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Chaos An Interdisciplinary Journal of Nonlinear Science

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

Benny Nogales, Sergio Curilef, Angel Ricardo Plastino

Published Jul 1, 2025• 1 citation
Publisher

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.

DOI
10.1063/5.0278593

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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.
Abstract We explore some entanglement-related features of systems consisting of two particles coupled through a central potential. We consider two types of systems, constituted by particles in one spatial dimension interacting through an attractive Dirac Delta potential, or through a harmonic quadratic potential. The degree of freedom corresponding to the system’s center of mass is either regarded as confined within a one dimensional box, or described by a localized Gaussian wave-packet. As a quantitative indicator of the degree of entanglement between the particles, we use the linear entropy of the one-particle marginal density matrix. In general, this quantity cannot be calculated analytically (except in some special cases related to the Harmonic interaction). Consequently, the linear entropy has to be evaluated numerically. Since the concomitant numerical problem consist in evaluating a multi-dimensional integral in four dimensions, the Monte Carlo integration method is an appropriate one for this task. We explore numerically how the system’s entanglement depends on the different interaction potentials, on the different types of confinement for the system’s center of mass, and on the associated parameters describing the size and geometry of the system.
We present a systematic study of entanglement control in one-dimensional systems of two distinguishable interacting particles in harmonic confinement. Using density matrix renormalization group calculations, we examine how the von Neumann entropy evolves under variations of mass ratio, trap-frequency asymmetry, interaction range, and localized external fields. Our results reveal a consistent picture connecting the real-space density overlap, the redistribution of quantum fluctuations, and the resulting entanglement. Within the parameter regimes investigated, mass imbalance produces more pronounced changes in entanglement than trap-frequency asymmetry. Furthermore, the competition between interaction range and field-induced polarization determines whether short- or long-range correlations dominate. By connecting the von Neumann entropy with measurable quantities such as the density difference Δ n ( x ) and the phase-space products 〈 x i 2 〉 〈 p x i 2 〉 , we establish experimentally accessible fingerprints of quantum correlation. Our analysis demonstrates how external and geometrical parameters can tune entanglement in confined two-particle models, providing a basis for exploring related control protocols in suitable experimental settings.
When two non-relativistic particles scatter in one dimension, they can become entangled. This entanglement process is constrained by the symmetries of the scattering system and the boundary conditions on the incoming state. Applying these constraints, three different mechanisms of entanglement can be identified: the superposition of reflected and transmitted modes, momentum correlations of the reflected mode due to inversion of the relative momentum, and momentum correlations in the transmitted and reflected modes due to dependence of the scattering amplitude on the relative momentum. We consider three standard potentials, the hard core, Dirac delta, and double Dirac delta, and show that the relative importance of these mechanisms depends on the interaction and on the properties of the incoming wavefunction. We find that even when the momenta distributions of the incoming articles are sharply peaked, entanglement due to the momentum correlations generated by reflection can be quite large for particles with unequal mass.
The D-dimensional harmonic system (i.e., a particle moving under the action of a quadratic potential) is, together with the hydrogenic system, the main prototype of the physics of multidimensional quantum systems. In this work, we rigorously determine the leading term of the Heisenberg-like and entropy-like uncertainty measures of this system as given by the radial expectation values and the Rényi entropies, respectively, at the limit of large D. The associated multidimensional position-momentum uncertainty relations are discussed, showing that they saturate the corresponding general ones. A conjecture about the Shannon-like uncertainty relation is given, and an interesting phenomenon is observed: the Heisenberg-like and Rényi-entropy-based equality-type uncertainty relations for all of the D-dimensional harmonic oscillator states in the pseudoclassical ( D → ∞ ) limit are the same as the corresponding ones for the hydrogenic systems, despite the so different character of the oscillator and Coulomb potentials.
We investigate finite-number effects in collisions between two states of an initially well-known number of identical bosons with contact interactions, oscillating in the presence of harmonic confinement in one dimension. We investigate two N/2N/2N/2 (interacting) ground states, which are initially displaced from the trap center, and the effects of varying interaction strength. The numerics focus on the simplest case of N=4N=4N=4. In the noninteracting case, such a system would display periodic oscillation with a half harmonic oscillator period (due to the left-right symmetry). With the addition of contact interactions between the bosons, collisions generate entanglement between each of the states and distribute energy into other modes of the oscillator. We study the system numerically via an exact diagonalization of the Hamiltonian with a finite basis, investigating left-right number uncertainty as our primary measure of entanglement. Additionally, we study the time evolution and equilibration of the single-body von Neumann entropy for both the attractive and repulsive cases. We identify parameter regimes for which attractive interactions create behavior qualitatively different from that of repulsive interactions, due to the presence of bound states (quantum solitons), and explain the processes behind this.
This article provides a convenient framework for quantitative evaluation of the entanglement generated when two structureless, distinguishable particles scatter nonrelativistically in one dimension. It explores how three factors determine the amount of entanglement generated: the momentum distributions of the incoming particles, their masses, and the interaction potential. Two important scales emerge, one set by the kinematics and one set by the dynamics. This method also provides two approximate analytic formulas useful for numerical evaluation of entanglement and reveals an interesting connection between purity, linear coordinate transformations, and momentum uncertainties.