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arXiv:1209.5299 · Journal of Physics A Mathematical and Theoretical

The relationship between entanglement, energy and level degeneracy in two-electron systems

A. P. Majtey, Angel Ricardo Plastino, J. S. Dehesa

Published Mar 1, 2012• 35 citations

Abstract

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.

DOI: 10.1088/1751-8113/45/11/115309

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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.

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.

Quantum Entanglement in

(d-1)

-Spherium

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 \emph{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.

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.

Controlling the properties of quantum matter is a central goal of condensed matter physics. In recent years, cavity embedding—that is, shaping the electromagnetic environment surrounding a material by placing it inside a cavity—has emerged as a new potential tuning knob. While such concepts were first successfully explored with ultracold atomic clouds, more recent experiments have demonstrated their feasibility in solid-state systems. Despite the differences between these platforms, they share a key feature: the coexistence of degrees of freedom with distinct levels of locality. On one hand, electrons or atoms exhibit local dynamics (hopping, interactions), while on the other, cavity modes are typically delocalized across the entire system. The study of this unconventional quantum many-body framework—investigated through aspects such as phase transitions, topology, and dynamical properties—constitutes the main focus of this thesis.