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Physical Review A

Spatial entanglement in two-electron atomic systems

Y.-C. Lin, Chih‐Yuan Lin, Y. K. Ho

Published Feb 13, 2013• 57 citations
Publisher

Abstract

Recently, there has been considerable interest in investigating quantum entanglement in two-electron systems, such as two-electron model atoms and quantum dots. In this work, the quantum entanglement of the helium atom and helium-like ions are explored using the two-electron wave functions constructed by the $B$-spline basis. As a measure of the spatial entanglement, the linear entropy of the reduced density matrix is investigated for the ground state and the singlet excited $1sn{s}^{1}S$ states, with $n=2$--10, of the helium atom. The variation of the spatial entanglement with the nuclear charges $Z=2$--15 for the helium-like ions is reported. Results are compared with the existing predictions.

DOI
10.1103/physreva.87.022316

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Recently, there has been considerable interest in investigating quantum entanglement in two-electron systems, such as two-electron model atoms and quantum dots. In this work, the quantum entanglement of the helium atom and helium-like ions are explored using the two-electron wave functions constructed by the BBB-spline basis. As a measure of the spatial entanglement, the linear entropy of the reduced density matrix is investigated for the ground state and the singlet excited 1sns1S1sn{s}^{1}S1sns1S states, with n=2n=2n=2--10, of the helium atom. The variation of the spatial entanglement with the nuclear charges Z=2Z=2Z=2--15 for the helium-like ions is reported. Results are compared with the existing predictions.
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.
In the present work, we present an investigation on quantum entanglement of the two-electron helium atom immersed in weakly coupled Debye plasmas, modeled by the Debye-Hückel, or screened Coulomb, potential to mimic the interaction between two charged particles inside the plasma. Quantum entanglement is related to correlation effects in a multi-particle system. In a bipartite system, a measurement made on one of the two entangled particles affects the outcome of the other particle, even if such two particles are far apart. Employing wave functions constructed with configuration interaction B-spline basis, we have quantified von Neumann entropy and linear entropy for a series of He 1,3Se and 1,3Po states in plasma-embedded helium atom.
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.
We explore the main entanglement properties exhibited by the eigenfunctions of two exactly soluble two-electron models, the Crandall atom and the Hooke atom, and compare them with the entanglement features of helium-like systems. We compute the amount of entanglement associated with the wavefunctions corresponding to the fundamental and first few excited states of these models. We investigate the dependence of the entanglement on the parameters of the models and on the quantum numbers of the eigenstates. It is found that the amount of entanglement of the system tends to increase with energy in both models. In addition, we study the entanglement of a few states of helium-like systems, which we compute using high-quality Kinoshita-like eigenfunctions. The dependence of the entanglement of helium-like atoms on the nuclear charge and on energy is found to be consistent with the trends observed in the previous two model systems.
Different entropies which can characterize the inseparable quantum correlation in a two-electron entangled artificial atom proposed by Moshinsky are analyzed. The analysis is based on an exact representation for the trace of the qqq-order one-body reduced density matrix in terms of the interparticle coupling. The entanglement spectrum of the model system and the possible role of the entropy-maximum concept to practical applications of matrices are discussed as well. The ground-state energy, readily obtained from the Schr\"odinger Hamiltonian, is recalculated within a density-matrix-functional representation for its components.