Pure Oscillator Research Articles

Addressing the non-perturbative regime of the quantum anharmonic oscillator by physics-informed neural networks

Abstract The use of deep learning in physical sciences has recently boosted the ability of researchers to tackle physical systems where little or no analytical insight is available. Recently, the Physics−Informed Neural Networks (PINNs) have been introduced as one of the most promising tools to solve systems of differential equations guided by some physically grounded constraints. In the quantum realm, such an approach paves the way to a novel approach to solve the Schrödinger equation for non-integrable systems. By following an unsupervised learning approach, we apply the PINNs to the anharmonic oscillator in which an interaction term proportional to the fourth power of the position coordinate is present. We compute the eigenenergies and the corresponding eigenfunctions while varying the weight of the quartic interaction. We bridge our solutions to the regime where both the perturbative and the strong coupling theory work, including the pure quartic oscillator. We investigate systems with real and imaginary frequency, laying the foundation for novel numerical methods to tackle problems emerging in quantum field theory.

SESAM-assisted Kerr-lens mode-locked Cr:ZnS laser.

Mode-locking in Cr:ZnS/Se lasers typically rely on Kerr-lensing (KLM) or a semiconductor saturable absorber mirror (SESAM). The former allows generation of shorter pulses, but, unlike the latter, does not support self-starting mode-locking. Here, we combine the advantages of these two techniques and demonstrate the SESAM-assisted KLM Cr:ZnS laser. Our self-starting oscillator generates up to 1 W of average power with 54 fs pulses at a central wavelength of 2360 nm. We identify a general limitation for further pulse shortening in SESAM mode-locked Cr:ZnS/Se lasers, which is related to the finite operation bandwidth of the semiconductor absorbers. In our experiment, we fully exploit the potential of commercially available GaSb SESAMs and fill their entire reflection bands. Furthermore, we compare the performance of a SESAM-assisted KLM laser with a pure KLM oscillator producing broadband, yet not self-starting, 33 fs pulses with 780 mW power. We also show that the choice of saturable absorbers has a negligible impact on the laser intensity noise, which is exceptionally low with sub-0.005% integrated noise.

Exploiting multi-stiffness combination inspired absorbers for simultaneous energy harvesting and vibration mitigation

In scenarios where wide-amplitude shock excitation is encountered, a routine bistable configuration struggles to achieve satisfactory energy transfer, and a sensible combination of multiple nonlinear oscillators with different stiffness mechanisms can be exploited to achieve optimal vibration mitigation performance. In this paper, a nonlinear energy conversion and multi-stiffness combination inspired dynamic vibration absorber (NECMC-DVA) is proposed with a magnetic-spring tunable nonlinear stiffness element as the substructure. Parametric studies are performed for different modes of magnet-spring tunable nonlinear stiffness unit arrays considering transient impulses and harmonic excitation. The NECMC-DVA device, which comprises bistable oscillators, cubic stiffness oscillators, and hybrid stiffness oscillators, is also experimentally demonstrated for its synergistic performance in vibration mitigation and energy harvesting. The performance of the NECMC-DVA is measured under transient impulses and harmonic excitations. It has been found that the NECMC-DVA, which consists of a bistable oscillator, a cubic stiffness oscillator, a hybrid stiffness oscillator, energy-capture elements, and motor sequences, achieves high-efficiency nonlinear energy transfer and significant shock robustness in wide-amplitude impulsive scenarios. The NECMC-DVA with hybrid energy sinks demonstrates more remarkable targeted energy transfer and energy conversion performance than its pure bistable oscillator counterpart. Furthermore, the NECMC-DVA prototype can achieve adaptive vibration control modulation and energy conversion management through motion control boards by monitoring ambient excitation signals. Therefore, the NECMC-DVA enables integrated active and passive mitigation and energy harvesting to simultaneously accommodate multi-source indeterminate vibration and shock environments.

On-Chip Quantization- and Correlation-Robust Jitter Measurement for Low Phase Noise and High Frequency Oscillators

For on-chip oscillator phase noise characterization and monitoring a common technique is the measurement of the equivalent cycle-to-cycle jitter. In this letter we show that the quantization introduced to the recorded jitter by the measurement with a time-to-digital converter can be prohibitive for calculating the phase noise, especially at high oscillation frequencies and for spectrally pure oscillators. A second drawback of this method is that supply voltage noise can mask the phase noise due to intrinsic device noise. To avoid these limitations, we investigate an approach where the oscillator under test is compared to an identical replica. The resulting time difference measurements are introduced as delta jitter. The validity of the method is proven in simulation and measurement of an on-chip jitter measurement macro designed in 16 nm FinFET CMOS technology. The measurement results indicate both robustness against quantization error and a self-canceling of correlated influences on phase noise in delta jitter measurements.

Asymptotics of eigenvalue sums when some turning points are complex

Recent work has shown a deep connection between semilocal approximations in density functional theory and the asymptotics of the sum of the Wentzel–Kramers–Brillouin (WKB) semiclassical expansion for the eigenvalues. However, all examples studied to date have potentials with only real classical turning points. But systems with complex turning points generate subdominant (SD) terms beyond those in the WKB series. The simplest case is a pure quartic oscillator. We show how to generalize the asymptotics of eigenvalue sums to include SD contributions to the sums, if they are known for the eigenvalues. These corrections to WKB greatly improve accuracy for eigenvalue sums, especially for many levels. We obtain further improvements to the sums through hyperasymptotics. For the lowest level, our summation method has error below 2 × 10−4. For the sum of the lowest ten levels, our error is less than 10−22. We report all results to many digits and include copious details of the asymptotic expansions and their derivation.

Hamiltonian matrix representation of harmonic oscillator system with Linear-Cubic-Quartic (LCQ) perturbation

In this paper, we analyze the matrix representation of the energy operator (Hamiltonian) of the harmonic oscillator system when this system is perturbed by an LCQ perturbation. The LCQ perturbation is simultaneously acted on this system, so the total Hamiltonian is a summation of the pure oscillator harmonic operator term (in the annihilation and creation operator statement) and the three perturbation terms. In this work, we use three different small parameters for each term of the perturbation for keeping the generality. We use the simple algebraic method by using the Dirac notation and the well-known base vector of harmonic oscillator to analyze this problem. Next, we present every term of this operator in matrix representation and then adding them for finding matrix representation for the total Hamiltonian.

Fast automatic frequency calibration assisted phase-locked highly stable optoelectronic oscillator

Highly stable, low phase noise microwave oscillators are essential for various applications. An optoelectronic oscillator (OEO) can overcome the short-term phase noise limitation of pure electronic oscillators at high oscillation frequency. Nonetheless, the long-term frequency stability should be addressed. To stabilize the frequency of OEO, a phase-locked loop (PLL) is widely used to synchronize the OEO to a stable reference. However, due to the narrow free-spectral-range (FSR) of the oscillation cavity of the OEO, the pull-in range of the PLL is limited. It is challenging to acquire phase-locking at startup and phase-relocking when mode-hopping of OEO occurs. Here, by using an automatic frequency calibration (AFC) assisted PLL, we attain a highly stable 10 GHz phase-locked OEO with robust phase-locking at startup and phase-relocking when mode-hopping of OEO occurs, for the first time. With the use of a fast digitally-controlled frequency shifter and a real-time frequency error detection unit in the AFC loop, the phase-locking and phase-relocking time are below 120 ms. Furthermore, it shows the phase noise of -135 dBc/Hz at 10 kHz offset, side-mode suppression ratio (SMSR) of 128 dBc, and Allan deviation of 4.8×10-11 at 5000 s for the phase-locked OEO. We thoroughly investigate the dynamics of the automatic frequency calibration, the phase-locking process, the phase-relocking after OEO mode-hopping, the system under vibration, and the frequency switching. Our approach is promising to generate a highly stable, low phase noise, and determinate frequency microwave signal, which can be used as a low phase noise reference for a microwave frequency synthesizer and high performance sampling clock for a data conversion system.

Uncommonly accurate energies for the general quartic oscillator

AbstractRecent advances in the asymptotic analysis of energy levels of potentials produce relative errors in eigenvalue sums of order 10−34, but few non‐trivial potentials have been solved numerically to such accuracy. We solve the general quartic potential (arbitrary linear combination of x2 and x4) beyond this level of accuracy using a basis of several hundred oscillator states. We list the lowest 20 eigenvalues for 9 such potentials. We confirm the known asymptotic expansion for the levels of the pure quartic oscillator, and extract the next two terms in the asymptotic expansion. We give analytic formulas for expansion in up to three even basis states. We confirm the virial theorem for the various energy components to similar accuracy. The sextic oscillator levels are also given. These benchmark results should be useful for extreme tests of approximations in several areas of chemical physics and beyond.

Application of Ateb and generalized trigonometric functions for nonlinear oscillators

In this paper, the new application of the generalized trigonometric function (GTF) and of the Ateb function in strong nonlinear dynamic systems is considered. It is found that the GTF and the Ateb function represent the closed-form solution of the purely nonlinear one-degree of freedom oscillator with specific initial conditions. Definition of the GTF and Ateb functions is introduced. In spite of the fact that both functions use the incomplete Beta function and its inverse form, the difference exists according to the definition of both of these functions. The correlation between these two types of functions is exposed. Main properties of the Ateb function and of the special GTF function with parameters $$a = \frac{1}{2}$$ and $$b = \frac{1}{\alpha +1}$$ , which are the solution of the pure nonlinear oscillator, are compared and the value of the functions are calculated. Special attention is directed toward the sine GTF and the cosine Ateb function. Advantages and disadvantages of the both type of solutions are discussed.

Role of intracellular calcium cycling in sinoatrial node myocyte phase-locking

The phase-locking behavior of the sinoatrial node (SAN) may be related to heart rate modulation, which can be physically studied from the perspective of nonlinear dynamics. However, previous studies have treated the myocyte as a pure membrane oscillator without considering its intracellular dynamics. Intracellular calcium cycling (ICC) plays a significant part in pacemaking, but its role in phase-locking is unknown. In the present work, we employ an updated SAN myocyte model incorporating ICC to reinvestigate this problem. The influences of ICC-relevant parameters are determined and explained. Some features which did not appear in older models arise in the presence of ICC. Moreover, we employ our previously proposed theoretical formulas to calculate the phase-locking range. The present work sheds light on the impact of sub-cellular-scale dynamics on the classical SAN phase-locking problem and provides some suggestions for regulating the pacemaking activity.

Numerical simulation of quantum anharmonic oscillator, embedded within an infinite square well potential, by matrix methods using Gnumeric spreadsheet

The quantum anharmonic oscillator has been solved numerically using matrix diagonalization technique. The interaction potential consisting of quadratic () and quartic (λx4) terms is embedded within an infinite square well potential of appropriate width, ‘a’ and its sine eigen functions are used as basis functions ‘N’ for the employed matrix method. The energy eigen values for the resultant Hamiltonian are solved in a free open source software (FOSS), Gnumeric, a simple worksheet environment. The numerical parameters ‘a’ and ‘N’ are optimized to converge to the expected energies for harmonic oscillator and those for anharmonic oscillator from perturbation theory for small values of physical parameter, ‘λ’. The pure quartic oscillator is studied for both small and large values of λ and validated with results obtained from other numerical techniques. The breakdown of perturbation approximation for large values of λ is also shown.

Steady state vibration of the periodically forced and damped pure nonlinear two-degrees-of-freedom oscillator

In the paper the pure nonlinear and damped two-mass oscillator excited with the periodical force is considered. The oscillator is modelled with a system of two coupled second order nonlinear and non-homogenous equations. Using the model two problems are investigated: one, the excitation force identification for the known vibrating response of the system, and the second, determination of vibrations of the system excited with the known periodical force. Using the steady-state motion of the nonlinear oscillator the method for identification of the excitation force is developed. For the pure nonlinear oscillator it is obtained that the forcing function has the form of the Ateb function. However, if the excitation force is known the procedure for computing of the steady-state vibration of the system is introduced. The solution corresponds to steady-state vibrations of the free oscillator, but the amplitude and phase are assumed to be time variable. The averaged solutions are obtained for pure nonlinear oscillator with additional linear elastic force and for the Van de r Pol oscillator. Analytically obtained solutions are compared with numerical ones. They are in good agreement.

Effective Negative Mass Nonlinear Acoustic Metamaterial with Pure Cubic Oscillator

Acoustic metamaterial, which can prohibit effectively the elastic wave propagation in the bandgap frequency range, has broad application prospects in the vibration and noise reduction areas. The Lindstedt–Poincaré method was utilized to analyze the dispersion curves of nonlinear metamaterial with a pure Duffing oscillator. The first-order perturbation solutions of acoustic and optical branches were obtained. Both the starting and cutoff frequencies of the bandgap are determined consequently. It was found that the soft/hard characteristics of pure Duffing oscillators could lead to the lower/upper movement of the starting and cutoff frequencies of the bandgap. By further researching the degraded linear system, the conclusion that actual nonlinear metamaterial bandgap region is wider than effective negative mass region is drawn and that both mass and stiffness ratio effect on the starting frequency is obtained. Effective positive mass can also lead to the vibration attenuation in bandgap. For nonlinear metamaterial, the translation effect of the external excitation amplitude on the bandgap range and the zero mass at the nonlinear bandgap cutoff frequency were discussed, and all above conclusion are identified by numerical analysis.

Relativistic Oscillators in Generalized Snyder Model

We present an exact solution of one-dimensional Klein–Gordon and Dirac oscillators subjected to the uniform electric field with Snyder–de Sitter model in the momentum space, known in quantum mechanics by the stark effect. The energy eigenvalues and eigenfunctions are determined for both cases. The pure relativistic oscillator is obtained as particular case by taking the limit when the electric field vanishes. We also have determined some formulas of thermodynamics properties for relativistic oscillators within this framework.

On the response of some discrete and continuous oscillatory systems with pure cubic nonlinearity: Exact solutions

This study is concerned with oscillatory systems with pure cubic nonlinearity, which are systematically presented, starting from a one degree of freedom oscillator, then focusing on chains with two and multi degrees of freedom, and ending with elastic systems (systems having an infinite number degrees of freedom). A one degree of freedom pure cubic oscillator is given as a reference case for the other systems under consideration as it has the exact closed-form solution for its free response. Similar normal modes and exact solutions for free response are investigated in chains consisting of two discrete masses attached with pure cubic springs in two configurations: when anchored at both ends, and anchored at one end only. The former class exhibits a pitchfork bifurcation related to their similar normal modes, and the way how this bifurcation is created and destroyed is demonstrated in terms of stiffness ratios. These chains are then enhanced with additional masses and springs of the same stiffness, and their similar normal modes are determined (modal shapes, frequencies and the exact closed-form solution for time responses). Finally, longitudinal vibration of a bar with a pure cubic relationship between the axial stress and strain are examined analytically, obtaining the exact solution for the first mode shape of a clamped–clamped and clamped–free bar. Their first modes are compared with those obtained for nonlinear chains with multiple masses as well as with the well-known results for linear longitudinal vibration of the bars with the same boundary conditions.

Instantons from perturbation theory

In quantum mechanics and quantum field theory perturbation theory generically requires the inclusion of extra contributions non-perturbative in the coupling, such as instantons, to reproduce exact results. We show how full non-perturbative results can be encoded in a suitable modified perturbative series in a class of quantum mechanical problems. We illustrate this explicitly in examples which are known to contain non-perturbative effects, such as the (supersymmetric) double-well potential, the pure anharmonic oscillator, and the perturbative expansion around a false vacuum.

A Simple Pure Water Oscillator

In this paper we analyze the properties of a pure water oscillator by considering the pure water lake as a well mixed two layered system. While there is heating of and evaporation from the shallow top layer, the temperature of the deep bottom layer is assumed to be constant. By exploiting the nonlinear dependence of the density of pure water on temperature, we describe two complementary mathematical models to capture the vertical instability resulting from the variation of the density of the top layer with temperature.

Refractive Index Mixing Rules and Excess Infrared Spectra of Binary Mixtures.

Three refractive index mixing rules, Arago-Biot, Lorentz-Lorenz, and Newton, are generalized to complex refractive index and used to define infrared (IR) spectra of the corresponding ideal liquid mixtures. Using the measured optical constants n and k for acetonitrile-water mixtures (Bertie and Lan, 1997) the excess absorbances, AE = Aobs - Aideal, are calculated. Relying upon the well-established properties of the acetonitrile-water mixtures, the interpretation of the excess absorbances is established that is essentially based on the understanding of a liquid as a set of oscillators. The set depends on the composition of the mixture and comprises oscillators as present in the pure components and oscillators perturbed by hydrogen bonding between unlike molecules. The main features of an excess spectrum can be established assuming chemical equilibria among various oscillators. The most informative parts of the spectrum of a yet unstudied binary system can well be observed and even qualitatively explained from the excess absorbance provided: first, a detailed vibrational study of the components has been done; and, second, it is well understood what actually is subtracted from Aobs. As examples, the binary mixtures of ethynylbenzene and tetrachloroethylene and 2-ethynylpyridine and tetrachloroethylene are considered.

Mechanical manifestations of bursting oscillations in slowly rotating systems

This study is concerned with certain mechanical systems that comprise discrete masses moving along slowly rotating objects. The corresponding equation of relative motion is derived, with the rotating motion creating slowly varying external excitation. Depending on the system parameters, two cases are distinguished: two-well and single-well potential, i.e. the Duffing bistable oscillator and a pure cubic oscillator. It is illustrated that both systems can exhibit bursting oscillations, consisting of fast oscillations around the slow flow. Their mechanisms are explained in terms of bifurcation theory: the first one with respect to the existence of certain saddle-node bifurcation points, and the second one by creation of a certain hysteresis loop. The exact expressions for the slow flow are derived, in the first case as a discontinuous curve, and in the second one as a continuous curve. The influence of the excitation magnitude, which is a potential control parameter, on the characteristics of bursting oscillations is numerically illustrated.

Amplitude dynamics favors synchronization in complex networks.

In this paper we study phase synchronization in random complex networks of coupled periodic oscillators. In particular, we show that, when amplitude dynamics is not negligible, phase synchronization may be enhanced. To illustrate this, we compare the behavior of heterogeneous units with both amplitude and phase dynamics and pure (Kuramoto) phase oscillators. We find that in small network motifs the behavior crucially depends on the topology and on the node frequency distribution. Surprisingly, the microscopic structures for which the amplitude dynamics improves synchronization are those that are statistically more abundant in random complex networks. Thus, amplitude dynamics leads to a general lowering of the synchronization threshold in arbitrary random topologies. Finally, we show that this synchronization enhancement is generic of oscillators close to Hopf bifurcations. To this aim we consider coupled FitzHugh-Nagumo units modeling neuron dynamics.