Surpassing fundamental limits of oscillators using nonlinear resonators

Hidden Vibrational Bistability Revealed by Intrinsic Fluctuations of a Carbon Nanotube

We demonstrate that a quiet state and large-amplitude self-sustained oscillations can coexist in a carbon nanotube subject to time-independent drive. A feature of the bistability is that it would be hysteresis free in the absence of noise, and the oscillatory state would not be seen. It is revealed by random switching between the stable states, which we observe in the time domain. We attribute the switching to fluctuations in the system and show that it displays Poisson statistics. We propose a minimalistic model that relates the emergence of the bistability to a nonmonotonic variation of nonlinear friction with the vibration amplitude. This new type of dynamical regime and the means to reveal it are generic and are of interest for various mesoscopic vibrational systems.

Coherent energy transfer in coupled nonlinear microelectromechanical resonators

Energy decay, describing the leakage of system energy to the environmental bath, is a universal behavior in oscillators. It has been utilized to elucidate energy transfer between vibrational modes of a resonator. In coupled resonators, achieving an ultra-low coupling rate is essential for observing energy interactions between resonators and environmental bath. Here, we observe periodic transient beating phenomenon by analyzing the transient responses of coupled nonlinear resonators with a coupling rate of 9.6 Hz. The energy transfer rate indicating the hybrid energy manipulation is impacted by asymmetry-induced energy localization and enhanced by nonlinearity. Time-resolved eigenstates, characterized by amplitude ratios, are employed as a quantitative tool to uncover the energy transfer and localization in coupled resonators under nonlinear operations. This work opens the possibilities to manipulate energy transfer, to probe energy localization, and to develop high-precision sensors utilizing the energy transfer between coupled nonlinear resonators.

Higher-order singularities in phase-tracked electromechanical oscillators

Singularities ubiquitously exist in different fields and play a pivotal role in probing the fundamental laws of physics and developing highly sensitive sensors. Nevertheless, achieving higher-order (≥3) singularities, which exhibit superior performance, typically necessitates meticulous tuning of multiple (≥3) coupled degrees of freedom or additional introduction of nonlinear potential energies. Here we propose theoretically and confirm using mechanics experiments, the existence of an unexplored cusp singularity in the phase-tracked (PhT) steady states of a pair of coherently coupled mechanical modes without the need for multiple (≥3) coupled modes or nonlinear potential energies. By manipulating the PhT singularities in an electrostatically tunable micromechanical system, we demonstrate an enhanced cubic-root response to frequency perturbations. This study introduces a new phase-tracking method for studying interacting systems and sheds new light on building and engineering advanced singular devices with simple and well-controllable elements, with potential applications in precision metrology, portable nonreciprocal devices, and on-chip mechanical computing.

Massively parallel electro-optic sampling of space-encoded optical pulses for ultrafast multi-dimensional imaging

High-speed and high-resolution imaging of surface profiles is critical for the investigation of various structures and mechanical dynamics of micro- and nano-scale devices. In particular, recent emergence of various nonlinear, transient and complex mechanical dynamics, such as anharmonic vibrations in mechanical resonators, has necessitated real-time surface deformation imaging with higher axial and lateral resolutions, speed, and dynamic range. However, real-time capturing of fast and complex mechanical dynamics has been challenging, and direct time-domain imaging of displacements and mechanical motions has been a missing element in studying full-field structural and dynamic behaviours. Here, by exploiting the electro-optic sampling with a frequency comb, we demonstrate a line-scan time-of-flight (TOF) camera that can simultaneously measure the TOF changes of more than 1000 spatial coordinates with hundreds megapixels/s pixel-rate and sub-nanometre axial resolution over several millimetres field-of-view. This unique combination of performances enables fast and precise imaging of both complex structures and dynamics in three-dimensional devices and mechanical resonators.

Tuning the nonlinearity of graphene mechanical resonators by Joule heating

As an inherent property of the device itself, nonlinearity in micro-/nano- electromechanical resonators is difficult to eliminate, and it has shown a wide range of applications in basic research, sensing and other fields. While many application scenarios require tunability of the nonlinearity, inherent nonlinearity of a mechanical resonator is difficult to be changed. Here, we report the experimental observation of a Joule heating induced tuning effect on the nonlinearity of graphene mechanical resonators. We fabricated multiple graphene mechanical resonators and detected their resonant properties by an optical interference method. The mechanical vibration of the resonators will enter from the linear to the nonlinear intervals if we enhance the external driving power to a certain value. We found that at a fixed drive power, the nonlinearity of a mechanical resonator can be tuned by applying a dc bias current on the resonator itself. The tuning mechanism could be explained by the nonlinear amplitude-frequency dependence theory. Our results may provide a research platform for the study of mechanical nonlinearity by using atomic-thin layer materials.

Precision Calibration of the Duffing Oscillator with Phase Control

The Duffing oscillator is a nonlinear extension of the ubiquitous harmonic oscillator and as such plays an outstanding role in science and technology. Experimentally, the system parameters are determined by a measurement of its response to an external excitation. When changing the amplitude or frequency of the external excitation, a sudden jump in the response function reveals the nonlinear dynamics prominently. However, this bistability leaves part of the full response function unobserved, which limits the precise measurement of the system parameters. Here, we exploit the often unknown fact that the response of a Duffing oscillator with nonlinear damping is a unique function of its phase. By actively stabilizing the oscillator's phase we map out the full response function. This phase control allows us to precisely determine the system parameters. Our results are particularly important for characterizing nanoscale resonators, where nonlinear effects are observed readily and which hold great promise for next generation of ultrasensitive force and mass measurements. We demonstrate our approach experimentally with an optically levitated particle in high vacuum.

Mode-localized accelerometer in the nonlinear Duffing regime with 75 ng bias instability and 95 ng/√Hz noise floor

Mode-localized sensors have attracted attention because of their high parametric sensitivity and first-order common-mode rejection to temperature drift. The high-fidelity detection of resonator amplitude is critical to determining the resolution of mode-localized sensors where the measured amplitude ratio in a system of coupled resonators represents the output metric. Operation at specific bifurcation points in a nonlinear regime can potentially improve the amplitude bias stability; however, the amplitude ratio scale factor to the input measurand in a nonlinear regime has not been fully investigated. This paper theoretically and experimentally elucidates the operation of mode-localized sensors with respect to stiffness perturbations (or an external acceleration field) in a nonlinear Duffing regime. The operation of a mode-localized accelerometer is optimized with the benefit of the insights gained from theoretical analysis with operation in the nonlinear regime close to the top critical bifurcation point. The phase portraits of the amplitudes of the two resonators under different drive forces are recorded to support the experimentally observed improvements for velocity random walk. Employing temperature control to suppress the phase and amplitude variations induced by the temperature drift, 1/f noise at the operation frequency is significantly reduced. A prototype accelerometer device demonstrates a noise floor of 95 ng/√Hz and a bias instability of 75 ng, establishing a new benchmark for accelerometers employing vibration mode localization as a sensing paradigm. A mode-localized accelerometer is first employed to record microseismic noise in a university laboratory environment.

Avoiding transduction-induced heating in suspended microchannel resonators using piezoelectricity

Calorimetry of single biological entities remains elusive. Suspended microchannel resonators (SMRs) offer excellent performance for real-time detection of various analytes and could hold the key to unlocking pico-calorimetry experiments. However, the typical readout techniques for SMRs are optical-based, and significant heat is dissipated in the sensor, altering the measurement and worsening the frequency noise. In this manuscript, we demonstrate for the first time full on-chip piezoelectric transduction of SMRs on which we focus a laser Doppler vibrometer to analyze its effect. We demonstrate that suddenly applying the laser to a water-filled SMR causes a resonance frequency shift, which we attribute to a local increase in temperature. When the procedure is repeated at increasing flow rates, the resonance frequency shift diminishes, indicating that convection plays an important role in cooling down the device and dissipating the heat induced by the laser. We also show that the frequency stability of the device is degraded by the laser source. In comparison to an optical readout scheme, a low-dissipative transduction method such as piezoelectricity shows greater potential to capture the thermal properties of single entities.

Nonlinear parametric oscillator: A tool for probing quantum fluctuations

Nanomechanical oscillators have, over the last few years, started probing regimes where quantum fluctuations are important. Here we consider a nonlinear parametric oscillator in the quantum domain. We show that in the classical subharmonic resonance zone, the quantum fluctuations are finite but greatly magnified depending on the strength of the nonlinear coupling. This should make such oscillators useful in probing quantum fluctuations.

Transduction of large optomechanical amplitudes with racetrack-loaded Mach-Zehnder interferometers

Chip-integrated photonic devices have stimulated development in areas ranging from telecommunications to optomechanics. Racetrack resonators have gained popularity for optomechanical transduction due to their high sensitivity and cavity finesse. However, they lack sufficient dynamic range to read out large amplitude mechanical resonators, which are preferred for sensing applications. We present a robust photonic circuit based on a Mach-Zehnder interferometer (MZI) combined with a racetrack resonator that increases linear range without compromising high transduction sensitivity. Optical and mechanical properties of combined MZI-racetrack devices are compared to lone racetracks with the same physical dimensions in the undercoupled, overcoupled and critical coupled regimes. We demonstrate an overall improvement in dynamic range, transduction responsivity, and mass sensitivity of up to 4x, 3x and 2.8x, respectively. Our highly phase sensitive MZI circuit also enables applications such as on-chip optical homodyning.

Dipole states and coherent interaction in surface-acoustic-wave coupled phononic resonators

Manipulation of mechanical motion at the micro-scale has been attracting continuous attention, leading to the successful implementation of various strategies with potential impact on classical and quantum information processing. We propose an approach based on the interplay between a pair of localised mechanical resonators and travelling surface acoustic waves (SAW). We demonstrate the existence of a two-sided interaction, allowing the use of SAW to trigger and control the resonator oscillation, and to manipulate the elastic energy distribution on the substrate through resonator coupling. Observation of the vectorial structure of the resonator motion reveals the existence of two coupling regimes, a dipole-dipole-like interaction at small separation distance versus a surface-mediated mechanical coupling at larger separation. These results illustrate the potential of this platform for coherent control of mechanical vibration at a resonator level, and reciprocally for manipulating SAW propagation using sub-wavelength elements.

Exotic states in a simple network of nanoelectromechanical oscillators

Synchronization of oscillators, a phenomenon found in a wide variety of natural and engineered systems, is typically understood through a reduction to a first-order phase model with simplified dynamics. Here, by exploiting the precision and flexibility of nanoelectromechanical systems, we examined the dynamics of a ring of quasi-sinusoidal oscillators at and beyond first order. Beyond first order, we found exotic states of synchronization with highly complex dynamics, including weak chimeras, decoupled states, traveling waves, and inhomogeneous synchronized states. Through theory and experiment, we show that these exotic states rely on complex interactions emerging out of networks with simple linear nearest-neighbor coupling. This work provides insight into the dynamical richness of complex systems with weak nonlinearities and local interactions.

A Temperature-Compensated Single-Crystal Silicon-on-Insulator (SOI) MEMS Oscillator with a CMOS Amplifier Chip

Self-sustained feedback oscillators referenced to MEMS/NEMS resonators have the potential for a wide range of applications in timing and sensing systems. In this paper, we describe a real-time temperature compensation approach to improving the long-term stability of such MEMS-referenced oscillators. This approach is implemented on a ~26.8 kHz self-sustained MEMS oscillator that integrates the fundamental in-plane mode resonance of a single-crystal silicon-on-insulator (SOI) resonator with a programmable and reconfigurable single-chip CMOS sustaining amplifier. Temperature compensation using a linear equation fit and look-up table (LUT) is used to obtain the near-zero closed-loop temperature coefficient of frequency (TCf) at around room temperature (~25 °C). When subject to small temperature fluctuations in an indoor environment, the temperature-compensated oscillator shows a >2-fold improvement in Allan deviation over the uncompensated counterpart on relatively long time scales (averaging time τ > 10,000 s), as well as overall enhanced stability throughout the averaging time range from τ = 1 to 20,000 s. The proposed temperature compensation algorithm has low computational complexity and memory requirement, making it suitable for implementation on energy-constrained platforms such as Internet of Things (IoT) sensor nodes.

Improving mechanical sensor performance through larger damping

Mechanical resonances are used in a wide variety of devices, from smartphone accelerometers to computer clocks and from wireless filters to atomic force microscopes. Frequency stability, a critical performance metric, is generally assumed to be tantamount to resonance quality factor (the inverse of the linewidth and of the damping). We show that the frequency stability of resonant nanomechanical sensors can be improved by lowering the quality factor. At high bandwidths, quality-factor reduction is completely mitigated by increases in signal-to-noise ratio. At low bandwidths, notably, increased damping leads to better stability and sensor resolution, with improvement proportional to damping. We confirm the findings by demonstrating temperature resolution of 60 microkelvin at 300-hertz bandwidth. These results open the door to high-performance ultrasensitive resonators in gaseous or liquid environments, single-cell nanocalorimetry, nanoscale gas chromatography, atmospheric-pressure nanoscale mass spectrometry, and new approaches in crystal oscillator stability.

Complex Dynamical Networks Constructed with Fully Controllable Nonlinear Nanomechanical Oscillators

Control of the global parameters of complex networks has been explored experimentally in a variety of contexts. Yet, the more difficult prospect of realizing arbitrary network architectures, especially analog physical networks that provide dynamical control of individual nodes and edges, has remained elusive. Given the vast hierarchy of time scales involved, it also proves challenging to measure a complex network's full internal dynamics. These span from the fastest nodal dynamics to very slow epochs over which emergent global phenomena, including network synchronization and the manifestation of exotic steady states, eventually emerge. Here, we demonstrate an experimental system that satisfies these requirements. It is based upon modular, fully controllable, nonlinear radio frequency nanomechanical oscillators, designed to form the nodes of complex dynamical networks with edges of arbitrary topology. The dynamics of these oscillators and their surrounding network are analog and continuous-valued and can be fully interrogated in real time. They comprise a piezoelectric nanomechanical membrane resonator, which serves as the frequency-determining element within an electrical feedback circuit. This embodiment permits network interconnections entirely within the electrical domain and provides unprecedented node and edge control over a vast region of parameter space. Continuous measurement of the instantaneous amplitudes and phases of every constituent oscillator node are enabled, yielding full and detailed network data without reliance upon statistical quantities. We demonstrate the operation of this platform through the real-time capture of the dynamics of a three-node ring network as it evolves from the uncoupled state to full synchronization.

Optically levitated nanoparticle as a model system for stochastic bistable dynamics

Nano-mechanical resonators have gained an increasing importance in nanotechnology owing to their contributions to both fundamental and applied science. Yet, their small dimensions and mass raises some challenges as their dynamics gets dominated by nonlinearities that degrade their performance, for instance in sensing applications. Here, we report on the precise control of the nonlinear and stochastic bistable dynamics of a levitated nanoparticle in high vacuum. We demonstrate how it can lead to efficient signal amplification schemes, including stochastic resonance. This work contributes to showing the use of levitated nanoparticles as a model system for stochastic bistable dynamics, with applications to a wide variety of fields.

Optimization of the Close-to-Carrier Phase Noise in a CMOS-MEMS Oscillator Using a Phase Tunable Sustaining-Amplifier

In this paper, the phase noise of a 24-MHz complimentary metal-oxide-semiconductor microelectromechanical systems (CMOS-MEMS) oscillator with zero-level vacuum package is studied. We characterize and analyze the nonlinear regime of each one of the modules that compose the oscillator (CMOS sustaining-amplifier and MEMS resonator). As we show, the presented resonator exhibits a high nonlinear behavior. Such a fact is exploited as a mechanism to stabilize the oscillation amplitude, allowing us to maintain the sustaining-amplifier working in the linear regime. Consequently, the nonlinear resonator becomes the main close-to-carrier phase noise source. The sustaining amplifier, which functions as a phase shifter, was developed such that MEMS operation point optimization could be achieved without an increase in circuitry modules. Therefore, the system saves on area and power, and is able to improve the phase noise 26 dBc/Hz (at 1-kHz carrier frequency offset).

Enhancement of Frequency Stability Using Synchronization of a Cantilever Array for MEMS-Based Sensors

Micro and nano electromechanical resonators have been widely used as single or multiple-mass detection sensors. Smaller devices with higher resonance frequencies and lower masses offer higher mass responsivities but suffer from lower frequency stability. Synchronization phenomena in multiple MEMS resonators have become an important issue because they allow frequency stability improvement, thereby preserving mass responsivity. The authors present an array of five cantilevers (CMOS-MEMS system) that are forced to vibrate synchronously to enhance their frequency stability. The frequency stability has been determined in closed-loop configuration for long periods of time by calculating the Allan deviation. An Allan deviation of 0.013 ppm (@ 1 s averaging time) for a 1 MHz cantilever array MEMS system was obtained at the synchronized mode, which represents a 23-fold improvement in comparison with the non-synchronized operation mode (0.3 ppm).

Effects of time-delayed feedback on the properties of self-sustained oscillators

Most self-sustained oscillations in biological systems and in technical applications are based on a feedback loop, and it is usually important to know how they will react when an external oscillatory force is applied. Here we investigate the effects that the introduction of a time delay in the feedback can have in the entrainment properties of self-sustained oscillators. To do this, we derive analytic expressions for the periodic trajectories and their asymptotic stability, for a generic external oscillatory force. This allows us to show that, for large quality factors, the resonance frequency does not depend on the feedback delay. When the external force is harmonic, it is shown that the largest entrainment range does not correspond to the time delay that gives the maximal response of the unforced oscillator. In fact, that delay gives the shortest entrainment range.

Ultrasensitive hysteretic force sensing with parametric nonlinear oscillators

We propose a method for linear detection of weak forces using parametrically driven nonlinear resonators. The method is based on a peculiar feature in the response of the resonator to a near resonant periodic external force. This feature stems from a complex interplay among the parametric drive, external force, and nonlinearities. For weak parametric drive, the response exhibits the standard Duffing-like single jump hysteresis. For stronger drive amplitudes, we find a qualitatively new double jump hysteresis which arises from stable solutions generated by the cubic Duffing nonlinearity. The additional jump exists only if the external force is present and the frequency at which it occurs depends linearly on the amplitude of the external force, permitting a straightforward ultrasensitive detection of weak forces. With state-of-the-art nanomechanical resonators, our scheme should permit force detection in the attonewton range.

Self-Sustained Micromechanical Oscillator with Linear Feedback

Autonomous oscillators, such as clocks and lasers, produce periodic signals without any external frequency reference. In order to sustain stable periodic motion, there needs to be an external energy supply as well as nonlinearity built into the oscillator to regulate the amplitude. Usually, nonlinearity is provided by the sustaining feedback mechanism, which also supplies energy, whereas the constituent resonator that determines the output frequency stays linear. Here, we propose a new self-sustaining scheme that relies on the nonlinearity originating from the resonator itself to limit the oscillation amplitude, while the feedback remains linear. We introduce a model for describing the working principle of the self-sustained oscillations and validate it with experiments performed on a nonlinear microelectromechanical oscillator.

Generating giant and tunable nonlinearity in a macroscopic mechanical resonator from a single chemical bond

Nonlinearity in macroscopic mechanical systems may lead to abundant phenomena for fundamental studies and potential applications. However, it is difficult to generate nonlinearity due to the fact that macroscopic mechanical systems follow Hooke's law and respond linearly to external force, unless strong drive is used. Here we propose and experimentally realize high cubic nonlinear response in a macroscopic mechanical system by exploring the anharmonicity in chemical bonding interactions. We demonstrate the high tunability of nonlinear response by precisely controlling the chemical bonding interaction, and realize, at the single-bond limit, a cubic elastic constant of 1 × 10(20) N m(-3). This enables us to observe the resonator's vibrational bi-states transitions driven by the weak Brownian thermal noise at 6 K. This method can be flexibly applied to a variety of mechanical systems to improve nonlinear responses, and can be used, with further improvements, to explore macroscopic quantum mechanics.

Devil's staircase in an optomechanical cavity

We study self-excited oscillations (SEOs) in an on-fiber optomechanical cavity. While the phase of SEOs randomly diffuses in time when the laser power injected into the cavity is kept constant, phase locking may occur when the laser power is periodically modulated in time. We investigate the dependence of phase locking on the amplitude and frequency of the laser-power modulation. We find that phase locking can be induced with a relatively low modulation amplitude provided that the ratio between the modulation frequency and the frequency of SEOs is tuned close to a rational number of relatively low hierarchy in the Farey tree. To account for the experimental results, a one-dimensional map, which allows evaluating the time evolution of the phase of SEOs, is theoretically derived. By calculating the winding number of the one-dimensional map, the regions of phase locking can be mapped in the plane of modulation amplitude and modulation frequency. Comparison between the theoretical predictions and the experimental findings yields a partial agreement.

Multistable internal resonance in electroelastic crystals with nonlinearly coupled modes

Nonlinear modal interactions have recently become the focus of intense research in micro- and nanoscale resonators for their use to improve oscillator performance and probe the frontiers of fundamental physics. However, our understanding of modal coupling is largely restricted to clamped-clamped beams, and lacking in systems with both geometric and material nonlinearities. Here we report multistable energy transfer between internally resonant modes of an electroelastic crystal plate and use a mixed analytical-numerical approach to provide new insight into these complex interactions. Our results reveal a rich bifurcation structure marked by nested regions of multistability. Even the simple case of two coupled modes generates a host of topologically distinct dynamics over the parameter space, ranging from the usual Duffing bistability to complex multistable behaviour and quasiperiodic motion.

Modelling the Size Effects on the Mechanical Properties of Micro/Nano Structures

Experiments on micro- and nano-mechanical systems (M/NEMS) have shown that their behavior under bending loads departs in many cases from the classical predictions using Euler-Bernoulli theory and Hooke's law. This anomalous response has usually been seen as a dependence of the material properties on the size of the structure, in particular thickness. A theoretical model that allows for quantitative understanding and prediction of this size effect is important for the design of M/NEMS. In this paper, we summarize and analyze the five theories that can be found in the literature: Grain Boundary Theory (GBT), Surface Stress Theory (SST), Residual Stress Theory (RST), Couple Stress Theory (CST) and Surface Elasticity Theory (SET). By comparing these theories with experimental data we propose a simplified model combination of CST and SET that properly fits all considered cases, therefore delivering a simple (two parameters) model that can be used to predict the mechanical properties at the nanoscale.

Low phase-noise autonomous parametric oscillator based on a 226.7 MHz AlN contour-mode resonator

We present the first parametric oscillator based on the use of a 226.7 MHz aluminum nitride contour-mode resonator. This topology enables an improvement in the phase noise of 16 dB at 1 kHz offset with respect to a conventional feedback-loop oscillator based on the same device. The recorded phase noise is -106 dBc/Hz at 1 kHz offset.

Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope

Parametric amplification, resulting from intentionally varying a parameter in a resonator at twice its resonant frequency, has been successfully employed to increase the sensitivity of many micro- and nano-scale sensors. Here, we introduce the concept of self-induced parametric amplification, which arises naturally from nonlinear elastic coupling between the degenerate vibration modes in a micromechanical disk-resonator, and is not externally applied. The device functions as a gyroscope wherein angular rotation is detected from Coriolis coupling of elastic vibration energy from a driven vibration mode into a second degenerate sensing mode. While nonlinear elasticity in silicon resonators is extremely weak, in this high quality-factor device, ppm-level nonlinear elastic effects result in an order-of-magnitude increase in the observed sensitivity to Coriolis force relative to linear theory. Perfect degeneracy of the primary and secondary vibration modes is achieved through electrostatic frequency tuning, which also enables the phase and frequency of the parametric coupling to be varied, and we show that the resulting phase and frequency dependence of the amplification follow the theory of parametric resonance. We expect that this phenomenon will be useful for both fundamental studies of dynamic systems with low dissipation and for increasing signal-to-noise ratio in practical applications such as gyroscopes.

Charge sensitivity enhancement via mechanical oscillation in suspended carbon nanotube devices

Single electron transistors (SETs) fabricated from single-walled carbon nanotubes (SWNTs) can be operated as highly sensitive charge detectors reaching sensitivity levels comparable to metallic radio frequency SETs (rf-SETs). Here, we demonstrate how the charge sensitivity of the device can be improved by using the mechanical oscillations of a single-walled carbon nanotube quantum dot. To optimize the charge sensitivity δQ, we drive the mechanical resonator far into the nonlinear regime and bias it to an operating point where the mechanical third order nonlinearity is canceled out. This way we enhance δQ, from 6 μe/(Hz)(1/2) for the static case to 0.97 μe/(Hz)(1/2) at a probe frequency of ∼1.3 kHz.

An analytical formulation for phase noise in MEMS oscillators

In recent years, there has been much interest in the design of low-noise MEMS oscillators. This paper presents a new analytical formulation for noise in a MEMS oscillator encompassing essential resonator and amplifier nonlinearities. The analytical expression for oscillator noise is derived by solving a second-order nonlinear stochastic differential equation. This approach is applied to noise modeling of an electrostatically addressed MEMS resonator-based square-wave oscillator in which the resonator and oscillator circuit nonlinearities are integrated into a single modeling framework. By considering the resulting amplitude and phase relations, we derive additional noise terms resulting from resonator nonlinearities. The phase diffusion of an oscillator is studied and the phase diffusion coefficient is proposed as a metric for noise optimization. The proposed nonlinear phase noise model provides analytical insight into the underlying physics and a pathway toward the design optimization for low-noise MEMS oscillators.

Nanotube mechanical resonators with quality factors of up to 5 million

Carbon nanotube mechanical resonators have attracted considerable interest because of their small mass, the high quality of their surfaces, and the pristine electronic states they host. However, their small dimensions result in fragile vibrational states that are difficult to measure. Here, we observe quality factors Q as high as 5 × 10(6) in ultra-clean nanotube resonators at a cryostat temperature of 30 mK, where we define Q as the ratio of the resonant frequency over the linewidth. Measuring such high quality factors requires the use of an ultra-low-noise method to rapidly detect minuscule vibrations, as well as careful reduction of the noise of the electrostatic environment. We observe that the measured quality factors fluctuate because of fluctuations of the resonant frequency. We measure record-high quality factors, which are comparable to the highest Q values reported in mechanical resonators of much larger size, a remarkable result considering that reducing the size of resonators is usually concomitant with decreasing quality factors. The combination of ultra-low mass and very large Q offers new opportunities for ultra-sensitive detection schemes and quantum optomechanical experiments.

Stabilization of a linear nanomechanical oscillator to its thermodynamic limit

The rapid development of micro- and nanomechanical oscillators in the past decade has led to the emergence of novel devices and sensors that are opening new frontiers in both applied and fundamental science. The potential of these devices is however affected by their increased sensitivity to external perturbations. Here we report a non-perturbative optomechanical stabilization technique and apply the method to stabilize a linear nanomechanical beam at its thermodynamic limit at room temperature. The reported ability to stabilize a nanomechanical oscillator to the thermodynamic limit can be extended to a variety of systems and increases the sensitivity range of nanomechanical sensors in both fundamental and applied studies.

High-sensitivity linear piezoresistive transduction for nanomechanical beam resonators

Highly sensitive conversion of motion into readable electrical signals is a crucial and challenging issue for nanomechanical resonators. Efficient transduction is particularly difficult to realize in devices of low dimensionality, such as beam resonators based on carbon nanotubes or silicon nanowires, where mechanical vibrations combine very high frequencies with miniscule amplitudes. Here we describe an enhanced piezoresistive transduction mechanism based on the asymmetry of the beam shape at rest. We show that this mechanism enables highly sensitive linear detection of the vibration of low-resistivity silicon beams without the need of exceptionally large piezoresistive coefficients. The general application of this effect is demonstrated by detecting multiple-order modes of silicon nanowire resonators made by either top-down or bottom-up fabrication methods. These results reveal a promising approach for practical applications of the simplest mechanical resonators, facilitating its manufacturability by very large-scale integration technologies.

Eliminating 1/f noise in oscillators

We study 1/f and narrow-bandwidth noise in precision oscillators based on high-quality factor resonators and feedback. The dynamics of such an oscillator are well described by two variables, an amplitude and a phase. In this description we show that low-frequency feedback noise is represented by a single noise vector in phase space. The implication of this is that 1/f and narrow-bandwidth noise can be eliminated by tuning controllable parameters, such as the feedback phase. We present parameter values for which the noise is eliminated and provide specific examples of noise sources for further illustration.

Nonlinear mode coupling and synchronization of a vacuum-trapped nanoparticle

We study the dynamics of a laser-trapped nanoparticle in high vacuum. Using parametric coupling to an external excitation source, the linewidth of the nanoparticle's oscillation can be reduced by three orders of magnitude. We show that the oscillation of the nanoparticle and the excitation source are synchronized, exhibiting a well-defined phase relationship. Furthermore, the external source can be used to controllably drive the nanoparticle into the nonlinear regime, thereby generating strong coupling between the different translational modes of the nanoparticle. Our work contributes to the understanding of the nonlinear dynamics of levitated nanoparticles in high vacuum and paves the way for studies of pattern formation, chaos, and stochastic resonance.

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.

Phase noise of oscillators with unsaturated amplifiers

We study the role of amplifier saturation in eliminating feedback noise in self-sustained oscillators. We extend previous works that use a saturated amplifier to quench fluctuations in the feedback magnitude, while simultaneously tuning the oscillator to an operating point at which the resonator nonlinearity cancels fluctuations in the feedback phase. We consider a generalized model which features an amplitude-dependent amplifier gain function. This allows us to determine the total oscillator phase noise in realistic configurations due to noise in both quadratures of the feedback, and to show that it is not necessary to drive the resonator to large oscillation amplitudes in order to eliminate noise in the phase of the feedback.

Graphene mechanical oscillators with tunable frequency

Oscillators, which produce continuous periodic signals from direct current power, are central to modern communications systems, with versatile applications including timing references and frequency modulators. However, conventional oscillators typically consist of macroscopic mechanical resonators such as quartz crystals, which require excessive off-chip space. Here, we report oscillators built on micrometre-size, atomically thin graphene nanomechanical resonators, whose frequencies can be electrostatically tuned by as much as 14%. Self-sustaining mechanical motion is generated and transduced at room temperature in these oscillators using simple electrical circuitry. The prototype graphene voltage-controlled oscillators exhibit frequency stability and a modulation bandwidth sufficient for the modulation of radiofrequency carrier signals. As a demonstration, we use a graphene oscillator as the active element for frequency-modulated signal generation and achieve efficient audio signal transmission.