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Journal of Physics A Mathematical and Theoretical

Quantum entanglement in two-electron atomic models

Daniel Manzano, Angel Ricardo Plastino, J. S. Dehesa, Tsuyoshi Koga

Published Jun 8, 2010• 65 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/43/27/275301

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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.
The entanglement properties of two-electron atomic systems have been the subject of considerable research activity in recent years. These studies are still somewhat fragmentary, focusing on numerical computations on particular states of systems such as helium, or on analytical studies of model systems such as the Moshinsky atom. Some general trends are beginning to emerge from these studies: the amount of entanglement tends to increase with energy and, in the case of excited states, entanglement does not necessarily tend to zero in the limit of vanishing interaction between the two constituting particles. A physical explanation of these properties, shared by the different two-electron models investigated so far, is still lacking. As a first step towards this goal, we perform here, via a perturbative approach, an analysis of entanglement in two-electron models that sheds new light on the physical origin of the aforementioned features and on their universal character.
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.
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.
There are very few systems of interacting particles (with continuous variables) for which the entanglement of the concomitant eigenfunctions can be computed in an exact, analytical way. Here we present analytical calculations of the amount of entanglement exhibited by s -states of spherium . This is a system of two particles (electrons) interacting via a Coulomb potential and confined to a ( d −1)-sphere (that is, to the surface of a d -dimensional ball). We investigate the dependence of entanglement on the radius R of the system, on the spatial dimensionality d , and on energy. We find that entanglement increases monotonically with R , decreases with d , and also tends to increase with the energy of the eigenstates. These trends are discussed and compared with those observed in other two-electron atomic-like models where entanglement has been investigated.
We present two methods of calculating the spatial entanglement of an interacting electron system within the framework of density-functional theory. These methods are tested on the model system of Hooke's atom for which the spatial entanglement can be exactly calculated. We analyze how the strength of the confining potential affects the spatial entanglement and how accurately the methods that we introduced reproduce the exact trends. We also compare the results to the outcomes of standard first-order perturbation methods. The accuracies of energies and densities when using these methods are also considered.