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arXiv:0710.5776 · Physical Review A

Entanglement generation in the scattering of one-dimensional particles

Nathan L. Harshman, Gary Hutton

Published Apr 15, 2008• 17 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.77.042310

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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.
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.
Einstein, Podolsky, and Rosen discussed their paradox in terms of measuring the positions or momenta of two particles. These degrees of freedom can become entangled upon scattering, but how much entanglement can be created in this process? Here we address this question using fully coherent calculations of bipartite scattering in three-dimensional space, quantifying entanglement by the inverse of the single particle purity. We show that the standard plane-wave description of scattering fails to capture the entanglement properties, due to the essential role of quantum uncertainty in the initial state. For a more realistic description of a scattering setup, we find that the entanglement scales linearly with the scattering cross section, including strong enhancement near shape resonances, for sufficiently narrow initial momentum dispersion. We highlight the differences between scattering in one and higher spatial dimensions and discuss how the generation of motional entanglement can be detected in experiments. Our results open the way to probing, controlling, and eventually using entanglement in quantum collisions.
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We analyze the entanglement generation in a one-dimensional scattering process. The two colliding particles have a Gaussian wave function and interact by hard-core repulsion. In our analysis, results on the entanglement of two-mode Gaussian states are used. The produced entanglement depends in a nonobvious way on the ratio of masses and initial widths for the two particles. The asymptotic wave function of the two particles and its associated ellipse yield additional geometric insight into these conditions. We discuss the difference from the quantitative analysis of the amount of entanglement generated by beam splitters with squeezed light.
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Abstract We give a quantitative evaluation of the entanglement dynamics in scattering events between two electrons interacting via Coulomb potential in 1D systems, We apply a method, specifically designed for indistinguishable particles, based on the calculation of the von Neumann entropy of the system density matrix. From the timedependent numerical solution of the two‐particle wavefunction we evaluate the entanglement evolution for different spin configurations. The procedure allows to understand the mechanisms that govern entanglement creation and its connection with the characteristic physical parameters and initial conditions of the system. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)