Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum

• URL: http://arxiv.org/abs/2502.02491v1
• Date: Tue, 04 Feb 2025 17:06:54 GMT
• Title: Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum
• Authors: Rutwig Campoamor-Stursberg, Francisco J. Herranz, Danilo Latini, Ian Marquette, Alfonso Blasco,
• Abstract summary: We consider the quantum analog of the generalized Zernike systems given by the Hamiltonian. This two-dimensional quantum model exhibits higher-order integrals of motion within the enveloping algebra of the Heisenberg algebra.
• Score: 0.0
• License:
• Abstract: We consider the quantum analog of the generalized Zernike systems given by the Hamiltonian: $$ \hat{\mathcal{H}} _N =\hat{p}_1^2+\hat{p}_2^2+\sum_{k=1}^N \gamma_k (\hat{q}_1 \hat{p}_1+\hat{q}_2 \hat{p}_2)^k , $$ with canonical operators $\hat{q}_i,\, \hat{p}_i$ and arbitrary coefficients $\gamma_k$. This two-dimensional quantum model, besides the conservation of the angular momentum, exhibits higher-order integrals of motion within the enveloping algebra of the Heisenberg algebra $\mathfrak h_2$. By constructing suitable combinations of these integrals, we uncover a polynomial Higgs-type symmetry algebra that, through an appropriate change of basis, gives rise to a deformed oscillator algebra. The associated structure function $\Phi$ is shown to factorize into two commuting components $\Phi=\Phi_1 \Phi_2$. This framework enables an algebraic determination of the possible energy spectra of the model for the cases $N=2,3,4$, the case $N=1$ being canonically equivalent to the harmonic oscillator. Based on these findings, we propose two conjectures which generalize the results for all $N\ge 2$ and any value of the coefficients $\gamma_k$, that they are explicitly proven for $N=5$. In addition, all of these results can be interpreted as superintegrable perturbations of the original quantum Zernike system corresponding to $N=2$ which are also analyzed and applied to the isotropic oscillator on the sphere, hyperbolic and Euclidean spaces.

Related papers

• Linearly coupled quantum harmonic oscillators and their quantum entanglement [0.0]
  It is shown that quantum entanglement depends on only one coefficient $R in (0,1), which includes all the parameters of the system under consideration. It has been shown that quantum entanglement can be very large at certain values of this coefficient.
  arXiv  Detail & Related papers  (2024-02-01T17:42:17Z)
• Non-standard quantum algebras and finite dimensional $\mathcal{PT}$-symmetric systems [0.0]
  We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard $U_z(sl(2, mathbb R))$ Hopf algebra deformation. We show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits.
  arXiv  Detail & Related papers  (2023-09-26T23:17:22Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Two-body Coulomb problem and $g^{(2)}$ algebra (once again about the Hydrogen atom) [77.34726150561087]
  It is shown that if the symmetry of the three-dimensional system is $(r, rho, varphi)$, the variables $(r, rho, varphi)$ allow a separation of variable $varphi$ and eigenfunctions. Thoses occur in the study of the Zeeman effect on Hydrogen atom.
  arXiv  Detail & Related papers  (2022-12-02T20:11:17Z)
• Exponential Unitary Integrators for Nonseparable Quantum Hamiltonians [0.0]
  Quantum Hamiltonians containing nonseparable products of non-commuting operators are problematic for numerical studies. We extend Chin's idea to represent nonseparable terms in terms of separable ones.
  arXiv  Detail & Related papers  (2022-11-15T14:09:15Z)
• Uncertainties in Quantum Measurements: A Quantum Tomography [52.77024349608834]
  The observables associated with a quantum system $S$ form a non-commutative algebra $mathcal A_S$. It is assumed that a density matrix $rho$ can be determined from the expectation values of observables. Abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables.
  arXiv  Detail & Related papers  (2021-12-14T16:29:53Z)
• Dynamics of Non-Gaussian Entanglement of Two Magnetically Coupled Modes [0.0]
  This paper surveys the quantum entanglement of two coupled harmonic oscillators via angular momentum. We study the effect of the anisotropy $ R=omega_12/omega_22 $, $omega_c$, asymmetry $ |n-m| $ and dynamics on the entanglement.
  arXiv  Detail & Related papers  (2021-08-09T15:52:34Z)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)
• Fermion and meson mass generation in non-Hermitian Nambu--Jona-Lasinio models [77.34726150561087]
  We investigate the effects of non-Hermiticity on interacting fermionic systems. We do this by including non-Hermitian bilinear terms into the 3+1 dimensional Nambu--Jona-Lasinio (NJL) model.
  arXiv  Detail & Related papers  (2021-02-02T13:56:11Z)
• Anharmonic oscillator: a solution [77.34726150561087]
  The dynamics in $x$-space and in $(gx)-space corresponds to the same energy spectrum with effective coupling constant $hbar g2$. A 2-classical generalization leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy.
  arXiv  Detail & Related papers  (2020-11-29T22:13:08Z)
• Unitary unfoldings of Bose-Hubbard exceptional point with and without particle number conservation [0.0]
  Non-Hermitian but $cal PT-$symmetric quantum system of an $N-$plet of bosons described by Bose-Hubbard Hamiltonian $H(gamma,v,c)$ is picked up. Non-Hermitian but $cal PT-$symmetric quantum system of an $N-$plet of bosons described by the three-parametric Bose-Hubbard Hamiltonian $H(gamma,v,c)$ is picked up.
  arXiv  Detail & Related papers  (2020-08-28T21:06:17Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.