The Fock-Darwin-Darboux system: eigenstates, information entropies and dispersion-like measures

TL;DR

This paper investigates the eigenstates, information entropies, and dispersion measures of the Fock-Darwin-Darboux system, a quantum model on curved space, providing analytical and numerical insights into its entropy characteristics.

Contribution

It extends the analysis of quantum information measures to the Fock-Darwin-Darboux system, a generalization involving curved space, and compares it with the classical Fock-Darwin system.

Findings

Analytical expressions for Shannon, Rényi, and Tsallis entropies are derived for the FD system.

Numerical analysis reveals the effects of curvature and magnetic field on entropy measures in the FDD system.

The Landau system on Darboux III space lacks infinite degeneracy of Landau levels.

Abstract

The Fock-Darwin (FD) quantum system describes the motion on the plane of a charged particle under the action of an isotropic oscillator potential together with a perpendicular constant magnetic field. When the isotropic oscillator is suppressed, the FD system leads to the Landau Hamiltonian with infinitely degenerate Landau levels. The Fock-Darwin-Darboux (FDD) system is the generalisation of the FD system to a particle moving on the Darboux III space, which is a conformally flat surface with non-constant negative curvature. We present a systematic study of some information-theoretic entropy and dispersion-like measures for these quantum systems. Since both systems are exactly solvable, analytical expressions for Shannon, R\'enyi and Tsallis entropies, among others, can be obtained. We show that for the FD system, its information-theoretic measures are formally the same as the ones for the harmonic oscillator, provided a modified effective frequency depending on the magnetic field is introduced. In the FDD case, the nonlinear nature of the underlying manifold precludes the existence of a simple closed form for the wave-function on momentum space, which is numerically analysed. We compare the numerical behaviour of the different entropy measures and we analyse the interplay arising in the FDD system between the curvature parameter and the magnetic field. In particular, it is shown that the Landau system on the Darboux III space has no infinitely degenerate Landau levels.