Tuning of Weyl point emergence in multi-terminal Josephson junctions using quantum point contacts

TL;DR

The paper investigates how the emergence of Weyl points in the Andreev-bound-state spectrum of a four-terminal Josephson junction can be electrically controlled using quantum point contacts. By modeling the system with a random-matrix scattering framework and applying Beenakker's determinant condition, it shows that increasing and balancing the number of conduction channels across all terminals markedly raises the probability of Weyl-point emergence (up to about 17% for a (2,2,2,2) configuration versus 4% for single-channel cases), while channel mixing can further enhance this probability. The results provide a practical route to tuning topological Weyl physics in multi-terminal superconducting devices and have implications for experimental detection via transverse conductance and related topological measurements.

Abstract

Multi-terminal Josephson junction with three or more superconductors is an attractive quantum system to emerge and tune exotic electronic states. In four terminal Josephson junctions, the Weyl physics, namely topologically protected zero energy state, emerges without assuming any exotic materials. In this study, we consider the four-terminal Josephson junction with the quantum point contact structures between the mesoscopic normal region and four superconducting terminals. The quantum point contacts can tune electrically the number of conduction channels. We theoretically investigate an effect of the increase of channels on the emergence of Weyl points. The increase of channels causes the increase of Andreev bound states in the system, which increase the emergence probability of Weyl points. When all terminals have two channels, the emergence probability is up to 17\%, which is about four times larger than that for all single channel junctions. We consider the balance of the number of conduction channels in the four terminals. When the number of channels is unbalanced, the increase of emergence probability is suppressed.

Figures (5)

• Figure 1: Model of the four terminal Josephson junction and the WPs in the energy spectrum of Andreev bound states. (a) Schematic of the four terminal Josephson junction based on the scattering matrix. The central block $\hat{s}_{\rm N}$ is the normal region, and the green terminals are superconducting terminals with phase $\varphi_j$. The QPC structures are embedded between the superconducting terminals and the normal region. The QPCs are represented by $\hat{s}_{{\rm Q},j}$. $\hat{s}_{\rm sys}$ is the combined normal region and the QPCs. $r_{\rm he}^{(1)}$ represents Andreev reflection. $\bm{a}_j$ ($\bm{a}_j^\prime$) and $\bm{b}_j$ ($\bm{b}_j^\prime$) are the incoming and outgoing waves to the normal region (QPCs), respectively. (b) A typical result of the Andreev spectrum on a plane crossing the four WPs at $T_j = 0.9$. $\Delta_0$ is the superconducting gap. At the positions of WPs, $(\varphi_1, \varphi_2, \varphi_3) \approx \pm (0.987 \pi, 0.582 \pi, -0.238 \pi)$ and $\pm (0.438 \pi, -0.720 \pi, 0.485 \pi)$, positive (red) and negative (blue) Andreev levels touch each other at $E=0$. (c) The Andreev levels at $T_j = 0.2$ when the WPs disappear by the pair annihilation. Hence, there are only two gapped points.
• Figure 2: Probability of WPs emergence when the QPC transmission probability is changed from $T(1,1,1,1)$ to $T(1,1,2,2)$. (a) WPs emergence probability map in the $T_3,T_4$ plane at $T_1 = T_2 = 1$. (b) Cross-sectional plot along the diagonal line in (a). The WP emergence probability increases from 4.32% to 8.57%.
• Figure 3: Probability of WPs emergence when the total number of conduction channels in systems increases from six to eight. (a) From $T(1,1,2,2)$ to $T(2,2,2,2)$ with $T_1 = T_2$. The WP emergence probability varies from 8.57% to 17.24%. (b) From $T(1,1,2,2)$ to $T(1,1,3,3)$ with $T_3 = T_4$. The WP emergence probability varies from 8.85% to 12.4%.
• Figure 4: Schematic illustration for the combinations of conduction channels. As a simplified diagram for ease of understanding, the QPC 1, 2 and the QPC 3, 4 are unified, respectively.
• Figure 5: WP emergence probability when a weak channel mixing due to non-adiabatic transport through the QPCs is considered at $T(1,1,T_3,T_4)$ with $T_3=T_4=1.25$ (red), $1.50$ (green), and $1.75$ (blue lines). The channel mixing parameter is $\theta = \theta_3 = \theta_4$.