Entanglement of electrons field-emitted from a superconductor

Under appropriate circumstances the electrons emitted from a superconducting tip can be entangled. We analyze these nonlocal correlations by studying the coincidences of the field-emitted electrons and show that electrons emitted in opposite directions violate Bell’s inequality. We scrutinize the interplay between the bosonic nature of Cooper pairs and the fermionic nature of electrons. We further discuss the feasibility of our analysis in the light of present experimental capabilities.

Figure 1

(Color online) Field emission of electrons and coincident detections. The size of the emitting region at the tip is $w$, and two of the emitted electrons are detected at ${\mathit{r}}_1$ and ${\mathit{r}}_2$.

Figure 2

(Color online) Normalized coincidences $Q\left(r,\theta \right)$ vs $\theta$, for normal $\Delta =0$ and superconducting $\Delta \ne 0$ emitters. Observe the large bunching peak at $\theta \simeq \pi$.

Figure 3

(Color online) Peak value of $Q\left(r,\theta \right)$ at $\theta =\pi$. The line at $Q=3/2$ indicates the entanglement threshold (above which the pair of electrons is entangled), and the one at $Q=\sqrt{2}/\left(\sqrt{2}-1\right)\simeq 3.41$ is that for the violation of Bell’s inequality (above which Bell’s inequality is violated). The parameters are the same as in Fig. 2, and $\xi \simeq 33.8{\lambda }_F$.