Quadratic Nonlinear Coupling Research Articles

Enhanced super-Heisenberg scaling precision by nonlinear coupling and postselection

In quantum metrology, the famous result of Heisenberg limit, scaling as 1/N (with N the number of probes), can be surpassed by considering nonlinear coupling measurement. In this work, we consider the most practice-relevant quadratic nonlinear coupling and show that the metrological precision can be enhanced from the 1/N32 super-Heisenberg scaling to 1/N2, by simply employing a pre- and post-selection (PPS) strategy, but not using any expensive quantum resources such as quantum entangled state of probes.

Acousto-optic cavity coupling in 2D phoxonic crystal with combined convex and concave holes

A two-dimensional cross-like phoxonic crystal (PxC) model is proposed, which exhibits simultaneously large complete photonic crystal (PtC) and phononic crystal (PnC) bandgaps. The most salient trait of the structure is the wide range of geometrical parameters compatible with large complete bandgaps. After geometrical optimization, photonic and phononic bandgaps with gap-to-midgap ratios of 11.5% and 90.7% are obtained, respectively. These values are close to the best topology-optimized reported values but are obtained with simple shapes compatible with nanoscale fabrication technology. These characteristics make the convex–concave topology a promising candidate for PxC devices. A cavity is then introduced by filling up one cross-like hole in the 7 × 7 super-cell. PtC and PnC bands with defects appear in the respective large complete bandgaps, confining phonons and photons in the same cavity. Acousto-optic (AO) coupling between photonic and phononic defect modes is further investigated by the finite element method, taking both photoelastic and moving interface mechanisms into consideration. The symmetries of both photonic and phononic modes play a dominant role in the coupling strength. Results show that the strongest linear coupling between a photonic transverse magnetic mode and phononic breathing mode is obtained due to the in-phase superposition in the x and y directions. A quadratic nonlinear coupling is observed when photonic modes are coupled with the phononic stretching mode due to the inverse superposition of x and y directions. Finally, the optomechanical coupling rates relative to zero-point motion are estimated.

Regenerative chatter identification in grinding using instantaneous nonlinearity indicator of servomotor current signal

Chatter is considered as one of the most important causes of instability in the precision grinding process. Suffering from the influence of nonlinearity and changed machining condition, regenerative chatter signal is nonlinear and non-stationary in nature, which makes the linear representations and stationary assumed methods not appropriate. Wavelet bicoherence is considered as an effective method for these nonlinear signals analysis. However, current wavelet bicoherence is often estimated by integrating over the finite time interval. This procedure may result to the loss of the time information; hence, it is not appropriate for chatter signal that cannot be assumed to be stationary. In this paper, an instantaneous nonlinearity indicator-based method is proposed for regenerative chatter identification in grinding by using the servomotor current signal. Firstly, the nonlinearity of servomotor current affected by regenerative chatter is discussed. After that, a non-stationary wavelet bicoherence is proposed for the quadratic nonlinear coupling detection of these nonlinear and non-stationary signals. On this basis, an instantaneous nonlinearity indicator is further established for regenerative chatter identification. The effectiveness of the proposed method is verified by simulations and experiments, and the results show that, compared with the commonly used wavelet bicoherence and continuous wavelet transform, this method can not only describe the non-stationary chatter signal in time-frequency domain but also extract the true nonlinear coupling component resulted from regenerative chatter.

Collective phase chaos in the dynamics of interacting oscillator ensembles

We study the chaotic behavior of order parameters in two coupled ensembles of self-sustained oscillators. Coupling within each of these ensembles is switched on and off alternately, while the mutual interaction between these two subsystems is arranged through quadratic nonlinear coupling. We show numerically that in the course of alternating Kuramoto transitions to synchrony and back to asynchrony, the exchange of excitations between two subpopulations proceeds in such a way that their collective phases are governed by an expanding circle map similar to the Bernoulli map. We perform the Lyapunov analysis of the dynamics and discuss finite-size effects.

Amplitude reduction of parametric resonance by dynamic vibration absorber based on quadratic nonlinear coupling

The paper proposes an amplitude reduction method for parametric resonance with a new type of dynamic vibration absorber utilizing quadratic nonlinear coupling. A main system with asymmetric nonlinear restoring force and harmonic excitation causes parametric resonance in the system. In contrast with autoparametric vibration absorber, the natural frequency of the vibration absorber is tuned to be in the neighborhood of twice that of the main system. For such a vibration absorber, we investigate the effect on the amplitude reduction for a parametrically excited main system. Analytical results using the method of multiple scales show that the amplitude of parametric resonance is reduced by the effect of the vibration absorber. The experimental results by a simple apparatus indicate that the parametric resonance is stabilized by the effects of both vibration absorber and Coulomb friction of the main system. Moreover, numerical results considering the Coulomb friction of the main system show that the amplitude of parametric resonance becomes close to zero by the proposed vibration absorber.

Evolution of ocean wave statistics in shallow water: Refraction and diffraction over seafloor topography

We present a stochastic model for the evolution of random ocean surface waves in coastal waters with complex seafloor topography. First, we derive a deterministic coupled‐mode model based on a forward scattering approximation of the nonlinear mild slope equation; this model describes the evolution of random, directionally spread waves over fully two‐dimensional topography, while accounting for wide angle refraction/diffraction, and quadratic nonlinear coupling. On the basis of the deterministic evolution equations, we derive transport equations for the wave statistical moments. This stochastic model evolves the complete wave cross‐correlation matrix and thus resolves spatially coherent wave interference patterns induced by topographic scattering as well as nonlinear energy transfers to higher and lower frequencies. In this paper we focus on the linear aspects of the interaction with seafloor topography. Comparison to analytic solutions and laboratory observations confirms that (1) the forward scattering approximation is suitable for realistic two‐dimensional topography, and (2) the combined effects of wide angle refraction and diffraction are accurately captured by the stochastic model.

Analysis on Nonlinear Coupled Vibrations of a Clamped Beam with a Tip-Mass Including Axial Inertia

Analytical results are presented on nonlinear coupled vibrations of a clamped beam with a tip-mass constrained by an axial spring. First, governing equations of the thin beam are derived including both effects of the axial inertia force and the geometrical nonlinearity of the beam. The modified Galerkin procedure is applied to the governing equation by introducing a coordinate function for the axial displacement considering the quadratic nonlinear coupling with the deflection. Nonlinear periodic responses are calculated with the harmonic balance method. Frequency responses of the principal resonance of the fundamental vibration mode are compared by changing the mass attached to the beam end. When the mass is not attached to the beam, the frequency response agrees well with the result neglecting the axial inertia. As the mass is increased, the response curve in comparatively large amplitude shifts to the lower frequency range, owing to the axial inertia of the tip-mass. The analytical result of the frequency response, taking account of the axial inertia, agrees well with the relevant experimental result of a post-buckled beam formerly presented, which verifies our analytical results.

Dynamics of structures with wideband autoparametric vibration absorbers: theory

This article analyses the dynamics of a resonantly excited single–degree–of–freedom linear system coupled to an array of nonlinear autoparametric vibration absorbers (pendulums). Each pendulum is also coupled to the neighbouring pendulums by linear elastic springs. The case of a 1:1:…:2 internal resonance between pendulums and the primary oscillator is studied for stationary (harmonic) and non–stationary (slow frequency sweep) excitations. The method of averaging is used to obtain amplitude equations that determine the first–order approximation to the nonlinear response of the system. The amplitude equations are analysed for their equilibrium as well as non–stationary solutions as a function of the parameters associated with the absorber pendulums. For stationary excitation, most steady–state solutions correspond to modes in which only one pendulum and the primary system execute coupled motions. Conditions for the existence of manifolds of equilibria are revealed when the averaged equations are expressed in modal coordinates. In the non–stationary case with linear frequency sweep through the primary resonance region, delays through pitchforks, smooth but rapid transitions through jumps, and transitions from one stable coupled–mode branch to another are studied using numerical simulations of the amplitude equations. The array of autoparametric pendulums is shown to effectively attenuate the large–amplitude resonant response of structures over a wide band of excitation frequencies.

Nonlinear interactions of slow and rapid rhythms in sympathetic nerve discharge.

We used bispectral analysis to characterize the nonlinear interactions of the respiratory-related (RR), cardiac-related (CR), or 10-Hz rhythms in sympathetic nerve discharge (SND) of urethan-anesthetized cats. Bispectral analysis investigates relationships among frequency triples in the same signal (inferior cardiac postganglionic SND) where the third frequency is the sum of the other two due to quadratic nonlinear coupling. Coupling of the RR and CR rhythms leading to the generation of new components (i.e., modulated frequencies) in SND occurred in 84% of the total cases, whereas the incidence was 71% for the RR and 10-Hz rhythms. The occurrence of such nonlinear interactions implies that the RR, CR, and 10-Hz rhythms are carried to common targets by the same postganglionic sympathetic neurons. Furthermore, we suggest that nonlinear interactions leading to the generation of new frequencies in SND may affect end-organ function beyond the level expected in simple cases of linear superposition of the primary rhythms. This suggestion is supported by our observation that strong coupling of the RR and CR rhythms resulted in appreciable power at the modulated frequencies.

Bispectral analysis of complex patterns of sympathetic nerve discharge.

Bispectral analysis was used to demonstrate quadratic nonlinear coupling (i.e., phase locking) of different frequency components in inferior cardiac sympathetic nerve discharge (SND) of urethan-anesthetized rats. The complex patterns of SND analyzed included mixtures of 1) the cardiac-related and 10-Hz rhythms, 2) the 10-Hz rhythm and irregular 2-to 6-Hz oscillations, and 3) the 10-Hz rhythm and a lower frequency non-cardiac-related rhythm near 4 Hz. In some cases, the bicoherence function (normalized bispectrum) showed no phase locking of these frequency components. Cases of nil bicoherence are equated with linear superposition of frequency components, which implies the existence of multiple and noninteractive central circuits. Increased complexity of SND was observed in other cases, as evidenced by significant phase locking of different frequency components with or without frequency locking. Frequency locking (higher frequency rhythm is a multiple of lower) was confirmed by constructing Lissajous orbital plots showing covariation of voltages in selectively filtered bands of SND. We equate frequency locking with nonlinear coupling of the central generators of different sympathetic nerve rhythms and phase locking without frequency locking possibly with nonlinearities arising at levels below noncoupled central rhythm generators.

Etudes d'interactions acoustiques non-linéaires dans les jets confinés, par la fonction de bicohérence

Self-sustained tones are encountered when a flow passes two orifice plates, sufficiently close together, inside a duct. The upstream orifice plate creates a confined jet which impinges on the edge of the downstream orifice plate. The shear layer oscillations (coherent structures) of the confined jet in the presence of a resonant acoustic field (pipe mode), constitute the main features of cavity tones. The tones are synchronized by the pipe modes. When the flow velocity is varied for a fixed geometry, frequency jumps from one mode to another are observed. Pressure spectra showing intense acoustic energy localised in narrow bands near pipe mode frequencies are measured. We emphasize the existence of transition flow regimes during which quadratic non-linear coupling between spontaneously excited modes can take place. A digital bispectral analysis (bicoherence function) is used to quantify the degree of nonlinear interactions of acoustic waves of finite amplitude in the so-called transition region

Reentrant hexagonal Turing structures

A new structural transition from stripes to hexagons is reported for the two-dimensional Brusselator model. Its existence is explained by the properties of the quadratic nonlinear coupling.

Non-linear vibratory interactions in systems of coupled beams

Small non-linear interactions of the type referred to as autoparametric may have a considerable effect on the forced oscillatory behaviour of structures. Under conditions of internal resonance these effects may bring about complex forms of response where the normal linear resonance of a directly excited mode is absorbed, and indirectly excited modes show simultaneous large non-synchronous responses. In this paper the forced vibration of a system of coupled beams is examined and it is shown that violent non-synchronous torsion and bending vibration observed in such a system may be explained by the existence of quadratic non-linear coupling terms and internal resonance effects between three and four modes. A finite degree of freedom model was formulated and the method of multiple scales was applied. For the case of three-mode interactions stationary solutions were obtained which gave very satisfactory agreement with measured responses, and showed the effects of detuning of the internal resonance. The analysis also gave a clear indication of the cause of strong four-mode interactions but did not yield stationary solutions in this case. Experimental results are included to illustrate the four-mode interactive response.

Planck distribution in a classical nonlinear coupled harmonic-oscillator system

A system ofN harmonic oscillators having a quadratic nonlinear coupling was numerically analysed. Solutions of the nonlinear problem are decomposed in terms of the normal modes of the linear system. A number of initial conditions were considered. We computed the energy in a mode as a function of time and the corresponding average energy. We found that, when initially all the energy of the mode system was in the lowest-frequency mode and the oscillators were at rest, the final time-averaged energy distribution showed that, as we increased the number of oscillators, the first fewlow odd modes received more energy on the average than neighboring even modes. When the energy was initially equipartitioned among the modes and the oscillators were initially at rest, we found that the time-averaged energy distribution of approximately the firstN/4 modes fell off monotonically with increasing frequency, with the other 3N/4 modes keeping essentially all of their energy. Finally, when the energy was initially distributed in a «Planck-like» distribution, the final time-averaged energy distribution was also a «Planck-like» distribution which was quite similar to the initial distribution. In fact, there was very little energy sharing among the modes in this case. When the space-time variation of the actual string was analysed, we found that the «Planck-like» distribution with the oscillators initially at rest was, at least to a good approximation, a «normal mode» of the nonlinear system.