Entanglement-related features of hydrogenic systems and other systems described by bound states of two interacting particles

Abstract

This paper delves into entanglement-related features of one-dimensional and three-dimensional systems comprising two particles interacting through an attractive potential, such as the delta, harmonic, and Coulomb ones. As a quantitative indicator of the amount of entanglement between the particles, we employ the linear entropy of the system's one-particle marginal density matrices. Except in some particular instances involving the harmonic potential, this quantity is not analytically tractable and requires numerical evaluation. Our aim is to elucidate some aspects of entanglement in hydrogenic systems. Hydrogenic systems, consisting of two particles interacting through the Coulomb potential, are of clear importance in physics and chemistry, but their entanglement properties have started to be explored only recently. To better understand entanglement in those systems, we first analyze one-dimensional systems interacting via Dirac delta and harmonic potentials. Insights gained from these one-dimensional cases provide valuable guidelines for studying entanglement in hydrogenic systems. We numerically investigate, for the interaction potentials already mentioned, and for different types of confinement for the center of mass, how the system's entanglement varies with the parameters that determine the size and geometry of the system's quantum state. We find that entanglement depends on a dimensionless quantity given by the quotient of two parameters characterizing the length scales associated with the interaction potential and the center of mass confinement. Entanglement approaches its maximum when the above-mentioned dimensionless quotient tends to its extreme values and adopts its minimum at an intermediate value of the dimensionless quotient. We find that the same general qualitative features of entanglement behavior are observed for different attractive interactions.