Minimal quantum dot based Kitaev chain with only local superconducting proximity effect

The possibility to engineer a Kitaev chain in quantum dots coupled via superconductors has recently emerged as a promising path toward topological superconductivity and possibly non-Abelian physics. Here we show that it is possible to avoid some of the main experimental hurdles on this path by using only local proximity effect on each quantum dot in a geometry that resembles a two-dot version of the proposal in Fulga et al. [New J. Phys. 15, 045020 (2013)]. There is no need for narrow superconducting couplers, additional Andreev bound states, or spatially varying magnetic fields; it suffices with spin-orbit interaction and a constant magnetic field in combination with control of the superconducting phase to tune the relative strengths of elastic cotunneling and an effective crossed-Andreev-reflection-like process generated by higher-order tunneling. We use a realistic spinful, interacting model and show that high-quality Majorana bound states can be generated already in a double quantum dot.

Figure 1

Setup consisting of two spinful QDs (1,2) with superconductivity induced by local tunneling to a bulk superconductor. The magnetic flux $\mathrm{\Phi }$ controls the phase difference between the induced superconducting pairing amplitudes, $\delta \varphi ={\varphi }_1-{\varphi }_2$. The QDs are coupled by both spin-conserving tunneling $t$ and (spin-orbit induced) spin-flip tunneling $t_{\mathrm{so}}$.

Figure 2

Calculated stability diagrams of the locally proximitized double QD using $V_z=1.25\mathrm{\Delta },U_l=2.5\mathrm{\Delta }$, and $U_{nl}=0.1\mathrm{\Delta }$. (a) $tanh\left(\delta E\right)$ as a function of the QD energy levels using the sweet-spot phase difference $\delta \varphi =\delta {\varphi }_{★}\approx 0.66\pi$, where $\delta E=E_{\mathrm{odd}}-E_{\mathrm{even}}$ is the energy difference between the odd and even parity ground states. The black box indicates the location of the antiparallel spin configuration, which is our focus. The labels indicate the (QD1, QD2) ground-state occupation for $\mathrm{\Delta }=t=t_{\mathrm{so}}=0$. (b) Parity of the ground state as a function of the QD energy levels in the antiparallel spin configuration, using the same parameters as in (a), except the superconducting phase difference which is varied with $\pm \pi /2$.

Figure 3

(a) $1-\mathrm{MP}$ as a function of $U_l$ and $V_z$, using $U_{nl}=0$. At each point, ${\varepsilon }_1,{\varepsilon }_2$, and $\delta \varphi$ are optimized to find a PMM sweet spot. The dashed lines represent the upper and lower bounds in Eq. (20), estimating where we can tune to a PMM sweet spot. For each marking in (a), a stability diagram indicating the ground state parity is shown in (b). The stability diagrams are centered around the optimized QD energy levels while using the optimized phase difference. Furthermore, the MP at each marking is shown above the corresponding stability diagram.

Figure 4

(a) $1-\mathrm{MP}$ as a function of $U_l$ and $U_{nl}$, using $V_z=1.25\mathrm{\Delta }$. At each point, ${\varepsilon }_1,{\varepsilon }_2$, and $\delta \varphi$ are optimized to find a PMM sweet spot. Furthermore, the region $U_{nl}>U_l$ is blurred since it is unphysical in a capacitive model [52]. (b) The MP at the sweet spot and the stability diagram showing the ground state parity at each marking in (a).

Figure 5

${\varepsilon }_1,{\varepsilon }_2$, and $\delta \varphi$ after sweet-spot optimization in the $(U_l,V_z)$ and $(U_l,U_{nl})$ planes. The values for $U_{nl}$ in the upper panels and for $V_z$ in the lower panels are the same as in Figs. 3 and 4.