Zeptonewton and attotesla per centimeter metrology with coupled oscillators.

We present the coupled oscillator: A new mechanism for signal amplification with widespread application in metrology. We introduce the mechanical theory of this framework and support it by way of simulations. We present a particular implementation of coupled oscillators: A microelectromechanical system (MEMS) that uses one large (∼100mm) N52 magnet coupled magnetically to a small (∼0.25mm), oscillating N52 magnet, providing a force resolution of 200zN measured over 1s in a noiseless environment. We show that the same system is able to resolve magnetic gradients of 130aT/cm at a single point (within 500µm). This technology, therefore, has the potential to revolutionize force and magnetic gradient sensing, including high-impact areas such cardiac and brain imaging.

The Metastable State of Fermi-Pasta-Ulam-Tsingou Models.

Classical statistical mechanics has long relied on assumptions such as the equipartition theorem to understand the behavior of the complicated systems of many particles. The successes of this approach are well known, but there are also many well-known issues with classical theories. For some of these, the introduction of quantum mechanics is necessary, e.g., the ultraviolet catastrophe. However, more recently, the validity of assumptions such as the equipartition of energy in classical systems was called into question. For instance, a detailed analysis of a simplified model for blackbody radiation was apparently able to deduce the Stefan-Boltzmann law using purely classical statistical mechanics. This novel approach involved a careful analysis of a "metastable" state which greatly delays the approach to equilibrium. In this paper, we perform a broad analysis of such a metastable state in the classical Fermi-Pasta-Ulam-Tsingou (FPUT) models. We treat both the α-FPUT and ß-FPUT models, exploring both quantitative and qualitative behavior. After introducing the models, we validate our methodology by reproducing the well-known FPUT recurrences in both models and confirming earlier results on how the strength of the recurrences depends on a single system parameter. We establish that the metastable state in the FPUT models can be defined by using a single degree-of-freedom measure-the spectral entropy (η)-and show that this measure has the power to quantify the distance from equipartition. For the α-FPUT model, a comparison to the integrable Toda lattice allows us to define rather clearly the lifetime of the metastable state for the standard initial conditions. We next devise a method to measure the lifetime of the metastable state tm in the α-FPUT model that reduces the sensitivity to the exact initial conditions. Our procedure involves averaging over random initial phases in the plane of initial conditions, the P1-Q1 plane. Applying this procedure gives us a power-law scaling for tm, with the important result that the power laws for different system sizes collapse down to the same exponent as Eα2→0. We examine the energy spectrum E(k) over time in the α-FPUT model and again compare the results to those of the Toda model. This analysis tentatively supports a method for an irreversible energy dissipation process suggested by Onorato et al.: four-wave and six-wave resonances as described by the "wave turbulence" theory. We next apply a similar approach to the ß-FPUT model. Here, we explore in particular the different behavior for the two different signs of ß. Finally, we describe a procedure for calculating tm in the ß-FPUT model, a very different task than for the α-FPUT model, because the ß-FPUT model is not a truncation of an integrable nonlinear model.

Modal engineering of electromagnetic circuits to achieve rapid settling times.

Inductive circuits and devices are ubiquitous and important design elements in many applications, such as magnetic drives, galvanometers, magnetic scanners, applying direct current (DC) magnetic fields to systems, radio frequency coils in nuclear magnetic resonance (NMR) systems, and a vast array of other applications. They are widely used to generate both DC and alternating current (AC) magnetic fields. Many of these applications require a rapid step and settling time, turning the DC or AC magnetic field on and off quickly. The inductive response normally makes this a challenging thing to do. In this article, we discuss open loop control algorithms for achieving rapid step and settling times in four general categories of applications: DC and AC systems where the system is either under- or over-damped. Each of these four categories requires a different algorithm, which we describe here. We show the operation of these drive methods using Simulink and Simscape modeling tools, analytical solutions to the underlying differential equations, and experimental results using an inductive magnetic coil and a Hall sensor. Finally, we demonstrate the application of these techniques to significantly reduce ringing in a standard NMR circuit. We intend this article to be practical, with useful, easy-to-apply algorithms and helpful tuning tricks.

Counterdiabatic driving in the classical ß-Fermi-Pasta-Ulam-Tsingou chain.

Shortcuts to adiabaticity (STAs) have been used to make rapid changes to a system while eliminating or minimizing excitations in the system's state. In quantum systems, these shortcuts allow us to minimize inefficiencies and heating in experiments and quantum computing protocols, but the theory of STAs can also be generalized to classical systems. We focus on one such STA, approximate counterdiabatic (ACD) driving, and numerically compare its performance in two classical systems: a quartic anharmonic oscillator and the ß Fermi-Pasta-Ulam-Tsingou lattice. In particular, we modify an existing variational technique to optimize the approximate driving and then develop classical figures of merit to quantify the performance of the driving. We find that relatively simple forms for the ACD driving can dramatically suppress excitations regardless of system size. ACD driving in classical nonlinear oscillators could have many applications, from minimizing heating in bosonic gases to finding optimal local dressing protocols in interacting field theories.

Analysis of a Casimir-driven parametric amplifier with resilience to Casimir pull-in for MEMS single-point magnetic gradiometry.

The Casimir force, a quantum mechanical effect, has been observed in several microelectromechanical system (MEMS) platforms. Due to its extreme sensitivity to the separation of two objects, the Casimir force has been proposed as an excellent avenue for quantum metrology. Practical application, however, is challenging due to attractive forces leading to stiction and device failure, called Casimir pull-in. In this work, we design and simulate a Casimir-driven metrology platform, where a time-delay-based parametric amplification technique is developed to achieve a steady-state and avoid pull-in. We apply the design to the detection of weak, low-frequency, gradient magnetic fields similar to those emanating from ionic currents in the heart and brain. Simulation parameters are selected from recent experimental platforms developed for Casimir metrology and magnetic gradiometry, both on MEMS platforms. While a MEMS offers many advantages to such an application, the detected signal must typically be at the resonant frequency of the device, with diminished sensitivity in the low frequency regime of biomagnetic fields. Using a Casimir-driven parametric amplifier, we report a 10,000-fold improvement in the best-case resolution of MEMS single-point gradiometers, with a maximum sensitivity of 6 Hz/(pT/cm) at 1 Hz. Further development of the proposed design has the potential to revolutionize metrology and may specifically enable the unshielded monitoring of biomagnetic fields in ambient conditions.

A system for probing Casimir energy corrections to the condensation energy.

In this article, we present a nanoelectromechanical system (NEMS) designed to detect changes in the Casimir energy. The Casimir effect is a result of the appearance of quantum fluctuations in an electromagnetic vacuum. Previous experiments have used nano- or microscale parallel plate capacitors to detect the Casimir force by measuring the small attractive force these fluctuations exert between the two surfaces. In this new set of experiments, we aim to directly detect the shifts in the Casimir energy in a vacuum due to the presence of the metallic parallel plates, one of which is a superconductor. A change in the Casimir energy of this configuration is predicted to shift the superconducting transition temperature (T c) because of the interaction between it and the superconducting condensation energy. In our experiment, we take a superconducting film, carefully measure its transition temperature, bring a conducting plate close to the film, create a Casimir cavity, and then measure the transition temperature again. The expected shifts are smaller than the normal shifts one sees in cycling superconducting films to cryogenic temperatures, so using a NEMS resonator in situ is the only practical way to obtain accurate, reproducible data. Using a thin Pb film and opposing Au surface, we observe no shift in T c >12 µK down to a minimum spacing of ~70 nm at zero applied magnetic field.

The ß Fermi-Pasta-Ulam-Tsingou recurrence problem.

We perform a thorough investigation of the first Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in the ß-FPUT chain for both positive and negative ß. We show numerically that the rescaled FPUT recurrence time Tr=tr/(N+1)3 depends, for large N, only on the parameter S≡Eß(N+1). Our numerics also reveal that for small |S|, Tr is linear in S with positive slope for both positive and negative ß. For large |S|, Tr is proportional to |S|-1/2 for both positive and negative ß but with different multiplicative constants. We numerically study the continuum limit and find that the recurrence time closely follows the |S|-1/2 scaling and can be interpreted in terms of solitons, as in the case of the KdV equation for the α chain. The difference in the multiplicative factors between positive and negative ß arises from soliton-kink interactions that exist only in the negative ß case. We complement our numerical results with analytical considerations in the nearly linear regime (small |S|) and in the highly nonlinear regime (large |S|). For the former, we extend previous results using a shifted-frequency perturbation theory and find a closed form for Tr that depends only on S. In the latter regime, we show that Tr∝|S|-1/2 is predicted by the soliton theory in the continuum limit. We then investigate the existence of the FPUT recurrences and show that their disappearance surprisingly depends only on Eß for large N, not S. Finally, we end by discussing the striking differences in the amount of energy mixing between positive and negative ß and offer some remarks on the thermodynamic limit.

Dynamical glass in weakly nonintegrable Klein-Gordon chains.

Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a nonintegrable perturbation creates a coupling network in action space which can be short or long ranged. We analyze the dynamics of observables which become the conserved actions in the integrable limit. We compute distributions of their finite time averages and obtain the ergodization time scale T_{E} on which these distributions converge to δ distributions. We relate T_{E} to the statistics of fluctuation times of the observables, which acquire fat-tailed distributions with standard deviations σ_{τ}^{+} dominating the means µ_{τ}^{+} and establish that T_{E}∼(σ_{τ}^{+})^{2}/µ_{τ}^{+}. The Lyapunov time T_{Λ} (the inverse of the largest Lyapunov exponent) is then compared to the above time scales. We use a simple Klein-Gordon chain to emulate long- and short-range coupling networks by tuning its energy density. For long-range coupling networks T_{Λ}≈σ_{τ}^{+}, which indicates that the Lyapunov time sets the ergodization time, with chaos quickly diffusing through the coupling network. For short-range coupling networks we observe a dynamical glass, where T_{E} grows dramatically by many orders of magnitude and greatly exceeds the Lyapunov time, which satisfies T_{Λ}≲µ_{τ}^{+}. This effect arises from the formation of highly fragmented inhomogeneous distributions of chaotic groups of actions, separated by growing volumes of nonchaotic regions. These structures persist up to the ergodization time T_{E}.

Erratum: Accelerated Search and Design of Stretchable Graphene Kirigami Using Machine Learning [Phys. Rev. Lett. 121, 255304 (2018)].

This corrects the article DOI: 10.1103/PhysRevLett.121.255304.

Behavior and breakdown of higher-order Fermi-Pasta-Ulam-Tsingou recurrences.

We numerically investigate the existence and stability of higher-order recurrences (HoRs), including super-recurrences, super-super-recurrences, etc., in the α and ß Fermi-Pasta-Ulam-Tsingou (FPUT) lattices for initial conditions in the fundamental normal mode. Our results represent a considerable extension of the pioneering work of Tuck and Menzel on super-recurrences. For fixed lattice sizes, we observe and study apparent singularities in the periods of these HoRs, speculated to be caused by nonlinear resonances. Interestingly, these singularities depend very sensitively on the initial energy and the respective nonlinear parameters. Furthermore, we compare the mechanisms by which the super-recurrences in the two models breakdown as the initial energy and respective nonlinear parameters are increased. The breakdown of super-recurrences in the ß-FPUT lattice is associated with the destruction of the so-called metastable state and thus with relaxation towards equilibrium. For the α-FPUT lattice, we find this is not the case and show that the super-recurrences break down while the lattice is still metastable and far from equilibrium. We close with comments on the generality of our results for different lattice sizes.

Accelerated Search and Design of Stretchable Graphene Kirigami Using Machine Learning.

Making kirigami-inspired cuts into a sheet has been shown to be an effective way of designing stretchable materials with metamorphic properties where the 2D shape can transform into complex 3D shapes. However, finding the optimal solutions is not straightforward as the number of possible cutting patterns grows exponentially with system size. Here, we report on how machine learning (ML) can be used to approximate the target properties, such as yield stress and yield strain, as a function of cutting pattern. Our approach enables the rapid discovery of kirigami designs that yield extreme stretchability as verified by molecular dynamics (MD) simulations. We find that convolutional neural networks, commonly used for classification in vision tasks, can be applied for regression to achieve an accuracy close to the precision of the MD simulations. This approach can then be used to search for optimal designs that maximize elastic stretchability with only 1000 training samples in a large design space of ∼4×10^{6} candidate designs. This example demonstrates the power and potential of ML in finding optimal kirigami designs at a fraction of iterations that would be required of a purely MD or experiment-based approach, where no prior knowledge of the governing physics is known or available.

Kirigami actuators.

Thin elastic sheets bend easily and, if they are patterned with cuts, can deform in sophisticated ways. Here we show that carefully tuning the location and arrangement of cuts within thin sheets enables the design of mechanical actuators that scale down to atomically-thin 2D materials. We first show that by understanding the mechanics of a single non-propagating crack in a sheet, we can generate four fundamental forms of linear actuation: roll, pitch, yaw, and lift. Our analytical model shows that these deformations are only weakly dependent on thickness, which we confirm with experiments on centimeter-scale objects and molecular dynamics simulations of graphene and MoS2 nanoscale sheets. We show how the interactions between non-propagating cracks can enable either lift or rotation, and we use a combination of experiments, theory, continuum computational analysis, and molecular dynamics simulations to provide mechanistic insights into the geometric and topological design of kirigami actuators.

Highly stretchable MoS2 kirigami.

We report the results of classical molecular dynamics simulations focused on studying the mechanical properties of MoS2 kirigami. Several different kirigami structures were studied based upon two simple non-dimensional parameters, which are related to the density of cuts, as well as the ratio of the overlapping cut length to the nanoribbon length. Our key findings are significant enhancements in tensile yield (by a factor of four) and fracture strains (by a factor of six) as compared to pristine MoS2 nanoribbons. These results, in conjunction with recent results on graphene, suggest that the kirigami approach may be generally useful for enhancing the ductility of two-dimensional nanomaterials.

Conductance signatures of electron confinement induced by strained nanobubbles in graphene.

We investigate the impact of strained nanobubbles on the conductance characteristics of graphene nanoribbons using a combined molecular dynamics - tight-binding simulation scheme. We describe in detail how the conductance, density of states, and current density of zigzag or armchair graphene nanoribbons are modified by the presence of a nanobubble. In particular, we establish that low-energy electrons can be confined in the vicinity of or within the nanobubbles by the delicate interplay among the pseudomagnetic field pattern created by the shape of the bubble, mode mixing, and substrate interaction. The coupling between confined evanescent states and propagating modes can be enhanced under different clamping conditions, which translates into Fano resonances in the conductance traces.

Transport properties of pristine few-layer black phosphorus by van der Waals passivation in an inert atmosphere.

Ultrathin black phosphorus is a two-dimensional semiconductor with a sizeable band gap. Its excellent electronic properties make it attractive for applications in transistor, logic and optoelectronic devices. However, it is also the first widely investigated two-dimensional material to undergo degradation upon exposure to ambient air. Therefore a passivation method is required to study the intrinsic material properties, understand how oxidation affects the physical properties and enable applications of phosphorene. Here we demonstrate that atomically thin graphene and hexagonal boron nitride can be used for passivation of ultrathin black phosphorus. We report that few-layer pristine black phosphorus channels passivated in an inert gas environment, without any prior exposure to air, exhibit greatly improved n-type charge transport resulting in symmetric electron and hole transconductance characteristics.

Functional renormalization group analysis of the half-filled one-dimensional extended Hubbard model.

We study the phase diagram of the half-filled one-dimensional extended Hubbard model at weak coupling using a novel functional renormalization group (FRG) approach. The FRG method includes in a systematic manner the effects of the scattering processes involving electrons away from the Fermi points. Our results confirm the existence of a finite region of bond charge density wave, also known as a "bond order wave" near U=2V and clarify why earlier g-ology calculations have not found this phase. We argue that this is an example in which formally irrelevant corrections change the topology of the phase diagram. Whenever marginal terms lead to an accidental symmetry, this generalized FRG method may be crucial to characterize the phase diagram accurately.