Entanglement of Nambu Spinors and Bell Inequality Test without Beam Splitters

The identification of electronic entanglement in solids remains elusive so far, which is owed to the difficulty of implementing spinor-selective beam splitters with tunable polarization direction. Here, we propose to overcome this obstacle by producing and detecting a particular type of entanglement encoded in the Nambu spinor or electron-hole components of quasiparticles excited in quantum Hall edge states. Because of the opposite charge of electrons and holes, the detection of the Nambu spinor translates into a charge-current measurement, which eliminates the need for beam splitters and assures a high detection rate. Conveniently, the spinor correlation function at fixed effective polarizations derives from a single current-noise measurement, with the polarization directions of the detector easily adjusted by coupling the edge states to a voltage gate and a superconductor, both having been realized in experiments. We show that the violation of Bell inequality occurs in a large parameter region. Our Letter opens a new route for probing quasiparticle entanglement in solid-state physics exempt from traditional beam splitters.

Figure 1

(a) Proposed setup constructed on the single chiral edge state (arrowed lines) in the integer quantum Hall regime. Quasiparticle entanglement is generated in the central region (hourglass area) by a periodic electrical potential $V(t)$. Two sources ($S_{1,2}$) and two drains ($D_{1,2}$) are all grounded. A grounded superconductor (${\mathrm{SC}}_{1,2}$) and a voltage gate ($V_{g1,2}$) are coupled to the outgoing channel on each side. (b) Representation of the Nambu spinor on the Bloch sphere. The north and south poles correspond to the electron ($|\mathbf{e}⟩$) and hole ($|\mathbf{h}⟩$) states, respectively. (c) The actions of ${\mathrm{SC}}_{1,2}$ and $V_{g1,2}$ inside the dashed box in (a) induce an effective rotation of the Nambu spinor.

Figure 2

Plot of $\mathcal{B}$ as a function of $\mathrm{\Delta }$ and $\delta V_g$ with (a) $\lambda =1$ and (b) $\lambda =0.9$. The dashed lines are the critical contours of $\mathcal{B}=2$. The relevant parameters are set as $v=1\times {10}^6\mathrm{m}/\mathrm{s}$, $L_s=300\mathrm{nm}$, and $L_g=100\mathrm{nm}$.