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Atoms

Quantum Entanglement and Shannon Information Entropy for the Doubly Excited Resonance State in Positronium Negative Ion

Chien-Hao Lin, Y. K. Ho

Published Sep 21, 2015• 13 citations
Publisher

Abstract

In the present work, we report an investigation on quantum entanglement in the doubly excited 2s2 1Se resonance state of the positronium negative ion by using highly correlated Hylleraas type wave functions, determined by calculation of the density of resonance states with the stabilization method. Once the resonance wave function is obtained, the spatial (electron-electron orbital) entanglement entropies (von Neumann and linear) can be quantified using the Schmidt decomposition method. Furthermore, Shannon entropy in position space, a measure for localization (or delocalization) for such a doubly excited state, is also calculated.

DOI
10.3390/atoms3030422

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In the present work, we report an investigation on quantum entanglement in the doubly excited 2s2 1Se resonance state of the positronium negative ion by using highly correlated Hylleraas type wave functions, determined by calculation of the density of resonance states with the stabilization method. Once the resonance wave function is obtained, the spatial (electron-electron orbital) entanglement entropies (von Neumann and linear) can be quantified using the Schmidt decomposition method. Furthermore, Shannon entropy in position space, a measure for localization (or delocalization) for such a doubly excited state, is also calculated.
In this work, we study the quantum entanglement for doubly excited resonance states in two-electron atomic systems such as the H - and Ps - ions and the He atom by using highly correlated Hylleraas type functions The resonance states are determined by calculation of density of resonance states with the stabilization method. The spatial (electron-electron orbital) entanglement entropies (linear and von Neumann) for the low-lying doubly excited states are quantified using the Schmidt-Slater decomposition method.
Quantum entanglement for two electrons in the excited states of helium-like atom or ions is investigated using two-electron wave functions constructed in the B-spline basis. As a measure of spatial (electron – electron orbital) entanglement, the von Neumann entropy and linear entropy of the reduced density matrix are calculated for the 1s2s 1,3S excited states for systems with some selected Z values from Z = 2 to 100. Results for the helium atom are compared with other available calculations. We have also investigated the entropies for these excited states when the nucleus charge is reduced from Z = 2 continuously to Z = 1. At such a critical charge, all the singly excited states of this system become unbound, and the linear entropies and the von Neumann entropies for the excited states approach 1/2 and 1, respectively; the limits for the entropies when one electron is bound to the nucleus, and the other is free.
Shannon entropy and Fisher information calculated from one-particle density distributions and von Neumann and linear entropies (the latter two as measures of entanglement) computed from the reduced one-particle density matrix are analyzed for the 1,3Se,1,3Po^{1,3}S^{e},\phantom{\rule{4pt}{0ex}}^{1,3}P^{o}1,3Se,1,3Po, and 1,3De^{1,3}D^{e}1,3De Rydberg series of He doubly excited states below the second ionization threshold. In contrast with the Shannon entropy, we find that both the Fisher information and entanglement measures are able to discriminate low-energy resonances pertaining to different 2(K,T)n2A_{2}(K,T)_{{n}_{2}}^{A}2​(K,T)n2​A​ series according to the Herrick-Sinano\ifmmode \breve{g}\else \u{g}\fi{}lu-Lin classification. Contrary to bound states, which show a clear and unique asymptotic value for both Fisher information and entanglement measures in their Rydberg series $1sn\ensuremath{\ell}$ for $n\ensuremath{\rightarrow}\ensuremath{\infty}$ (which implies a loss of spatial entanglement), the variety of behaviors and asymptotic values of entanglement above the noninteracting limit value in the Rydberg series of doubly excited states 2(K,T)n2A_{2}(K,T)_{{n}_{2}}^{A}2​(K,T)n2​A​ indicates a signature of the intrinsic complexity and remnant entanglement in these high-lying resonances even with infinite excitation ${n}_{2}\ensuremath{\rightarrow}\ensuremath{\infty}$, for which all known attempts of resonance classifications fail in helium.
Shannon entropies and Fisher information calculated from one-particle density distributions and von Neumann and linear entropies (the latter two as a measure of entanglement) computed from the reduced one-particle density matrix are analyzed for the 1 , 3 s e , 1,3 P 0 and 1,3 D e Rydberg series of He doubly excited states below the second ionization threshold. We find that both Fisher information and entanglement measures are able to discriminate resonances pertaining to different (K,T) A series.
We provide benchmark values for Shannon information entropies in position space for the ground state and ls2s 1Se excited state of helium confined with finite confinement potentials by employing the highly correlated Hylleraas-type wave functions. For the excited state, a “tilt” (small oscillation) on the curve of Shannon entropy as a function of width size for the confinement potential is observed. Justified by the behavior of the electron density, the localization or delocalization of the helium wave functions confined with repulsive and attractive finite oscillator (FO) potentials are examined.