Harmonic Oscillator Research Articles

Alchemical harmonic approximation based potential for iso-electronic diatomics: Foundational baseline for Δ-machine learning.

We introduce the alchemical harmonic approximation (AHA) of the absolute electronic energy for charge-neutral iso-electronic diatomics at fixed interatomic distance d0. To account for variations in distance, we combine AHA with this ansatz for the electronic binding potential, E(d)=(Eu-Es)Ec-EsEu-Esd/d0+Es, where Eu, Ec, Es correspond to the energies of the united atom, calibration at d0, and the sum of infinitely separated atoms, respectively. Our model covers the two-dimensional electronic potential energy surface spanned by distances of 0.7-2.5Å and differences in nuclear charge from which only one single point (with elements of nuclear charge Z1, Z2, and distance d0) is drawn to calibrate Ec. Using reference data from pbe0/cc-pVDZ, we present numerical evidence for the electronic ground-state of all neutral diatomics with 8, 10, 12, and 14 electrons. We assess the validity of our model by comparison to legacy interatomic potentials (harmonic oscillator, Lennard-Jones, and Morse) within the most relevant range of binding (0.7-2.5Å) and find comparable accuracy if restricted to single diatomics and significantly better predictive power when extrapolating to the entire iso-electronic series. We also investigated Δ-learning of the electronic absolute energy using our model as a baseline. This baseline model results in a systematic improvement, effectively reducing training data needed for reaching chemical accuracy by up to an order of magnitude from ∼1000 to ∼100. By contrast, using AHA+Morse as a baseline hardly leads to any improvement and sometimes even deteriorates the predictive power. Inferring the energy of unseen CO converges to a prediction error of ∼0.1 Ha in direct learning and ∼0.04 Ha with our baseline.

The functional role of oscillatory dynamics in neocortical circuits: A computational perspective

The dynamics of neuronal systems are characterized by hallmark features such as oscillations and synchrony. However, it has remained unclear whether these characteristics are epiphenomena or are exploited for computation. Due to the challenge of selectively interfering with oscillatory network dynamics in neuronal systems, we simulated recurrent networks of damped harmonic oscillators in which oscillatory activity is enforced in each node, a choice well supported by experimental findings. When trained on standard pattern recognition tasks, these harmonic oscillator recurrent networks (HORNs) outperformed nonoscillatory architectures with respect to learning speed, noise tolerance, and parameter efficiency. HORNs also reproduced a many characteristic features of neuronal systems, such as the cerebral cortex and the hippocampus. In trained HORNs, stimulus-induced interference patterns holistically represent the result of comparing sensory evidence with priors stored in recurrent connection weights, and learning-induced weight changes are compatible with Hebbian principles. Implementing additional features characteristic of natural networks, such as heterogeneous oscillation frequencies, inhomogeneous conduction delays, and network modularity, further enhanced HORN performance without requiring additional parameters. Taken together, our model allows us to give plausible a posteriori explanations for features of natural networks whose computational role has remained elusive. We conclude that neuronal systems are likely to exploit the unique dynamics of recurrent oscillator networks whose computational superiority critically depends on the oscillatory patterning of their nodal dynamics. Implementing the proposed computational principles in analog hardware is expected to enable the design of highly energy-efficient and self-adapting devices that could ideally complement existing digital technologies.

The Science of Nanostructure Acoustic Vibrations.

Ultrafast excitation of nanoparticles can excite the acoustic vibrational modes of the structure that correlate with the expansion coordinates. These modes are frequently seen in transient absorption experiments on metal nanoparticle samples and occasionally for semiconductors. The aim of this review is to give an overview of the physical chemistry of nanostructure acoustic vibrations. The issues discussed include the excitation mechanism, how to calculate the mode frequencies using continuum mechanics, and the factors that control vibrational damping. Recent results that demonstrate that the high frequencies inherent to the acoustic modes of nanomaterials trigger a viscoelastic response in surrounding liquids are also discussed, as well as vibrational coupling between nanostructures and mode hybridization within the nanostructures. Mode hybridization provides a way of manipulating the lifetimes of the acoustic modes, which is potentially useful for applications such as mass sensing.

AgGaS2 and Derivatives: Design, Synthesis, and Optical Properties.

Silver gallium sulfide (AgGaS2) is a ternary A(I)B(III)X(VI)2-type semiconductor featuring a direct bandgap and high chemical stability. Structurally resembling diamond, AgGaS2 has gained considerable attention as a highly promising material for nonlinear optical applications such as second harmonic generation and optical parametric oscillation. In attempts to expand the research scope, on the one hand, AgGaS2-derived bulk materials with similar diamond-like configurations have been investigated for the enhancement of nonlinear optics performance, especially the improvement of laser-induced damage thresholds and/or nonlinear coefficients; on the other hand, nanoscale AgGaS2 and its derivatives have been synthesized with sizes as low as the exciton Bohr radius for the realization of potential applications in the fields of optoelectronics and lighting. This review article focuses on recent advancements and future opportunities in the design of both bulk and nanocrystalline AgGaS2 and its derivatives, covering structural, electronic, and chemical aspects. By delving into the properties of AgGaS2 in bulk and nanocrystalline states, this review aims to deepen the understanding of chalcopyrite materials and maximize their utilization in photon conversion and beyond.

Dephasing enabled fast charging of quantum batteries

We propose and analyze a universal method to obtain fast charging of a quantum battery by a driven charger system using controlled, pure dephasing of the charger. While the battery displays coherent underdamped oscillations of energy for weak charger dephasing, the quantum Zeno freezing of the charger energy at high dephasing suppresses the rate of transfer of energy to the battery. Choosing an optimum dephasing rate between the regimes leads to a fast charging of the battery. We illustrate our results with the charger and battery modeled by either two-level systems or harmonic oscillators. Apart from the fast charging, the dephasing also renders the charging performance more robust to detuning between the charger, drive, and battery frequencies for the two-level systems case.

Damping Optimization and Energy Absorption of Mechanical Metamaterials for Enhanced Vibration Control Applications: A Critical Review.

Metamaterials are pushing the limits of traditional materials and are fascinating frontiers in scientific innovation. Mechanical metamaterials (MMs) are a category of metamaterials that display properties and performances that cannot be realized in conventional materials. Exploring the mechanical properties and various aspects of vibration and damping control is becoming a crucial research area. Their geometries have intricate features inspired by nature, which make them challenging to model and fabricate. The fabrication of MMs has become possible because of the emergence of additive manufacturing (AM) technology. Mechanical vibrations in engineering applications are common and depend on inertia, stiffness, damping, and external excitation. Vibration and damping control are important aspects of MM in vibrational environments and need to be enhanced and explored. This comprehensive review covers different vibration and damping control aspects of MMs fabricated using polymers and other engineering materials. Different morphological configurations of MMs are critically reviewed, covering crucial vibration aspects, including bandgap formation, energy absorption, and damping control to suppress, attenuate, isolate, and absorb vibrations. Bandgap formation using different MM configurations is presented and reviewed. Furthermore, studies on the energy dissipation and absorption of MMs are briefly discussed. In addition, the vibration damping of various lattice structures is reviewed along with their analytical modeling and experimental measurements. Finally, possible research gaps are highlighted, and a general systematic procedure to address these areas is suggested for future research. This review paper may lay a foundation for young researchers intending to start and pursue research on additive-manufactured MM lattice structures for vibration control applications.

Deviation of a system of nonreciprocally coupled harmonic oscillators from a conservative system

Deviation of a system of nonreciprocally coupled harmonic oscillators from a conservative system

Non-Markovian Feedback for Optimized Quantum Error Correction

Bosonic codes allow the encoding of a logical qubit in a single component device, utilizing the infinitely large Hilbert space of a harmonic oscillator. In particular, the Gottesman-Kitaev-Preskill code has recently been demonstrated to be correctable well beyond the break-even point of the best passive encoding in the same system. Current approaches to quantum error correction (QEC) for this system are based on protocols that use feedback, but the response is based only on the latest measurement outcome. In our work, we use the recently proposed feedback-GRAPE (gradient-ascent pulse engineering with feedback) method to train a recurrent neural network that provides a QEC scheme based on memory, responding in a non-Markovian way to the full history of previous measurement outcomes, optimizing all subsequent unitary operations. This approach significantly outperforms current strategies and paves the way for more powerful measurement-based QEC protocols. Published by the American Physical Society 2025

The even-order harmonic oscillator perturbed by a decreasing scalar potential

This paper continues a previous work studying the perturbation L = H + V on ℝ, with H = (-1)h ½d2h/dx2h + x2h, where h ∈ ℕ*, and V is a decreasing scalar potential. It is assumed that |V(n)(x)| ≤ cn(1 + x2)-s/2 for x ∈ ℝ, s ∈ ℝ*+ \ {1}, and n ∈ ℕ. Let λk denote the kth eigenvalue of H, and suppose the eigenvalues of L near λk can be expressed as λk + μk. The authors prove an asymptotic formula for the fluctuation {μk}, which is given by a transformation of V. This paper specifically examines the case where s = 1.

The time-dependent harmonic oscillator revisited

Abstract We point out a rather effective approach for solving the time-dependent harmonic oscillator $\ddot q=-\omega^2 q$ under various regularity assumptions. Where $\omega(t)$ is $C^1$ this is reduced to Hamilton equation for the angle variable $\psi$ alone (the action variable $I$ is obtained by quadrature). The fixed point theorem for the integral equation equivalent to the generic Cauchy problem for $\psi(t)$ 
yields a sequence $\{\psi^{(h)}\}_{h\in \mathbb{N}_0}$ converging to $\psi$ rather fast; if $\omega$ varies slowly or little, already $\psi^{(0)}$ approximates $\psi$ well for rather long time lapses. The discontinuities of $\omega$, if any, determine those of $\psi,I$. 
The zeroes of $q,\dot q$ are investigated via Riccati equations. Our approach may simplify the study of: upper and lower bounds 
on the solutions; the stability of the trivial one; parametric resonance when $\omega(t)$ is periodic; the adiabatic invariance of $I$; asymptotic expansions in a slow time parameter $\varepsilon$; time-dependent driven and damped parametric oscillators; etc.

HOMAc: A Parameterization of the Harmonic Oscillator Model of Aromaticity (HOMA) That Includes Antiaromaticity.

The harmonic oscillator model of aromaticity (HOMA) offers a straightforward route to quantifying aromaticity that requires no other information than the bond lengths of the conjugated ring in question. Given that such information is often readily obtainable from quantum-chemical calculations, it is pertinent to improve this parametrized model as much as possible. Here, a new version of HOMA is presented where, atypically, the corresponding parameters are derived from the actual bond lengths of both aromatic and antiaromatic (rather than nonaromatic) reference compounds, as calculated with a high-level method. The resulting model, which we denote HOMAc, covers CC, CN, NN, and CO bonds and is tested at eight different levels of theory for 45 (single-ring, multi-ring, carbocyclic, N,O-heterocyclic) molecules across the aromatic-antiaromatic spectrum. Thereby, it is found that HOMAc provides a description of aromaticity and antiaromaticity in better accord with magnetic, energetic, and π-delocalization-based reference data than does the standard parametrization of HOMA. Altogether, the results highlight the possibility to realize more reliable geometry-based probing of (anti)aromaticity with the use of HOMAc and with substantial freedom in the choice of quantum-chemical method.

Dunkl-Schrödinger equation in higher dimensions

Abstract This paper presents analytical solutions for eigenvalues and eigenfunctions of the Schrödinger equation in higher dimensions, incorporating the Dunkl operator. Two fundamental quantum mechanical problems are examined in their exact forms: the d-dimensional harmonic oscillator and the Coulomb potential. In order to obtain analytical solutions to these problems, both Cartesian and polar coordinate systems were employed. Firstly, the Dunkl-Schrödinger equation is derived in d-dimensional Cartesian coordinates, and then for the isotropic harmonic potential interaction, its solutions are given. Subsequently, using polar coordinates the angular and radial parts of the Dunkl-Schrödinger equation are obtained. It is demonstrated that the system permits the separation of variables in both coordinate systems, with the resulting separated solutions expressed through Laguerre and Jacobi polynomials. Then, the radial Dunkl-Schrödinger equation is solved using the isotropic harmonic, pseudoharmonic, and Coulomb potentials. The eigenstates and eigenvalues are obtained for each case and the behavior of the energy eigenvalue functions are illustrated graphically with the reduced probability densities.

Self-consistent M1 radiative transitions of excited Bc and heavy quarkonia with different polarizations in the light-front quark model

In this study, we investigate the properties of pseudoscalar and vector charmonia, bottomonia, and Bc mesons using the light-front quark model, focusing on the M1 radiative transition. For that purpose, we conduct a variational analysis with a quantum chromodynamics (QCD)-motivated effective Hamiltonian, employing a trial wave function expanded in the harmonic oscillator basis functions up to the 3S state. We fit the model parameters to mass spectra and decay constants, obtaining reasonable agreement with experimental data and correctly reflecting the hierarchy of mass spectra and decay constants. In analyzing the M1 radiative transition, we consider both good (μ=+) and transverse (μ=⊥) current components with both longitudinal (h=0) and transverse (h=±1) polarizations, demonstrating that the results from both components of currents and polarizations are identical. Self-consistency is achieved by substituting M with M0 when computing the operators for decay constants and radiative transitions. We also find that the difference between longitudinal and transverse polarizations of the observables may quantify the anisotropy of the model wave function. Our results on radiative transitions align reasonably well with experimental data, lattice QCD, and theoretical predictions. Furthermore, we also provide predictions for Bc mesons that can be tested in experiments. Published by the American Physical Society 2025

Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance

Inerter dampers are effectively employed to mitigate and dampen structural vibrations in slender or high-rise buildings. The simple viscous inerter damper, with a viscous dashpot placed in series with an inerter, is designed to create resonant vibration damping, although the damper itself is without an internal resonance. The apparent resonant behavior is instead obtained by increasing the damper inertance until the two lowest modes of the considered building model interact, whereafter the viscous coefficient is adjusted until the desired response mitigation is achieved. The present modal interaction tuning requires that the reduced-order single-mode dynamic model of the building includes both inertia and flexibility from the (other) modes otherwise discarded by the model reduction. While the inertia correction adjusts the modal mass of the inerter damper, the corresponding flexibility introduces the apparent damper stiffness that creates the desired damper resonance. Thus, the accurate representation of other modes is essential for the design and resonant tuning of the simple viscous inerter damper. The resonant damper performance by the non-resonant viscous inerter damper is illustrated by a numerical example with a 20-story building model, for which the desired resonant modal interaction requires an inertance of almost ten times the entire translational building mass.

The Group-Algebraic Formalism of Quantum Probability and Its Applications in Quantum Statistical Mechanics.

We show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quantum system must respect certain commutation relations, we take the group generated by a Lie algebra constructed with these commutation relations as the bridge, so that the classical characteristic function defined in a Euclidean space is transformed to a normalized, non-negative definite function defined in this group. Indeed, on the quantum side, this group-theoretical characteristic function is equivalent to the density matrix; hence, it can be adopted to represent the state of a quantum ensemble. It is also found that this new representation may have significant advantages in applications. As two examples, we show its effectiveness and convenience in solving the quantum-optical master equation for a harmonic oscillator coupled with its thermal environment, and in simulating the quantum cat map, a paradigmatic model for quantum chaos. Other related issues are reviewed and discussed as well.

Research on vibration damping of integral squeeze film damper based on fan rotor tester of turbofan

Research on vibration damping of integral squeeze film damper based on fan rotor tester of turbofan

Dynamic analysis of viscoelastic functionally graded nanoplate

In this article, the dynamic behavior of nanoplates under time-dependent load is investigated, focusing on functionally graded viscoelastic materials and nanoscale effects. Eringen’s nonlocal elasticity theory is utilized to examine mechanical response of the nanoplate. Hamilton’s principle is utilized to derive the equations of motion, taking into account both kinetic and potential energy aspects. The obtained complex partial differential equations are then solved employing Navier method and provides an efficient way to obtain analytical solutions. The study initially performed a free vibration analysis for functionally graded nanoplate, comparing the obtained results with those available in the literature to validate the developed method. Following this validation, a parametric analysis was conducted to examine the influence of both nonlocal parameter, which accounts for nanoscale effects, and power law exponent governing material gradation on free vibration behavior of functionally graded nanoplate. Finally, as the original contribution of this study, a damped forced vibration analysis was carried out within the scope of the parametric study, investigating the effects of power law exponents, viscoelastic parameters, nonlocal parameters, and various geometric properties on functionally graded viscoelastic nanoplates’ the displacement-time relationship and maximum displacements.

A partition function estimator.

We propose an estimator that allows us to calculate the value of a simple system's partition function using finite sampling. The core idea is to neglect the contribution from high energy microstates, which are difficult to be sampled properly, and then calculate a volume correction term to compensate for this. As a proof of concept, the estimator is applied to calculate the partition function for several model systems, ranging from a simple harmonic oscillator to a Lennard-Jones fluid with hundreds of particles. Our results agree well with the numerically exact solutions or reference data, demonstrating that efficiently estimating partition functions for the studied example cases is possible and computationally affordable.

A glimpse of matrix mechanics via the harmonic oscillator

In this paper, we provide a glimpse of the original formulation of quantum mechanics (a.k.a. matrix mechanics), as proposed by Heisenberg, Born, and Jordan in 1925, by showing how it solves the quantum harmonic oscillator. This presentation can shed light on the relationship between different formulations and give students a deeper understanding of quantum mechanics.

Influence of Nanodiamonds on Mechanical, Acoustic, Vibration Damping and Flame Retardancy in Jute Fiber-Reinforced Epoxy Composite Panels Used in Indoor Applications

Influence of Nanodiamonds on Mechanical, Acoustic, Vibration Damping and Flame Retardancy in Jute Fiber-Reinforced Epoxy Composite Panels Used in Indoor Applications