Anisotropic Harmonic Oscillator Research Articles

Kohnʼs theorem and Galilean symmetry

The relation between the separability of a system of charged particles in a uniform magnetic field and Galilean symmetry is revisited using Duvalʼs “Bargmann framework”. If the charge-to-mass ratios of the particles are identical, ea/ma=ϵ for all particles, then the Bargmann space of the magnetic system is isometric to that of an anisotropic harmonic oscillator. Assuming that the particles interact through a potential which only depends on their relative distances, the system splits into one representing the center of mass plus a decoupled internal part, and can be mapped further into an isolated system using Niedererʼs transformation. Conversely, the manifest Galilean boost symmetry of the isolated system can be “imported” to the oscillator and to the magnetic systems, respectively, to yield the symmetry used by Gibbons and Pope to prove the separability. For vanishing interaction potential the isolated system is free and our procedure endows all our systems with a hidden Schrödinger symmetry, augmented with independent internal rotations. All these properties follow from the cohomological structure of the Galilei group, as explained by Souriauʼs “décomposition barycentrique”.

The time-dependent forced anisotropic oscillator in noncommutative phase space

Wave functions of the time-dependent forced anisotropic harmonic oscillator in noncommutative phase space are derived using the linear transformation and unitary transformation methods. The energy spectrum is given for the stationary system. Further, quantum fluctuations and the squeezing effect are investigated. It is found that the anisotropic property of the harmonic oscillator in noncommutative space has the squeezing effect.

BOSE–EINSTEIN CONDENSATES IN A HOMOGENEOUS GRAVITATIONAL FIELD

The behavior of a Bose–Einstein condensate in a homogeneous gravitational field is analyzed. We consider two different trapping potentials. Firstly, the gas is inside a finite container. The effects of the finiteness of the height of the container in connection with the presence of a homogeneous gravitational field are mathematically analyzed and the resulting energy eigenvalues are deduced and used to obtain the corresponding partition function and ensuing thermodynamical properties. Secondly, the trapping potential is an anisotropic harmonic oscillator and the effects of the gravitational field and of the zero-point energy on the condensation temperature are also considered. These results are employed in order to put forward an experiment which could test the so-called Einstein Equivalence Principle.

THE QUANTUM H4 INTEGRABLE SYSTEM

The quantum H4 integrable system is a 4D system with rational potential related to the non-crystallographic root system H4 with 600-cell symmetry. It is shown that the gauge-rotated H4 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H4, is in algebraic form: it has polynomial coefficients in front of the derivatives. Any eigenfunction is a polynomial multiplied by ground-state function (factorization property). Spectra correspond to one of the anisotropic harmonic oscillators. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector α = (1, 5, 8, 12).

Effective Theory of Electron-Hole Exchange in Semiconductor Quantum Dots

Semiconductor self-assembled quantum dots (QDs) are expected to become building blocks of novel optoelectronic devices capable of generating single photons and entangled photon pairs (EPPs) on demand. The major barrier to generation of EPPs is the so-called "bright exciton splitting" originating from the long-range electron-hole exchange interaction. A k · p effective mass theory of electron-hole exchange in semiconductor quantum dots is presented (Phys. Rev. B 81, 045311). The matrix element responsible for the "bright" exciton splitting in the effective exciton Hamiltonian is identified and analyzed. An excitonic fine structure for a model quantum dot with quasi-two-dimensional anisotropic harmonic oscillator (2DLAHO) confining potential is considered as a function of the shape and external fields.

Superintegrability and higher order polynomial algebras

We present a method to obtain higher order integrals and polynomial algebras for two-dimensional quantum superintegrable systems separable in Cartesian coordinates from ladder operators. All systems with a second- and a third-order integral of motion separable in Cartesian coordinates were studied. The integrals of motion of two of them do not generate a cubic algebra. We construct for these Hamiltonians a higher order polynomial algebra from their ladder operators. We obtain quintic and seventh-order polynomial algebras. We also give for the polynomial algebras of order 7 realizations in terms of deformed oscillator algebras. These realizations and finite-dimensional unitary representations allow us to obtain the energy spectrum. We also apply the construction to the caged anisotropic harmonic oscillator and a system involving the fourth Painlevé transcendent.

Theory of exciton fine structure in semiconductor quantum dots: Quantum dot anisotropy and lateral electric field

Theory of exciton fine structure in semiconductor quantum dots and its dependence on quantum-dot anisotropy and external lateral electric field is presented. The effective exciton Hamiltonian including long-range electron-hole exchange interaction is derived within the $k\ensuremath{\cdot}p$ effective-mass approximation. The exchange matrix elements of the Hamiltonian are expressed explicitly in terms of electron and hole envelope functions. The matrix element responsible for the ``bright'' exciton splitting is identified and analyzed. An excitonic fine structure for a model quantum dot with quasi-two-dimensional anisotropic harmonic oscillator confining potential is analyzed as a function of the shape anisotropy, size, and applied lateral electric field.

ATOMIC COHERENT STATES AS THE EIGENSTATES OF A TWO-DIMENSIONAL ANISOTROPIC HARMONIC OSCILLATOR IN A UNIFORM MAGNETIC FIELD

In the newly constructed entangled state representation embodying quantum entanglement of Einstein, Podolsky and Rosen, the usual wave function of atomic coherent state ∣τ〉 = exp (μJ+-μ*J-)∣j, -j〉 turns out to be just proportional to a single-variable ordinary Hermite polynomial of order 2j, where j is the spin value. We then prove that a two-dimensional time-independent anisotropic harmonic oscillator in a uniform magnetic field possesses energy eigenstates which can be classified as the states ∣τ〉 in terms of the spin values j.

SU(3) symmetry in the triaxially deformed harmonic oscillator

An anisotropic harmonic oscillator Hamiltonian can be brought into invariant form under $\mathrm{SU}(3)$ transformations by applying nonlinear transformations to the oscillator bosons. The classification of the single-particle levels based on this covering group predicts magic numbers for the triaxial oscillator. It is shown that when the deformation $|\ensuremath{\delta}|$ is not too large, the physical operators are approximated by the group operators. Estimation is carried out for the alignment of orbital angular momentum in a triaxial field.

Reduction of superintegrable systems: The anisotropic harmonic oscillator

We introduce a 2N-parametric family of maximally superintegrable systems in N dimensions, obtained as a reduction of an anisotropic harmonic oscillator in a 2N-dimensional configuration space. These systems possess closed bounded orbits and integrals of motion which are polynomial in the momenta. They generalize known examples of superintegrable models in the Euclidean plane.

Superintegrability of the caged anisotropic oscillator

We study the caged anisotropic harmonic oscillator, which is a new example of a superintegrable or accidentally degenerate Hamiltonian. The potential is that of the harmonic oscillator with rational frequency ratio (l:m:n) but additionally with barrier terms describing repulsive forces from the principal planes. This confines the classical motion to a sector bounded by the principal planes, or a cage. In three degrees of freedom, there are five isolating integrals of motion, ensuring that all bound trajectories are closed and strictly periodic. Three of the integrals are quadratic in the momenta, the remaining two are polynomials of order 2(l+m−1) and 2(l+n−1). In the quantum problem, the eigenstates are multiply degenerate, exhibiting l2m2n2 copies of the fundamental pattern of the symmetry group SU(3).

Microscopic, quantum derivation of cranking model for nuclear collective rotation: harmonic oscillator case

In this article, the conventional semiclassical one-dimensional cranking model (CR), which is commonly used to investigate rotational structures of deformed nuclei, is derived from microscopic, quantum first principles for the harmonic oscillator case. The space-fixed particle coordinates are canonically transformed to an Euler angle and a set of 3N – 1 intrinsic coordinates to decompose the nuclear Hamiltonian into intrinsic and collective rotational components plus a Coriolis-centrifugal term that couples the intrinsic and rotational motions. To overcome the difficulties associated with finding an appropriate set of intrinsic coordinates, the rotational component in the transformed Hamiltonian is expressed in terms of the space-fixed coordinates and momenta by taking the commutator of the original Hamiltonian with the Euler angle, and by choosing an explicit expression for the Euler angle in terms of space-fixed particle coordinates. The intrinsic component in the transformed Hamiltonian is then the difference between the original Hamiltonian and the rotational component. The nuclear wave function is chosen as the product of an intrinsic function and an eigenfunction of the angular momentum operator (as in the unified rotational model). The Hamiltonian and Schrodinger equation for the intrinsic system become functions of the angular-momentum quantum number and intrinsic operators that are expressed in terms of space-fixed particles coordinates and momenta. The intrinsic Schrodinger equation is then reduced to that of a one-body operator using Hartree–Fock mean-field approximation. The intrinsic mean-field Hamiltonian is chosen to be an anisotropic harmonic oscillator Hamiltonian, and the Hartree–Fock mean-field equation is unitarily transformed to an equation resembling that of the CR but with oscillator frequencies and angular velocity that are microscopically and quantum mechanically determined. The unitary transformation is selected such that the model predicts the kinematic rigid-body moment of inertia, as does the CR when self-consistency condition is used.PACS Nos.: 21.60.Ev, 21.60.Fw, 21.60.Jz

Commensurate anisotropic oscillator, SU(2) coherent states and the classical limit

We demonstrate a formally exact quantum–classical correspondence between the stationary coherent states associated with the commensurate anisotropic two-dimensional harmonic oscillator and the classical Lissajous orbits. Our derivation draws upon the earlier work of Louck et al (1973 J. Math. Phys. 14 692) wherein they have provided a non-bijective canonical transformation that maps, within a degenerate eigenspace, the commensurate anisotropic oscillator on to the isotropic oscillator. This mapping leads, in a natural manner, to a Schwinger realization of SU(2) in terms of the canonically transformed creation and annihilation operators. Through the corresponding coherent states built over a degenerate eigenspace, we directly effect the classical limit via the expectation values of the underlying generators. Our work completely accounts for the fact that the SU(2) coherent state in general corresponds to an ensemble of Lissajous orbits.

Two ultracold atoms in a completely anisotropic trap

As a limiting case of ultracold atoms trapped in deep optical lattices, we consider two interacting atoms trapped in a general anisotropic harmonic oscillator potential, and obtain exact solutions of the Schrödinger equation for this system. The energy spectra for different geometries of the trapping potential are compared.

Anisotropic harmonic oscillator, non-commutative Landau problem and exotic Newton–Hooke symmetry

We investigate the planar anisotropic harmonic oscillator with explicit rotational symmetry as a particle model with non-commutative coordinates. It includes the exotic Newton–Hooke particle and the non-commutative Landau problem as special, isotropic and maximally anisotropic, cases. The system is described by the same (2+1)-dimensional exotic Newton–Hooke symmetry as in the isotropic case, and develops three different phases depending on the values of the two central charges. The special cases of the exotic Newton–Hooke particle and non-commutative Landau problem are shown to be characterized by additional, so(3) or so(2,1) Lie symmetry, which reflects their peculiar spectral properties.

Noetherian first integrals

From time to time one finds claims in the literature that first integrals/invariants of Lagrangian systems are nonnoetherian. Such claims diminish the contribution of Noether in the topic of integrability. We provide an explicit demonstration of noetherian symmetries associated with the integrals which have been termed nonnoetherian. To further emphasise our point we construct the noetherian first integrals/invariants, which are associated with symmetries linear in the velocities, for the two-dimensional autonomous isotropic harmonic oscillator and the autonomous anisotropic oscillator and illustrate the roles which the invariants can play in the description of the classical motion. We relate these symmetries to the corresponding problem in quantum mechanics. Further we show that the complete symmetry group of this anisotropic harmonic oscillator has the same representation as that of the corresponding isotropic oscillator. As a concluding example we show that a symmetry claimed to be nonnoetherian is trivially Noetherian.

Single particle calculations for a Woods–Saxon potential with triaxial deformations, and large Cartesian oscillator basis (new version code)

We present a new version of the computer program which solves the Schrödinger equation of the stationary states for an average nuclear potential of Woods–Saxon type. In this work, we take specifically into account triaxial (i.e. ellipsoidal) nuclear surfaces. The deformation is specified by the usual Bohr parameters. The calculations are carried out in two stages. In the first, one calculates the representative matrix of the Hamiltonian in the Cartesian oscillator basis. In the second stage one diagonalizes this matrix with the help of subroutines of the EISPACK library. This new version calculates all the eigenvalues up to a given cutoff energy, and gives the components of the corresponding eigenfunctions. For a more convenient handling, these results are stored simultaneously in the computer memory, and on a files. Program summary Title of program:Triaxial2007 Catalogue identifier:ADSK_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADSK_v2_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Summary of revision:One input file instead two. Reduced number of input parameters. Storage of eigenvalues and eigenvectors in memory in a very simple way which makes the code very convenient to the user. Reasons for the new version: More convenient handling of the eigenvectors Catalogue number old version: ADSK Catalogue number new version:ADSK_v2_0 Journal: Computer Physics Commun. 156 (2004) 241–282 Licensing provisions: none Computer: PC Pentium 4, 2600 MHz Hard disk: 40 Gb RAM: 256 Mb Swap file: 4 Gb Operating system: WINDOWS XP Software used: Compaq Visual FORTRAN (with full optimizations in the settings project options) Programming language used:Fortran 77/90 (double precision) Number of bits in a word: 32 No. of lines in distributed program, including test data, etc.:4058 No. of bytes in distributed program, including test data, etc.:75 590 Distribution format:tar.gz Nature of the problem: The single particle energies and the single particle wave functions are calculated from one-body Hamiltonian including a central field of Woods–Saxon type, a spin–orbit interaction, and the Coulomb potential for the protons. We consider only ellipsoidal (triaxial) shapes. The deformation of the nuclear shape is fixed by the usual Bohr parameters ( β , γ ) . Method of solution: The representative matrix of the Hamiltonian is built by means of the Cartesian basis of the anisotropic harmonic oscillator, and then diagonalized by a set of subroutines of the EISPACK library. Two quadrature methods of Gauss are employed to calculate, respectively, the integrals of the matrix elements of the Hamiltonian, and the integral defining the Coulomb potential. Restrictions: There are two restrictions for the code: The number of the major shells of the basis does not have to exceed N max = 26 . For the largest values of N max (∼23–26), the diagonalization takes the major part of the running time, but the global run-time remains reasonable. Typical running time: (With full optimization in the project settings of the Compaq Visual Fortran on Windows XP) With N max = 23 , for the neutrons case, and for both parities, the running time is about 40 sec on the P4 computer at 2.6 GHz. In this case, the calculation of the matrix elements takes only about 17 sec. If all unbound states are required, the runtime becomes larger.

Confined quantum systems: spectra of weakly bound electrons in a strongly anisotropicoblate harmonic oscillator potential

The spectrum and the electron density distribution of two, three, and four electrons confinedby a strongly anisotropic oblate-type harmonic oscillator potential with small values ofω have been studied for all spin states by using the quantum chemicalconfiguration interaction (CI) method employing large Cartesian anisotropicGaussian basis sets. The convergence of the calculated energy spectrumwith respect to different-size basis sets shows that an accuracy better than2 × 10−6 has been obtained for the three-electron harmonic oscillator quantum dot by using a basisset including up to a k-type function, but that in order to achieve similar accuracy for thefour-electron quantum dot the basis set has to be supplemented with functions as high asan m-type function. The analysis of the two leading configurations in the CI wavefunctionsshows that large electron correlation prevails for the low lying states of the systems studied,particularly for the states with low spin multiplicity. Two types of electronic mode areidentified, namely, the circular mode and the breathing mode. It is shown thatthe states having an excitation into the circular mode are energetically morefavourable than those into the breathing mode when the strength of confinementω is small.

Does the Phenomenological Approach Contradict the Quantum Theory of Exciton-Polariton Spatial Dispersion?

This paper clarifies the controversial issue over 40 years between the quantum approach by Pekar and the phenomenological approach by Ginzburg about the exciton-polariton spatial dispersion theory. For an isotropic nongyrotropic medium, the analytical explicit function of the impermeability tensor \(\eta _{ij} (\omega ,\mathop k\limits^ \to )\) (the inverse of the permittivity tensor) is obtained from the anisotropic undamped-wave harmonic oscillator model. After expanding \(\eta _{ij} (\omega ,\mathop k\limits^ \to )\) with respect to small parameters within it rather than to wavevector \(\mathop k\limits^ \to\), the approximate refractive indices can be determined from the eigenvalue equation. By this treatment, the phenomenological approach is proved to be the approximation of the quantum approach near the resonance frequency. The condition for the approximation is discussed.

A Connection Between Geometry and Dynamical Systems

This paper deals with a geometric and systematic approach to the integration of a nonlinear dynamical system: the anisotropic harmonic oscillator in a radial quartic potential. We study this system from a different angle: a) We show, using a Lax-type representation of the Hamilton’s equations of motion, that the system is linearized in the jacobian variety of a smooth genus 2 hyperelliptic curve. b) We find via Kowalewski-Painlevé analysis the principal balances of the hamiltonian vector field defined by the hamiltonian and we show that the system is algebraic complete integrable. c) We also describe an explicit embedding of the abelian variety which completes the generic invariant surface, into projective space. d) We give a direct proof that the abelian variety obtained in this paper is dual to Prym variety and can also be seen as a double unramified cover of the jacobian variety of an hyperelliptic curve of genus 2. e) We show that at some special values of the parameters λ1 and λ2, we can describe elliptic solutions which are associated with two-gap elliptic solitons of the Korteweg-de Vries equation.