Emission of entangled Kramers pairs from a helical mesoscopic capacitor

The realization of single-electron sources in integer quantum Hall systems has paved the way for exploring electronic quantum optics experiments in solid-state devices. In this paper, we characterize a single Kramers pair emitter realized by a driven antidot embedded in a two-dimensional topological insulator, where spin-momentum locked edge states can be exploited for generating entanglement. Contrary to previous proposals, the antidot is coupled to both edges of a quantum spin Hall bar, thus enabling this mesoscopic capacitor to emit an entangled two-electron state. We study the concurrence $\mathcal{C}$ of the emitted state and the efficiency $\mathcal{F}$ of its emission as a function of the different spin-preserving and spin-flipping tunnel couplings of the antidot with the edges. We show that the efficiency remains very high ($\mathcal{F}\ge 50\%$) even for maximally entangled states ($\mathcal{C}=1$). We also discuss how the entanglement can be probed by means of noise measurements and violation of the Clauser-Horne-Shimony-Holt inequality.

Figure 1

Antidot (central gray area) realized in a narrow QSH bar (yellow) coupled to both the two edges in a two-terminal configurations [left (L) and right (R) gray rectangles]. Tunnel junction 2 is created at $x=0, y=0$ with tunnel junction 1 at $y=\pi R, z=0$. Spin-up and spin-down edge states are depicted in red and blue, respectively. The antidot is driven by a time-dependent potential $U\left(t\right)$.

Figure 2

Plot of the concurrence $\mathcal{C}$ as a function of the parameters of the second tunnel junction $p_2$ and $f_2$, with fixed values $p_1=0.15$ and $f_1=0.05$.

Figure 3

Plot of the efficiency $\mathcal{F}$ as a function of the parameters of the second tunnel junction $p_2$ and $f_2$, with fixed values $p_1=0.15$ and $f_1=0.05$.

Figure 4

Plot of the two-terminal noise ${\mathcal{P}}_0$ in units of $e^2\mathrm{\Omega }/\pi$ as a function of the parameters of the second tunnel junction $p_2$ and $f_2$, with fixed values $p_1=0.15$ and $f_1=0.05$.

Figure 5

Setup for detection of the violation of the CHSH inequlity. The injected two-particle state is mixed by two additional QPCs which act as beam splitters; the mixing is tuned via the QPC tunneling parameters ${\theta }_L$ and ${\theta }_R$. After the QPCs the particles are collected in a four terminal geometry.