Classical Oscillator Research Articles

Frequency Control and Coherent Excitation Transfer in a Nanostring-resonator Network

Coupling, synchronization, and non-linear dynamics of resonator modes are omnipresent in nature and highly relevant for a multitude of applications ranging from lasers to Josephson arrays and spin torque oscillators. Nanomechanical resonators are ideal candidates to study these effects on a fundamental level and to realize all-mechanical platforms for information processing and storage. For larger resonator networks, however, this requires the ability to tune the mode frequencies selectively and to operate the resonators in the strong coupling regime. Here, we present a proof-of-principle realization of a resonator network consisting of two high-quality nanostring resonators, coupled mechanically by a shared support. First, we demonstrate that we can control the fundamental mode frequencies of both nanostrings independently by a strong drive tone resonant with one of the higher harmonics of the network, rendering local control gates redundant. The tuning mechanism relies on an effective increase of the pre-stress in a highly excited nanostring, known as geometric nonlinearity. Using this selective frequency control of the individual nanomechanical resonators, we investigate the coherent dynamics of the resonator network, which is a classical model system showing several of the characteristic features of strongly coupled quantum systems. In particular, we demonstrate mode splitting, classical Rabi oscillations, as well as adiabatic and diabatic transitions between the coupled states representing the classical analog of Landau-Zener tunneling. Therefore, this coupling and tuning concept opens the path to a selective phonon transfer between two spatially separated mechanical resonators.

Chain of Physically Related Independent Mechanical Oscillators

A lumped-parameter mechanical system consisting of a chain of physically related rigid bodies, each of which has one rotational degree of freedom, is considered. It is shown that the inertialess elastic elements connecting the absolutely rigid bodies of the chain can be chosen so that the mechanical structure acquires the properties of a so-called absolute mechanical filter. The motion of any inertial element of this system is described by the equation of a classical harmonic oscillator with one degree of freedom. Using the system considered as an example, it is shown that there is a relationship between the models of classical and quantum mechanics. From the positions of modern classical mechanics, this lumped-parameter system confirms the well-known Einstein’s hypothesis theoretical physics that a rigid body is a system of independent oscillators.

Simulating quantum dynamical phenomena using classical oscillators: Landau-Zener-St\xfcckelberg-Majorana interferometry, latching modulation, and motional averaging

A quantum system can be driven by either sinusoidal, rectangular, or noisy signals. In the literature, these regimes are referred to as Landau-Zener-Stückelberg-Majorana (LZSM) interferometry, latching modulation, and motional averaging, respectively. We demonstrate that these pronounced and interesting effects are also inherent in the dynamics of classical two-state systems. We discuss how such classical systems are realized using either mechanical, electrical, or optical resonators. In addition to the fundamental interest of such dynamical phenomena linking classical and quantum physics, we believe that these are attractive for the classical analogue simulation of quantum systems.

Macroscopic quantum escape of Bose-Einstein condensates: Analysis of experimentally realizable quasi-one-dimensional traps

The variational JWKB method is used to determine experimentally accessible macroscopic quantum tunneling regimes of quasibound Bose-Einstein condensates in two quasi-one-dimensional trap configurations. The potentials can be created by magnetic and optical traps: a symmetric trap from two offset Gaussian barriers and a tilt trap from a linear gradient and Gaussian barrier. Scaling laws in barrier parameters, ranging from inverse polynomial to square root times exponential, are calculated and used to elucidate different dynamical regimes, such as when classical oscillations dominate tunneling rates in the symmetric trap. The symmetric trap is found to be versatile, with tunneling times at and below 1 s, able to hold ${10}^{3}--{10}^{4}$ atoms, and realizable for atoms ranging from rubidium to lithium, with unadjusted scattering lengths. The tilt trap produces subsecond tunneling times, is able to hold a few hundred atoms of lighter elements such as lithium, and requires the use of Feshbach resonance to reduce scattering lengths. To explore a large parameter space, an extended Gaussian variational ansatz is used, which can approximate large traps with Thomas-Fermi profiles. Nonlinear interactions in the Gross-Pitaevskii equation are shown to produce additional effective mean-field barriers, affecting scaling laws.

Discrete dynamical stabilization of a naturally diverging mass in a harmonically time-varying potential

We numerically investigate the stability and linear oscillatory behavior of a naturally diverging mass whose potential energy is harmonically modulated. It is known that in the Kapitza limit, i.e. when the period of modulation is much smaller than the diverging time, the collapsing mass can be dynamically stabilized and behave like an effective classic harmonic oscillator. We find that in the regime where the period of modulation is larger than the collapsing time of the mass, dynamical stabilization is still possible but in a discrete fashion. Only almost-periodic vibrational modes, or Floquet forms (FFs), are allowed that are located in independent stability stripes in the modulation parameter space. Reducing the FFs to their periodic eigenfunctions, one can transform the original equation of motion to a dimensionless Schrödinger stationary wave equation with a harmonic potential. This transformation allows for an analytical prediction of the stability stripes and the modal shapes of the vibrating mass. These results shed new light on the stability of linear dynamical systems, analytical solutions of Mathieu equations and on the relations between Initial and Boundary Value Problems.

Analytical solutions for the quantum parametric oscillator from corresponding classical dynamics via a complex Riccati equation

The time-dependent Schrödinger equation for quadratic Hamiltonians has Gaussian wave packets as exact solutions. For the parametric oscillator with frequency ω(t), the width of these wave packets must be time-dependent. This time-dependence can be determined via the solution of a complex nonlinear Riccati equation or an equivalent real nonlinear Ermakov equation. All quantum dynamical properties of the system can easily be constructed from these solutions, e.g., uncertainties of position and momentum, their correlations, tunnelling currents, ground state energies, etc. In addition, the link to the corresponding classical dynamics is supplied via linearization of the Riccati equation to a complex Newtonian equation, actually representing two equations of the same kind: one for the real and one for the imaginary part. If the solution of one part is known, the missing (linear independent) solution of the other can be obtained via a conservation law for the motion in the complex plane. Knowing these two solutions, the solution of the Ermakov equation can be determined immediately plus the explicit expressions for all the quantum dynamical properties mentioned above. Furthermore, these two solutions also provide the time-dependent Green function for the systems propagation for any given initial condition, not only for Gaussians. It is also possible to obtain the corresponding Wigner function and generalized creation and annihilation operators in this way. In searching for problems with analytical solutions, the crucial point is to identify systems that have either a wave packet solution with time-dependent width (in closed form) that obeys a Riccati equation, or a classical parametric oscillator with frequency ω(t) that still enables an analytical solution of the Newtonian equation. Comparing the wave packets with the solution of the diffusion equation – also a Gaussian with time-dependent width – leads to a real Riccati equation. A first attempt at “complexifying” this equation turns out to be problematic. Nevertheless, it provides a clue to time-dependent frequencies inversely proportional to time that lead to Newtonian equations with analytical solutions. Following a detailed analysis of the corresponding quantum dynamics and energetics, possible extensions of the method are indicated.

Generalized Sturmian Functions used for a discrete wavelet construction

In this paper we discuss the implementation of scaling functions and wavelets based on Generalized Sturmian Functions (GSF). When dealing with finite dimensional spaces, the completeness relation of the GSF basis generates localized functions in two coordinates. By fixing one of the coordinates on particular points one defines the corresponding scaling functions. Wavelets are defined in a similar way by fixing, on a complementary set of functions, one of the coordinates on another set of particular points. This procedure allows for a multiscale decomposition of any signal. When the chosen points are the zeros of the th GSF, scaling functions generate a -dimensional space and happen to be orthogonal. We use GSF associated to the classical damped harmonic oscillator to build scaling functions and wavelets, and apply them to represent eye-tracking signals.

Classical Drude Polarizable Force Field Model for Methyl Phosphate and Its Interactions with Mg2.

Phosphate groups are essential components of nucleic acids and proteins, whose interactions with solvent, metal ions, and ionic side chains help control folding and binding. Methyl phosphate (MP) represents a simple analog of phosphate moieties that are post-translation modifications in proteins and present at the termini of nucleic acids, among other environments. In the present study, we optimized parameters for use in polarizable molecular dynamics simulations of MP in its mono- and dianionic forms, MP- ≡ CH3HPO4- and MP2- ≡ CH3PO42-, along with P i2- ≡ HPO42-, in the context of the classical Drude oscillator model. Parameter optimization was done in a manner consistent with the remainder of the Drude molecular mechanics force field, choosing atomic charges and polarizabilities to reproduce molecular properties from quantum mechanics as well as experimental hydration free energies. Optimized parameters were similar to existing dimethyl phosphate parameters, with a few significant differences. The developed parameters were then used to compute magnesium binding affinities in aqueous solution, using alchemical molecular dynamics free energy simulations. Good agreement with experiment was obtained, and outer sphere binding was shown to be predominant for MP- and MP2-.

Vibration control of vortex-induced vibrations of a bridge deck by a single-side pounding tuned mass damper

This paper proposes a new method to mitigate vortex-induced vibrations (VIVs) of a bridge deck using a single-side pounding tuned mass damper (SS-PTMD). The SS-PTMD is a passive control device and comprises an undamped tuned mass with a pounding boundary covered with viscoelastic (VE) materials layer. A nonlinear force model for describing impact behavior of VE materials is used to simulate the response of a single degree of freedom (SDOF) system controlled by a SS-PTMD. The free pounding experiments are performed to determine the model parameters of impact force and validate the simulation method. The optimal design of SS-PTMD for SDOF system subjected to sinusoidal excitation is carried out by numerical optimization, and the optimized SS-PTMD is applied to control the vertical VIVs of a bridge deck. The control performance is experimentally examined by elastically mounted section model tests in wind tunnel. In addition, the classical wake oscillator model is used to predict the behavior of the coupled fluid-structure system under VIV and explore the control performance of the SS-PTMD. The experimental results show that the maximum response of the bridge deck model was reduced by 94% when a SS-PTMD with mass ratio of 2% was applied. It is also shown that nonlinearity in vortex shedding forces has little influence on control performance of SS-PTMD optimized under sinusoidal excitation.

Structural Evolution and Microwave Dielectric Properties of xZn0.5Ti0.5NbO4-(1- x)Zn0.15Nb0.3Ti0.55O2 Ceramics.

Structure and microwave properties of xZn0.5Ti0.5NbO4-(1 - x)Zn0.15Nb0.3Ti0.55O2 ceramics in the range of x = 0.0-1.0 were investigated. Rietveld refinement analysis and Raman spectra show that rutile- and orthorhombic-type solid solutions formed at 0-0.2 and 0.65-1, a composite at 0.2-0.64. In the solid solution regions, chemical bonds are enlarged. In this case, the Zn/Ti/Nb-O1 bond covalency and bond susceptibility are reduced, and lattice energy and thermal expansion coefficient increase along with x increases, which is mainly responsible for the development of microwave dielectric properties. Furthermore, far-infrared spectra and a classical oscillator model were used to discuss the intrinsic dielectric properties in detail. Temperature stable ceramic was obtained for x = 0.516: εr ∼ 46.11, Q × f ∼ 27 031 GHz, and τf ∼ -1.51 ppm/°C, which is promising for microwave applications.

Origin of magnetoresistance suppression in thin γ−MoTe2

We use both classical magnetotransport and quantum oscillation measurements to study the thickness evolution of the extremely large magnetoresistance (XMR) material and type-II Weyl semimetal candidate, $\gamma$-MoTe$_2$, protected from oxidation. We find that the magnetoresistance is systematically suppressed with reduced thickness. This occurs concomitantly with both a decrease in carrier mobility and increase in electron-hole imbalance. We model the two effects separately and conclude that the XMR effect is more sensitive to the former.

Application of Synchrosqueezed Wavelet Transforms for Extraction of the Oscillatory Parameters of Subsynchronous Oscillation in Power Systems

The most classical subsynchronous oscillation (SSO) mode extraction methods have some shortcomings, such as lower mode identification and poor anti-noise properties. Thus, this paper proposes a new time-frequency analysis method, namely, synchrosqueezed wavelet transforms (SWT). SWT combines the advantages of empirical mode decomposition (EMD) and wavelet, which has the adaptability of EMD, and improve the ability of anti-mode mixing on EMD and wavelet. Thus, better anti-noise property and higher mode identification can be achieved. Firstly, the SSO signal is transformed by SWT and its time-frequency spectrum is obtained. Secondly, the attenuation characteristic of each intrinsic mode type (IMT) component in its time-frequency spectrum is analyzed by an automatic identification algorithm, and determine which IMT component needs reconstruction. After that, the selected IMT components with divergent characteristic are reconstructed. Thirdly, high-accuracy detection for mode parameter identification is achieved by the Hilbert transform (HT). Simulation and application examples prove the effectiveness of the proposed method.

Understanding photoluminescence of metal nanostructures based on an oscillator model

Scattering and absorption properties of metal nanostructures have been well understood based on the classic oscillator theory. Here, we demonstrate that photoluminescence of metal nanostructures can also be explained based on a classic model. The model shows that inelastic radiation of an oscillator resembles its resonance band after external excitation, and is related to the photoluminescence from metallic nanostructures. The understanding based on the classic oscillator model is in agreement with that predicted by a quantum electromagnetic cavity model. Moreover, by correlating a two-temperature model and the electron distributions, we demonstrate that both one-photon and two-photon luminescence of the metal nanostructures undergo the same mechanism. Furthermore, the model explains most of the emission characteristics of the metallic nanostructures, such as quantum yield, spectral shape, excitation polarization and power dependence. The model based on an oscillator provides an intuitive description of the photoluminescence process and may enable rapid optimization and exploration of the plasmonic properties.

Combining the polarizable Drude force field with a continuum electrostatic Poisson-Boltzmann implicit solvation model.

In this work, we have combined the polarizable force field based on the classical Drude oscillator with a continuum Poisson-Boltzmann/solvent-accessible surface area (PB/SASA) model. In practice, the positions of the Drude particles experiencing the solvent reaction field arising from the fixed charges and induced polarization of the solute must be optimized in a self-consistent manner. Here, we parameterized the model to reproduce experimental solvation free energies of a set of small molecules. The model reproduces well-experimental solvation free energies of 70 molecules, yielding a root mean square difference of 0.8 kcal/mol versus 2.5 kcal/mol for the CHARMM36 additive force field. The polarization work associated with the solute transfer from the gas-phase to the polar solvent, a term neglected in the framework of additive force fields, was found to make a large contribution to the total solvation free energy, comparable to the polar solute-solvent solvation contribution. The Drude PB/SASA also reproduces well the electronic polarization from the explicit solvent simulations of a small protein, BPTI. Model validation was based on comparisons with the experimental relative binding free energies of 371 single alanine mutations. With the Drude PB/SASA model the root mean square deviation between the predicted and experimental relative binding free energies is 3.35 kcal/mol, lower than 5.11 kcal/mol computed with the CHARMM36 additive force field. Overall, the results indicate that the main limitation of the Drude PB/SASA model is the inability of the SASA term to accurately capture non-polar solvation effects. © 2018 Wiley Periodicals, Inc.

Determining Eigenfrequencies and Homogeneous Widths of Lines of Intermolecular Vibrations in Water and in Aqueous Solutions of Hydrogen Peroxide Using Raman Spectroscopy

We have examined low-frequency Raman spectra of intermolecular vibrations of weak aqueous solutions of hydrogen peroxide and water. The differences between the observed Raman frequencies and the data from the literature on the IR-absorption frequencies of the same vibrations, as well as the interrelations between the frequencies and the widths of Lorentzian contours that approximate the vibrational spectra of the dynamic susceptibility, have been discussed. Based on a model of free damped vibrations of a classical oscillator with an inhomogeneous broadening, we have explained these effects for the first time. The homogeneous line widths and the damping times of the observed intermolecular vibrations have been determined. The eigenfrequencies of these vibrations have been calculated, and they have been shown to agree well with the data from the literature on the frequencies of IR absorption of water. We have shown that these parameters of intermolecular vibrations for water and for aqueous solutions of hydrogen peroxide differ from each other.

Chaos in a second-order non-autonomous Wien-bridge oscillator without extra nonlinearity

PurposeThe purpose of this paper is to develop a simple chaotic circuit. The circuit can be fabricated by less discrete electronic components, within which complex dynamical behaviors can be generated.Design/methodology/approachA second-order non-autonomous inductor-free chaotic circuit is presented, which is obtained by introducing a sinusoidal voltage stimulus into the classical Wien-bridge oscillator. The proposed circuit only has two dynamic elements, and its nonlinearity is realized by the saturation characteristic of the operational amplifier in the classical Wien-bridge oscillator. After that, its dynamical behaviors are revealed by means of bifurcation diagram, Lyapunov exponent and phase portrait and further confirmed using the 0-1 test method. Moreover, an analog circuit using less discrete electronic components is implemented, and its experimental results are measured to verify the numerical simulations.FindingsThe equilibrium point located in a line segment varies with time evolution, which leads to the occurrence of periodic, quasi-periodic and chaotic behaviors in the proposed circuit.Originality/valueUnlike the previously published works, the significant values of the proposed circuit with simple topology are inductor-free realization and without extra nonlinearity, which make the circuit can be used as a paradigm for academic teaching and experimental illustraction for chaos.

Bistability and displacement fluctuations in a quantum nanomechanical oscillator

Remarkable features have been predicted for the mechanical fluctuations at the bistability transition of a classical oscillator coupled capacitively to a quantum dot [Phys. Rev. Lett. 115, 206802 (2015)]. These results have been obtained in the regime $\omega$0 $\ll$ kBT $\ll$ $\Gamma$, where $\omega$0, T , and $\Gamma$ are the mechanical resonating frequency, the temperature, and the tunneling rate, respectively. A similar behavior could be expected in the quantum regime of $\Gamma$ $\ll$ kBT $\ll$ $\omega$0. We thus calculate the energy and displacement fluctuation spectra and study their behavior as a function of the electro-mechanical coupling constant when the system enters the Frank-Condon regime. We find that, in analogy with the classical case, the energy fluctuation spectrum and the displacement spectrum widths show a maximum for values of the coupling constant at which a mechanical bistability establishes.

Polarizable Force Field for Molecular Ions Based on the Classical Drude Oscillator.

Development of accurate force field parameters for molecular ions in the context of a polarizable energy function based on the classical Drude oscillator is a crucial step toward an accurate polarizable model for modeling and simulations of biological macromolecules. Toward this goal we have undertaken a hierarchical approach in which force field parameter optimization is initially performed for small molecules for which experimental data exists that serve as building blocks of macromolecular systems. Small molecules representative of the ionic moieties of biological macromolecules include the cationic ammonium and methyl substituted ammonium derivatives, imidazolium, guanidinium and methylguanidinium, and the anionic acetate, phenolate, and alkanethiolates. In the present work, parameters for molecular ions in the context of the Drude polarizable force field are optimized and compared to results from the nonpolarizable additive CHARMM general force field (CGenFF). Electrostatic and Lennard-Jones parameters for the model compounds are developed in the context of the polarizable SWM4-NDP water model, with emphasis on assuring that the hydration free energies are consistent with previously reported parameters for atomic ions. The final parameters are shown to be in good agreement with the selected quantum mechanical (QM) and experimental target data. Analysis of the structure of water around the ions reveals substantial differences between the Drude and additive force fields indicating the important role of polarization in dictating the molecular details of aqueous solvation. The presented parameters represent the foundation for the charged functionalities in future generations of the Drude polarizable force field for biological macromolecules as well as for drug-like molecules.

Computational Re-design of Synthetic Genetic Oscillators for Independent Amplitude and Frequency Modulation.

To perform well in biotechnology applications, synthetic genetic oscillators must be engineered to allow independent modulation of amplitude and period. This need is currently unmet. Here, we demonstrate computationally how two classic genetic oscillators, the dual-feedback oscillator and the repressilator, can be re-designed to provide independent control of amplitude and period and improve tunability-that is, a broad dynamic range of periods and amplitudes accessible through the input "dials." Our approach decouples frequency and amplitude modulation by incorporating an orthogonal "sink module" where the key molecular species are channeled for enzymatic degradation. This sink module maintains fast oscillation cycles while alleviating the translational coupling between the oscillator's transcription factors and output. We characterize the behavior of our re-designed oscillators over a broad range of physiologically reasonable parameters, explain why this facilitates broader function and control, and provide general design principles for building synthetic genetic oscillators that are more precisely controllable.

A Novel Wake Oscillator Model for Vortex-Induced Vibrations Prediction of A Cylinder Considering the Influence of Reynolds Number

It is well known that the Reynolds number has a significant effect on the vortex-induced vibrations (VIV) of cylinders. In this paper, a novel in-line (IL) and cross-flow (CF) coupling VIV prediction model for circular cylinders has been proposed, in which the influence of the Reynolds number was comprehensively considered. The Strouhal number linked with the vortex shedding frequency was calculated through a function of the Reynolds number. The coefficient of the mean drag force was fitted as a new piecewise function of the Reynolds number, and its amplification resulted from the CF VIV was also taken into account. The oscillating drag and lift forces were modelled with classical van der Pol wake oscillators and their empirical parameters were determined based on the lock-in boundaries and the peak-amplitude formulas. A new peak-amplitude formula for the IL VIV was developed under the resonance condition with respect to the mass-damping ratio and the Reynolds number. When compared with the results from the experiments and some other prediction models, the present model could give good estimations on the vibration amplitudes and frequencies of the VIV both for elastically-mounted rigid and long flexible cylinders. The present model considering the influence of the Reynolds number could generally provide better results than that neglecting the effect of the Reynolds number.