]. If an array has appreciable frequency dispersion, global sensor responsivity gets reduced. However, if the elements of the array are made into a self-sustained oscillators and synchronized with one another, then the array responsivity will recover due to a reduction in phase noise [3]. 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 of synchronization in microor nanomechanical systems. However, these do not, in fact, demonstrate the phenomenon as conventionally defined [3] -that is, the phase locking of weakly coupled selfsustained oscillators. Shim et al.[27] reported synchronization of the driven excitations in coupled resonat...

Quantum Synchronization of a Driven Self-Sustained Oscillator

“…Optomechanical systems [17] appear to offer a particularly promising approach. Recent experiments have reported classical synchronization of nanomechanical oscillators [18,19], the quantum many-body dynamics of an array of identical optomechanical cells has been predicted to show synchronized behavior [9], and quantitative measures for quantum synchronization based on the Heisenberg uncertainty principle have been applied to two and many coupled optomechanical cells [10].…”

Synchronization in a semiclassical Kuramoto model

“…Several experimental works have shown that synchronization can be observed in the laboratory within Josepshon junction arrays [40], nanomechanical systems [17], or optomechanical systems [41]. All of these systems share one prominent property: They behave quantum mechanically at sufficiently low temperatures.…”

“…However, there is not doubt about the fundamental importance of studying quantum fluctuations within the emergence of synchronized states [9][10][11][12][13][14][15][16]. Moreover, the Kuramoto model has been implemented on circuits and microand nanomechanical structures [17,18], systems that have already met the quantum domain [19,20]. At the quantum level, synchronization, understood as the emergence of a coherent behavior from an incoherent situation in the absence of external fields, is reminiscent of phenomena such as Bose-Einstein condensation and has been observed in interacting condensates of quasiparticles [21,22].…”

Direct and parametric synchronization of a graphene self-oscillator

“…Indeed, their strong nonlinearity, tunability, and convenient time scales make detailed experimental studies of synchronization possible, including the observation of features such as phase slipping, phase locking, phase inertia, and phase oscillation. 4,[7][8][9][10] Compared to top-down fabricated NEMS devices, graphene nanomechanical systems offer enhanced nonlinear response due to their extreme aspect ratio. This enables new experimental studies of parametric synchronization and phaseoscillation dynamics, which are the topics of this letter.…”

“…[3][4][5][6] To study these phenomena experimentally, NanoElectroMechanical Systems (NEMS) have been proposed as representative model systems. Indeed, their strong nonlinearity, tunability, and convenient time scales make detailed experimental studies of synchronization possible, including the observation of features such as phase slipping, phase locking, phase inertia, and phase oscillation.…”