Hybridization of topologically distinct quartet modes in three-terminal graphene Josephson junctions

TL;DR

This work demonstrates direct, phase-resolved spectroscopy of high-order Cooper multiplets in a graphene-based three-terminal Josephson junction. By mapping Andreev-bound-state dispersions across a two-dimensional phase torus using a superconducting tunnel probe, the authors identify resonances consistent with Cooper quartets and reveal quantized winding trajectories with two distinct quartet branches. A dedicated theoretical framework, including Dyson-equation/RPA analyses and a Cooper-quartet diagram, reproduces the observed phase-space structure and explains the observed avoided crossings via coherent hybridization. The results establish multiterminal graphene junctions as a versatile platform for engineering synthetic Andreev band structures with nontrivial topology and highlight the potential for phase-controlled, high-order superconducting transport in quantum devices.

Abstract

Multiterminal Josephson junctions offer a powerful playground for exploring exotic superconducting and topological phenomena beyond the reach of conventional two-terminal devices. In this work, we present the direct spectroscopic observation of Cooper quartet resonances, a signature of correlated tunneling of two Cooper pairs across the device, in a graphene three-terminal Josephson junction (3TJJ). Using tunneling spectroscopy, we visualize how Andreev bound states (ABS) evolve across a two-dimensional superconducting phase space, controlled by the two independent phase differences in the 3TJJ. These measurements reveal sharp local minima in the differential conductance spectra locked in a specific phase condition of superconducting phase variables. The resulting quantized trajectories around the compact torus of the superconducting phase variables reveal an underlying topological winding in the multipair transport. To interpret our results, we develop a theoretical model that connects the observed quartet resonances to the coherent hybridization of multiple ABS branches, a hallmark of the rich pairing process enabled by multiterminal geometries. Our results highlight the potential of multiterminal superconducting devices to host engineered superconducting states and pave the way for new approaches to topological band structure design based on phase-controlled, higher-order superconducting transport.

Figures (15)

• Figure 1: Schematics of Cooper quartet processes and experimental device. (a) $Q_L$, $Q_R$, and $Q_B$ schematics represent three-terminal quartet processes where electron-hole conversion occurs twice at either $S_L$, $S_R$, or $S_B$, respectively. $Q_L$, $Q_R$, and $Q_B$ have phase relations as $2\varphi_L-\varphi_R-\varphi_B$, $-\varphi_L+2\varphi_R-\varphi_B$ and $-\varphi_L-\varphi_R+2\varphi_B$, respectively. (b) Scanning electron microscope (SEM) image of a representative three-terminal JJ. The magnified SEM image on the right highlights the three superconducting terminals denoted as $S_i$ ($i = L, B, R$). A fourth Al terminal is weakly coupled to the device forming a tunnel probe $S_T$. (c) Schematic representation of the device and the measurement setup. (d) Tunneling conductance $G$ as a function of phases $\varphi_L$ and $\varphi_R$ at bias voltage $V_t$ = 0 V. $\varphi_L$ and $\varphi_R$ are obtained by applying $I_L$ and $I_R$ to the flux bias lines with step sizes of 0.25 $\mu$A and 1 $\mu$A, respectively.
• Figure 2: Tunneling spectroscopy in three-terminal JJ. (a-c) Tunneling conductance $G$ as a function of independent phases $\varphi_L$ and $\varphi_R$ governed by fluxes $\Phi_L$ and $\Phi_R$ at various bias voltages $V_t$ = 50 $\mu$V (a), $V_t$ = 100 $\mu$V (b), and $V_t$ = 150 $\mu$V (c). The colormaps are obtained by varying $I_L$ and $I_R$ with a step sized of 0.25 $\mu$A and 1 $\mu$A respectively. (d) Theoretically calculated tunneling conductance as a function of phases $\varphi_L$ and $\varphi_R$ at dimensionless probe bias voltage $V_{t,eff} = |eV_t|/2\Delta$.
• Figure 3: Hybridization of quartet modes. (a) Tunneling conductance $G$ as a function of independent phases $\varphi_L$ and $\varphi_R$ at the bias voltage $V_t$ = 100 $\mu$V. The colormap is obtained by varying $I_L$ and $I_R$ with a step sized of 0.25 $\mu$A and 1 $\mu$A, respectively. White dashed lines (diamond shape) indicate two modes of the quartet resonance with quartet phase variables $\chi_q = 2\varphi_L-\varphi_R$ and $\chi_q = -\varphi_L+2\varphi_R$. (b) Diagonal line cuts along $\varphi_L$ = $t$ and $\varphi_R$ = $-t+\alpha$ with the variable $\varphi_L$ = $t$ plotted on the x-axis and the tunneling conductance $G$ on the y-axis for $\alpha = 0$ (blue) and $\alpha = \pi/3$ (orange). Local minina of $G$ can be obtained at each value of $\alpha$. (c) The scatter plot shows $\varphi_L$ ($n$) as a function of $\alpha$. The minima of $G$ and the corresponding phases $\varphi_L$ ($n$) is obtained by varying $\alpha~(\pi)$ (between -1 to 1) with a step size of $1/100$. (d) Illustration of two families of Cooper quartet resonances with quartet phase variables $\chi_q = 2\varphi_L-\varphi_R$ and $\chi_q = -\varphi_L+2\varphi_R$ indicating slope 2 (red) and slope 1/2 (blue), respectively.
• Figure S1: Tunneling gap. (a) Tunneling conductance $G$ as a function of the bias voltage $V_t$ in device 1. (b) Tunneling conductance $G$ as a function of the bias voltage $V_t$ for device 2.
• Figure S2: Tunneling spectroscopy in three-terminal JJ at $V_t \geq \Delta$. (a-d) Tunneling conductance $G$ as a function of independent phases $\varphi_L$ and $\varphi_R$ governed by fluxes $\Phi_L$ and $\Phi_R$ at constant $V_t$’s ($V_t \geq \Delta$) for device 1. The colormaps are obtained by varying $I_L$ and $I_R$ with a step sized of 0.25 $\mu$A and 1 $\mu$A respectively. The colormaps are measured at $T$ = 200 mK and back gate voltage $V_g$ = 0 V.