Coulomb Scattering in a 2D Interacting Electron Gas and Production of EPR Pairs

We propose a setup to generate nonlocal spin Einstein-Podolsky-Rosen pairs via pair collisions in a 2D interacting electron gas, based on constructive two-particle interference in the spin-singlet channel at the $\pi /2$ scattering angle. We calculate the scattering amplitude via the Bethe-Salpeter equation in the ladder approximation and small $r_s$ limit and find that the Fermi sea leads to a substantial renormalization of the bare scattering process. From the scattering length, we estimate the current of spin-entangled electrons and show that it is within experimental reach.

Figure 1

(a) Proposed setup: two quantum point contacts filter electrons from two reservoirs with initial momenta ${\mathbf{p}}_1\simeq -{\mathbf{p}}_2$. The two detectors (with an aperture angle $2\delta \theta$) are placed such that only electrons that collide (shaded area) at a scattering angle around $\pi /2$ are registered. Because of interference, the scattering amplitude vanishes at $\pi /2$ for the spin-triplet states, allowing only the spin-entangled singlets to be collected: one electron of the singlet state in detector 1 and its partner in detector 2. The scattering cross section and the electron flux could be measured via an AFM tip [8]. (b) Scattering parameters: $\mathbf{P}={\mathbf{p}}_1+{\mathbf{p}}_2={\mathbf{p}}_1^{\prime }+{\mathbf{p}}_2^{\prime }$ is the total momentum, $\mathbf{p}=({\mathbf{p}}_1-{\mathbf{p}}_2)/2$ and ${\mathbf{p}}^{\prime }=({\mathbf{p}}_1^{\prime }-{\mathbf{p}}_2^{\prime })/2$ are the relative momenta, and $\theta =\angle (\mathbf{p},{\mathbf{p}}^{\prime })$ is the scattering angle between them. The initial (${\mathbf{p}}_1,{\mathbf{p}}_2$ ) and final (${\mathbf{p}}_1^{\prime },{\mathbf{p}}_2^{\prime }$) momenta are connected by a circle of radius $p=p^{\prime }$ due to energy and momentum conservation.

Figure 2

Plots of scattering quantities obtained from the $t$ matrix (4) for $r_s=0.86$ (GaAs) and $k_{B}T/E_F={10}^{-2}$. (a) Angular dependence of the scattering length $\lambda (\theta )$. We compare $\lambda$ to its Born approximation given by $t\simeq V$ and to the bare scattering of two particles (no Fermi sea) given by the exact result ($t_C$) or by the first order ($V_C$). (b) Dependence of $\lambda (\pi /2)$ on the sheet density $n$ (see $r_s=m{e}_0^2/{\hslash }^2\sqrt{\pi n}$ in top axis). (c) Ratio $R(\theta ,\delta \theta )$ of triplets/singlets detected at a scattering angle $\theta$ for an aperture $\delta \theta =5°$.