Universality of Entanglement Creation in Low-Energy Two-Dimensional Scattering

Abstract

We prove that the entanglement created in the low-energy scattering of two particles in two dimensions is given by a universal coefficient that is independent of the interaction potential. This is strikingly different from the three dimensional case, where it is proportional to the total scattering cross section. Before the collision the state is a product of two normalized Gaussians. We take the purity as the measure of the entanglement after the scattering. We give a rigorous computation, with error bound, of the leading order of the purity at low-energy. For a large class of potentials, that are not assumed to be spherically symmetric, we prove that the low-energy behaviour of the purity, $\mathcal P$, is universal. It is given by $\mathcal P= 1- \frac{1}{(\ln (\sigma/\hbar))^2} \mathcal E$, where $\sigma$ is the variance of the Gaussians and the entanglement coefficient, $\mathcal E$, depends only on the masses of the particles and not on the interaction potential. The entanglement depends strongly in the difference of the masses. It takes its minimum when the masses are equal, and it increases rapidly with the difference of the masses.