Eur. Phys. J. Plus (2022) 137: 451
https://doi.org/10.1140/epjp/s13360-022-02627-5

Regular Article

Quantum-information theory of a Dirichlet ring with Aharonov–Bohm field

Department of Applied Physics and Astronomy, University of Sharjah, P.O. Box 27272, Sharjah, United Arab Emirates

^a oolendski@sharjah.ac.ae

Received: 8 February 2022
Accepted: 20 March 2022
Published online: 11 April 2022

Abstract

Shannon quantum-information entropies $S_{\rho ,\gamma }({\varphi }_{\mathit{AB}},r_0)$, Fisher informations $I_{\rho ,\gamma }({\varphi }_{\mathit{AB}},r_0)$, Onicescu energies $O_{\rho ,\gamma }({\varphi }_{\mathit{AB}},r_0)$ and Rényi entropies $R_{\rho ,\gamma }({\varphi }_{\mathit{AB}},r_0;\alpha )$ are calculated both in the position (subscript $\rho$) and momentum ($\gamma$) spaces as functions of the inner radius $r_0$ for the two-dimensional Dirichlet unit-width annulus threaded by the Aharonov–Bohm (AB) flux ${\varphi }_{\mathit{AB}}$. Small (huge) values of $r_0$ correspond to the thick (thin) rings with extreme of $r_0=0$ describing the dot. Discussion is based on the analysis of the corresponding position ${\Psi }_{\mathit{nm}}({\varphi }_{\mathit{AB}},r_0;\mathbf{r})$ and momentum ${\Phi }_{\mathit{nm}}({\varphi }_{\mathit{AB}},r_0;\mathbf{k})$ waveforms, with n and m being principal and magnetic quantum indices, respectively: the former allows an analytic expression at any AB field whereas for the latter it is true at the flux-free configuration, ${\varphi }_{\mathit{AB}}=0$, only. It is shown, in particular, that the position Shannon entropy $S_{{\rho }_{\mathit{nm}}}({\varphi }_{\mathit{AB}},r_0)$ [Onicescu energy $O_{{\rho }_{\mathit{nm}}}({\varphi }_{\mathit{AB}},r_0)$] grows logarithmically [decreases as $1/r_0$] with large $r_0$ tending to the same asymptote $S_{\rho }^{\mathrm{asym}}=ln(4\pi r_0)-1$ [$O_{\rho }^{\mathrm{asym}}=3/(4\pi r_0)$] for all orbitals whereas their Fisher counterpart $I_{{\rho }_{\mathit{nm}}}({\varphi }_{\mathit{AB}},r_0$) approaches in the same regime the m-independent limit mimicking in this way the energy spectrum variation with $r_0$, which for the thin structures exhibits quadratic dependence on the principal index. Frequency of the fading oscillations of the radial parts of the wave vector functions ${\Phi }_{\mathit{nm}}({\varphi }_{\mathit{AB}},r_0;\mathbf{k})$ increases with the inner radius what results in the identical $r_0\gg 1$ asymptote for all momentum Shannon entropies $S_{{\gamma }_{\mathit{nm}}}({\varphi }_{\mathit{AB}};r_0)$ with the alike n and different m. The same limit causes the Fisher momentum components $I_{\gamma }({\varphi }_{\mathit{AB}},r_0)$ to grow exponentially with $r_0$. Based on these calculations, properties of the complexities $e^{S}O$ are addressed too. Among many findings on the Rényi entropy, it is proved that the lower limit ${\alpha }_{\mathit{TH}}$ of the semi-infinite range of the dimensionless coefficient $\alpha$, where the momentum component of this one-parameter entropy exists, is not influenced by the radius; in particular, the change of the topology from the simply, $r_0=0$, to the doubly, $r_0>0$, connected domain is unable to change ${\alpha }_{\mathit{TH}}=2/5$. AB field influence on the measures is calculated too. Parallels are drawn to the geometry with volcano-shape confining potential and similarities and differences between them are discussed.