Lowest excited configuration of harmonium

Abstract

The harmonium model has long been regarded as an exactly solvable laboratory bench for quantum chemistry [W. Heisenberg, Z. Phys. 38, 411 (1926)]. For studying correlation energy, only the ground state of the system has received consideration heretofore. This is a spin singlet state. In this work we exhaustively study the lowest excited (spin triplet) harmonium state, with the main purpose of revisiting the relation between entanglement measures and correlation energy for this quite different species. The task is made easier by working with Wigner quasiprobabilities on phase space.

Entanglement in N-harmonium: bosons and fermions

“…This has been possible because for these harmonic systems we have been able to calculate analytically not only the one-body reduced matrix for bosons and fermions, but also the von Neumann entropy in the bosonic case as well as the linear entropy in the fermionic case. In doing so, we complement and extend to harmonic systems with an arbitrary number of particles the study of entanglement recently done for various two-electron models [13,[21][22][23][24][25] as well as some helium-like systems [6,14,15,17] and certain quantum complex networks [49].…”

“…For completeness, let us mention that the reader will have no difficulty in verifying that ρ 1 (r, r) -the diagonal of (29)-coincides (except for the difference in the normalization conventions) with the one obtained in [25,Sect. V] by the Wigner function method.…”

“…For the case of an assembly of bosons we will compute the von Neumann entropy of the one-body reduced density matrix finding explicitly the occupation numbers of the system, as described in the next section. Recently, there has been a renewed interest in formulating the reduced density matrix theory using Wigner quasidistributions [25,39,40]. Our treatment for bosons will be done by using the Wigner function in phase space, what is a natural procedure for these systems.…”

“…Up to now, however, only entanglement of some models of two bound electrons have been determined. We refer to the 2-harmonium (or Moshinsky) [21][22][23][24][25], Crandall and Hooke [13] atoms. In all these models the electron confinement is harmonic, and the electron-electron interaction is of harmonic (2-harmonium), r −1 12 (Crandall), and Coulombic (Hooke) type.…”

Quantum correlations in one-dimensional Wigner molecules

“…In particular, the study of the manybody properties of various quantum composite systems is nowadays one of the most active areas of theoretical physics [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. Investigating quantum correlations in systems of interacting particles trapped in external potentials is not only important in view of the context of quantum information technology [24] but also a key to improving our understanding of quantum matter.…”