Two-dimensional Harmonic Oscillator Research Articles

Induced magnetization in anisotropic environment

The non-Markovian dynamics of a two-dimensional charged harmonic oscillator linearly coupled to a neutral bosonic heat bath is investigated in a linear-polarized electric field. The analytical expressions for the time-dependent and asymptotic magnetic moment are derived for the Markovian and non-Markovian dynamics. It is predicted that the liner-polarized electric field generates strong orbital currents and magnetization in a symmetric harmonic oscillator embedded into anisotropic heat bath. The role of the mixed dissipative kernel in magnetization is analyzed.

Observation of Boyer-Wolf Gaussian modes

Stable laser resonators support three fundamental families of transverse modes: the Hermite, Laguerre, and Ince Gaussian modes. These modes are crucial for understanding complex resonators, beam propagation, and structured light. We experimentally observe a new family of fundamental laser modes in stable resonators: Boyer-Wolf Gaussian modes. By studying the isomorphism between laser cavities and quadratic Hamiltonians, we design a laser resonator equivalent to a quantum two-dimensional anisotropic harmonic oscillator with a 2:1 frequency ratio. The generated Boyer-Wolf Gaussian modes exhibit a parabolic structure and show remarkable agreement with our theoretical predictions. These modes are also eigenmodes of a 2:1 anisotropic gradient refractive index medium, suggesting their presence in any physical system with a 2:1 anisotropic quadratic potential. We identify a transition connecting Boyer-Wolf Gaussian modes to Weber nondiffractive parabolic beams. These new modes are foundational for structured light, and open exciting possibilities for applications in laser micromachining, particle micromanipulation, and optical communications.

A two-dimensional harmonic oscillator confined in a circle in the presence of a constant electric field: an informational approach

In this work, we study an electron subjected to a harmonic oscillator potential confined in a circle of radius r0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0$$\\end{document} and in the presence of a constant electric field. We obtain energies and eigenfunctions for three different confinement radii as a function of the electric field strength. We have used the linear variational method by constructing the trial function as a linear combination of two-dimensional confined harmonic oscillator wave functions. We calculate the radial standard deviation as a measure of the dispersion of the probability density. We also computed the Shannon entropy and Fisher information, in configuration and momentum spaces, as localization-delocalization measures for three different confinement radii and as a function of the electric field strength. We find that Shannon entropy and Fisher information are more reliable than variance in determining electron location. The behaviour of Shannon entropy and Fisher information curves is shown to depend on each specific state under study.Graphical abstract

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.

Two-dimensional open quantum system in dissipative bosonic heat bath, external magnetic field, and two time-dependent electric fields.

Non-Markovian dynamics of a charged particle in a two-dimensional harmonic oscillator linearly coupled to a neutral bosonic heat bath is investigated in an external uniform magnetic field and two perpendicular time-dependent electric fields. The analytical expressions for the time-dependent and asymptotic angular momentum are derived for the Markovian and non-Markovian dynamics. The dependence of the angular momentum on the frequency of the electric field, cyclotron frequency, collective frequency, and anisotropy of the heat bath is studied. The angular momentum (or magnetization) of a charged particle can be ruled by varying the frequency of the electric field.

Experimental realization of entangled coherent states in two-dimensional harmonic oscillators of a trapped ion

Entangled coherent states play pivotal roles in various fields such as quantum computation, quantum communication, and quantum sensing. We experimentally demonstrate the generation of entangled coherent states with the two-dimensional motion of a trapped ion system. Using Raman transitions with appropriate detunings, we simultaneously drive the red and blue sidebands of the two transverse axes of a single trapped ion and observe multi-periodic entanglement and disentanglement of its spin and two-dimensional motion. Then, by measuring the spin state, we herald entangled coherent states of the transverse motions of the trapped ion and observe the corresponding modulation in the parity of the phonon distribution of one of the harmonic oscillators. Lastly, we trap two ions in a linear chain and realize Mølmer–Sørensen gate using two-dimensional motion.

Trajectories of two-dimensional harmonic oscillators in a rotating frame: application to Foucault pendulum problem

Trajectories of two-dimensional harmonic oscillators in a rotating frame: application to Foucault pendulum problem

Non-Hermitian two-dimensional harmonic oscillator in noncommutative phase-space

In this paper, we extend the result of [A. Fring et al. J. Phys. A 43 (2010) 345–401] in Noncommutative phase-space (NCPS). We compute the non-Hermitian Hamiltonian of a harmonic oscillator in NCPS. We construct a new [Formula: see text]-symmetry in NCPS and prove that the system does not possess a broken [Formula: see text]-regime. We then compute the eigenvalue spectrum of the non-Hermitian Hamiltonian of the system.

Landau levels for a Weyl pair in a monolayer medium and thermal quantities

In this paper, we consider a Weyl pair under the effect of an external uniform magnetic field in a monolayer medium without considering any charge-charge interaction between the particles. Choosing the interaction of the particles with the magnetic field in the symmetric gauge we seek for an analytical solution of the corresponding form of a one-time fully-covariant two-body Dirac equation derived from quantum electrodynamics via the action principle. As it is usual with two-body problems, we separate the relative motion and center of mass motion coordinates. Assuming the center of mass is at rest, we derive a matrix equation in terms of the relative motion coordinates without considering any group theoretical method. This equation gives a wave equation in exactly soluble form and accordingly we obtain the spinor components and complete energy eigen-states (in closed form) for such a spinless composite structure. Our results not only give exact Landau levels for such a Weyl pair in a monolayer medium but also show the considered system behaves as a two-dimensional harmonic oscillator. Furthermore, our findings give exactly the excited states of a Weyl particle under the effect of uniform external magnetic field in a monolayer graphene sheet and there is no imprint to distinguish these modes from each other. This means that the performed experiments based on Landau levels for a monolayer graphene sheet may actually involve many-body effects. Our results provide a suitable basis to analyze the associated thermal quantities and accordingly we discuss the thermal properties by determining free energy, total energy, entropy and specific heat for the composite system in question.

Analog of Foucault precession in two-dimensional quantum harmonic oscillators with Berry curvature

Analog of Foucault precession in two-dimensional quantum harmonic oscillators with Berry curvature

Elliptic billiard with harmonic potential: Classical description.

The classical dynamics of the isotropic two-dimensional harmonic oscillator confined by an elliptic hard wall is discussed. The interplay between the harmonic potential with circular symmetry and the boundary with elliptical symmetry does not spoil the separability in elliptic coordinates; however, it generates nontrivial energy and momentum dependencies in the billiard. We analyze the equimomentum surfaces in the parameter space and classify the kinds of motion the particle can have in the billiard. The winding numbers and periods of the rotational and librational trajectories are analytically calculated and numerically verified. A remarkable finding is the possibility of having degenerate rotational trajectories with the same energy but different second constant of motion and different caustics and periods. The conditions to get these degenerate trajectories are analyzed. Similarly, we show that obtaining two different rotational trajectories with the same period and second constant of motion but different energy is possible.

Quantum Harmonic Oscillators in One and Two Dimensions

The Schrödinger equation is a significant achievement on the development of quantum mechanics. By solving the Schrödinger equation, the fundamental behaviors and properties of a microscopic particle can be found in one- to three-dimensions. The article focuses on the derivations of quantum harmonic oscillators in one dimension by solving second order-differential equation. The wave functions, probability densities under different energy levels are presented. The results can be used to estimate different forms of continuous potential experienced by an oscillator. By calculating the uncertainty relation of the oscillator under one specific excited state, the general relation can be confirmed. Based on the output, by splitting the variables, the two-dimensional harmonic oscillator can also be derived. The degenerate energy levels are presented as . Numerical simulations are made to visualize those results and suggests the relation of energy levels and number of maxima of the probability densities.

Chaotic and integrable magnetic fields in one-dimensional hybrid Vlasov–Maxwell equilibria

The construction of kinetic equilibrium states is important for studying stability and wave propagation in collisionless plasmas. Thus, many studies over the past decades have been focused on calculating Vlasov–Maxwell equilibria using analytical and numerical methods. However, the problem of kinetic equilibrium of hybrid models is less studied, and self-consistent treatments often adopt restrictive assumptions ruling out cases with irregular and chaotic behaviour, although such behaviour is observed in spacecraft observations of space plasmas. In this paper, we develop a one-dimensional (1-D), quasineutral, hybrid Vlasov–Maxwell equilibrium model with kinetic ions and massless fluid electrons and derive associated solutions. The model allows for an electrostatic potential that is expressed in terms of the vector potential components through the quasineutrality condition. The equilibrium states are calculated upon solving an inhomogeneous Beltrami equation that determines the magnetic field, where the inhomogeneous term is the current density of the kinetic ions and the homogeneous term represents the electron current density. We show that the corresponding 1-D system is Hamiltonian, with position playing the role of time, and its trajectories have a regular, periodic behaviour for ion distribution functions that are symmetric in the two conserved particle canonical momenta. For asymmetric distribution functions, the system is nonintegrable, resulting in irregular and chaotic behaviour of the fields. The electron current density can modify the magnetic field phase space structure, inducing orbit trapping and the organization of orbits into large islands of stability. Thus, the electron contribution can be responsible for the emergence of localized electric field structures that induce ion trapping. We also provide a paradigm for the analytical construction of hybrid equilibria using a rotating two-dimensional harmonic oscillator Hamiltonian, enabling the calculation of analytic magnetic fields and the construction of the corresponding distribution functions in terms of Hermite polynomials.

Radial wave function of 2D and 3D quantum harmonic oscillator

One dimensional quantum harmonic oscillator is well studied in elementary textbooks of quantum mechanics. The wave function of one-dimensional oscillator harmonic can be written in term of Hermite polynomial. Due to the symmetry of the spring energy, the wave functions of two-dimensional and three-dimensional harmonic oscillators can be written as products of the one-dimensional case. Because of that, the wave functions of two- and three-dimensional cases are focused on cartesian coordinates. In this article, we utilize polar and spherical coordinates to describe the wave function of two- and three-dimensional harmonic oscillators, respectively. The radial part of the wave functions can be written in term of associated Laguerre polynomials.

The B<sub>2</sub> Harmonic Oscillator with Reflections and Superintegrability

The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group B<sub>2</sub>, which is the symmetry group of the square. The angular momentum operator is also modified with reflections. The wavefunctions are known to be built up from Jacobi and Laguerre polynomials. This paper introduces a fourth-order differential-difference operator commuting with the Hamiltonian but not with the angular momentum operator; a specific instance of superintegrability. The action of the operator on the usual orthogonal basis of wavefunctions is explicitly described. The wavefunctions are classified according to the representations of the group: four of degree one and one of degree two. The identity representation encompasses the wavefunctions invariant under the group. The paper begins with a short discussion of the modified Hamiltonians associated to finite reflection groups, and related raising and lowering operators. In particular, the Hamiltonian for the symmetric groups describes the Calogero-Sutherland model of identical particles on the line with harmonic confinement.

On the two-dimensional time-dependent anisotropic harmonic oscillator in a magnetic field

A charged harmonic oscillator in a magnetic field, Landau problems, and an oscillator in a noncommutative space share the same mathematical structure in their Hamiltonians. We have considered a two-dimensional anisotropic harmonic oscillator with arbitrarily time-dependent parameters (effective mass and frequencies), placed in an arbitrarily time-dependent magnetic field. A class of quadratic invariant operators (in the sense of Lewis and Riesenfeld) have been constructed. The invariant operators (Î) have been reduced to a simplified representative form by a linear canonical transformation [the group Sp(4,R)]. An orthonormal basis of the Hilbert space consisting of the eigenvectors of Î is obtained. In order to obtain the solutions of the time-dependent Schrödinger equation corresponding to the system, both the geometric and dynamical phase-factors are constructed. A generalized Peres–Horodecki separability criterion (Simon’s criterion) for the ground state corresponding to our system has been demonstrated.

Exactly solvable model behind Bose-Hubbard dimers, Ince-Gauss beams, and aberrated optical cavities

The authors present a widely applicable, solvable model (an isotropic two-dimensional harmonic oscillator subject to two simple perturbations) and show that it underlies a broad range of physical phenomena displaying a topological transition, including aberrated optical cavities that support Ince-Gauss beams as their modes and the Bose-Hubbard dimer describing two coupled superfluids.

Chaos In 2D Bohmian Trajectories of Commensurate Harmonics Oscillators

Particle trajectories guided by the wave function are well-defined through Bohmian mechanics, which is a causal interpretation of quantum mechanics. Periodic and chaotic behaviours could be exhibited from the certain classical integrable systems that have been shown within this framework. In this study, we developed Mathematica programs to plot the Bohmian trajectories and Lyapunov exponents. These programs serve as computer experiments for numerical generation and illustration of the results. We show that the behaviours of commensurate two-dimensional harmonic oscillator systems are dependent on ratios of frequency.

Some Dynamic Aspects of a Sextic Galactic Potential in a Rotating Reference Frame

This work aims to explore some dynamic aspects of the problem of star motion that is impacted by the rotation of the galaxy, which we model as a bisymmetric potential based on a two-dimensional harmonic oscillator with sextic perturbations. We demonstrate analytically that the motion is non-integrable when certain conditions are met. The analytical results for the non-integrability are confirmed by showing the irregularity of the behavior of the motion through utilizing the Poincaré surface of a section as a numerical method. The motion equilibrium positions are detected, and their stability is discussed. We show that the force generated by the rotating frame acts as a stabilizer for the maximum equilibrium points. We display graphically that the size of the stability regions relies on the angular velocity magnitude for the frame. Through the application of Lyapunov’s theorem, periodic solutions can be constructed which are close to the equilibrium positions. Furthermore, we demonstrate that there are one or two families of periodic solutions relying on whether the equilibrium point is a saddle or stable, respectively.

Collapse-revival of entanglement in a non-commutative harmonic oscillator revealed via coherent states and path integral

Abstract We develop an approach using coherent states and path integral to investigate the dynamics of entanglement in a simple two-dimensional non-commutative harmonic oscillator. We start by employing a Bopp shift to convert the Hamiltonian describing the system into a commutative equivalent one. This allows us to construct coherent states and calculate the propagator in standard way. By deriving the explicit expression of the time-dependent coherent states and considering its connection with the number states, we provide exact results for evaluating the degree of entanglement between the ground state and any excited state through the purity function. The interesting emerging result is that, as long as the non-commutativity parameter is non-zero, our system exhibits the phenomenon of collapse and revival of entanglement.