Motional entanglement in low-energy collisions near shape resonances

TL;DR

The study addresses how much motional entanglement can be generated in low-energy three-dimensional collisions near shape resonances. It develops fully coherent 3D scattering calculations with finite initial uncertainty and quantifies entanglement via the inverse of the single-particle purity, linking entanglement to the scattering cross section and phase shifts. A key finding is a linear relationship between asymptotic entanglement and cross section for sufficiently narrow momentum dispersion, with strong enhancement near shape resonances; plane-wave descriptions are insufficient. The work outlines experimental prospects for detecting collisional entanglement and sets the stage for probing entanglement generation in collisions and its potential applications in quantum control and sensing.

Abstract

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.

Figures (4)

• Figure 1: (a) 3D collision of two structureless free particles, initially with momenta $\pm{p}_{0}\hat{z}$, positions $\mp\frac{{q}_0}{2}\hat{z}$ and same spatial dispersion $\vec{a}$ that spreads under time of flight. (b) Entanglement measure $\mathcal{K}$ for three initial Gaussian wave packets, and linear fit of $\mathcal{K}$ to the overlap of scattered and non-scattered wavefunctions versus time of flight over single particle mass [$q_0=0$, $p_0=0.8$, under potential $V_0=8$ in Eq. \ref{['eq-138']}], see End Matter for the fit parameters.
• Figure 2: (a) Entanglement $\mathcal{K}$ vs mean incident momentum $p_0$, near a $p$-wave shape resonance ($p_{res}=0.87$ for $V_0=8$) for different spatial (momentum) dispersions $\vec{a}$ ($\Delta {p}$) and initial locations $q_0$. (b-c): Cross section $\sigma$ (black solid lines) and comparison to linear fits, cf. Eq. \ref{['eq:sigma-fit']} [colored symbols with labels $(\Delta p,q_0)$] vs (b) collision momentum for $V_0=8$ and (c) potential depth with fixed $p_0=0.8$.
• Figure 3: Cross section $\sigma$ (black solid lines) and comparison to linear fits, cf. Eq. \ref{['eq:sigma-fit']} (colored symbols) near a $p$-wave shape resonance for (a) a single potential $V_0=8.8$ ($p_{res}=0.64$) and (b) different potentials (Table \ref{['tab:table1']}) with $\sigma_{fit}$ evaluated at $p_0=p_{res}$ (vertical dashed lines), for $q_0=5$, $\vec{a}=(a_{\perp},a_{\perp},20)$, $a_{\perp}=4,10,20$, cf. Fig. \ref{['f2']} for the correspondence with $\Delta p$.
• Figure 4: Entanglement in 1D scattering: $\mathcal{K}(t\to+\infty)$ vs potential depth $V_0$, calculated with $p_0=0.8$ and $a=5,10,20,40$ (solid curves with color darkening as $a$ increases). $\mathcal{K}$ converges onto $1/(T^2+R^2)$ (dashed pink curve, $T+R=1$).