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Journal of Physics B Atomic Molecular and Optical Physics

Quantum entanglement in helium

J. S. Dehesa, Tsuyoshi Koga, Rafael J. Yáñez, Angel Ricardo Plastino, Rodolfo O. Esquivel

Published Dec 15, 2011• 73 citations
Publisher

Abstract

We compute the entanglement of the ground state and several singlet and triplet excited states of the helium atom using high-quality, state-of-the-art wavefunctions. The behaviour of the entanglement of the helium eigenstates is similar to that observed in some exactly soluble two-electron systems. In particular, the amount of entanglement exhibited by the eigenstates tends to increase with energy.

DOI
10.1088/0953-4075/45/1/015504

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We compute the entanglement of the ground state and several singlet and triplet excited states of the helium atom using high-quality, state-of-the-art wavefunctions. The behaviour of the entanglement of the helium eigenstates is similar to that observed in some exactly soluble two-electron systems. In particular, the amount of entanglement exhibited by the eigenstates tends to increase with energy.
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.
The thesis aims to study the applicability of techniques of Quantum Information theory to complex systems, like molecules and biomolecules. Recently, concepts of Quantum Information theory have been applied to the study of phenomena involving molecules, for instance the interaction light-biomolecules. In particular, the entanglement has been proposed to quantify the non-classicality of interactions. The computation of molecular structures and interactions, on the other hand, has been studied for decades mainly to understand reactions? kinetic and energetic characteristics. It is not straightforward to apply these methods to the computation of quantum correlations. We have therefore explored the computability of such effects using Quantum Information techniques. We started with the study of the electronic correlations in the Helium atom, that is the simplest non trivial case, in order to evaluate the computational problems and to develop approximate methods suitable to more complex systems. We start recalling the definition and basic properties of entanglement, as it is considered in Quantum Information theory. We illustrate how it can be measured and the difficulties one encounters when identical particles are involved, then we concentrate on the case of Helium in particular. In order to study the entanglement in Helium, we treat separately the singlet and the triplet configurations. The method we use consists in computing the reduced density matrix and then the entropy. The possible choices of coordinates, basis functions and computation strategies are discussed, together with their impact on the algorithms. The reduced, single-electron von Neumann and linear entropy for several low-energy eigenstates of Helium are computed by means of a simple configuration-interaction variational method.
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.