Proposal for an on-demand source of polarized electrons into the edges of a topological insulator

We propose a device that allows for the emission of pairs of spin-polarized electrons into the edge states of a two-dimensional topological insulator. Charge and spin emission is achieved using a periodically driven quantum dot weakly coupled to the edge states of the host topological insulator. We present calculations of the emitted time-dependent charge and spin currents of such a dynamical scatterer using the Floquet scattering matrix approach. Experimental signatures of spin-polarized two-particle emission can be found in noise measurements. Here a new form of noise suppression, named ${\mathbb{Z}}_2$ antibunching, is introduced. Additionally, we propose a setup in which entanglement of the emitted electrons is generated. This entanglement is based on a postselection procedure and becomes manifest in a violation of a Clauser-Horne-Shimony-Holt inequality.

Figure 1

Sketch of the Spin Particles Source. The quantum dot is supposed to be of circular shape with a perimeter of length $L$. The Kramers' partners are indicated by the ket states $|\uparrow \rangle$ and $|\downarrow \rangle$ (see main text for further details).

Figure 2

Zero-temperature emitted charge as a function of the height of the driving step $U_0$. The various lines correspond to different values of the reflection probability of the QPC $|{\lambda }_{\mathrm{pb}}{|}^2$: $|{\lambda }_{\mathrm{pb}}{|}^2=0.95$ (solid red line), $|{\lambda }_{\mathrm{pb}}{|}^2=0.5$ (dashed blue line), and $|{\lambda }_{\mathrm{pb}}{|}^2=0.1$ (dotted green line). Please note that the steps are given in units of twice the electron charge. Upper inset: The density of states of the QD for various values of $|{\lambda }_{\mathrm{pb}}|$. Lower inset: Model of driving we consider: the driving of the gate is switched on at time $t_0$ and the potential is changed by a quantity $U_0$.

Figure 3

After the emission, the two spin-polarized electrons are separated spatially, propagating along the upper and the lower edge. A second QPC (C) allows for interference of the two particles. The QPC is tuned in a regime, where only forward scattering processes are allowed (${\lambda }_{\mathrm{pb}}=0$ for C). No reabsorption processes are considered.

Figure 4

The SpPS creates two particles propagating along the edges of the 2D TI. Two QPCs allow for an exchange of these two particles, redistributing them to two parties “Alice” and “Bob.” Postselection after these two QPCs yields a nonvanishing concurrence. Two additional QPCs, the analogs of “local” polarizers, are characterized by angles ${\vartheta }_{\mathrm{A}/\mathrm{B}}$ allowing for violation of the CHSH inequality.

Figure 5

The concurrence after postselection of simultaneous detection events for symmetric QPCs as a function of the spin-preserving amplitude ${\gamma }_{\mathrm{sp}}$ for different values of the spin-flipping amplitude ${\gamma }_{\mathrm{sf}}$, ranging from ${\gamma }_{\mathrm{sf}}=0$ to ${\gamma }_{\mathrm{sf}}=3$ in steps of 0.5. The arrow indicates the direction of increasing ${\gamma }_{\mathrm{sf}}$. Inset: Concurrence as a function of ${\gamma }_{\mathrm{sf}}$ and ${\gamma }_{\mathrm{pf}}$.

Figure 6

The quantum point contact (QPC) connecting the edge states to the quantum dot. Propagation along the edges is spin polarized, such that on the upper edge spin-up [spin-down] propagates to the right [left] and vice-versa on the lower edge. A constriction in the bulk structure makes the edge states overlapping which allows for different scattering processes described by the Hamiltonians (1) and (A1).

Figure 7

In the figure are shown the processes contributing to the time-dependent scattering matrix Eqs. (B10). Panel (a) shows the back-reflection, proportional to ${\lambda }_{\mathrm{pb}}$, panels (b) and (c) (green line) are the forward spin conserving processes proportional to ${\lambda }_{\mathrm{ff}}$, and panels (d) and (e) (red line) are the forward spin flipping processes proportional to ${\lambda }_{\mathrm{pf}}$. Each time the edge states are arriving to the QPC they get a contribution proportional to ${\lambda }_{\mathrm{pb}}$.