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Journal of Physics B Atomic Molecular and Optical Physics

Excited state entanglement on a two-electron system near the ionization threshold

Omar Osenda, Pablo Serra

Published Mar 10, 2008• 24 citations
Publisher

Abstract

We have investigated the critical properties of the lowest-energy triplet state of the spherical helium atom. Using finite-size scaling methods we calculate critical charge and critical exponents for both the energy and the von Neumann entropy near the ionization threshold. We show that the scaling properties of the energy and the von Neumann entropy for this excited state are qualitatively different from those obtained for the ground state. These scaling properties are quantified in terms of critical exponents; therefore, the analysis is applicable to other few-fermion systems.

DOI
10.1088/0953-4075/41/6/065502

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We have investigated the critical properties of the lowest-energy triplet state of the spherical helium atom. Using finite-size scaling methods we calculate critical charge and critical exponents for both the energy and the von Neumann entropy near the ionization threshold. We show that the scaling properties of the energy and the von Neumann entropy for this excited state are qualitatively different from those obtained for the ground state. These scaling properties are quantified in terms of critical exponents; therefore, the analysis is applicable to other few-fermion systems.
The critical behavior of entanglement near a quantum phase transition has been studied intensely over the past few years. Few-body quantum systems show critical behavior near the ionization point. We investigate the scaling properties of the von Neumann entropy for an atomiclike system near the ionization threshold. Using finite size scaling methods we calculate the critical charge and the critical exponent associated to the von Neumann entropy. The parallelism between the behavior of entanglement near a quantum phase transition and the behavior of the von Neumann entropy in a critical few-body quantum system is analyzed.
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 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.
Abstract In the present work, we present an investigation of Collins Conjecture, a hypothesis made by D. M. Collins in 1993 relating correlation energy and entanglement entropy, by calculating the ground state and singly‐excited triplet‐spin 1s2s 3S and 1s3s 3S state energies of the helium iso‐electronic sequence, with Z = 2–15. By using extensive orthonormal configuration interaction (CI) type wave functions with B‐Spline basis up to about 6000 terms, linear entropy and von Neumann entropy for the abovementioned atomic systems are determined. Together with the Hartree‐Fock energies obtained following a self‐consistent field theory, we have found that there exist linearly proportionalities between the renormalized correlation energies and entanglement entropies of both linear and von Neumann, showing a support for Collins Conjecture applicable to the ground and singly‐excited triplet‐spin states in the helium sequence, for a range of finite Z‐values.
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.
Abstract In this work we have performed state-of-the-art configuration-interaction (CI) calculations to determine the linear and von Neumann entanglement entropies for the helium-like systems with varying nuclear charge Z in the range . The focus of the work resides on determining accurate entanglement values for 2-electron systems with the lowest computational cost through compact CI-wave functions. Our entanglement results for the helium atom fully agree with the results obtained with higher quality wave functions of the Kinoshita type (Dehesa [5]). We find that the correlation energy is linearly related to the entanglement measures associated with the linear and von Neumann entropies of the single-particle reduced density matrizes, which sheds new light on the physical implications of entanglement in helium-like systems. Moreover, we report CI-wave-function-based benchmark results for the entanglement values for all members of the helium isoelectronic series with an accuracy similar to that of Kinoshita-type wave functions. Finally, we give parametric expressions of the linear and von Neumann entanglement measures for two-electron systems as Z varies from 1 to 10.