Exploring the energy spectrum of a four-terminal Josephson junction: Towards topological Andreev band structures

TL;DR

The paper addresses realizing and characterizing topological Andreev bands in high-dimensional synthetic space by studying a phase-controlled four-terminal Josephson junction. It combines 3D phase-resolved tunneling spectroscopy with a minimal four-terminal/three-dot model to reveal tri-Andreev molecule formation and predict Weyl nodes in the ABS spectrum. The work shows that independent tuning of three superconducting phases yields Weyl-node–like zero-energy crossings and topologically nontrivial bands, robust over a broad parameter range, with transparency-driven hybridization shaping the spectra. This establishes a practical platform for exploring topological ABSs in multiterminal superconducting devices and points toward microwave-based spectroscopic routes to resolve Weyl physics in these systems.

Abstract

Hybrid multiterminal Josephson junctions (JJs) are expected to harbor a novel class of Andreev bound states (ABSs), including topologically nontrivial states in four-terminal devices. In these systems, topological phases emerge when ABSs depend on at least three superconducting phase differences, resulting in a three-dimensional (3D) energy spectrum characterized by Weyl nodes at zero energy. Here, we realize a four-terminal JJ in a hybrid Al/InAs heterostructure, where ABSs form a synthetic 3D band structure. We probe the energy spectrum using tunneling spectroscopy and identify spectral features associated with the formation of a tri-Andreev molecule, a bound state whose energy depends on three superconducting phases and, therefore, is able to host topological ABSs. The experimental observations are well described by a numerical model. The calculations predict the appearance of four Weyl nodes at zero energy within a gap smaller than the experimental resolution. These topological states are theoretically predicted to remain stable within an extended region of the parameter space, well accessible by our device. These findings establish an experimental foundation to study high-dimensional synthetic band structures in multiterminal JJs, and to realize topological Andreev bands.

Figures (14)

• Figure 1: (a) False-colored scanning electron micrograph of the four-terminal device under study, showing the three-loop geometry. (b) Close-up view of the scattering area taken prior to the gate deposition. Gates structures are drawn on the image (yellow). (c) Tunneling spectroscopy measurement, showing the differential conductance $G$ as a function of bias $V_{\rm{sd}}$ and the current $I_{\rm{L}}$ flowing in flux line L. (d) Differential conductance map measured at $V_{\rm{sd}} = -175~\mu \rm{V}$ as a function of the flux-line currents $I_{\rm{L}}$ and $I_{\rm{R}}$. (e) As in (d), but measured as a function of the magnetic fluxes $\Phi_{\rm{L}}$ and $\Phi_{\rm{R}}$ after the current-to-flux remapping (see text). The schematics on the right of (d) and (e) represent the orientations of the corresponding maps with respect to the cubic unit cell in the 3D flux space.
• Figure 2: (a) Tunneling differential conductance $G$ measured as a function of DC voltage bias $V_{\rm{sd}}$ and magnetic flux $\Phi_{\rm{L}}$, with $\Phi_{\rm{M}} = \Phi_{\rm{R}} = 0$. (b) As in (a), but varying $\Phi_{\rm{M}}$ with $\Phi_{\rm{L}} = \Phi_{\rm{R}} = 0$. (c) Tunneling differential conductance measured at fixed $V_{\rm{sd}} = -200$$\mu \rm{V}$ and plotted in the 3D flux unit cell for values $G \geq 0.095 \times 2e^2/h$. The unit cell axes are also labeled using the crystallographic-like notation used to define specific directions in the flux space. (d,e) Flux-flux maps of two cube faces [as indicated by the colored squares in (c)], measured at $V_{\rm{sd}} = -200$$\mu \rm{V}$, showing avoided crossing induced by the formation of bi-Andreev molecules. (f,g) $G$ measured as a function of $V_{\rm{sd}}$ along the directions (110) and (101), as indicated by the white arrows in (d) and (e), respectively.
• Figure 3: (a) Slicing the cubic unit cell (see schematic) by measuring $\Phi_{\rm{L}}$-$\Phi_{\rm{R}}$ maps at different values of $\Phi_{\rm{M}}$, with voltage bias $V_{\rm{sd}}$ kept fixed at $-175~\mu \rm{V}$. (b,c) Cuts measured as a function of $V_{\rm{sd}}$ along the $(101)$ and $(10\overline{1})$ direction (see gray arrows in (a) at $\Phi_M = 0.5~\Phi_0$). In the schematics on the right of the plots, the currents flowing in the two outer loops of device are sketched, revealing that mutual inductive effects on the center loop are expected along $\Phi_{101}$ and not along $\Phi_{10\overline{1}}$.
• Figure 4: (a) Schematic of the model comprising three two-level quantum dots (blue, red, green) coupled to each other and to four superconducting terminals (T1-T4). Andreev bound states (ABSs) form in the quantum dots and have energies dispersing as a function of the superconducting phase differences $\phi_i$. The model parameters (see definitions in the text) are: $\epsilon = 0.005\Delta$, $\Gamma = 0.37\Delta$ and $t= 0.12\Delta$. (b) Phase-effective flux relation including mutual inductive coupling between the loops when the other phases $\phi_{\rm{L}}$ and $\phi_{\rm{R}}$ are kept constant to zero (dashed gray line) and when they are equal to $\phi_{\rm{M}}$ (solid purple line). (c) Simulated energy spectrum along $\Phi_{\rm{L}}$, with $\Phi_{\rm{M}} = \Phi_{\rm{R}} = 0$. The energy axis $E$ is shifted by $\Delta = 175~\mu \rm{eV}$ to align to the experimental data. (d) Simulated flux-flux map at $E = 0$ corresponding to the measurement shown in Fig.\ref{['fig3']}(a) for $\Phi_{\rm{M}} = 0.5~\Phi_0$. (e) Conceptual illustration of the hybridization among three degenerate energy levels resulting in bonding, nonbonding and antibonding states separated by energy gaps, as in a tri-atomic molecule. (f) Simulated energy spectra along $\Phi_{10\overline{1}}$ for different $\Phi_{\rm{M}}$ values, showing that hybridization between ABSs opens energy gaps at the crossing points between different levels. The colored squares refer to the arrows in Fig. \ref{['fig5']}(f).
• Figure 5: Experimental (a) and simulated (b) zero-energy plane (at fixed $V_{\rm{sd}}=-175~\mu \rm{V}$ or $E=0$, respectively) along a diagonal of the cubic unit cell (top-left schematic). Experimental (c) and simulated (d) energy spectrum along the diagonal direction of (a,b), i.e., along the flux direction $\Phi_{111}$ [gray arrow in (a)]. (e,f) As in (a,b), but along another diagonal plane of the cubic unit cell $\Phi_{10 \overline{1}}$-$\Phi_{\rm{M}}$ (bottom-left schematic). Experimental (g) and simulated (h) energy spectrum along the flux direction $\Phi_{1\overline{1}\overline{1}}$ [gray arrow in (e)]