Dynamical Coulomb Blockade and Spin-Entangled Electrons

We consider the creation of mobile and nonlocal spin-entangled electrons from tunneling of a BCS-superconductor (SC) to two normal leads of finite resistivity. The resulting dynamical Coulomb blockade effect, which we describe phenomenologically in terms of an electromagnetic environment, is shown to be enhanced for tunneling of two electrons from a Cooper pair into the same lead compared to the desired pair-split process where each electron enters a different lead. Conversely, this latter process is suppressed by a finite separation between the tunneling points on the SC.

Figure 1

Entangler setup: A BCS bulk superconductor (SC) with chemical potential ${\mu }_S$ is tunnel coupled (amplitude $t_0$) via points ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ of the SC to two Fermi liquid leads $1,2$ with resistance $R_{1,2}$. The leads are held at the same chemical potential ${\mu }_l$ such that a bias voltage $\mu ={\mu }_S-{\mu }_l$ is applied between the SC and the two leads via the voltage source $V$. The tunnel junctions $1,2$ have capacitances $C_{1,2}$.

Figure 2

Current ratio $I_2/I_1$ (entangler efficiency) and $I_1$ in the regime $\mu \ll \Delta ,{\omega }_R$ and $\Delta \gg E_c,{\omega }_R$ as a function of $4/g=4R/R_Q$. Chosen parameters: $E_c=0.1\mathrm{m}\mathrm{e}\mathrm{V}$, $k_F\delta r=10$, $\Gamma =0.1$, and $\mu =5\mu \mathrm{e}\mathrm{V}$ (left plot), $\mu =15\mu \mathrm{e}\mathrm{V}$ (right plot). For a 2D SC, $I_1$ and $I_1/I_2$ can be multiplied by 10.