One-dimensional pseudoharmonic oscillator: classical remarks and quantum-information theory

Abstract

Motion along semi-infinite straight line in a potential that is a combination of positive quadratic and inverse quadratic functions of the position is considered with the emphasis on the analysis of its quantum-information properties. Classical measure of symmetry of the potential is proposed and its dependence on the particle energy and the factor $\mathfrak{a}$ describing a relative strength of its constituents is described; in particular, it is shown that a variation of the parameter $\mathfrak{a}$ alters the shape from the half-harmonic oscillator (HHO) at $\mathfrak{a}=0$ to the perfectly symmetric one of the double frequency oscillator (DFO) in the limit of huge $\mathfrak{a}$. Quantum consideration focuses on the analysis of information-theoretical measures, such as standard deviations, Shannon, Rényi and Tsallis entropies together with Fisher information, Onicescu energy and non--Gaussianity. For doing this, among others, a method of calculating momentum waveforms is proposed that results in their analytic expressions in form of the confluent hypergeometric functions. Increasing parameter $\mathfrak{a}$ modifies the measures in such a way that they gradually transform into those corresponding to the DFO what, in particular, means that the lowest orbital saturates Heisenberg, Shannon, Rényi and Tsallis uncertainty relations with the corresponding position and momentum non--Gaussianities turning to zero. A simple expression is derived of the orbital-independent lower threshold of the semi-infinite range of the dimensionless Rényi/Tsallis coefficient where momentum components of these one-parameter entropies exist which shows that it varies between $1/4$ at HHO and zero when $\mathfrak{a}$ tends to infinity. Physical interpretation of obtained mathematical results is provided.

arXiv comment40 pages, 11 figures

References in corpus (37)

• Measuring entanglement entropy through the interference of quantum many-body twins
• Quantum thermalization through entanglement in an isolated many-body system
• Entropic Uncertainty Relations and their Applications
• Probing entanglement entropy via randomized measurements
• Probing entanglement in a many-body-localized system
• Entropic uncertainty relations - A survey
• Non-Gaussian Quantum States and Where to Find Them
• Formulation of the uncertainty relations in terms of the Renyi entropies
• Resource theory of quantum non-Gaussianity and Wigner negativity
• Quantifying the non-Gaussian character of a quantum state by quantum relative entropy
• Quantifying non-Gaussianity for quantum information
• Preparing for the quantum revolution -- what is the role of higher education?
• Heun Functions and Some of Their Applications in Physics
• Achieving a quantum smart workforce
• Quantum-memory-assisted entropic uncertainty relations
• Continuous-variable entropic uncertainty relations
• A remark on rational isochronous potentials
• On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles
• Effect of the surface curvature on the magnetic moment and persistent currents in two-dimensional quantum rings and dots
• Spiked harmonic oscillators
• On the Role of Information Theoretic Uncertainty Relations in Quantum Theory
• Matrix elements for a generalized spiked harmonic oscillator
• Experimental Detection of the Correlation Rényi Entropy in the Central Spin Model
• Exact Analytical Solution of the N-dimensional Radial Schrodinger Equation with Pseudoharmonic Potential via Laplace Transform Approach
• Gaussian optimizers for entropic inequalities in quantum information
• Rényi and Tsallis entropies: three analytic examples
• Quantum information measures of the Aharonov-Bohm ring in uniform magnetic fields
• Rényi and Tsallis entropies of the Aharonov-Bohm ring in uniform magnetic fields
• Uncertainty Relations for coarse-grained measurements: an overview
• Quantifying the nonlinearity of a quantum oscillator
• Magnetic field control of the intraband optical absorption in two-dimensional quantum rings
• Comparative analysis of information measures of the Dirichlet and Neumann two-dimensional quantum dots
• Rényi and Tsallis entropies of the Dirichlet and Neumann one-dimensional quantum wells
• Information and quantum theories: an analysis in one-dimensional systems
• Quantum information measures of the Dirichlet and Neumann hyperspherical dots
• Quantum-information theory of a Dirichlet ring with Aharonov-Bohm field
• Comment on "On the realisation of quantum Fisher information"

Cited by in corpus (3)

• Remarks on the Study of Electronic Properties and Photoionization Process in Rotating 2D Quantum Rings
• Quantum-information theory of magnetic field influence on circular dots with different boundary conditions
• Informational analysis of the confinement of an electron in an asymmetric double quantum dot