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arXiv:2411.10358 · Physical Review A

Dynamics of correlations and entanglement generation in electron-molecule inelastic scattering

Martin Mendez, Federico M. Pont

Published May 7, 2025• 2 citations
arXivHTMLPDF

Abstract

The dynamics and processes involved in particle-molecule scattering, including nuclear dynamics, are described and analyzed using various quantum information quantities throughout the different stages of the scattering. The main process studied and characterized with the information quantities is the interatomic Coulombic electronic capture, an inelastic process that can lead to dissociation of the target molecule. The analysis focuses on a one-dimensional transversely confined NeHe molecule model used to simulate the scattering between an electron ${e}^{\ensuremath{-}}(\text{particle})$ and a ${\text{NeHe}}^{+}$ ion (molecule). The time-independent Schr\"odinger equation is solved using the finite element method with a self-developed Julia package FEMTISE to compute potential energy curves and the parameters of the interactions between particles. The time-dependent Schr\"odinger equation is solved using the multiconfiguration time-dependent Hartree algorithm. The time-dependent electronic and nuclear probability densities are calculated for different electron incoming energies, evidencing elastic and inelastic processes that can be correlated to changes in von Neumann entropy, von Neumann mutual information, and Shannon mutual information. The expectation value of the position of the particles, as well as their standard deviations, are analyzed along the whole dynamics and related to the entanglement during the collision and after the process is over, thus highlighting the dynamics of entanglement generation. It is shown that the correlations generated in the collision are partially retained only when the inelastic process is active.

DOI
10.1103/physreva.111.052411

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The dynamics and processes involved in particle-molecule scattering, including nuclear dynamics, are described and analyzed using various quantum information quantities throughout the different stages of the scattering. The main process studied and characterized with the information quantities is the interatomic Coulombic electronic capture, an inelastic process that can lead to dissociation of the target molecule. The analysis focuses on a one-dimensional transversely confined NeHe molecule model used to simulate the scattering between an electron e\ensuremath−(particle){e}^{\ensuremath{-}}(\text{particle})e\ensuremath−(particle) and a NeHe+{\text{NeHe}}^{+}NeHe+ ion (molecule). The time-independent Schr"odinger equation is solved using the finite element method with a self-developed Julia package FEMTISE to compute potential energy curves and the parameters of the interactions between particles. The time-dependent Schr"odinger equation is solved using the multiconfiguration time-dependent Hartree algorithm. The time-dependent electronic and nuclear probability densities are calculated for different electron incoming energies, evidencing elastic and inelastic processes that can be correlated to changes in von Neumann entropy, von Neumann mutual information, and Shannon mutual information. The expectation value of the position of the particles, as well as their standard deviations, are analyzed along the whole dynamics and related to the entanglement during the collision and after the process is over, thus highlighting the dynamics of entanglement generation. It is shown that the correlations generated in the collision are partially retained only when the inelastic process is active.
We study the Born-Oppenheimer dynamics within a model for a coupled electron-nuclear motion. Differential Shannon entropies are calculated from the time-dependent probability densities of the combined system and, using single particle densities, entropies for the electronic and nuclear degrees of freedom are derived. These functions provide information on details of the wave packet motion. From the entropies, we determine the mutual information which characterizes particle correlations. This quantity is compared to other measures of electron-nuclear entanglement. Numerical results are interpreted within an analytically solvable approach, and it is documented how these functions depend on properties of the Born-Oppenheimer wave function and, in particular, how dynamical effects like wave packet focusing and dispersion influence the correlation between the particles.
Applications of information-theoretic measures to a time-dependent coupled electron-nuclear system to analyze the dynamics and correlation between both particles are presented. For this, differential Shannon entropies that are derived from time-dependent coordinate-space and momentum-space probability densities are calculated. Two distinct scenarios are investigated: one exhibiting adiabatic Born-Oppenheimer dynamics and the other involving strong non-adiabatic transitions. The total and single-particle entropies, as well as the mutual information are analyzed and compared to semi-analytical expressions. The results reveal that in the adiabatic regime, correlations manifest differently in coordinate and momentum spaces, which is related to the formation of nodes. In the non-adiabatic case, entropies can be decomposed into state-specific contributions, revealing information about the transition between adiabatic states.
We perform the quantitative evaluation of the entanglement dynamics in scattering events between two indistinguishable electrons interacting via the Coulomb potential in one- and two-dimensional semiconductor nanostructures. We apply a criterion based on the von Neumann entropy and the Schmidt decomposition of the global state vector suitable for systems of identical particles. From the time-dependent numerical solution of the two-particle wave function of the scattering carriers we compute their entanglement evolution for different spin configurations: two electrons with the same spin, with different spin, and singlet and triplet spin states. The procedure allows us to evaluate the mechanisms that govern entanglement creation and their connection with the characteristic physical parameters and initial conditions of the system. The cases in which the evolution of entanglement is similar to the one obtained for distinguishable particles are discussed.
We study the influence of nodal structures in two-dimensional quantum mechanical densities on wave packet entanglement. This is motivated by our recent study [Entropy, 25, 970 (2023)], which showed that the mutual information derived from the momentum-space probability density of a coupled two-particle system exhibits an unusual time dependence, which is not encountered if the position-space density is employed in the calculation. In studying a model density, here, we identify cases where the mutual information increases with the number of nodes in the wave function and approaches a finite value, whereas in this limit, the linear correlation vanishes. The results of the analytical model are then applied to interpret the correlation measures for coupled electron-nuclear dynamics, which are treated by numerically solving the time-dependent Schrödinger equation.
We calculate differential Shannon entropies derived from time-dependent coordinate-space and momentum-space probability densities. This is performed for a prototype system of a coupled electron-nuclear motion. Two situations are considered, where one is a Born-Oppenheimer adiabatic dynamics, and the other is a diabatic motion involving strong non-adiabatic transitions. The information about coordinate- and momentum-space dynamics derived from the total and single-particle entropies is discussed and interpreted with the help of analytical models. From the entropies, we derive mutual information, which is a measure for the electron-nuclear correlation. In the adiabatic case, it is found that such correlations are manifested differently in coordinate- and momentum space. For the diabatic dynamics, we show that it is possible to decompose the entropies into state-specific contributions.
Quantum Information Theory is a new field with potential implications for the conceptual foundations of Quantum Mechanics through density matrices. In particular, information entropies in Hilbert space representation are highly advantageous in contrast with the ones in phase space representation since they can be easily calculated for large systems. In this work, novel von Neumann conditional, mutual, and joint entropies are employed to analyze the dissociation process of small molecules, Cl2 and HCl, by using the spectral decomposition of the first reduced density matrix in natural atomic orbital-based representation which allows us to assure rotational invariance, N- and v-representability in the Atoms-in-Molecules (AIM) scheme. Quantum information entropies permit to analyze the dissociation process through quantum mechanics concepts such as electron correlation and entanglement, showing interesting critical points which are not present in the energy profile, such as charge depletion and accumulation, along with bond breaking regions.