Entanglement Detection from Conductance Measurements in Carbon Nanotube Cooper Pair Splitters

Spin-orbit interaction provides a spin filtering effect in carbon nanotube based Cooper pair splitters that allows us to determine spin correlators directly from current measurements. The spin filtering axes are tunable by a global external magnetic field. By a bending of the nanotube, the filtering axes on both sides of the Cooper pair splitter become sufficiently different that a test of entanglement of the injected Cooper pairs through a Bell-like inequality can be implemented. This implementation does not require noise measurements, supports imperfect splitting efficiency and disorder, and does not demand a full knowledge of the spin-orbit strength. Using a microscopic calculation we demonstrate that entanglement detection by violation of the Bell-like inequality is within the reach of current experimental setups.

Figure 1

Double quantum dot CPS based on a bent CNT in an external magnetic field $\mathbf{B}$. Because of $\mathbf{B}$, SOI, and the bending angle ${\theta }_{\mathrm{CNT}}$ of the CNT, the spin-valley degeneracy of the QD levels is lifted, and the resulting 4 levels (boxes) are spin polarized as indicated by the arrows (see also Fig. 2). The superconductor SC injects Cooper pairs (hourglass shape) that split onto the QDs and provide a current to the normal leads N that is modulated by the spin projections of the QDs (tunable by the gates $V_{L,R}$) and can be used to determine the spin correlators for the Bell inequality.

Figure 2

Values $Q$ of the Bell equation (5) (left panel) for an ideal bent CNT-CPS as a function of in-plane $\mathbf{B}$-field rotation angle $\theta$, for the lowest valence band orbitals in a CNT of chirality (18, 10), $|\mathbf{B}|=0.4\mathrm{T}$, ${\theta }_{\mathrm{CNT}}=30°$, $\alpha =-0.08\mathrm{meV}$, $\beta =-0.15\mathrm{meV}$, and QD lengths of 200 nm. For this situation, $|\mathbf{B}|/|{\mathbf{B}}_{\mathrm{SOI}}|\approx {\mu }_{B}g|\mathbf{B}|/2|\alpha -\beta |=0.34$. The horizontal lines mark the threshold $Q=2$ and the maximal possible $Q=2\sqrt{2}$. The right panels show the $\theta$ dependence of the level energies of both QDs. The spectra are identical up to the shift by ${\theta }_{\mathrm{CNT}}$ marked by the vertical dashed lines. The arrows indicate the spin polarizations in a global spin basis, as used for the determination of $Q$.

Figure 3

Results from the microscopic calculation of a CPS, based on a zigzag CNT of chirality (20, 0) with a bending angle ${\theta }_{\mathrm{CNT}}\approx 30°$, in a field of $|\mathbf{B}|=0.5\mathrm{T}$ (see the Supplemental Material [24]). The SOI energies $\alpha =-0.10\mathrm{meV}$ and $\beta =-0.40\mathrm{meV}$ lead to $|\mathbf{B}|/|{\mathbf{B}}_{\mathrm{SOI}}|\approx {\mu }_{B}g|\mathbf{B}|/2|\alpha -\beta |=0.10$. (a) Map of the conductance product $G_L{G}_R$ (units of $e^4/h^2$) as a function of QD gate voltages $V_{L,R}$ at $\theta =25°$. The 4 levels of each QD give rise to the 4 resonances labeled by $\pm a_{\tau }$, $\pm b_{{\tau }^{\prime }}$. Inside the black squares, the CPS acts as a spin-valley filter for the projections $P_{\pm a_{\tau }}{P}_{\pm b_{{\tau }^{\prime }}}$, and integrating the signal within each black square yields the corresponding observable. (b) $V_{L,R}$ values marking the positions of the resonances of the levels $\pm a_{\tau }$ and $\pm b_{{\tau }^{\prime }}$ of the two QDs as a function of $\theta$. The curves are identical up to the shift by ${\theta }_{\mathrm{CNT}}$. Levels in the same valley $\tau$ see the same field ${\mathbf{B}}_{\mathrm{eff}}^{\tau }$ and are identified by having the same curvature as a function of $\theta$. (c) $Q$ as a function of $\theta$ for the conductances $G$ given as subscripts of $Q$ in the figure legend. The $Q$ values are obtained by analyzing data as shown in panel (a) by the method described in the text. The yellow shaded region marks the allowed range of violation of the Bell inequality for the spin singlets in the steady state.