Quantum information measures of the Aharonov-Bohm ring in uniform magnetic fields

Abstract

Shannon quantum information entropies $S_{ρ,γ}$, Fisher informations $I_{ρ,γ}$, Onicescu energies $O_{ρ,γ}$ and complexities $e^SO$ are calculated both in position (subscript $ρ$) and momentum ($γ$) spaces for azimuthally symmetric 2D nanoring that is placed into combination of transverse uniform magnetic field $\bf B$ and Aharonov-Bohm (AB) flux $ϕ_{AB}$ and whose potential profile is modeled by superposition of quadratic and inverse quadratic dependencies on radius $r$. Increasing intensity $B$ flattens momentum waveforms $Φ_{nm}({\bf k})$ and in the limit of infinitely large fields they turn to zero, what means that the position wave functions $Ψ_{nm}({\bf r})$, which are their Fourier counterparts, tend in this limit to the $δ$-functions. Position (momentum) Shannon entropy depends on the field $B$ as a negative (positive) logarithm of $ω_{eff}\equiv\left(ω_0^2+ω_c^2/4\right)^{1/2}$, where $ω_0$ determines the quadratic steepness of the confining potential and $ω_c$ is a cyclotron frequency. This makes the sum ${S_ρ}_{nm}+{S_γ}_{nm}$ a field-independent quantity that increases with the principal $n$ and azimuthal $m$ quantum numbers and does satisfy entropic uncertainty relation. Position Fisher information does not depend on $m$, linearly increases with $n$ and varies as $ω_{eff}$ whereas its $n$ and $m$ dependent Onicescu counterpart ${O_ρ}_{nm}$ changes as $ω_{eff}^{-1}$. The products ${I_ρ}_{nm}{I_γ}_{nm}$ and ${O_ρ}_{nm}{O_γ}_{nm}$ are $B$-independent quantities. A dependence of the measures on the ring geometry is discussed. It is argued that a variation of the position Shannon entropy or Onicescu energy with the AB field uniquely determines an associated persistent current as a function of $ϕ_{AB}$ at $B=0$. An inverse statement is correct too.