Harmonic Potential Research Articles

Compression of a line of spheres or bubbles in a transverse confining potential

ABSTRACT A line of spheres or bubbles, confined in a cylindrically symmetric harmonic potential, buckles under axial compression. Increasing compression results in a sequence of initially planar, and eventually non-planar, structures. We present a catalogue of these, drawing on both experiments and computer simulations. The latter make use of the theory of Morse and Witten to model bubble–bubble interaction. Numerical results are summarised in stability diagrams, of potential relevance to the creation of chiral photonic crystals from self-assembled packings of spheres.

Generalised Ornstein–Uhlenbeck process: memory effects and resetting

Abstract In this work we consider a generalised Ornstein-Uhlenbeck process for a stochastically driven particle in an harmonic potential which is governed by a Fokker-Planck equation in the presence of a memory kernel. We analyse the probability density function, the mean and the mean squared displacement by employing the subordination approach connecting the operational time of the process with the (generalised) laboratory time. We provide analytical results for the mean and the mean squared displacement in case of a power-law memory kernel which corresponds to the fractional Ornstein-Uhlenbeck process. The generalised Ornstein-Uhlenbeck process in the presence of Poissonian resetting is also investigated by using the renewal equation approach, and the nonequilibrium stationary state approached in the long time limit is obtained. The analytical results are confirmed by numerical simulations based on the coupled Langevin equations.

Phase transitions in chromatin: Mesoscopic and mean-field approaches.

By means of a minimal physical model, we investigate the interplay of two phase transitions at play in chromatin organization: (1) liquid-liquid phase separation within the fluid solvating chromatin, resulting in the formation of biocondensates; and (2) the coil-globule crossover of the chromatin fiber, which drives the condensation or extension of the chain. In our model, a species representing a domain of chromatin is embedded in a binary fluid. This fluid phase separates to form a droplet rich in a macromolecule (B). Chromatin particles are trapped in a harmonic potential to reproduce the coil and globular phases of an isolated polymer chain. We investigate the role of the droplet material B on the radius of gyration of this polymer and find that this radius varies nonmonotonically with respect to the volume fraction of B. This behavior is reminiscent of a phenomenon known as co-non-solvency: a polymer chain in a good solvent (S) may collapse when a second good solvent (here B) is added in low quantity and expands at higher B concentration. In addition, the presence of finite-size effects on the coil-globule transition results in a qualitatively different impact of the droplet material on polymers of various sizes. In the context of genetic regulation, our results suggest that the size of chromatin domains and the quantity of condensate proteins are key parameters to control whether chromatin may respond to an increase in the quantity of chromatin-binding proteins by condensing or expanding.

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.

SIMP dark matter during reheating

Strongly interacting massive particle (SIMP) has become one of the promising dark matter (DM) candidates due to its capability of addressing the small-scale anomaly, where the final DM abundance is set via the freeze-out of 3→ 2 or 4→ 2 annihilation process involving solely the dark sector particles. In this work, we explore the freeze-out of SIMP DM during the inflationary reheating epoch. During reheating, the radiation energy density evolves differently based on the shape of inflaton potential and spin of its decay products than the standard radiation-dominated picture; as a result, in this scenario, the freeze-out temperature varies distinctly with DM mass compared to the standard case. Large entropy injection due to inflaton decay demands a smaller cross-section to satisfy the observed relic than the standard radiation-dominated freeze-out case. The required cross-section, satisfying the relic density constraint and the maximum allowed thermally averaged cross-section by the unitarity of the S-matrix, set an upper limit on the DM mass. The upper bound on the mass of the dark matter for 3→2 (4→2) is 1 GeV (7 MeV), assuming a radiation-dominated background. Interstingly, these limits get relaxed to 106 (104) GeV for 3→2 (4→2) SIMP dark matter for quadratic inflaton potential. We find that a small amount of DM parameter space survives for reheating with quadratic inflaton potential after considering the lower bound of reheating temperature, put by the latest CMB observation depending on the inflationary models. In the case of the quartic inflaton potential, the allowed DM parameter space gets reduced compared to the quadratic case.

A New Paradigm in Quantum Fields: The Quantum Oscillator with Semi-Quanta (IQuO) (Part One)

A New Paradigm in Quantum Fields: The Quantum Oscillator with Semi-Quanta (IQuO) (Part One)

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.

Superconductivity and its Properties: Cooper Pairs, Meissner Effect, London Moment, Gravitomagnetic and Gravitoelectric Processes

The following properties of superconductivity are considered in this work: the creation and destruction of Cooper pairs, “mass defect”, Meissner effect, London moment, gravitomagnetic and gravitoelectric processes. The conducted research has shown that for explanation of the above-mentioned properties of superconductivity it is necessary to investigate the characteristics of physical vacuum. With this aim, the following physical models of physical vacuum are analyzed in the work. The Feynman model of creating of virtual photons by quantum objects, explaining the creation and destruction of Cooper pairs, “mass defect” and partly Meissner effect. The model of physical vacuum consisting of quantum oscillators by A. Einstein and O. Stern and the model of magnetic field by L. I. Sedov. The latter two models are basic for interpretation of Meissner effect, London moment, gravitomagnetic and gravitoelectric processes.

Variation in (Hyper)Polarizability of H2 Molecule in Bond Dissociation Processes Under Spatial Confinement.

We report the results of calculations of the linear polarizability and second hyperpolarizability of the H2 molecule in the bond dissociation process. These calculations were performed for isolated molecules, as well as molecules under spatial confinement. The spatial confinement was modeled using the external two-dimensional (cylindrical) harmonic oscillator potential. In contrast to the recently investigated polar LiH molecule, it was shown that the spatial confinement significantly diminishes the linear and nonlinear response of H2 for each interatomic (H-H) distance.

Solutions of the Generalized Dunkl-Schrödinger Equation for Harmonic and Coulomb Potentials in two Dimensions

Solutions of the Generalized Dunkl-Schrödinger Equation for Harmonic and Coulomb Potentials in two Dimensions

Modeling the tail risk of crude oil futures using a quantum approach

This study proposes a quantum approach to measure the tail risk of crude oil futures. Based on the quantum harmonic oscillator (QHO) model, we first model the log return process of crude oil futures. From the equilibrium distribution of QHO model, we calculate the well-known tail risk measures such as Value-at-Risk (VaR) and Expected Shortfall (ES); we confirm that this quantum approach provides precise estimates of in-sample and out-of-sample VaR and ES for crude oil futures. The findings suggest the QHO model for log return process outperforms the geometric Brownian motion for price process and the historical simulation approach. The excellence of QHO model in explaining the tail risk of crude oil futures can be attributed to its fast angular frequency and high probability allocation in excited states. Crude oil market participants can use this study’s analytical framework to precisely measure and manage potential risks in the marketplace. Regulatory authorities can evaluate the degree of crude oil market instability through the current market state of crude oil futures characterized by the QHO model parameters while devising and revising crude oil market regulations. Future research can examine quantum models with (1) different assumptions, e.g., position-dependent diffusion and time-varying equilibrium return, and (2) different potential functions, e.g., infinite and finite square well.

Fringing effects in the operation of harmonic backing potential Kelvin probes

In this paper we present the effects of fringing on the measurement of the tip-sample capacitance for Kelvin probes driven by a harmonic backing potential. Moreover, we show that in this case it is still possible to obtain good estimations of operational parameters that allow for the fully controlled operation of harmonically driven Kelvin probes.

Many-body Stark resonances by the complex absorbing potential method

The resonances of many-body Stark Hamiltonians are characterized by the complex absorbing potential method. Namely, the resonances are shown to be the limit points of complex discrete eigenvalues of many-body Stark Hamiltonians with quadratic complex potential when the coefficient of the complex potential tends to zero. The proof is based on the complex distortion outside a cone. Pseudodifferential operators are used in the proof of the main estimate.

Impurity-induced vortex lattice melting and turbulence in rotating Bose–Einstein condensates

Abstract We numerically investigate the impact of various impurities on rotating Bose–Einstein condensates confined within two-dimensional harmonic and Gaussian distributed square lattice potentials. Without impurities, the rotating condensates display an organized square lattice pattern of vortices due to the influence of Gaussian distributed square lattice potential. The introduction of impurity potentials disrupts this lattice structure, inducing a phase transition from an ordered state to a disordered state. Our analysis encompasses both static and dynamic types of impurities. The static impurities are implemented using a randomly varying potential with a spatially random amplitude. The transformation of the vortex lattice structure, in this case, relies on the strength and lattice constant of the impurity potential. For dynamical impurities, we employ a Gaussian obstacle that orbits around the condensate at a specific distance from its center. In this scenario, the vortex lattice melting occurs beyond a certain threshold radius and frequency of oscillation of the rotating obstacle. We characterize the melting of the vortex lattice due to impurities using various quantities, such as the structure factor and angular momentum. Notably, in the vortex-melted state, the angular momentum follows a power-law dependence with an exponent of approximately 1.73, regardless of the type of impurity. Finally, we demonstrate the signature of the presence of a turbulent state within the vortex-melted state generated by both static and dynamical impurities.

A novel scheme for modelling dissipation (gain) and thermalization in open quantum systems

Abstract In this letter, we introduce a novel method for investigating dissipation (gain) and thermalization in an open quantum system. In this method, the quantum system is coupled linearly with a copy of itself or with another system described by a finite number of bosonic operators. The time-dependent coupling functions play a fundamental role in this scheme. To demonstrate the efficiency and significance of the method, we apply it to some ubiquitous open quantum systems. Firstly, we investigate a quantum oscillator in the presence of a thermal bath at the inverse temperature β, obtaining the reduced density matrix, the Husimi distribution function, and the quantum heat distribution function accurately. The results are consistent with existing literature by appropriate choices for the time-dependent coupling function. To illustrate the generalizability of this method to systems interacting with multiple thermal baths, we study the interaction of a quantum oscillator with two thermal baths at different temperatures and obtain compatible results. Subsequently, we analyze a two-level atom with energy or phase dissipation and derive the spontaneous emission and the pure dephasing processes consistently using the new method. Finally, we investigate the Markovianity in a dissipative two-level system.

Measurements of the Rovibrationally Dependent Lifetimes of C2 in the 13Σg+ and f3Σg- States through the VUV-Pump-UV-Probe Scheme.

The dicarbon molecule, C2, is one of the most important diatomic species in various gaseous environments. Despite extensive spectroscopic studies in the last two centuries, the radiative and photodissociative properties of C2 in its highly excited electronic states are still largely unexplored, particularly in the short vacuum ultraviolet (VUV) region. In this study, the lifetimes of C2 for rotational levels in the recently identified 13Σg+ state up to the vibrational level ν' = 4 and in the f3Σg- state up to ν' = 2 are measured for the first time with a VUV-pump-UV-probe photoionization scheme. It is found that all rovibrational levels in the f3Σg- state have lifetimes shorter than the dynamical limit of ∼4.7 ns of the present method, and the lifetimes in the 13Σg+ state show obvious and complex dependence on the rotational and vibrational quantum levels. Generally, the lifetime in the 13Σg+ state becomes shorter at higher rotational and vibrational levels, i.e., the longest lifetime measured in the ν' = 0 level is ∼18 ns, while all rotational levels in the ν' = 4 level are found to have shorter lifetimes than ∼4.7 ns. This implies that predissociation might occur and become more prominent at the higher energy levels in the 13Σg+ state. The prominent dependence of the lifetime on the rotational, vibrational and electronic level indicates complex dynamical processes in the highly excited states of C2.

Interaction Between Gravitational Waves and Trapped Bose–Einstein Condensates

Inspired by recent proposals for detecting gravitational waves by using Bose–Einstein condensates (BECs), we investigated the interplay between these two phenomena. A gravitational wave induces a phase shift in the fidelity amplitude of the many-body quantum state. We investigated the enhancement of the phase shift in the case of Bose condensates confined by an anisotropic harmonic potential, considering both ideal and interacting BECs.

Few-to-many-particle crossover of pair excitations in a superfluid

Motivated by recent advances in the creation of few-body atomic Fermi gases with attractive interactions, we study theoretically the few-to-many-particle crossover of pair excitations, which for large particle numbers evolve into a mode that describes amplitude fluctuations of the superfluid order parameter (the “Higgs” mode). Our analysis is based on the hypothesis that salient aspects of the excitation spectrum are captured by interactions between time-reversed pair states in a harmonic oscillator potential. Microscopically, this assumption leads to a Richardson-type pairing model, which is integrable and thus allows a systematic quantitative study of the few-to-many-particle crossover with only minor numerical effort. We first establish a parity effect in the ground-state energy, i.e., a spectral convexity in the energy of open-shell configurations compared to their closed-shell neighbors, which is quantified by a so-called Matveev-Larkin parameter discussed for mesoscopic superconductors, which generalizes the pairing gap to mesoscopic ensembles and which behaves quantitatively differently in a few-body and a many-body regime. The crossover point for this quantity is widely tunable as a function of interaction strength. We then compute the excitation spectrum and demonstrate that the pair excitation energy shows a minimum that deepens with increasing particle number and shifts to smaller interaction strengths, consistent with the finite-size precursor of a quantum phase transition to a superfluid state. We extract a critical finite-size scaling exponent that characterizes the decrease of the gap with increasing particle number. Published by the American Physical Society 2024

On the construction of the coherent states of the semiconfined harmonic oscillator with a position-dependent effective mass

Abstract We present details of the method for constructing the coherent states of the quantum harmonic oscillator model that exhibits the semiconfinement effect due to a specific change of its mass by position. Its energy spectrum completely overlaps with the Hermite oscillator energy spectrum, whereas the wavefunctions of the stationary states are expressed via the generalized Laguerre polynomials. Two different methods have been applied to compute the coherent states. They are generalized and Barut-Girardello coherent states methods. In both cases, the exact expressions have been obtained. We also analyzed some limit cases, under which the constructed coherent states completely recover the coherent states of the Hermite oscillator.

Irradiated Coherent States in Magnetoresistance of 2D Electron Systems

Herein, a novel theoretical approach on the microwave‐induced resistance oscillations based on the coherent states of the quantum harmonic oscillator is reported. Within the theoretical model of microwave‐induced electron orbits, the coherent state expression for the quantum harmonic oscillator under a time‐dependent force (radiation) is calculated. Thus, it is found that the principle of minimum uncertainty of coherent states, involving time and energy, is the main responsible of photo‐oscillations and zero‐resistance states. Thus, essential experimental results of this remarkable effect are shed light (and explained) on. The origin of photo‐oscillations, their position with respect to the magnetic field, and their periodicity can be distinguished.