Stabilization of a linear nanomechanical oscillator to its thermodynamic limit

Engineering error correcting dynamics in nanomechanical systems

Nanomechanical oscillators are an alternative platform for computation in harsh environments. However, external perturbations arising from such environments may hinder information processing by introducing errors into the computing system. Here, we simulate the dynamics of three coupled Duffing oscillators whose multiple equilibrium states can be used for information processing and storage. Our analysis reveals that, within experimentally relevant parameters, error correcting dynamics can emerge, wherein the system's state is robust against random external impulses. We find that oscillators in this configuration have several surprising and attractive features, including dynamic isolation of resonators exposed to extreme impulses and the ability to correct simultaneous errors.

Ultrahigh-quality-factor micro- and nanomechanical resonators using dissipation dilution

Mechanical resonators are widely used in sensors, transducers and optomechanical systems, where mechanical dissipation sets the ultimate limit to performance. Over the past 15 years, the quality factors in strained mechanical resonators have increased by four orders of magnitude, surpassing the previous state of the art achieved in bulk crystalline resonators at room temperature and liquid helium temperatures. In this Review, we describe how these advances were made by leveraging 'dissipation dilution'-where dissipation is reduced through a combination of static tensile strain and geometric nonlinearity in dynamic strain. We then review the state of the art in strained nanomechanical resonators and discuss the potential for even higher quality factors in crystalline materials. Finally, we detail current and future applications of dissipation-diluted mechanical resonators.

Ultrasensitive optomechanical strain sensor

We demonstrate an ultrasensitive optomechanical strain sensor based on a SiN membrane and a Fabry-Perot cavity, enabling the measurements of both static and dynamic strain by monitoring reflected light fluctuations using a single-frequency laser. The SiN membrane offers high-quality-factor mechanical resonances that are sensitive to minute strain fluctuations. The two-beam Fabry-Perot cavity is constructed to interrogate the motion state of the SiN membrane. A static strain resolution of 4.00 nɛ is achieved by measuring mechanical resonance frequency shifts of the SiN membrane. The best dynamic resolution is 4.47 pɛHz-1/2, which is close to that of the sensor using high-finesse cavity and optical frequency comb, overcoming the dependence of ultrasensitive strain sensors on narrow-linewidth laser and high-finesse cavity with frequency locking equipment. This work opens up a promising avenue for a new generation of ultrasensitive strain sensors.

Ultrahigh Quality Factor of a Levitated Nanomechanical Oscillator

A levitated nanomechanical oscillator under ultrahigh vacuum is highly isolated from its environment. It has been predicted that this isolation leads to very low mechanical dissipation rates. However, a gap persists between predictions and experimental data. Here, we levitate a silica nanoparticle in a linear Paul trap at room temperature, at pressures as low as 7×10^{-11}  mbar. We measure a dissipation rate of 2π×69(22)  nHz, corresponding to a quality factor exceeding 10^{10}, more than 2 orders of magnitude higher than previously shown. A study of the pressure dependence of the particle's damping and heating rates provides insight into the relevant dissipation mechanisms.

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.

Fundamental limits and optimal estimation of the resonance frequency of a linear harmonic oscillator

All physical oscillators are subject to thermodynamic and quantum perturbations, fundamentally limiting measurement of their resonance frequency. Analyses assuming specific ways of estimating frequency can underestimate the available precision and overlook unconventional measurement regimes. Here we derive a general, estimation-method-independent Cramer Rao lower bound for a linear harmonic oscillator resonance frequency measurement uncertainty, seamlessly accounting for the quantum, thermodynamic and instrumental limitations, including Fisher information from quantum backaction- and thermodynamically-driven fluctuations. We provide a universal and practical maximum-likelihood frequency estimator reaching the predicted limits in all regimes, and experimentally validate it on a thermodynamically-limited nanomechanical oscillator. Low relative frequency uncertainty is obtained for both very high bandwidth measurements (≈ 10-5 for τ = 30 μs ) and measurements using thermal fluctuations alone (<10-6). Beyond nanomechanics, these results advance frequency-based metrology across physical domains.

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.

Frequency Stabilization of Nanomechanical Resonators Using Thermally Invariant Strain Engineering

Microfabricated mechanical resonators enable precision measurement techniques from atomic force microscopy to emerging quantum applications. The resonance frequency-based physical sensing combines high precision with long-term stability. However, widely used Si₃N₄ resonators suffer from frequency sensitivity to temperature due to the differential thermal expansion vs the Si substrates. Here we experimentally demonstrate temperature- and residual stress-insensitive 16.51 MHz tuning fork nanobeam resonators with nonlinear clamps defining the stress and frequency by design, achieving a low fractional frequency sensitivity of (2.5 ± 0.8) × 10^-6 K^-1, a 72× reduction. On-chip optical readout of resonator thermomechanical fluctuations allows precision frequency measurement without any external excitation at the thermodynamically limited frequency Allan deviation of ≈7 Hz/Hz^1/2 and (relative) bias stability of ≈10 Hz (≈ 0.6 × 10^-6) above 1 s averaging, remarkably, on par with state-of-the-art driven devices of similar mass. Both the resonator stabilization and the passive frequency readout can benefit a wide variety of micromechanical sensors.

Shot-Noise-Limited Nanomechanical Detection and Radiation Pressure Backaction from an Electron Beam

Detecting nanomechanical motion has become an important challenge in science and technology. Recently, electromechanical coupling to focused electron beams has emerged as a promising method adapted to ultralow scale systems. However the fundamental measurement processes associated with such complex interaction remain to be explored. Here we report a highly sensitive detection of the Brownian motion of μm-long semiconductor nanowires (InAs). The measurement imprecision is found to be set by the shot noise of the secondary electrons generated along the electromechanical interaction. By carefully analyzing the nanoelectromechanical dynamics, we demonstrate the existence of a radial backaction process that we identify as originating from the momentum exchange between the electron beam and the nanomechanical device, which is also known as radiation pressure.

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.

Thermodynamics Far from the Thermodynamic Limit

Understanding how small systems exchange energy with a heat bath is important to describe how their unique properties can be affected by the environment. In this contribution, we apply Landsberg's theory of temperature-dependent energy levels to describe the progressive thermalization of small systems as their spectrum is perturbed by a heat bath. We propose a mechanism whereby the small system undergoes a discrete series of excitations and isentropic spectrum adjustments leading to a final state of thermal equilibrium. This produces standard thermodynamic results without invoking system size. The thermal relaxation of a single harmonic oscillator is analyzed as a model example of a system with a quantized spectrum than can be embedded in a thermal environment. A description of how the thermal environment affects the spectrum of a small system can be the first step in using environmental factors, such as temperature, as parameters in the design and operation of nanosystem properties.

A low-frequency chip-scale optomechanical oscillator with 58 kHz mechanical stiffening and more than 100th-order stable harmonics

For the sensitive high-resolution force- and field-sensing applications, the large-mass microelectromechanical system (MEMS) and optomechanical cavity have been proposed to realize the sub-aN/Hz^1/2 resolution levels. In view of the optomechanical cavity-based force- and field-sensors, the optomechanical coupling is the key parameter for achieving high sensitivity and resolution. Here we demonstrate a chip-scale optomechanical cavity with large mass which operates at ≈77.7 kHz fundamental mode and intrinsically exhibiting large optomechanical coupling of 44 GHz/nm or more, for both optical resonance modes. The mechanical stiffening range of ≈58 kHz and a more than 100^th-order harmonics are obtained, with which the free-running frequency instability is lower than 10⁻⁶ at 100 ms integration time. Such results can be applied to further improve the sensing performance of the optomechanical inspired chip-scale sensors.

Real-Time Measurement of Nanotube Resonator Fluctuations in an Electron Microscope

Mechanical resonators based on low-dimensional materials provide a unique platform for exploring a broad range of physical phenomena. The mechanical vibrational states are indeed extremely sensitive to charges, spins, photons, and adsorbed masses. However, the roadblock is often the readout of the resonator, because the detection of the vibrational states becomes increasingly difficult for smaller resonators. Here, we report an unprecedentedly sensitive method to detect nanotube resonators with effective masses in the 10^-20 kg range. We use the beam of an electron microscope to resolve the mechanical fluctuations of a nanotube in real-time for the first time. We obtain full access to the thermally driven Brownian motion of the resonator, both in space and time domains. Our results establish the viability of carbon nanotube resonator technology at room temperature and pave the way toward the observation of novel thermodynamics regimes and quantum effects in nanomechanics.

Nano-Optomechanical Systems for Gas Chromatography

Microgas chromatography (GC) is promising for portable chemical analysis. We demonstrate a nano-optomechanical system (NOMS) as an ultrasensitive mass detector in gas chromatography. Bare, native oxide, silicon surfaces are sensitive enough to monitor volatile organic compounds at ppm levels, while simultaneously demonstrating chemical selectivity. The NOMS is able to sense GC peaks from derivatized metabolites at physiological concentrations. This is an important milestone for small-molecule quantitation assays in next generation metabolite analyses for applications such as disease diagnosis and personalized medicine. The optical microring, which plays an important role in the nanomechanical signal transduction mechanism, can also be used as an analyte concentration sensor. Different adsorption kinetics regimes are realized at different temperatures allowing temporary condensation of the analyte onto the sensor surfaces. This effect amplifies the signal, resulting in a 1 ppb level limit of detection, without partition enhancement from absorbing media. This sensitivity bodes well for NOMS as universal, ultrasensitive detectors in micro-GC, breath analysis, and other chemical-sensing applications.

High-throughput measurement of single-cell growth rates using serial microfluidic mass sensor arrays

Methods to rapidly assess cell growth would be useful for many applications, including drug susceptibility testing, but current technologies have limited sensitivity or throughput. Here we present an approach to precisely and rapidly measure growth rates of many individual cells simultaneously. We flow cells in suspension through a microfluidic channel with 10–12 resonant mass sensors distributed along its length, weighing each cell repeatedly over the 4–20 min it spends in the channel. Because multiple cells traverse the channel at the same time, we obtain growth rates for >60 cells/h with a resolution of 0.2 pg/h for mammalian cells and 0.02 pg/h for bacteria. We measure the growth of single lymphocytic cells, mouse and human T cells, primary human leukemia cells, yeast, Escherichia coli and Enterococcus faecalis. Our system reveals subpopulations of cells with divergent growth kinetics and enables assessment of cellular responses to antibiotics and antimicrobial peptides within minutes.

Correlated anomalous phase diffusion of coupled phononic modes in a sideband-driven resonator

The dynamical backaction from a periodically driven optical cavity can reduce the damping of a mechanical resonator, leading to parametric instability accompanied by self-sustained oscillations. Here we study experimentally and theoretically new aspects of the backaction and the discrete time-translation symmetry of a driven system using a micromechanical resonator with two nonlinearly coupled vibrational modes with strongly differing frequencies and decay rates. We find self-sustained oscillations in both the low- and high-frequency modes. Their frequencies and amplitudes are determined by the nonlinearity, which also leads to bistability and hysteresis. The phase fluctuations of the two modes show near-perfect anti-correlation, a consequence of the discrete time-translation symmetry. Concurrently, the phase of each mode undergoes anomalous diffusion. The phase variance follows a power law time dependence, with an exponent determined by the 1/f-type resonator frequency noise. Our findings enable compensating for the fluctuations using a feedback scheme to achieve stable frequency downconversion.

Dynamical backaction from a periodically driven cavity can reduce the damping of a mechanical resonator causing parametric instability. Here, the authors observe simultaneous self-sustained oscillations in both the mechanical and cavity modes and their correlated phase diffusion.

Frequency fluctuations in silicon nanoresonators

Frequency stability is key to the performance of nanoresonators. This stability is thought to reach a limit with the resonator's ability to resolve thermally induced vibrations. Although measurements and predictions of resonator stability usually disregard fluctuations in the mechanical frequency response, these fluctuations have recently attracted considerable theoretical interest. However, their existence is very difficult to demonstrate experimentally. Here, through a literature review, we show that all studies of frequency stability report values several orders of magnitude larger than the limit imposed by thermomechanical noise. We studied a monocrystalline silicon nanoresonator at room temperature and found a similar discrepancy. We propose a new method to show that this was due to the presence of frequency fluctuations, of unexpected level. The fluctuations were not due to the instrumentation system, or to any other of the known sources investigated. These results challenge our current understanding of frequency fluctuations and call for a change in practices.

Suspended photonic waveguide devices

This article describes recent research at the U.S. Naval Research Laboratory that focuses on the use of micro- and nanomachining techniques for photonic waveguide devices. By selectively etching a sacrificial layer that the waveguide core is supported by, in whole or in part, the waveguide obtains enhanced properties and functionality, such as mechanical flexibility, index contrast, birefringence, and evanescent field depth. We describe how these properties enable unique waveguide applications in areas such as cavity optomechanics, displacement sensing, electro-optics, and nonlinear optics.

Thermomechanical Two-Mode Squeezing in an Ultrahigh-Q Membrane Resonator

We realize a quantum-compatible multimode interaction in an ultrahigh Q mechanical resonator via a reservoir-mediated parametric coupling. We use this interaction to demonstrate nondegenerate parametric amplification and thermomechanical noise squeezing, finding excellent agreement with a theoretical model of this interaction over a large dynamic range. This realization of strong multimode nonlinearities in a mechanical platform compatible with quantum-limited optical detection and cooling makes this a powerful system for nonlinear approaches to quantum metrology, transduction between optical and phononic fields, and the quantum manipulation of phononic degrees of freedom.

High-speed multiple-mode mass-sensing resolves dynamic nanoscale mass distributions

Simultaneously measuring multiple eigenmode frequencies of nanomechanical resonators can determine the position and mass of surface-adsorbed proteins, and could ultimately reveal the mass tomography of nanoscale analytes. However, existing measurement techniques are slow (<1 Hz bandwidth), limiting throughput and preventing use with resonators generating fast transient signals. Here we develop a general platform for independently and simultaneously oscillating multiple modes of mechanical resonators, enabling frequency measurements that can precisely track fast transient signals within a user-defined bandwidth that exceeds 500 Hz. We use this enhanced bandwidth to resolve signals from multiple nanoparticles flowing simultaneously through a suspended nanochannel resonator and show that four resonant modes are sufficient for determining their individual position and mass with an accuracy near 150 nm and 40 attograms throughout their 150-ms transit. We envision that our method can be readily extended to other systems to increase bandwidth, number of modes, or number of resonators.

Interplay of driving and frequency noise in the spectra of vibrational systems

We study the spectral effect of the fluctuations of the vibration frequency. Such fluctuations play a major role in nanomechanical and other mesoscopic vibrational systems. We find that, for periodically driven systems, the interplay of the driving and frequency fluctuations results in specific spectral features. We present measurements on a carbon nanotube resonator and show that our theory allows not only the characterization of the frequency fluctuations but also the quantification of the decay rate without ring-down measurements. The results bear on identifying the decoherence of mesoscopic oscillators and on the general problem of resonance fluorescence and light scattering by 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.

An integrated low phase noise radiation-pressure-driven optomechanical oscillator chipset

High-quality frequency references are the cornerstones in position, navigation and timing applications of both scientific and commercial domains. Optomechanical oscillators, with direct coupling to continuous-wave light and non-material-limited f × Q product, are long regarded as a potential platform for frequency reference in radio-frequency-photonic architectures. However, one major challenge is the compatibility with standard CMOS fabrication processes while maintaining optomechanical high quality performance. Here we demonstrate the monolithic integration of photonic crystal optomechanical oscillators and on-chip high speed Ge detectors based on the silicon CMOS platform. With the generation of both high harmonics (up to 59th order) and subharmonics (down to 1/4), our chipset provides multiple frequency tones for applications in both frequency multipliers and dividers. The phase noise is measured down to −125 dBc/Hz at 10 kHz offset at ~400 μW dropped-in powers, one of the lowest noise optomechanical oscillators to date and in room-temperature and atmospheric non-vacuum operating conditions. These characteristics enable optomechanical oscillators as a frequency reference platform for radio-frequency-photonic information processing.

Heralded preparation and readout of entangled phonons in a photonic crystal cavity

We propose a realistic protocol for the preparation and readout of mechanical Bell states in an optomechanical system. The proposal relies on parameters characterizing a photonic crystal cavity mode, coupled to two localized flexural modes of the structure, but equally applies to other optomechanical systems in the same parameter range. The nonclassical states are heralded via optical postselection and revealed in specific interference patterns characterizing the emission at the cavity frequency.