Impurity entanglement through electron scattering in a magnetic field

We study the entanglement of magnetic impurities in an environment of electrons through successive scattering while an external magnetic field is applied. We show that the dynamics of the problem can be approximately described by a reduced model of three interacting spins, which reveals an intuitive view on how spins can be entangled by controlled electron scattering. The role of the magnetic field is rather crucial. Depending on the initial state configuration, the magnetic field can either increase or decrease the resulting entanglement but more importantly it can allow the impurities to be maximally entangled.

Figure 1

(a) Two embedded impurities in a ring are entangled via successive electron scattering. In addition an external magnetic field is applied. (b) A minimal model of three interacting spins.

Figure 2

Entanglement dynamics (concurrence) of the two impurities within model (3) for (a) initially aligned ${\mathcal{C}}_{\uparrow \uparrow }\left(t\right)$ and (b) anti-aligned ${\mathcal{C}}_{\uparrow \downarrow }\left(t\right)$ impurities and different values of the applied magnetic field. Inset: Amplitude ${\mathcal{C}}_{\uparrow \uparrow }$ vs the magnetic field.

Figure 3

Entanglement dynamics (concurrence) ${\mathcal{C}}_{\mathrm{t}}\left(t\right)$ of the full model (1) with $\tilde{t}=5J$ and $L=16$ for two values of the magnetic field $B=0$ and $B=-0.5J$ starting from an initially entangled triplet state. The dashed line indicates the minimum of entanglement in the absence of magnetic field.

Figure 4

Entanglement dynamics of the full model (1) with $L=16$ for two initially aligned impurities obtained via numerical diagonalization for different electron initial states (see text) and impurity distances $N$. The magnetic field is chosen $B=-J/L$, so that $\Omega =0$.