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

Abstract

Shannon quantum information entropies $S_{\rho ,\gamma }$, Fisher informations $I_{\rho ,\gamma }$, Onicescu energies $O_{\rho ,\gamma }$ and complexities $e^{S}O$ are calculated both in position (subscript $\rho$) and momentum ($\gamma$) spaces for azimuthally symmetric 2D nanoring that is placed into combination of transverse uniform magnetic field $\mathbf{B}$ and Aharonov-Bohm (AB) flux ${\varphi }_{AB}$ and whose potential profile is modeled by superposition of quadratic and inverse quadratic dependencies on radius $r$. Increasing intensity $B$ flattens momentum waveforms ${\Phi }_{nm}(\mathbf{k})$ and in the limit of infinitely large fields they turn to zero, what means that the position wave functions ${\Psi }_{nm}(\mathbf{r})$, which are their Fourier counterparts, tend in this limit to the $\delta$-functions. Position (momentum) Shannon entropy depends on the field $B$ as a negative (positive) logarithm of ${\omega }_{eff}\equiv {\left({\omega }_0^2+{\omega }_c^2/4\right)}^{1/2}$, where ${\omega }_0$ determines the quadratic steepness of the confining potential and ${\omega }_c$ is a cyclotron frequency. This makes the sum ${S_{\rho }}_{nm}+{S_{\gamma }}_{nm}$ 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 ${\omega }_{eff}$ whereas its $n$ and $m$ dependent Onicescu counterpart ${O_{\rho }}_{nm}$ changes as ${\omega }_{eff}^{-1}$. The products ${I_{\rho }}_{nm}{I_{\gamma }}_{nm}$ and ${O_{\rho }}_{nm}{O_{\gamma }}_{nm}$ 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 ${\varphi }_{AB}$ at $B=0$. An inverse statement is correct too.

11 figures, 1 table

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