Entanglement of Two Impurities through Electron Scattering

We study how two magnetic impurities embedded in a solid can be entangled by an injected electron scattering between them and by subsequent measurement of the electron’s state. We start by investigating an ideal case where only the electronic spin interacts successively through the same unitary operation with the spins of the two impurities. We find conditions for the impurity spins to be maximally entangled with a significant success probability. We then consider a more realistic description which includes both the forward and backscattering amplitudes. In this scenario, we obtain the entanglement between the impurities as a function of the interaction strength of the electron-impurity coupling. We find that our scheme allows us to entangle the impurities maximally with a significant probability.

Figure 1

Setup for our scheme to entangle two magnetic impurities of spin $1/2$ in a solid through electron scattering. The electron flies along a one-dimensional chain and interacts magnetically its with the impurities to entangle them. In a realistic scenario the electron may also spatially scatter from the impurities.

Figure 2

Plots of the probability of success $P$ (dashed line for aligned and dotted line for antialigned impurity spins) and entanglement $E$ (solid line for aligned and line with circles for antialigned impurity spins) obtained between the impurity spins subject to success (detection of the electron spin to be down) in the ideal case of the electron successively interacting with the two impurities through the same unitary operation as a function of $Jt$.

Figure 3

Realistic case: Plots of the probability $P$ (dashed line) of detecting the transmitted electron in the spin-down state, and the amount of entanglement $E$ (solid line) obtained between the impurity spins in that case, as a function of the interaction strength $J\rho ({ϵ}_F)$, for the realistic scenario of the electron successively scattering off the two impurities. Our results were calculated numerically to 14th order in the coupling constant, ensuring an accuracy of at least one part in ${10}^{-8}$.