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

Abstract

Shannon quantum information entropies S ρ,γ (φ AB , r 0 ), Fisher informations I ρ,γ (φ AB , r 0 ), Onicescu energies O ρ,γ (φ AB , r 0 ) and Rényi entropies R ρ,γ (φ AB , r 0 ; α) are calculated both in the position (subscript ρ) and momentum (γ) spaces as functions of the inner radius r 0 for the two-dimensional Dirichlet unit-width annulus threaded by the Aharonov-Bohm (AB) flux φ 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 Ψ nm (φ AB , r 0 ; r) and momentum Φ nm (φ AB , r 0 ; 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, φ AB = 0, only. It is shown, in particular, that the position Shannon entropy S ρnm (φ AB , r 0 ) [Onicescu energy O ρnm (φ AB , r 0 )] grows logarithmically [decreases as 1/r 0 ] with large r 0 tending to the same asymptote S asym ρ = ln(4πr 0 ) − 1 [O asym ρ = 3/(4πr 0 )] for all orbitals whereas their Fisher counterpart I ρnm (φ 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 Φ nm (φ AB , r 0 ; k) increases with the inner radius what results in the identical r 0 ≫ 1 asymptote for all momentum Shannon entropies S γnm (φ AB ; r 0 ) with the alike n and different m. The same limit causes the Fisher momentum components I γ (φ 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 α T H of the semi-infinite range of the dimensionless coefficient α, 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 α T H = 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.