Harmonic Potential Research Articles

The thermodynamic properties of diatomic gases. A rigorous analytical approach

The paper reports a closed-form analytical representation of the thermodynamic functions of diatomic molecules in the ground electronic state. For this purpose, analytical formulas for vibrational and rotational partition functions are derived. A rigorous method is proposed for factorizing the vibrational and rotational contributions to the ro-vibrational partition function using an average vibrational quantum number. The behavior of thermodynamic functions as a function of temperature is discussed for carbon monoxide as a test system. The reliability of the analytical method is verified by comparison with experimental data and with the direct summation method, which uses theoretical values of energy levels.

Modulational instability of a harmonically trapped quantum droplet

We investigate the modulational instability (MI) of a harmonically trapped quantum droplet. The MI is the key mechanism for the formation of soliton trains in diverse physical media, as a result of the interplay between the intrinsic nonlinearity and the kinetic-energy term. Here, we analytically get the time-dependent criterion for MI of a trapped quantum droplets with Lee-Huang-Yang (LHY) term. It is shown that the external harmonic potential can dramatically change the condition of modulational instability. Moreover, we find that there exists a critical time tc, that is the biggest time before instability of droplet will set in. We find the tc depends on the harmonic trapping frequency k. Meanwhile, the results show that the trapping frequency k also determine the solitons’ types, a single peak soliton or a soliton train can be excited by the different k. Our theoretical predicts are confirmed by the direct numerical calculation of the generalized Gross–Pitaevskii equation with LHY correction term.

Optical Third Harmonic Generation of Spherical Quantum Dots under Inversely Quadratic Hellmann Plus Inversely Quadratic Potential

Abstract The third harmonic generation (THG) coefficient for the spherical quantum dot (QD) system with Inversely Quadratic Hellmann Plus Inversely Quadratic Potential (IQHIQP) is investigated theoretically, considering the regulation of the quantum size, confinement potential depth and external environment. The numerical simulation results indicate that the THG coefficient can reach the order of 10-12 m2/V2, which strongly relies on the tunable factor, with its resonant peak experiencing a red-shift or blue-shift. Interestingly, the effect of temperature on the THG coefficient in terms of peak location and size is consistent with the quantum dot radius but contrasts with the hydrostatic pressure. Thus, it is crucial to focus on the influence of internal and external parameters on nonlinear optical effects, and to implement the theory in practical experiments and the manufacture of optoelectronic devices.

Frequency dependent polarizability of confined harmonic oscillators

Abstract The D-dimensional Schr ̈odinger equation for a confined harmonic potential perturbed by a time-dependent dipole field is analysed. The model uses exact confined solutions for the harmonic potential whereas the time-dependent field is treated using first-order time-dependent perturbation theory. Exact analytic solutions are obtained in terms of generalised Laguerre functions. This system has applications to nanostructures where there are strong confinement potentials in some dimensions and example calculations are carried out to illustrate the effect of the dimension.

The Partition of Unity Finite Element Method for the Schrödinger Equation

Abstract A Schrödinger equation for the system’s wavefunctions in a parallelepiped unit cell subject to Bloch-periodic boundary conditions must be solved repeatedly in quantum mechanical computations to derive the materials’ properties. Recent studies have demonstrated how enriched finite element type Galerkin methods can substantially lower the number of degrees of freedom necessary to produce accurate solutions with respect to the standard plane-waves method. In particular, the flat-top partition of unity finite element method enriched with the radial eigenfunctions of the one-dimensional Schrödinger equation offers a very effective way of solving the three-dimensional Schrödinger eigenvalue problem. We investigate the theoretical properties of this approximation method, its well-posedness and stability, we prove its convergence and derive suitable bound for the ℎ- and 𝑝-refinement in the L 2 L^{2} and energy norm for both the eigenvalues and the eigenfunctions. Finally, we confirm these theoretical results by applying this method to the eigenvalue problem of the one-electron Schrödinger equation with the harmonic potential, for which the exact solution is known.

Stable stripe and vortex solitons in two-dimensional spin-orbit coupled Bose–Einstein condensates

We present a flexible manipulation and control of solitons via Bose–Einstein condensates. In the presence of Rashba spin–orbit coupling and repulsive interactions within a harmonic potential, our investigation reveals the numerical local solutions within the system. By manipulating the strength of repulsive interactions and adjusting spin–orbit coupling while maintaining a zero-frequency rotation, diverse soliton structures emerge within the system. These include plane-wave solitons, two distinct types of stripe solitons, and odd petal solitons with both single and double layers. The stability of these solitons is intricately dependent on the varying strength of spin–orbit coupling. Specifically, stripe solitons can maintain a stable existence within regions characterized by enhanced spin–orbit coupling while petal solitons are unable to sustain a stable existence under similar conditions. When rotational frequency is introduced to the system, solitons undergo a transition from stripe solitons to a vortex array characterized by a sustained rotation. The rotational directions of clockwise and counterclockwise are non-equivalent owing to spin–orbit coupling. As a result, the properties of vortex solitons exhibit significant variation and are capable of maintaining a stable existence in the presence of repulsive interactions.

Control of the oscillation frequency of a vortex cluster in the trapped polariton condensate

An optically trapped exciton–polariton condensate forms states corresponding to excited energy levels of the confining potential. Recently, it was shown that non-uniformity of the ring-shaped trapping potential leads to the simultaneous occupation of two split energy levels. This results in the formation of an oscillating vortex cluster with periodically changing signs of topological charges. Here, we experimentally demonstrate the control of the topological charge oscillation frequency by tuning the ellipticity of the excitation profile. Our observations are corroborated using the linear Schrödinger equation for a two-dimensional quantum harmonic oscillator. Our findings open a promising avenue for the investigation of optical vorticity properties and their applications in controllable settings.

Molecular Study on the DKP Equation in (1 + 3) Dimensions with Isotropic Oscillator plus Inverse Quadratic Potential in Non-Commutative Space

Molecular Study on the DKP Equation in (1 + 3) Dimensions with Isotropic Oscillator plus Inverse Quadratic Potential in Non-Commutative Space

Deformation state of a graphene sheet within the framework of the continuum moment-membrane theory of elasticity

The paper proposes an approach to finding the stress-strain state (SSS) of structures containing graphene, a novel nanomaterial that has currently found а wide range of practical applications in nanoelectromechanical systems. Graphene is a 2D basic building block for other carbon structures such as membranes, sheets, nanotubes, etc. To describe the SSS of a graphene sheet, the phenomenological continuum moment-membrane theory of plates is used, from which, due to the fact that graphene is an ultrathin material, the concept of thickness is excluded. The physical elasticity relationships of a graphene sheet are expressed through its rigidity characteristics, determined using the harmonic potential of interatomic interactions in carbon. A differential formulation and the corresponding variational formulation are given for the problem of static deformation and determination of the natural frequencies and modes of vibration of a graphene sheet. The variational formulation is based on the Lagrange principle and is implemented numerically using the finite element method. Finite element relations are constructed taking into account moment effects of the behavior of a graphene sheet. For approximation, a 4-node rectangular finite element is used. Numerical solutions to several problems of static deformation of a graphene sheet under conditions of a plane stress state and transverse bending are presented, and the analysis of its natural vibrations is also performed. Good convergence of numerical simulation results in all considered problems is demonstrated. The obtained numerical solutions are essential in designing and calculating resonators in which ultrathin nanostructures are used. The establishment of the fact that a graphene sheet has a high intrinsic frequency falling in the GHz region (for example, quartz resonators are characterized by megahertz frequencies) opens up new prospects for using graphene itself as an ultrasensitive nanomechanical resonator for detecting small masses and ultrasmall displacements.

Shannon entropy measurements for quantum oscillator system in the presence of a spiral dislocation

Shannon entropy measurements for quantum oscillator system in the presence of a spiral dislocation

Absorption and fluorescence spectroscopy of cold proflavine ions isolated in the gas phase.

Proflavine, a fluorescent cationic dye with strong absorption in the visible, has been proposed as a potential contributor to diffuse interstellar bands (DIBs). To investigate this hypothesis, it is essential to examine the spectra of cold and isolated ions for comparison. Here, we report absorption spectra of proflavine ions, trapped in a liquid-nitrogen-cooled ion trap filled with helium-buffer gas, as well as fluorescence spectra to provide further information on the intrinsic photophysics. We find absorption- and fluorescence-band maxima at 434.2 ± 0.1 and 434.7 ± 0.3nm, corresponding to a Stokes shift of maximum 48cm-1, which indicates minor differences between ground-state and excited-state geometries. Based on time-dependent density functional theory, we assign the emitting state to S2 as its geometry closely resembles that of S0, whereas the S1 geometry differs from that of S0. As a result, simulated spectra involving S1 exhibit long Franck-Condon progressions, absent in the experimental spectra. The latter displays well-resolved vibrational features, assigned to transitions involving in-plane vibrational modes where the vibrational quantum number changes by one. Dominant transitions are associated with vibrations localized on the NH2 moieties. Experiments repeated at room temperature yield broader spectra with maxima at 435.5 ± 1nm (absorption) and 438.0 ± 1nm (fluorescence). We again conclude that prevalent fluorescence arises from S2, i.e., anti-Kasha behavior, in agreement with previous work. Excited-state lifetimes are 5 ± 1ns, independent of temperature. Importantly, we exclude the possibility that a narrow DIB at 436.4nm originates from cold proflavine cations as the band is redshifted compared to our absorption spectra.

Weak-valued correlation functions: Insights and precise readout strategies

The correlation function in quantum systems plays a vital role in decoding their properties and gaining insights into physical phenomena. Its interpretation corresponds to the propagation of particle excitations between space-time, similar in spirit to the idea of quantum weak measurement in terms of recording the system information by interaction. By defining weak-valued correlation function, we propose the basic insights and the universal methods for recording them on the apparatus through weak measurement. To demonstrate the feasibility of our approach, we perform numerical experiments of perturbed quantum harmonic oscillators, addressing the intricate interplay between the coupling strength and the number of ensemble copies. Additionally, we extend our protocol to the domain of quantum field theory, where joint weak values encode crucial information about the correlation function. Hopefully, this comprehensive investigation can advance our understanding of the fundamental nature of the correlation function and weak measurement in quantum theories. Published by the American Physical Society 2024

ℏ 4 quantum corrections to semiclassical transmission probabilities

The combination of vibrational perturbation theory with the replacement of the harmonic oscillator quantization condition along the reaction coordinate with an imaginary action to be used in the uniform semiclassical approximation for the transmission probability has been shown in recent years to be a practical method for obtaining thermal reaction rates. To date, this theory has been developed systematically only up to second order in perturbation theory. Although it gives the correct leading order term in an ℏ2 expansion, its accuracy at lower temperatures, where tunneling becomes important, is not clear. In this paper, we develop the theory to fourth order in the action. This demands developing the quantum perturbation theory up to sixth order. Remarkably, we find that the fourth order theory gives the correct ℏ4 term in the expansion of the exact thermal rate. The relative magnitude of the fourth order correction as compared to the second order term objectively indicates the accuracy of the second order theory. We also extend the previous modified second order theory to the fourth order case, creating an ℏ2 modified potential for this purpose. The resulting theory is tested on the standard examples—symmetric and asymmetric Eckart potentials and a Gaussian potential. The modified fourth order theory is remarkably accurate for the asymmetric Eckart potential.

The spectra masses for heavy pentaquark using generalized fractional of the extended Nikiforov-Uvaro method

The Nikiforov-Uvarov method is an efficient technique for solving of heavy diquark systems. It has been used to derive analytic-exact energy eigenvalues and eigenfunctions in fractional forms, which are useful in describing such systems. The potentials employed include the Cornell potential, harmonic potential, and spin-spin interaction; have been updated with respect to previous studies. Mass spectra of heavy pentaquarks were also calculated using this method. Compared to previous studies, the present results exhibit good experimental data agreement and are improved. We deduce that the fractional models contribute greatly to the heavy pentaquark masses.

Duality between the quantum inverted harmonic oscillator and inverse square potentials

In this paper we show how the quantum mechanics of the inverted harmonic oscillator (IHO) can be mapped to the quantum mechanics of a particle in a super-critical inverse square potential (ISP). We demonstrate this by relating both of these systems to the Berry–Keating system with Hamiltonian H=(xp+px)/2 . It has long been appreciated that the quantum mechanics of the ISP has an ambiguity in choosing a boundary condition near the origin and we show how this ambiguity is mapped to the IHO system. Imposing a boundary condition requires specifying a distance scale where it is applied and changes to this scale come with a renormalisation group (RG) evolution of the boundary condition that ensures observables do not directly depend on the scale (which is arbitrary). Physical scales instead emerge as RG invariants of this evolution. The RG flow for the ISP is known to follow limit cycles describing the discrete breaking of classical scale invariance in a simple example of a quantum anomaly, and we find that limit cycles also occur for the IHO. However, unlike the ISP where the continuous scaling symmetry is explicit, in the case of the IHO it is hidden and occurs because the Hamiltonian is part of a larger su(1,1) spectrum generating algebra. Our map does not require the boundary condition to be self-adjoint, as can be appropriate for systems that involve the absorption or emission of particles.

Superposition and interference states in q-deformed quantum oscillator

Superposition and interference states in q-deformed quantum oscillator

Effect of retarded friction and added mass on the swimming speed of a vibrating two-sphere

A theoretical expression is derived for the mechanical contribution to the mean swimming speed of a vibrating two-sphere, consisting of two spheres connected by an elastic spring, and immersed in a viscous incompressible fluid. The spring provides a harmonic potential for oscillations about a mean distance between centers. The system is made to oscillate at a chosen frequency by activating forces which sum to zero. The mechanical contribution to the resulting mean swimming velocity is calculated from the mechanical equations of motion and the corresponding impedance matrix of linear response. The frequency-dependent pair friction coefficients are found from approximate expressions derived earlier. The mechanical contribution is calculated to second order in the amplitude of stroke as a function of the scaling number, a dimensionless combination of size, frequency, and kinematic viscosity. Retarded friction and added mass determine the functional behavior.

Energy-Conversion Device Using a Quantum Engine with the Work Medium of Two-Atom Entanglement.

Although entanglement is considered as an essential resource for quantum information processing, whether entanglement helps for energy conversion or output in the quantum regime is still lack of experimental witness. Here, we report on an energy-conversion device operating as a quantum engine with the working medium acted by two entangled ions confined in a harmonic potential. The two ions are entangled by virtually coupling to one of the vibrational modes shared by the two ions, and the quantum engine couples to a quantum load, which is another shared vibrational mode. We explore the energy conversion efficiency of the quantum engine and investigate the useful energy (i.e., the maximum extractable work) stored in the quantum load by tuning the two ions in different degrees of entanglement as well as detecting the change of the phonons in the load. Our observation provides, for the first time, quantitative evidence that entanglement fuels the useful energy produced by the quantum engine, but not helpful for the energy conversion efficiency. We consider that our results may be useful to the study of quantum batteries for which one of the most indexes is the maximum extractable energy.

Energetics of the dissipative quantum oscillator

In this paper, we discuss some aspects of the energetics of a quantum Brownian particle placed in a harmonic trap, also known as the dissipative quantum oscillator. Based on the fluctuation–dissipation theorem, we analyze two distinct notions of thermally-averaged energy that can be ascribed to the oscillator. These energy functions, respectively dubbed hereafter as the mean energy and the internal energy, are found to be inequivalent for arbitrary system–bath coupling strength when the dissipation kernel has a non-zero memory timescale, as in the case of a Drude bath. Remarkably, both the energy functions satisfy the quantum counterpart of energy equipartition theorem, but with different probability distributions on the frequency domain of the heat bath. Moreover, the Gibbs approach to thermodynamics provides us with yet another thermally-averaged energy function. In the weak-coupling limit, all the above-mentioned energy expressions reduce to ϵ=ħω02coth(ħω02kBT), which is the familiar result. We generalize our analysis to the case of the three-dimensional dissipative charged oscillator placed in a spatially-uniform magnetic field.

Dynamics of Cosmological Scalar Fields Revisited

This paper reviews the dynamics of a single isotropic and homogeneous scalar field φ(t) in the context of cosmological models. A non-standard approach to the solution of the Einstein–Klein–Gordon equations is described which uses the scalar field as the evolution parameter for cosmic dynamics. General conclusions about the qualitative behaviour of the solutions can be drawn, and examples of how to obtain explicit solutions for some cosmological models of interest are given. For arbitrary potentials, analytical results can be obtained from the slow-roll approximation by using a series expansion for the Hubble parameter H[φ], from which a quantitative estimate for the number of e-folds of expansion is obtained. This approach is illustrated with the examples of quadratic potentials and hilltop models, with special consideration of Higgs-type potentials. The GUT-scale is shown to come out of such a model quite naturally. Finally, it is discussed how to find scalar potentials giving rise to a predetermined scalar-field behaviour and the associated evolution of the scale factor.