Anisotropic Harmonic Oscillator Research Articles

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.

Describing geophysical turbulence with a Schrödinger–Coriolis equation in velocity space

In this paper, we examine the predictions of the scale-relativity approach for a turbulent fluid in rotation. We first show that the time derivative of the governing Navier–Stokes equation in the usual x-space can be transformed into a Schrödinger-like equation in velocity space with an external vectorial field to account for the rotation, together with a local velocity harmonic oscillator (VHO) potential in the v-space. The coefficients of this VHO are given by the second order x-derivatives of the pressure. We can then give formulas for the velocity and acceleration probability distribution functions (PDF). Using a simple model of anisotropic harmonic oscillator, we compare our predictions with relevant data from both direct numerical simulations (DNS) and oceanic drifter velocity measurements. We find a good agreement of the predicted acceleration PDF with that observed from drifters and some possible support in DNS for the existence of gaps in the local velocity PDF, expected in the presence of a Coriolis force.

Three-dimensional approach applied to quasistationary states of deformed α emitters

We apply a three-dimensional (3D) approach to investigate the quasistationary states of well-deformed $\ensuremath{\alpha}$ emitters. With a splitting of the anisotropic 3D potential into internal and external parts at a separation surface, the 3D $\ensuremath{\alpha}$-cluster decay width is determined by the initial wave function of a true bound state of an anisotropic harmonic oscillator potential and a nonresonance scattering wave function of Coulomb potential. Substantial difference between the one-dimensional (1D) and 3D decay width is found for typical $\ensuremath{\alpha}$ emitters with large quadrupole and hexadecapole deformations.

Poisson Bracket Formulation for a Dissipative Two- Dimensional Anisotropic Harmonic Oscillator with Fractional Derivatives: Analysis and Applications

Poisson Bracket Formulation for a Dissipative Two- Dimensional Anisotropic Harmonic Oscillator with Fractional Derivatives: Analysis and Applications

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.

On higher-dimensional superintegrable systems: a new family of classical and quantum Hamiltonian models

We introduce a family of n-dimensional Hamiltonian systems which, contain, as special reductions, several superintegrable systems as the Tremblay–Turbiner–Winternitz system, a generalized Kepler potential and the anisotropic harmonic oscillator with Rosochatius terms. We conjecture that there exist special values in the space of parameters, apart from those leading to known cases, for which this new Hamiltonian family is superintegrable.

Entanglement in phase-space distribution for an anisotropic harmonic oscillator in noncommutative space

The bipartite Gaussian state, corresponding to an anisotropic harmonic oscillator in a noncommutative space (NCS), is investigated with the help of Simon’s separability condition (generalized Peres-Horodecki criterion). It turns out that, to exhibit the entanglement between the noncommutative co-ordinates, the parameters (mass and frequency) have to satisfy a unique constraint equation. We have considered the most general form of an anisotropic oscillator in NCS, with both spatial and momentum noncommutativity. The system is transformed to the usual commutative space (with usual Heisenberg algebra) by a well-known Bopp shift. The system is transformed into an equivalent simple system by a unitary transformation, keeping the intrinsic symplectic structure (\(Sp(4,{\mathbb {R}})\)) intact. Wigner quasiprobability distribution is constructed for the bipartite Gaussian state with the help of a Fourier transformation of the characteristic function. It is shown that the identification of the entangled degrees of freedom is possible by studying the Wigner quasiprobability distribution in phase space. We have shown that the co-ordinates are entangled only with the conjugate momentum corresponding to other co-ordinates.

Non-linear ladder operators and coherent states for the 2:1 oscillator

The 2:1 two-dimensional anisotropic quantum harmonic oscillator is considered and new sets of states are defined by means of normal-ordering non-linear operators through the use of non-commutative binomial theorems as well as solving recurrence relations. The states generated are good candidates for the natural generalisation of the coherent states of the two-dimensional isotropic oscillator. The two-dimensional non-linear generalised ladder operators lead to several chains of states which are connected in a non trivial way. The uncertainty relations of the defining chain of states are calculated and it is found that they admit a resolution of the identity and the spatial distribution of the wavefunction produces Lissajous figures in correspondence with the classical 2:1 oscillator.

Conformal generation of an exotic rotationally invariant harmonic oscillator

An exotic rotationally invariant harmonic oscillator (ERIHO) is constructed by applying a non-unitary isotropic conformal bridge transformation (CBT) to a free planar particle. It is described by the isotropic harmonic oscillator Hamiltonian supplemented by a Zeeman type term with a real coupling constant $g$. The model reveals the Euclidean ($|g|<1$) and Minkowskian ($|g|>1$) phases separated by the phases $g=+1$ and $g=-1$ of the Landau problem in the symmetric gauge with opposite orientation of the magnetic field. A hidden symmetry emerges in the system at rational values of $g$. Its generators, together with the Hamiltonian and angular momentum produce non-linearly deformed $\mathfrak{u}(2)$ and $\mathfrak{gl}(2,{\mathbb R})$ algebras in the cases of $0<|g|<1$ and $\infty>|g|>1$, which transmute one into another under the inversion $g\rightarrow -1/g$. Similarly, the true, $\mathfrak{u}(2)$, and extended conformal, $\mathfrak{gl}(2,{\mathbb R})$, symmetries of the isotropic Euclidean oscillator ($g=0$) interchange their roles in the isotropic Minkowskian oscillator ($|g|=\infty$), while two copies of the $\mathfrak{gl}(2,{\mathbb R})$ algebra of analogous symmetries mutually transmute in Landau phases. We show that the ERIHO system is transformed by a peculiar unitary transformation into the anisotropic harmonic oscillator generated, in turn, by anisotropic CBT. The relationship between the ERIHO and the subcritical phases of the harmonically extended Landau problem, as well as with a plane isotropic harmonic oscillator in a uniformly rotating reference frame, is established.

Entanglement induced by noncommutativity: anisotropic harmonic oscillator in noncommutative space

Quantum entanglement, induced by spatial noncommutativity, is investigated for an anisotropic harmonic oscillator. Exact solutions for the system are obtained after the model is re-expressed in terms of canonical variables, by performing a particular Bopp’s shift to the noncommutating degrees of freedom. Employing Simon’s separability criterion, we find that the states of the system are entangled provided a unique function of the (mass and frequency) parameters obeys an inequality. Entanglement of Formation for this system is also computed and its relation to the degree of anisotropy is discussed. It is worth mentioning that, even in a noncommutative space, entanglement is generated only if the harmonic oscillator is anisotropic. Interestingly, the Entanglement of Formation saturates for higher values of the deformation parameter \(\theta \), that quantifies spatial noncommutativity.

Anharmonicity in light nuclei near drip lines

Single-particle energy levels of nucleons moving in an anisotropic harmonic oscillator potential along with a spin-orbit and an orbit-orbit interaction have been calculated. The model proposed herein incorporates shapes generated by higher multipolar anisotropies other than the quadruple one. The gaps among the energy levels are strongly dependent on the degree of anisotropy and hence, affect the nucleonic separation energies. For large anisotropies, this characteristic disturbs the occurrences of the usual magic numbers and the sequential ordering of the shells characterized by principal quantum number, N, associated with the elementary shell model. For large asymmetries, the states characterized by higher N, may come down and lie below the energies of the states with lower N and vice versa. This feature of the model provides a natural explanation of intruder states or the island of inversion observed experimentally. For certain choices of the parameters, the large gaps in single nucleonic separation energies could occur at neutron numbers, 10,18,30, and 32 as observed, respectively, in 20Ne, 28Ne, 52Ti, and 54Ti. For neutron as well as proton number 14, the magnitude of this energy gap depends strongly on the choice of the strength of the spin-orbit and orbit-orbit terms. The model provides a simple understanding of the observed fact that some nuclei with proton or neutron number 14 sometimes exhibit this gap but not others. It also provides an interesting insight of having a low-laying (1/2 − ) state in 7He, and confirms the ground state spin and parity of 11Be to be (1/2+) and those of 9He to be (1/2+), both of which have been contentious. This investigation indicated the density distribution, i.e., the shapes of many of nuclei near drip line to be anisotropic having a symmetry about their body-fixed z-axis.

Symmetry algebra of the planar anisotropic quantum harmonic oscillator with rational ratio of frequencies

The symmetry algebra of the two-dimensional quantum harmonic oscillator with rational ratio of frequencies is identified as a non-linear extension of the u(2) algebra. The finite dimensional representation modules of this algebra are studied and the energy eigenvalues are de- termined using algebraic methods of general applicability to quantum superintegrable systems.

Nonlinear extension of the u(3) algebra as the symmetry algebra of the three-dimensional anisotropic quantum harmonic oscillator with rational ratios of frequencies and the Nilsson model

The symmetry algebra of the N-dimensional anisotropic quantum har- monic oscillator with rational ratios of frequencies is constructed by a method of general applicability to quantum superintegrable systems. The special case of the 3-dim oscillator is studied in more detail, because of its relevance in the description of superdeformed nuclei and nuclear and atomic clusters. In this case the symmetry algebra turns out to be a nonlinear extension of the u(3) algebra. A generalized angular momentum operator useful for labeling the degenerate states is constructed, clarifying the connection of the present formalism to the Nilsson model.

Decoherence of a Damped Anisotropic Harmonic Oscillator under Magnetic Field Effects in a Two-Dimensional Noncommutative Phase-Space

In this paper, decoherence of a damped anisotropic harmonic oscillator in the presence of a magnetic field is studied in the framework of the Lindblad theory of open quantum systems in noncommutative phase-space. General fundamental conditions that should follow our quantum mechanical diffusion coefficients appearing in the master equation are kindly derived. From the master equation, the expressions of density operator, the Wigner distribution function, the expectation and variance with respect to coordinates and momenta are obtained. Based on these quantities, the total energy of the system is evaluated and simulations show its dependency to phase-space structure and its improvement due to noncommutativity effects and the environmental temperature as well. In addition, we also evaluate the decoherence time scale and show that it increases with noncommutativity phase-space effects as compared to the commutative case. It turns out from simulations that this time scale is significantly improved under magnetic field effects.

Symmetries of Anisotropic Harmonic Oscillators with Rational Ratios of Frequencies and their Relations to U(N) and O(N+1)

The concept of bisection of a harmonic oscillator or hydrogen atom, vised in the past in establishing the connection between U(3) and 0(4), is generalized into multisection (trisection, tetrasection, etc). It is then shown that all symmetries of the N-dimensional anisotropic harmonic oscillator with rational ratios of frequencies (RHO), some of which are underlying the structure of superdeformed and hyperdeformed nuclei, can be obtained from the U(N) symmetry of the corresponding isotropic oscillator with an appropriate combination of multisections. Furthermore, it is seen that bisections of the N-dimensional hydrogen atom, which possesses an 0(N+1) symmetry, lead to the U(N) symmetry, so that further multisections of the hydrogen atom lead to the symmetries of the N-dim RHO. The opposite is in general not true, i.e. multisections of U(N) do not lead to 0(N+1) symmetries, the only exception being the occurence of 0(4) after the bisection of U(3).

Quantum Lissajous Scars.

A quantum scar-an enhancement of a quantum probability density in the vicinity of a classical periodic orbit-is a fundamental phenomenon connecting quantum and classical mechanics. Here we demonstrate that some of the eigenstates of the perturbed two-dimensional anisotropic (elliptic) harmonic oscillator are strongly scarred by the Lissajous orbits of the unperturbed classical counterpart. In particular, we show that the occurrence and geometry of these quantum Lissajous scars are connected to the anisotropy of the harmonic confinement, but unlike the classical Lissajous orbits the scars survive under a small perturbation of the potential. This Lissajous scarring is caused by the combined effect of the quantum (near) degeneracies in the unperturbed system and the localized character of the perturbation. Furthermore, we discuss experimental schemes to observe this perturbation-induced scarring.

Coherent States for the Isotropic and Anisotropic 2D Harmonic Oscillators

In this paper we introduce a new method for constructing coherent states for 2D harmonic oscillators. In particular, we focus on both the isotropic and commensurate anisotropic instances of the 2D harmonic oscillator. We define a new set of ladder operators for the 2D system as a linear combination of the x and y ladder operators and construct the S U ( 2 ) coherent states, where these are then used as the basis of expansion for Schrödinger-type coherent states of the 2D oscillators. We discuss the uncertainty relations for the new states and study the behaviour of their probability density functions in configuration space.

Canonical quantization of anisotropic Bianchi I cosmology from scalar vector tensor Brans Dicke gravity

We applied a generalized scalar-vector-tensor Brans Dicke gravity model to study canonical quantization of an anisotropic Bianchi I cosmological model. Regarding an anisotropic Harmonic Oscillator potential we show that the corresponding Wheeler de Witt wave functional of the system is described by Hermit polynomials. We obtained a quantization condition on the ADM mass of the cosmological system which raises versus the quantum numbers of the Hermit polynomials. Our calculations show that the inflationary expansion of the universe can be originate from the big bang with no naked singularity due to the uncertainty principle.

A generalized statistical complexity based on Rényi entropy of a noncommutative anisotropic oscillator in a homogeneous magnetic field

We calculate the shape Rényi and generalized Rényi complexity of a noncommutative anisotropic harmonic oscillator in a homogeneous magnetic field. To do so, we first obtain the Rényi entropy in position and momentum spaces of the exact normalized wave functions. We observe that shape Rényi and generalized Rényi complexities are monotone functions of noncommutative parameter ([Formula: see text]) in some short range in position space. We analyze the effect of the noncommutative parameter, the magnetic field and the anisotropy on shape Rényi and generalized Rényi complexities.

Nuclear mass parameters and moments of inertia in a folded-Yukawa mean-field approach

Collective moments of inertia as well as quadrupole and octupole mass parameters are calculated in the perturbative cranking approximation using the folded-Yukawa mean-field potential written in Cartesian coordinates. The resulting single-particle Hamiltonian is required, as a minimal symmetry, to be invariant under z-signature and time-reversal transformations. The deformation of the nucleus is defined through a shape parametrization in cylindrical coordinates, with the so-called Funny–Hills and Trentalange–Koonin–Sierk shapes as typical and performant examples. To take pairing correlations into account, the standard set of BCS equations is solved with an approximation of constant pairing strength. The numerical program determining the quadrupole–octupole mass tensor and the moments of inertia, written in Fortran, is constructed according to the here presented study as an extension and application of the “yukawa” code for the diagonalization of the folded-Yukawa mean-field potential in the basis of a harmonic-oscillator in Cartesian coordinates, published in this journal in 2016. Program summaryProgram Title: yukmomsProgram Files doi:http://dx.doi.org/10.17632/9ygk4bgkfr.1Licensing provisions: GPLv3Programming language: FortranNature of problem: The moments of inertia and the quadrupole and octupole mass parameters are calculated within the perturbative cranking approximation. As an input, the folded-Yukawa mean-field is diagonalized in the harmonic-oscillator deformed basis to generate the single-particle energies and wave functions. Nuclear shapes are given by means of well known Funny–Hills or Trentalange–Koonin–Sierk parametrizations able to describe the elongation, mass asymmetry, non-axiality and neck of the deformed nucleus.Solution method: Having diagonalized the matrix corresponding to the folded-Yukawa single-nucleon Hamiltonian expressed in the basis of the anisotropic harmonic oscillator basis, one then calculates the matrix elements of the one-body angular-momentum as well as the quadrupole and octupole-moments operators. These matrix elements together with the solutions of the coupled BCS equations are essential input to calculate the desired collective masses and moments of inertia of atomic nuclei.