Phase Control of Majorana Bound States in a Topological $\mathsf{X}$ Junction

Topological superconductivity supports exotic Majorana bound states (MBS) which are chargeless zero-energy emergent quasiparticles. With their non-Abelian exchange statistics and fractionalization of a single electron stored nonlocally as a spatially separated MBS, they are particularly suitable for implementing fault-tolerant topological quantum computing. While realizing MBS has focused on one-dimensional systems, the onset of topological superconductivity requires delicate parameter tuning and geometric constraints pose significant challenges for their control and demonstration of non-Abelian statistics. To overcome these challenges, building on recent experiments in planar Josephson junctions (JJs), we propose a MBS platform of $\mathsf{X}$-shaped JJs. This versatile implementation reveals how external flux control of the superconducting phase difference can generate and manipulate multiple MBS pairs to probe non-Abelian statistics. The underlying topological superconductivity exists over a large parameter space, consistent with materials used in our fabrication of such $\mathsf{X}$ junctions, as an important step towards scalable topological quantum computing.

Figure 1

(a) $\mathsf{X}$ junction ($\mathsf{X}\mathrm{J}$) schematic formed by epitaxial superconducting ($S$) regions (blue) covering a 2DEG (yellow). A crossed $\mathsf{X}$ channel with the angle of $2\theta$, defined between the $S$ leads, can be tuned into the topological regime with MBS ${\gamma }_1,\dots ,{\gamma }_4$ (stars) at its ends by the phase differences among the $S$ regions with an in-plane magnetic field along $y$ direction. All MBS pair combinations are obtained through modulating the phase differences between ${\phi }_1$, ${\phi }_2$, ${\phi }_3$, and ${\phi }_4$. (b) SEM image for the $\mathsf{X}\mathrm{J}$ with schematic external fluxes ${\mathrm{\Phi }}_1$ (between $S_1,S_2$), ${\mathrm{\Phi }}_2(S_2,S_3)$, and ${\mathrm{\Phi }}_3(S_3,S_4)$. (c) MBS exchange with fluxes, $\tau$ is the switching time, and ${\mathrm{\Phi }}_0$ is the magnetic flux quantum.

Figure 2

(a)–(c) Energy spectra at $B_y=0.4\mathrm{T}$ as a function of $\varphi$ for the diagonal, long- and short-edge MBS in the $\mathsf{X}\mathrm{J}$ with half-central angle $\theta =0.1\pi$, as seen in Fig. 1. Red lines: evolution of finite-energy states into MBS inside the topological gap. The $\varphi$ values between the two green lines (${\varphi }_0$) give the topologically nontrivial states. (d)–(f) Magnetic field dependence of the energy spectrum with superconducting phases $({\phi }_1,\dots ,{\phi }_4)=(\pi ,\pi ,0,0)$, (0, $\pi$, 0, 0), and (0, 0, $\pi$, 0), respectively. The red lines retain the meaning as in (a)–(c). The materials and geometry parameters are specified in the main text.

Figure 3

(a)–(e) Schematic of exchanging MBS in the $\mathsf{X}\mathrm{J}$ with external flux control, following the protocol from Fig. 1. ${\varphi }_0$ is the phase difference supporting all three MBS types shown in Fig. 2. (f)–(j) Evolution of the calculated MBS probability densities from the diagonal (a) to a long-edge MBS (b) through continuously changing ${\phi }_1$ from $\pi$ to 0, but fixing ${\phi }_2=\pi$, ${\phi }_3=0$, and ${\phi }_4=0$. (k)–(o) Evolution of the calculated MBS probability densities from diagonal (c) to short-edge MBS (d) by changing ${\phi }_2$ from $\pi$ to 0, but fixing ${\phi }_1=0$, ${\phi }_3=\pi$, and ${\phi }_4=0$. It is not important if the corners do not coincide with the ends of diagonals. The $\mathrm{Al}/\mathrm{InAs}$ parameters are taken from Fig. 2 with $B_y=0.2\mathrm{T}$.

Figure 4

(a) Schematic of four simultaneous MBS in the $\mathsf{X}\mathrm{J}$ with superconducting phases $({\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4)=(\pi ,0,\pi ,0)$. (b) Magnetic field dependence of the low-energy spectrum for (a). (c) Calculated probability densities for the two lowest-energy states in (b) with $B_y=0.2\mathrm{T}$. The parameters are taken from Fig. 2.