Showing papers on "Parametric oscillator published in 2022"

Spectroscopic observation of the crossover from a classical Duffing oscillator to a Kerr parametric oscillator

Abstract: We study microwave response of a Josephson parametric oscillator consisting of a superconducting transmission-line resonator with an embedded DC-superconducting quantum interference device (-SQUID). The DC-SQUID allows to control the magnitude of a Kerr nonlinearity over the ranges where it is smaller or larger than the photon loss rate. Spectroscopy measurements reveal the change in the microwave response from a classical Duffing oscillator to a Kerr parametric oscillator in a single device. In the single-photon Kerr regime, we observe parametric oscillations with a well-defined phase of either 0 or $\ensuremath{\pi}$, whose probability can be controlled by an externally injected signal.

The Stability Analysis of a Vibrating Auto-Parametric Dynamical System Near Resonance

Abstract: This paper examines a new vibrating dynamical motion of a novel auto-parametric system with three degrees of freedom. It consists of a damped Duffing oscillator as a primary system attached to a damped spring pendulum as a secondary system. Lagrange’s equations are utilized to acquire the equations of motion according to the number of the system’s generalized coordinates. The perturbation technique of multiple scales is applied to provide the solutions to these equations up to a higher order of approximations, with the aim of obtaining more accurate novel results. The categorizations of resonance cases are presented, in which the case of primary external resonance is examined to demonstrate the conditions of solvability of the steady-state solutions and the equations of modulation. The time histories of the achieved solutions, the resonance curves in terms of the modified amplitudes and phases, and the regions of stability are outlined for various parameters of the considered system. The non-linear stability, in view of both the attained stable fixed points and the criterion of Routh–Hurwitz, is investigated. The results of this paper will be of interest for specialized research that deals with the vibration of swaying buildings and the reduction in the vibration of rotor dynamics, as well as studies in the fields of mechanics and space engineering.

Preheating in Palatini Higgs inflation on the lattice

Abstract: We study preheating following Higgs inflation in the Palatini formulation of gravity. We numerically evolve perturbations of the radial mode of the Higgs field and that of three scalars modeling the gauge bosons. We compare the two non-perturbative mechanisms of growth of excitations — parametric resonance and tachyonic instability — and confirm that the latter plays the dominant role. Our results provide further evidence that preheating in Palatini Higgs inflation happens within a single oscillation of the Higgs field about the bottom of its potential, consistent with the approximation of an instantaneous preheating.

Effect of geometric imperfections and circumferential symmetry on the internal resonances of cylindrical shells

Abstract: In the present work the resonant response of an imperfect cylindrical shell is investigated using Donnell’s nonlinear shallow shell theory. For this, a reduced order multi degree of freedom model is obtained by discretizing the partial differential equation of motion using the Galerkin procedure and a consistent modal expansion which captures the nonlinear modal interactions and couplings between two asymmetric vibration modes with near commensurable natural frequencies in a 1:2 ratio. As a result of the circumferential symmetry each mode exhibits a 1:1 internal resonance, leading to a possible 1:1:2:2 multiple internal resonances. These modes are coupled through quadratic and cubic nonlinearities arising from the shell curvature and nonlinear strain–displacement relation. The existence and stability of solutions and their bifurcations are investigated using numerical continuation methods for bifurcation analysis and their stability are studied using Floquet theory. It is known that geometric imperfections have a strong influence on the response of thin shell structures. Here, a detailed parametric analysis shows the influence of different forms of geometric imperfections on the shell natural frequencies and bifurcations in the main resonance region. Several branches of solutions due to multiple bifurcations are detected leading to dynamic jumps under increasing and decreasing frequency sweep. Steady-state harmonic and quasi-periodic responses resulting from Neimark–Sacker bifurcations are detected. The reduced order model demonstrates the influence of the geometric imperfection shape and magnitude on the bifurcation scenario and the energy transfer among the four interacting modes.

Three-wave mixing traveling-wave parametric amplifier with periodic variation of the circuit parameters

Abstract: We report on the implementation of a near-quantum-limited, traveling-wave parametric amplifier that uses three-wave mixing (3WM). To favor amplification by 3WM, we use superconducting nonlinear asymmetric inductive element (SNAIL) loops, biased with a dc magnetic flux. In addition, we equip the device with dispersion engineering features to create a stopband at the second harmonic of the pump and suppress the propagation of the higher harmonics that otherwise degrade the amplification. With a chain of 440 SNAILs, the amplifier provides up to 20 dB gain and a 3-dB bandwidth of 1 GHz. The added noise by the amplifier is found to be less than one photon.

Broadband Josephson parametric amplifier using lumped-element transmission line impedance matching architecture

Abstract: We present a fishbone-like lumped-element artificial transmission line to overcome impedance mismatch in a reflection-type Josephson parametric amplifier between a nonlinear resonator and an external transmission line. Using this easily prepared architecture, we design and fabricate a broadband Josephson parametric amplifier, which has gain in an excess of 20 dB with a bandwidth of hundreds of MHz. Furthermore, by varying the working point of the device, the operating frequency of amplification can be tuned in a wide frequency range of 1 GHz while the amplifier operates in the mode of either three-wave mixing or four-wave mixing. Such a parametric amplifier is suitable for engineering applications of superconducting circuit quantum electrodynamics.

Effect of dry friction on a parametric nonlinear oscillator

Abstract: Parametrically excited oscillators are used in several domains, in particular to improve the dynamical behaviour of systems like in the case of the parametric amplification or parametric energy harvesting. Although dry friction is often omitted during system modelling due to the complexity of its nonsmooth nature, it is sometimes necessary to account for this kind of damping to adequately represent the system motion. In this paper, it is proposed to investigate the effect of dry friction on the dynamical behaviour of a nonlinear parametric oscillator. Using the pendulum case as example, the problem is formulated according to a Mathieu-Duffing equation. Semi-analytical developments using the harmonic balance method and the method of varying amplitudes are used to find the solutions of this equation and their stability. These results are validated thanks to a comparison with time integration simulations. Effects of initial conditions on the basins of attractions of the solutions are also studied using these simulations. It is found that trivial and nontrivial solutions of the oscillator including dry friction are not connected, giving birth to isolated periodic solutions branches. Thus, both initial displacement and phase between the excitation and the oscillator displacement must be carefully chosen to reach periodic solutions. Finally, a method based on the energy principle is used to find the critical forcing amplitude and frequency needed to obtain the birth of nontrivial solutions for the nonlinear parametric oscillator including dry friction.

On the modeling of a parametric cubic–quintic nonconservative Duffing oscillator via the modified homotopy perturbation method

Abstract: Abstract This paper is devoted to obtain an approximate solution to the damped quintic–cubic nonlinear Duffing–Mathieu equation via a modified homotopy perturbation method (HPM). The modification under consideration deals with the improvement of the HPM with the exponential decay parameter. This scheme allows us to get a solution to the damped nonlinear Duffing–Mathieu equation, which the classical HPM failed to obtain. It is found that the solutions and the characteristic curves are affected by the presence of the damping force. The frequency-amplitude characteristics of a symbiotic solution are confirmed as well as the stability condition is carried out in the (non)-resonance cases. All the calculations are done via Mathematica. The comparison between both of the numerical and analytical solutions showed a very good agreement. Illustrated graphs are plotted for a superior realization of periodic motions in the Duffing–Mathieu oscillator. Nonlinear behaviors of each oscillation motion have been characterized through frequency curves.

Frequency-Dependent Squeezing from a Detuned Squeezer.

Abstract: Frequency-dependent squeezing is a promising technique to overcome the standard quantum limit in optomechanical force measurements, e.g., gravitational wave detectors. For the first time, we show that frequency-dependent squeezing can be produced by detuning an optical parametric oscillator from resonance. Its frequency-dependent Wigner function is reconstructed quantum tomographically and exhibits a rotation by 39°, along which the noise is reduced by up to 5.5 dB. Our setup is suitable for realizing effective negative-mass oscillators required for coherent quantum noise cancellation.

Periodic solution of the parametric Gaylord's oscillator with a non-perturbative approach

Abstract: The vibration of a regular rigid bar without sliding over a solid annular surface of a specified radius can be considered by a parametric Gaylord's oscillator. The governing equation was the result of a strong nonlinear oscillation without having a natural frequency. The present work is concerned with obtaining the approximate solution and amplitude-frequency equation of the parametric Gaylord's equation via an easier process. The non-perturbative approach was applied twice to analyze the present oscillator. Two steps are used, the first is to transform Gaylord's oscillator to the parametric pendulum equation having a natural frequency. The second step is to establish the amplitude-frequency relationship which was taken out in terms of the Bessel functions. A periodic analytic solution is obtained, in the presence or without the parametric force. The frequency at the resonance case is established without a perturbation for the first time. The stability condition is established and discussed graphically. The analytic solution was also validated by comparing it with its corresponding numerical data which showed a very good agreement. In a word, by dissection of the behavior of strong nonlinearity oscillators, the non-perturbative technique is characterized by its ease and simplicity along with high accuracy when compared to other perturbative methods.

Primary parametric amplification in a weakly forced Mathieu equation

Abstract: The present study deals with the response of a forced Mathieu equation with damping, with weak harmonic direct excitation at the same frequency as the parametric excitation. A second-order perturbation analysis using the method of multiple scales unfolds parametric amplification at primary resonance. The parametric effect on the primary resonance behavior occurs with a slow time scale of second order, although the effect on the steady-state response is of order one. As the parametric excitation level increases, the response at primary resonance stretches before becoming unbounded and unstable. Analytical expressions for predicting the response amplitudes are presented and compared with numerical results for a specific set of system parameters. Dependence of the amplification behavior, and indeed possible deamplification, on parameters is examined. The effect of parametric excitation on the response phase behavior is also presented.

A haloscope amplification chain based on a traveling wave parametric amplifier.

Abstract: In this paper, we will describe the characterization of an RF amplification chain based on a traveling wave parametric amplifier. The detection chain is meant to be used for dark matter axion searches, and thus, it is coupled to a high Q microwave resonant cavity. A system noise temperature Tsys = (3.3 ± 0.1) K is measured at a frequency of 10.77 GHz, using a novel calibration scheme, allowing for measurement of Tsys exactly at the cavity output port.

Terahertz amplifiers based on gain reflectivity in cuprate superconductors

Abstract: We demonstrate that parametric driving of suitable collective modes in cuprate superconductors results in a reflectivity $R>1$ for frequencies in the low terahertz regime. We propose to exploit this effect for the amplification of coherent terahertz radiation in a laser-like fashion. As an example, we consider the optical driving of Josephson plasma oscillations in a monolayer cuprate at a frequency that is blue-detuned from the Higgs frequency. Analogously, terahertz radiation can be amplified in a bilayer cuprate by driving a phonon resonance at a frequency slightly higher than the upper Josephson plasma frequency. We show this by simulating a driven-dissipative $U(1)$ lattice gauge theory on a three-dimensional lattice, encoding a bilayer structure in the model parameters. We find a parametric amplification of terahertz radiation at zero and nonzero temperature.

A three-dimensional Josephson parametric amplifier

Abstract: A Josephson parametric amplifier (JPA) is executed in a three-dimensional (3D) microwave cavity by coupling it to a superconducting quantum interference device (SQUID) that is embedded in a two-dimensional resonator. The JPA is activated in a three-wave mixing configuration by injecting ac magnetic flux, at twice the 3D cavity frequency, into the SQUID. An 8.3 GHz cavity is measured in a non-degenerate phase-insensitive configuration which yields gains in excess of 40 dB, where a 20 dB gain results in an operational bandwidth of 0.4 MHz, a 1 dB compression point of −115 dBm with half a quantum of added noise.

Parametric excitation suppression in a floating cylinder via dynamic vibration absorbers: a comparative analysis

Abstract: Abstract Parametric excitation in the pitch/roll degrees of freedom (DoFs) can induce dynamic instability in floating cylinder-type structures such as spar buoys, floating offshore wind or wave energy converters. At certain frequency and amplitude ranges of the input waves, parametric coupling between the heave and pitch/roll DoFs results in undesirable large amplitude rotational motion. One possible remedy to mitigate the existence of parametric resonance is the use of dynamic vibration absorbers. Two prominent types of dynamic vibration absorbers are tuned mass dampers (TMDs) and nonlinear energy sinks (NESs), which have contrasting properties with regard to their amplitude and frequency dependencies when absorbing kinetic energy from oscillating bodies. This paper investigates the suppression of parametric resonance in floating bodies utilizing dynamic vibration absorbers, comparing the performance of TMDs against NESs for a test case considering a floating vertical cylinder. In addition to the type of dynamic vibration absorber utilized, the paper also examines the DoF which it acts on, comparing the benefits between attaching the vibration absorber to the primary (heave) DoF or the secondary (pitch) DoF. The results show that the TMD outperforms the NES and that it is more effective to attach the vibration absorber to the heave DoF when eliminating parametric resonance in the pitch DoF.

Gravitational dynamics in Higgs inflation: Preinflation and preheating with an auxiliary field

Abstract: The dynamics of both the preinflationary and the preheating epochs for a model consisting of a Higgs inflaton plus an additional auxiliary field are studied in full General Relativity. The minimally coupled auxiliary field allows for parametric-type resonances that successfully transfer energy from the inflaton condensate to particle excitations in both fields. Depending on the interaction strengths of the fields, the broad resonance periods lead to structure formation consisting of large under/over-densities, and possibly the formation of compact objects. Moreover, when confronting the same model to multi-field inhomogeneous preinflation, the onset of inflation is shown to be a robust outcome. At relatively large Higgs values, the non-minimal coupling acts as a stabilizer, protecting the dynamics of the inflaton, and significantly reducing the impact of perturbations in other fields and matter sectors. These investigations further confirm the robustness of Higgs inflation to multi-field inhomogeneous initial conditions, while putting in evidence the formation of complex structures during the reheating.

The response of an inerter-based dynamic vibration absorber with a parametrically excited centrifugal pendulum

Abstract: The inerter has been integrated into various vibration mitigation devices, whose mass amplification effect could enhance the suppression capabilities of these devices. In the current study, the inerter is integrated with a pendulum vibration absorber, referred to as inerter pendulum vibration absorber (IPVA). To demonstrate its efficacy, the IPVA is integrated with a linear, harmonically forced oscillator seeking vibration mitigation. A theoretical investigation is conducted to understand the nonlinear response of the IPVA. It is shown that the IPVA operates based on a nonlinear energy transfer phenomenon wherein the energy of the linear oscillator transfers to the pendulum vibration absorber as a result of parametric resonance of the pendulum. The parametric instability is predicted by the harmonic balance method along with Floquet theory. A perturbation analysis shows that a pitchfork bifurcation and period doubling bifurcation are necessary and sufficient conditions for the parametric resonance to occur. An arc-length continuation scheme is used to predict the boundary of parametric instability in the parameter space and verify the perturbation analysis. The effects of various system parameters on the parametric instability are examined. Finally, the IPVA is compared with a linear benchmark and an autoparametric vibration absorber, and shows more efficacious vibration suppression.

Capture Into Resonance in Nonlinear Oscillatory Systems with Decaying Perturbations

Abstract: We study the influence of oscillatory perturbations on nonlinear nonisochronous oscillatory systems in the plane. We assume that the perturbation amplitude decays and the frequency is unboundedly increasing in time. We study capture into resonance in the case where the amplitude of the system unboundedly increases and the frequency adjusts to the perturbation frequency. We discuss the existence, stability, and asymptotic behavior of resonance solutions at long times. We propose the technique based on averaging method and construction of the Lyapunov functions. The results obtained are applied to the Duffing oscillator with decaying parametric perturbations.

Study of the Influence of Nonlinear Moments upon Intensity of Parametric Roll

Abstract: Hydrodynamical analysis of the conditions for the occurrence of chaotic ship roll, leading in some cases to the capsizing of the vessel, showed that such conditions are most likely to occur in the zone of the main parametric resonance of the roll when its period is sequentially doubled, and subharmonic oscillations turn into chaotic ones. This circumstance necessitates special attention to the regime of parametric roll resonance, issues of its occurrence, development, and establishment as well as to the methods of calculation of its amplitudes. In the present paper, the study of the parametric ship roll is conducted on the basis of the Lugovsky formula. An account is taken of the additional nonlinear moments M¯X23 and M¯X24, obtained through the application of the small parameter method. Presented are the calculation results for the parametric roll of five different ships performing motions at various course angles both with and without account of the aforementioned nonlinear moments. Demonstrated therewith is a significant influence of the nonlinear moments upon the maximum amplitudes of the parametric roll, especially in the case of beam waves.

Parametric resonance in a conservative system of coupled nonlinear oscillators

Abstract: We study a dimer in a periodic potential well, which is a conservative but nonintegrable system. This seemingly simple system exhibits a surprisingly rich dynamics. Using a systematic asymptotic analysis, we demonstrate that the translation mode of the dimer (center of mass motion) may induce a parametric resonance of the oscillatory mode. No external forcing occurs, thus this system belongs to the class of autoparametric systems. When the dimer energy is such that both particles are trapped in neighboring potential wells, we derive the relevant amplitude equations for the eigenmodes (center of mass motion and relative motion) and show that they are integrable. In the opposite limit, when the dimer slides along the external potential so that the center of mass motion is basically a translation, we also exhibit autoparametric amplification of the relative motion. In both cases our calculations provide reliable estimates of the relevant parameters for the autoparametric resonance to appear. Moreover, the comparison between the numerical integration of the actual system and the asymptotic analysis evidences an excellent quantitative agreement.

Stability Analysis for Spatial Autoparametric Resonances of Framed Structures

Abstract: The spatial dynamic instabilities of framed structures due to autoparametric resonances have been seldom investigated in the published literature. Based on the finite element method (FEM), the spatial parametric vibration equations are established for general framed structures. The Newmark’s method and the energy-growth exponent (EGE) are used to determine the stability of the spatial autoparametric resonances of framed structures. A portal frame model is used to conduct a spatial (out-of-plane) autoparametric resonance experiment. The numerical results of the autoparametric resonances are found to agree with those of the test, which proves the validity of the present theoretical formulation. A numerical example for autoparametric resonance stability analysis of a spatial frame is presented to firstly predict the three instability modes of autoparametric resonances, i.e. global unidirectional translational instability, bidirectional (diagonal) translational instability and torsional instability. When the excitation frequency is approximately twice the modal frequency of spatial vibration of a framed structure, spatial dynamic instability will occur due to autoparametric resonance. A small excitation force can cause a strong autoparametric resonance of the framed structure. The potential risk of spatial dynamic instability is revealed for the framed structures under periodic loads.

Sub-three-cycle pulses at 2 µm from a degenerate optical parametric amplifier.

Abstract: In this work we present a compact two-stage optical parametric amplifier (OPA) pumped at degeneracy by the fundamental of a Yb:KGW laser system. The output pulses span from 1.7 to 2.5 µm (120-176 THz) and are compressed to a sub-20 fs duration. This parametric amplifier exploits the broad phase-matching bandwidth at the degeneracy point in bismuth triborate (BiBO) and periodically poled lithium tantalate (PPLT). The result drastically expands the availability of ultrashort pulses with few-microjoule energy from near-infrared (NIR) to even longer wavelengths in the mid-infrared (MIR) spectral region.