Noise-enabled precision measurements of a duffing nanomechanical resonator

Tunable Stochastic State Switching in 2D MoS2 Nanomechanical Resonators with Nonlinear Mode Coupling and Internal Resonance

Coupled nanomechanical resonators have unveiled fascinating physical phenomena, including phonon-cavity coupling, coupled energy decay pathway, avoided crossing, and internal resonance. Despite these discoveries, the mechanisms and control techniques of nonlinear mode coupling phenomena with internal resonances require further exploration. Here, we report on the observation of stochastic switching between the two resonance states with coupled 1:1 internal resonance, for resonant two-dimensional (2D) molybdenum disulfide (MoS2) nanoelectromechanical systems (NEMS), which is directly driven to the critical coupling regime without parametric pumping. We further demonstrate that the probability of state switching is linearly tunable from ∼0% to ∼100% by varying the driving voltage. Furthermore, we gradually increase the white noise amplitude and show that the probability of obtaining the higher-energy state decreases, and the stochastic switching phenomenon eventually disappears. The results provide insights into the dynamics of coupled NEMS resonators and open up new possibilities for sensing and stochastic computing.

Barrier crossing in a viscoelastic medium under active noise: Predictions of Kramers' flux-over-population method

The biochemical activity inside a cell has recently been suggested to act as a source of hydrodynamic fluctuations that can speed up or slow down enzyme catalysis [Tripathi et al., Commun. Phys. 5, 101 (2022).] The idea has been tested against and largely corroborated by simulations of activated barrier crossing in a simple fluid in the presence of thermal and athermal noise. The present paper attempts a wholly analytic solution to the same noise-driven barrier crossing problem but generalizes it to include viscoelastic memory effects of the kind likely to be present in cellular interiors. A calculation of the model's barrier crossing rate, using Kramers' flux-over-population formalism, reveals that in relation to the case where athermal noise is absent, athermal noise always accelerates barrier crossing, though the extent of enhancement depends on the duration τ0 over which the noise acts. More importantly, there exists a critical τ0-determined by the properties of the medium-at which Kramers' theory breaks down and, on approach to which, the rate grows significantly. The possibility of such a giant enhancement is potentially open to experimental validation using optically trapped nanoparticles in viscoelastic media that are acted on by externally imposed colored noise.

Proliferation of unstable states and their impact on stochastic out-of-equilibrium dynamics in two coupled Kerr parametric oscillators

Networks of nonlinear parametric resonators are promising candidates as Ising machines for annealing and optimization. These many-body out-of-equilibrium systems host complex phase diagrams of coexisting stationary states. The plethora of states manifest via a series of bifurcations, including bifurcations that proliferate purely unstable solutions. Here we demonstrate that the latter take a fundamental role in the stochastic dynamics of the system. Specifically, they determine the switching paths and the switching rates between stable solutions. We demonstrate experimentally the impact of the added unstable states on noise-activated switching dynamics in a network of two coupled parametric 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.

Effects of stochastic forces on the nonlinear behaviour of a silicon nitride membrane nanoelectromechanical resonator

In this work, we present the effects of stochastic force generated by white noise on the nonlinear dynamics of a circular silicon nitride membrane. By tuning the membrane to the Duffing nonlinear region, detected signals switching between low- and high-amplitudes have been observed. They are generated by noise-assisted random jumps between bistable states at room temperature and exhibit high sensitivity to the driving frequency. Through artificially heating different mechanical vibration modes by external input of white noise, the switching rate exhibits exponential dependence on the effective temperature and follows with Kramer's law. Furthermore, both the measured switching rate and activation energy exhibit sensitivity to the width of the hysteresis window in nonlinear response and the driving force, which is in qualitative agreement with the theoretical descriptions. Besides, white noise-induced hysteresis window squeezing and bifurcation point shifting have also been observed, which are attributed to the stochastic force modulation of the spring constant of the membrane. These studies are carried out in an all-electric operating scheme at room temperature, paving the way for the exploration of probability distribution-based functional elements that can be massively integrated on-chip.

Nonlinear Stiffness and Nonlinear Damping in Atomically Thin MoS2 Nanomechanical Resonators

We report on experimental measurements and quantitative analyses of nonlinear dynamic characteristics in ultimately thin nanomechanical resonators built upon single-layer, bilayer, and trilayer (1L, 2L, and 3L) molybdenum disulfide (MoS₂) vibrating drumhead membranes. This synergistic study with calibrated measurements and analytical modeling on observed nonlinear responses has led to the determination of nonlinear damping and stiffness coefficients at cubic and quintic orders for these two-dimensional (2D) resonators operating in the very high frequency (VHF) band (up to ∼90 MHz). We find that the quintic force can be ∼20% of the Duffing force at larger amplitudes, and thus, it generally cannot be ignored in a nonlinear dynamics analysis. This study provides the first quantification of nonlinear damping and frequency detuning characteristics in 2D semiconductor nanomechanical resonators and elucidates their origins and dependency on engineerable parameters, setting a foundation for future exploration and utilization of the rich nonlinear dynamics in 2D nanomechanical systems.

Mechanical Bistability in Kerr-modified Cavity Magnomechanics

Bistable mechanical vibration is observed in a cavity magnomechanical system, which consists of a microwave cavity mode, a magnon mode, and a mechanical vibration mode of a ferrimagnetic yttrium-iron-garnet sphere. The bistability manifests itself in both the mechanical frequency and linewidth under a strong microwave drive field, which simultaneously activates three different kinds of nonlinearities, namely, magnetostriction, magnon self-Kerr, and magnon-phonon cross-Kerr nonlinearities. The magnon-phonon cross-Kerr nonlinearity is first predicted and measured in magnomechanics. The system enters a regime where Kerr-type nonlinearities strongly modify the conventional cavity magnomechanics that possesses only a radiation-pressure-like magnomechanical coupling. Three different kinds of nonlinearities are identified and distinguished in the experiment. Our Letter demonstrates a new mechanism for achieving mechanical bistability by combining magnetostriction and Kerr-type nonlinearities, and indicates that such Kerr-modified cavity magnomechanics provides a unique platform for studying many distinct nonlinearities in a single experiment.

Superconducting Cavity Electromechanics: The Realization of an Acoustic Frequency Comb at Microwave Frequencies

We present a nonlinear multimode superconducting electroacoustic system, where the interplay between superconducting kinetic inductance and piezoelectric strong coupling establishes an effective Kerr nonlinearity among multiple acoustic modes at 10 GHz that could hardly be achieved via intrinsic mechanical nonlinearity. By exciting this multimode Kerr system with a single microwave tone, we further demonstrate a coherent electroacoustic frequency comb and provide theoretical understanding of multimode nonlinear interaction in the superstrong coupling limit. This nonlinear superconducting electroacoustic system sheds light on the active control of multimode resonator systems and offers an enabling platform for the dynamic study of microcombs at microwave frequencies.

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.

Mechanical memory operations in piezotransistive GaN microcantilevers using Au nanoparticle-enhanced photoacoustic excitation

Nonlinear oscillations in micro- and nanoelectromechanical systems have emerged as an exciting research area in recent years due to their promise in realizing low-power, scalable, and reconfigurable mechanical memory and logic devices. Here, we report ultralow-power mechanical memory operations utilizing the nonlinear oscillation regime of GaN microcantilevers with embedded piezotransistive AlGaN/GaN heterostructure field effect transistors as highly sensitive deflection transducers. Switching between the high and low oscillatory states of the nonlinear oscillation regime was demonstrated using a novel phase-controlled opto-mechanical excitation setup, utilizing a piezo actuator and a pulsed laser as the primary and secondary excitation sources, respectively. Laser-based photoacoustic excitation was amplified through plasmonic absorption in Au nanoparticles deposited on a transistor. Thus, the minimum switching energy required for reliable memory operations was reduced to less than a picojoule (pJ), which translates to one of the lowest ever reported, when normalized for mass.

Solution to the Kramers barrier crossing problem caused by two noises: Thermal noise and Poisson white noise

We consider the escape of a particle trapped in a metastable potential well and acted upon by two noises. One of the noises is thermal and the other is Poisson white noise, which is non-Gaussian. Using path integral techniques, we find an analytic solution to this generalization of the classic Kramers barrier crossing problem. Using the "barrier climbing" path, we calculate the activation exponent. We also derive an approximate expression for the prefactor. The calculated results are compared with the simulations, and a good agreement between the two is found. Our results show that, unlike in the case of thermal noise, the rate depends not just on the barrier height but also on the shape of the whole barrier. A comparison between the simulations and the theory also shows that a better approximation for the prefactor is needed for agreement for all values of the parameters.

Noise-induced switching from a symmetry-protected shallow metastable state

We consider escape from a metastable state of a nonlinear oscillator driven close to triple its eigenfrequency. The oscillator can have three stable states of period-3 vibrations and a zero-amplitude state. Because of the symmetry of period-tripling, the zero-amplitude state remains stable as the driving increases. However, it becomes shallow in the sense that the rate of escape from this state exponentially increases, while the system still lacks detailed balance. We find the escape rate and show how it scales with the parameters of the oscillator and the driving. The results facilitate using nanomechanical, Josephson-junction based, and other mesoscopic vibrational systems for studying, in a well-controlled setting, the rates of rare events in systems lacking detailed balance. They also describe how fluctuations spontaneously break the time-translation symmetry of a driven oscillator.

Frequency stabilization and noise-induced spectral narrowing in resonators with zero dispersion

Mechanical resonators are widely used as precision clocks and sensitive detectors that rely on the stability of their eigenfrequencies. The phase noise is determined by different factors including thermal noise, frequency noise of the resonator and noise in the feedback circuitry. Increasing the vibration amplitude can mitigate some of these effects but the improvements are limited by nonlinearities that are particularly strong for miniaturized micro- and nano-mechanical systems. Here we design a micromechanical resonator with non-monotonic dependence of the eigenfrequency on energy. Near the extremum, where the dispersion of the eigenfrequency is zero, the system regains certain characteristics of a linear resonator, albeit at large amplitudes. The spectral peak undergoes narrowing when the noise intensity is increased. With the resonator serving as the frequency-selecting element in a feedback loop, the phase noise at the extremum amplitude is ~3 times smaller than the minimal noise in the conventional nonlinear regime.

New periodic-chaotic attractors in slow-fast Duffing system with periodic parametric excitation

A new type of responses called as periodic-chaotic motion is found by numerical simulations in a Duffing oscillator with a slowly periodically parametric excitation. The periodic-chaotic motion is an attractor, and simultaneously possesses the feature of periodic and chaotic oscillations, which is a new addition to the rich nonlinear motions of the Duffing system including equlibria, periodic responses, quasi-periodic oscillations and chaos. In the current slow-fast Duffing system, we find three new attractors in the form of periodic-chaotic motions. These are called the fixed-point chaotic attractor, the fixed-point strange nonchaotic attractor, and the critical behavior with the maximum Lyapunov exponent fluctuating around zero. The system periodically switches between one attractor with a fixed single-well potential and the other with time-varying two-well potentials in every period of excitation. This behavior is apparently the mechanism to generate the periodic-chaotic motion.

Resonantly Induced Friction and Frequency Combs in Driven Nanomechanical Systems

We propose a new mechanism of friction in resonantly driven vibrational systems. The form of the friction force follows from the time- and spatial-symmetry arguments. We consider a microscopic mechanism of this resonant force in nanomechanical systems. The friction can be negative, leading to the onset of self-sustained oscillations of the amplitude and phase of forced vibrations, which result in a frequency comb in the power spectrum.

High-Frequency Stochastic Switching of Graphene Resonators Near Room Temperature

Stochastic switching between the two bistable states of a strongly driven mechanical resonator enables detection of weak signals based on probability distributions, in a manner that mimics biological systems. However, conventional silicon resonators at the microscale require a large amount of fluctuation power to achieve a switching rate in the order of a few hertz. Here, we employ graphene membrane resonators of atomic thickness to achieve a stochastic switching rate of 4.1 kHz, which is 100 times faster than current state-of-the-art. The (effective) temperature of the fluctuations is approximately 400 K, which is 3000 times lower than the state-of-the-art. This shows that these membranes are potentially useful to transduce weak signals in the audible frequency domain. Furthermore, we perform numerical simulations to understand the transition dynamics of the resonator and use analytical expressions to investigate the relevant scaling parameters that allow high-frequency, low-temperature stochastic switching to be achieved in mechanical resonators.

Strong negative nonlinear friction from induced two-phonon processes in vibrational systems

Self-sustained vibrations in systems ranging from lasers to clocks to biological systems are often associated with the coefficient of linear friction, which relates the friction force to the velocity, becoming negative. The runaway of the vibration amplitude is prevented by positive nonlinear friction that increases rapidly with the amplitude. Here we use a modulated electromechanical resonator to show that nonlinear friction can be made negative and sufficiently strong to overcome positive linear friction at large vibration amplitudes. The experiment involves applying a drive that simultaneously excites two phonons of the studied mode and a phonon of a faster decaying high-frequency mode. We study generic features of the oscillator dynamics with negative nonlinear friction. Remarkably, self-sustained vibrations of the oscillator require activation in this case. When, in addition, a resonant force is applied, a branch of large-amplitude forced vibrations can emerge, isolated from the branch of the ordinary small-amplitude response.

Negative linear friction is known to lead to self-sustained vibrations in many systems. Here, the authors show that when nonlinear negative friction in an electromechanical oscillator becomes larger than its positive linear counterpart such self-sustained vibrations require activation.

Driven nonlinear nanomechanical resonators as digital signal detectors

Because of their nonlinearity, vibrational modes of resonantly driven nanomechanical systems have coexisting stable states of forced vibrations in a certain range of the amplitude of the driving force. Depending on its phase, which encodes binary information, a signal at the same frequency increases or decreases the force amplitude. The resulting force amplitude can be outside the range of bistability. The values of the mode amplitude differ significantly on the opposite sides of the bistability region. Therefore the mode amplitude is very sensitive to the signal phase. This suggests using a driven mode as a bi-directional bifurcation amplifier, which switches in the opposite directions depending on the signal phase and provides an essentially digital output. We study the operation of the amplifier near the critical point where the width of the bistability region goes to zero and thus the threshold of the signal amplitude is low. We also develop an analytical technique and study the error rate near the threshold. The results apply to a broad range of currently studied systems and extend to micromechanical systems and nonlinear electromagnetic cavities.

Measuring Frequency Fluctuations in Nonlinear Nanomechanical Resonators

Advances in nanomechanics within recent years have demonstrated an always expanding range of devices, from top-down structures to appealing bottom-up MoS₂ and graphene membranes, used for both sensing and component-oriented applications. One of the main concerns in all of these devices is frequency noise, which ultimately limits their applicability. This issue has attracted a lot of attention recently, and the origin of this noise remains elusive to date. In this article we present a very simple technique to measure frequency noise in nonlinear mechanical devices, based on the presence of bistability. It is illustrated on silicon-nitride high-stress doubly clamped beams, in a cryogenic environment. We report on the same T/ f dependence of the frequency noise power spectra as reported in the literature. But we also find unexpected damping fluctuations, amplified in the vicinity of the bifurcation points; this effect is clearly distinct from already reported nonlinear dephasing and poses a fundamental limit on the measurement of bifurcation frequencies. The technique is further applied to the measurement of frequency noise as a function of mode number, within the same device. The relative frequency noise for the fundamental flexure δ f/ f₀ lies in the range 0.5-0.01 ppm (consistent with the literature for cryogenic MHz devices) and decreases with mode number in the range studied. The technique can be applied to any type of nanomechanical structure, enabling progress toward the understanding of intrinsic sources of noise in these devices.

Energy Dissipation Pathways in Few-Layer MoS2 Nanoelectromechanical Systems

Free standing, atomically thin transition metal dichalcogenides are a new class of ultralightweight nanoelectromechanical systems with potentially game-changing electro- and opto-mechanical properties, however, the energy dissipation pathways that fundamentally limit the performance of these systems is still poorly understood. Here, we identify the dominant energy dissipation pathways in few-layer MoS₂ nanoelectromechanical systems. The low temperature quality factors and resonant frequencies are shown to significantly decrease upon heating to 293 K, and we find the temperature dependence of the energy dissipation can be explained when accounting for both intrinsic and extrinsic damping sources. A transition in the dominant dissipation pathways occurs at T ~ 110 K with relatively larger contributions from phonon-phonon and electrostatic interactions for T > 110 K and larger contributions from clamping losses for T < 110 K. We further demonstrate a room temperature thermomechanical-noise-limited force sensitivity of ~8 fN/Hz^1/2 that, despite multiple dissipation pathways, remains effectively constant over the course of more than four years. Our results provide insight into the mechanisms limiting the performance of nanoelectromechanical systems derived from few-layer materials, which is vital to the development of next-generation force and mass sensors.

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.

Phononic Frequency Comb via Intrinsic Three-Wave Mixing

Optical frequency combs have resulted in significant advances in optical frequency metrology and found wide applications in precise physical measurements and molecular fingerprinting. A direct analogue of frequency combs in the phononic or acoustic domain has not been reported to date. In this Letter, we report the first clear experimental evidence for a phononic frequency comb. We show that the phononic frequency comb is generated through the intrinsic coupling of a driven phonon mode with an autoparametrically excited subharmonic mode. The experiments depict the comb generation process evidenced by a spectral response consisting of equally spaced discrete and phase coherent comb lines. Through systematic experiments at different drive frequencies and amplitudes, we portray the well-connected process of phononic frequency comb formation and define the attributes to control the features associated with comb formation in such a system. In addition to the demonstration of frequency comb, the interplay between the nonlinear resonances and the well-known Duffing phenomenon is also observed.

Steady-state mechanical squeezing in a double-cavity optomechanical system

We study the physical properties of double-cavity optomechanical system in which the mechanical resonator interacts with one of the coupled cavities and another cavity is used as an auxiliary cavity. The model can be expected to achieve the strong optomechanical coupling strength and overcome the optomechanical cavity decay, simultaneously. Through the coherent auxiliary cavity interferences, the steady-state squeezing of mechanical resonator can be generated in highly unresolved sideband regime. The validity of the scheme is assessed by numerical simulation and theoretical analysis of the steady-state variance of the mechanical displacement quadrature. The scheme provides a platform for the mechanical squeezing beyond the resolved sideband limit and solves the restricted experimental bounds at present.

An electromechanical Ising Hamiltonian

Solving intractable mathematical problems in simulators composed of atoms, ions, photons, or electrons has recently emerged as a subject of intense interest. We extend this concept to phonons that are localized in spectrally pure resonances in an electromechanical system that enables their interactions to be exquisitely fashioned via electrical means. We harness this platform to emulate the Ising Hamiltonian whose spin 1/2 particles are replicated by the phase bistable vibrations from the parametric resonances of multiple modes. The coupling between the mechanical spins is created by generating two-mode squeezed states, which impart correlations between modes that can imitate a random, ferromagnetic state or an antiferromagnetic state on demand. These results suggest that an electromechanical simulator could be built for the Ising Hamiltonian in a nontrivial configuration, namely, for a large number of spins with multiple degrees of coupling.

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.

Scaling laws for the bifurcation escape rate in a nanomechanical resonator

We report on experimental and theoretical studies of the fluctuation-induced escape time from a metastable state of a nanomechanical Duffing resonator in a cryogenic environment. By tuning in situ the nonlinear coefficient γ we could explore a wide range of the parameter space around the bifurcation point, where the metastable state becomes unstable. We measured in a relaxation process the distribution of the escape times. We have been able to verify its exponential distribution and extract the escape rate Γ. We investigated the scaling of Γ with respect to the distance to the bifurcation point and γ, finding an unprecedented quantitative agreement with the theoretical description of the stochastic problem. Simple power scaling laws turn out to hold in a large region of the parameter space, as anticipated by recent theoretical predictions. These unique findings, implemented in a model dynamical system, are relevant to all systems experiencing underdamped saddle-node bifurcation.

Nonlinear tuning of microresonators for dynamic range enhancement

This paper investigates the development of a novel framework and its implementation for the nonlinear tuning of nano/microresonators. Using geometrically exact mechanical formulations, a nonlinear model is obtained that governs the transverse and longitudinal dynamics of multilayer microbeams, and also takes into account rotary inertia effects. The partial differential equations of motion are discretized, according to the Galerkin method, after being reformulated into a mixed form. A zeroth-order shift as well as a hardening effect are observed in the frequency response of the beam. These results are confirmed by a higher order perturbation analysis using the method of multiple scales. An inverse problem is then proposed for the continuation of the critical amplitude at which the transition to nonlinear response characteristics occurs. Path-following techniques are employed to explore the dependence on the system parameters, as well as on the geometry of bilayer microbeams, of the magnitude of the dynamic range in nano/microresonators.

Work fluctuations in a nonlinear micromechanical oscillator driven far from thermal equilibrium

We explore fluctuation relations in a periodically driven micromechanical torsional oscillator. In the linear regime where the modulation is weak, we verify that the ratio of the work variance to the mean work is constant, consistent with conventional fluctuation theorems. We then increase the amplitude of the periodic drive so that the response becomes nonlinear and two nonequilibrium oscillation states coexist. Due to interstate transitions, the work variance exhibits a peak at the driving frequency at which the occupation of the two states is equal. Moreover, the work fluctuations depend exponentially on the inverse noise intensity. Our data are consistent with recent theories on systems driven into bistability that predict generic behaviors different from conventional fluctuation theorems.

Cascaded optical transparency in multimode-cavity optomechanical systems

Electromagnetically induced transparency has great theoretical and experimental importance in many areas of physics, such as atomic physics, quantum optics and, more recent, cavity optomechanics. Optical delay is the most prominent feature of electromagnetically induced transparency, and in cavity optomechanics, the optical delay is limited by the mechanical dissipation rate of sideband-resolved mechanical modes. Here we demonstrate a cascaded optical transparency scheme by leveraging the parametric phonon-phonon coupling in a multimode optomechanical system, where a low damping mechanical mode in the unresolved-sideband regime is made to couple to an intermediate, high-frequency mechanical mode in the resolved-sideband regime of an optical cavity. Extended optical delay and higher transmission as well as optical advancing are demonstrated. These results provide a route to realize ultra-long optical delay, indicating a significant step towards integrated classical and quantum information storage devices.

Nonlinear dynamics of a doubly clamped carbon nanotube resonator considering surface stress

A new study on the dynamic performances of doubly clamped carbon nanotube resonators taking account of the surface effect has been performed.

Nanomechanical gas sensing with nonlinear resonant cantilevers

Cantilevers play an important role as linear transducers in nanoscience, with nanomechanical detection of mass and stress as a clear example. We performed gas sensing experiments with a standard functionalized cantilever driven strongly into the regime of nonlinear oscillations. We compared the cantilever response to the selective adsorption of ethanol vapour in the nonlinear regime, to the ones obtained in the conventional linear static and dynamic sensing modes. In the nonlinearly driven cantilever, the adsorption and desorption-induced frequency shifts were enhanced by over a factor of three compared to resonant sensing with the same mode in the linear regime. This demonstrates a route towards gas detectors that exploit nonlinearity to enhance the responsivity, which can be implemented with standard cantilever devices.

Duffing oscillation-induced reversal of magnetic vortex core by a resonant perpendicular magnetic field

Nonlinear dynamics of the magnetic vortex state in a circular nanodisk was studied under a perpendicular alternating magnetic field that excites the radial modes of the magnetic resonance. Here, we show that as the oscillating frequency is swept down from a frequency higher than the eigenfrequency, the amplitude of the radial mode is almost doubled to the amplitude at the fixed resonance frequency. This amplitude has a hysteresis vs. frequency sweeping direction. Our result showed that this phenomenon was due to a Duffing-type nonlinear resonance. Consequently, the amplitude enhancement reduced the vortex core-switching magnetic field to well below 10 mT. A theoretical model corresponding to the Duffing oscillator was developed from the Landau–Lifshitz–Gilbert equation to explore the physical origin of the simulation result. This work provides a new pathway for the switching of the magnetic vortex core polarity in future magnetic storage devices.

Stochastic switching of cantilever motion

The cantilever is a prototype of a highly compliant mechanical system and has an instrumental role in nanotechnology, enabling surface microscopy, and ultrasensitive force and mass measurements. Here we report fluctuation-induced transitions between two stable states of a strongly driven microcantilever. Geometric nonlinearity gives rise to an amplitude-dependent resonance frequency and bifurcation occurs beyond a critical point. The cantilever response to a weak parametric modulation is amplified by white noise, resulting in an optimum signal-to-noise ratio at finite noise intensity. This stochastic switching suggests new detection schemes for cantilever-based instrumentation, where the detection of weak signals is mediated by the fluctuating environment. For ultrafloppy, cantilevers with nanometer-scale dimensions operating at room temperature--a new transduction paradigm emerges that is based on probability distributions and mimics nature.

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.

Single-layer MoS(2) mechanical resonators

Mechanical resonators are fabricated from freely suspended single-layer MoS2 . Their dynamics  have been studied by optical interferometry. These resonators behave as membranes with resonance frequencies in between 10 and 30 MHz and quality factors in between 16 and 109. We also demonstrate clear signatures of nonlinear resonance in these atomically thin resonators.

Orbital trajectory of an acoustic bubble in a cylindrical resonator

Acoustic cavitation-induced microbubbles in a cylindrical resonator filled with water tend to concentrate into ring patterns due to the cylindrical geometry of the system. The shape of these ring patterns is directly linked to the Bjerknes force distribution in the resonator. Experimental observations showed that cavitation bubbles located in the vicinity of this ring may exhibit a spiraling behavior around the pressure nodal line. This spiraling phenomenon is numerically studied, the conditions for which a single cavitation bubble follows an orbital trajectory are established, and the influences of the acoustic pressure amplitude and the initial bubble radius are investigated.

Intervention-based stochastic disease eradication

Disease control is of paramount importance in public health, with infectious disease extinction as the ultimate goal. Although diseases may go extinct due to random loss of effective contacts where the infection is transmitted to new susceptible individuals, the time to extinction in the absence of control may be prohibitively long. Intervention controls are typically defined on a deterministic schedule. In reality, however, such policies are administered as a random process, while still possessing a mean period. Here, we consider the effect of randomly distributed intervention as disease control on large finite populations. We show explicitly how intervention control, based on mean period and treatment fraction, modulates the average extinction times as a function of population size and rate of infection spread. In particular, our results show an exponential improvement in extinction times even though the controls are implemented using a random Poisson distribution. Finally, we discover those parameter regimes where random treatment yields an exponential improvement in extinction times over the application of strictly periodic intervention. The implication of our results is discussed in light of the availability of limited resources for control.

Stationary energy probability density of oscillators driven by a random external force

We derive rigorous analytical results for the stationary energy probability density function of linear and nonlinear oscillators driven by additive Gaussian noise. Our study focuses on two cases: (i) a harmonic oscillator subjected to Gaussian colored noise with an arbitrary correlation function and (ii) nonlinear oscillators with a general potential driven by Gaussian white noise. We also derive analytical expressions for the stationary moments of the energy and investigate the partition of the mean energy between kinetic and potential energy. To illustrate our general results, we consider specifically the case of exponentially correlated noise for (i) and power-law and bistable potentials for (ii). Our theoretical results are substantiated by Langevin simulations.

Complex dynamics in nanosystems

Complex dynamics associated with multistability have been studied extensively in the past but mostly for low-dimensional nonlinear dynamical systems. A question of fundamental interest is whether multistability can arise in high-dimensional physical systems. Motivated by the ever increasing widespread use of nanoscale systems, we investigate a prototypical class of nanoelectromechanical systems: electrostatically driven Si nanowires, mathematically described by a set of driven, nonlinear partial differential equations. We develop a computationally efficient algorithm to solve the equations. Our finding is that multistability and complicated structures of basins of attraction are common types of dynamics, and the latter can be attributed to extensive transient chaos. Implications of these phenomena to device operations are discussed.

Switching dynamics and linear response spectra of a driven one-dimensional nonlinear lattice containing an intrinsic localized mode

An intrinsic localized mode (ILM) represents a localized vibrational excitation in a nonlinear lattice. Such a mode will stay in resonance as the driver frequency is changed adiabatically until a bifurcation point is reached, at which point the ILM switches and disappears. The dynamics behind switching in such a many body system is examined here through experimental measurements and numerical simulations. Linear response spectra of a driven micromechanical array containing an ILM were measured in the frequency region between two fundamentally different kinds of bifurcation points that separate the large amplitude ILM state from the two low amplitude vibrational states. Just as a natural frequency can be associated with a driven harmonic oscillator, a similar natural frequency has been found for a driven ILM via the beat frequency between it and a weak, tunable probe. This finding has been confirmed using numerical simulations. The behavior of this nonlinear natural frequency plays important but different roles as the two bifurcation points are approached. At the upper transition its frequency coalesces with the driver and the resulting bifurcation is very similar to the saddle-node bifurcation of a single driven Duffing oscillator, which is treated in an Appendix. The lower transition occurs when the four-wave mixing partner of the natural frequency of the ILM intersects the topmost extended band mode of the same symmetry. The properties of linear local modes associated with the driven ILM are also identified experimentally for the first time and numerically but play no role in these transitions.

Noise-induced current switching in semiconductor superlattices: observation of nonexponential kinetics in a high-dimensional system

We report on measurements of first-passage-time distributions associated with current switching in weakly coupled GaAs/AlAs superlattices driven by shot noise, a system that is both far from equilibrium and high dimensional. Static current-voltage (I-V) characteristics exhibit multiple current branches and bistability; precision, high-bandwidth current switching data are collected in response to steps in the applied voltage to final voltages V1 near the end of a current branch. For a range of V1 values, the measured switching times vary stochastically. At short times (≲10  μs), the switching time distributions decay exponentially, while at longer times the distributions develop nonexponential tails that follow an approximate power law over several decades. The power law decay behavior is attributed to the presence of multiple switching pathways, which may arise from small spatial variations in the superlattice growth parameters.

Singular response of bistable systems driven by telegraph noise

We show that weak periodic driving can exponentially strongly change the rate of escape from a potential well of a system driven by telegraph noise. The analysis refers to an overdamped system, where escape requires that the noise amplitude θ exceed a critical value θ(c). For θ close to θ(c), the exponent of the escape rate displays a nonanalytic dependence on the amplitude of an additional low-frequency modulation. This leads to giant nonlinearity of the response of a bistable system to periodic modulation. Also studied is the linear response to periodic modulation far from θ(c). We analyze the scaling of the logarithm of the escape rate with the distance to the saddle-node and pitchfork bifurcation points. The analytical results are in excellent agreement with numerical simulations.

Collective quantum jumps of Rydberg atoms

We study an open quantum system of atoms with a long-range Rydberg interaction, laser driving, and spontaneous emission. Over time, the system occasionally jumps between a state of low Rydberg population and a state of high Rydberg population. The jumps are inherently collective, and in fact, exist only for a large number of atoms. We explain how entanglement and quantum measurement enable the jumps, which are otherwise classically forbidden.

Homoclinic orbits and chaos in a pair of parametrically driven coupled nonlinear resonators

We study the dynamics of a pair of parametrically driven coupled nonlinear mechanical resonators of the kind that is typically encountered in applications involving microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS). We take advantage of the weak damping that characterizes these systems to perform a multiple-scales analysis and obtain amplitude equations, describing the slow dynamics of the system. This picture allows us to expose the existence of homoclinic orbits in the dynamics of the integrable part of the slow equations of motion. Using a version of the high-dimensional Melnikov approach, developed by G. Kovačič and S. Wiggins [Physica D 57, 185 (1992)], we are able to obtain explicit parameter values for which these orbits persist in the full system, consisting of both Hamiltonian and non-Hamiltonian perturbations, to form so-called Šilnikov orbits, indicating a loss of integrability and the existence of chaos. Our analytical calculations of Šilnikov orbits are confirmed numerically.

Nonlinear damping in mechanical resonators made from carbon nanotubes and graphene

The theory of damping is discussed in Newton's Principia and has been tested in objects as diverse as the Foucault pendulum, the mirrors in gravitational-wave detectors and submicrometre mechanical resonators. In general, the damping observed in these systems can be described by a linear damping force. Advances in nanofabrication mean that it is now possible to explore damping in systems with one or more atomic-scale dimensions. Here we study the damping of mechanical resonators based on carbon nanotubes and graphene sheets. The damping is found to strongly depend on the amplitude of motion, and can be described by a nonlinear rather than a linear damping force. We exploit the nonlinear nature of damping in these systems to improve the figures of merit for both nanotube and graphene resonators. For instance, we achieve a quality factor of 100,000 for a graphene resonator.

Quantum entanglement between a nonlinear nanomechanical resonator and a microwave field

We consider a theoretical model for a nonlinear nanomechanical resonator coupled to a superconducting microwave resonator. The nanomechanical resonator is driven parametrically at twice its resonance frequency, while the superconducting microwave resonator is driven with two tones that differ in frequency by an amount equal to the parametric driving frequency. We show that the semiclassical approximation of this system has an interesting fixed-point bifurcation structure. In the semiclassical dynamics a transition from stable fixed points to limit cycles is observed as one moves from positive to negative detuning. We show that signatures of this bifurcation structure are also present in the full dissipative quantum system and further show that the bifurcation structure leads to mixed-state entanglement between the nanomechanical resonator and the microwave cavity in the dissipative quantum system that is a maximum close to the semiclassical bifurcation. Quantum signatures of the semiclassical limit cycles are presented.

Ideal mean-field transition in a modulated cold atom system

We show that an atomic system in a periodically modulated optical trap displays an ideal mean-field symmetry-breaking transition. The symmetry is broken with respect to time translation by the modulation period. We describe experimental observations and develop a full microscopic theory of the observed critical phenomena. The transition is explained as resulting from the interplay of the long-range interatomic interaction and nonequilibrium fluctuations in the strongly modulated system. The observations, including anomalous fluctuations in the symmetry broken phase, are fully described by the theory.

Damping of nanomechanical resonators

We study the transverse oscillatory modes of nanomechanical silicon nitride strings under high tensile stress as a function of geometry and mode index m≤9. Reproducing all observed resonance frequencies with classical elastic theory we extract the relevant elastic constants. Based on the oscillatory local strain we successfully predict the observed mode-dependent damping with a single frequency-independent fit parameter. Our model clarifies the role of tensile stress on damping and hints at the underlying microscopic mechanisms.