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

Technical Summary: Hybridization of Topologically Distinct Quartet Modes in Three-Terminal Graphene Josephson Junctions

Problem Statement
Multiterminal Josephson junctions (MTJJs) offer a synthetic multi-dimensional phase space where Andreev bound states (ABS) form complex band structures, hosting exotic phenomena such as higher-order Cooper multiplets. Among these, Cooper quartets—processes where two Cooper pairs coherently split across three terminals—represent a hallmark of multiterminal entanglement. While theoretical proposals suggest strategies to identify these states via current-voltage characteristics or supercurrent-phase measurements, existing experimental evidence remains largely indirect. Prior approaches rely on integrated transport signatures that obscure the fine structure of the underlying ABS spectrum, lacking the resolution to unambiguously resolve the dispersion of ABS associated with higher-order Cooper processes. A spectroscopic approach capable of isolating ABS dispersions over a multi-dimensional phase space is required to directly probe these quartet states and their topological properties.

Methodology
The authors employed a combined experimental and theoretical approach using a graphene-based three-terminal Josephson junction (3TJJ).

• Device Fabrication: The device consists of an hBN/graphene/hBN van der Waals heterostructure with edge-contacted superconducting terminals made of Ti/Al. A fourth Al terminal, weakly coupled without a Ti adhesion layer, serves as a superconducting tunneling probe ($S_T$).
• Measurement Technique: The study utilized phase-resolved tunneling spectroscopy. The differential conductance ($G = dI_t/dV_t$) was measured between the tunnel probe and the central node of the junction. The superconducting phases of the left ($\varphi_L$) and right ($\varphi_R$) terminals were independently controlled via external magnetic fluxes, while the middle terminal phase was fixed ($\varphi_B=0$).
• Bias Regimes: Measurements were conducted across various bias voltages ($V_t$). For $|eV_t| < \Delta$ (where $\Delta$ is the superconducting gap), the probe directly measures sub-gap ABS. For $|eV_t| > 2\Delta$, the probe accesses nonequilibrium ABS via quasiparticle injection and relaxation, allowing spectroscopy of in-gap states even at high bias.
• Theoretical Modeling: To interpret the data, the authors developed a theoretical model based on a Random Phase Approximation (RPA) for a disordered normal-metal–insulator–superconductor (NIS) interface. This model calculates nonlocal Andreev reflection using Green's function formalism, accounting for multiple scattering events and the constructive interference conditions required for quartet formation.

Key Results

1. Direct Spectroscopic Observation of Quartet Resonances: The differential conductance maps as a function of $\varphi_L$ and $\varphi_R$ revealed sharp local minima (resonances) locked to specific phase conditions. As the bias voltage approached the gap edge ($V_t \approx \Delta/1.8$), these resonances realigned along specific linear trajectories defined by the quartet phase variables: $\chi_q = 2\varphi_L - \varphi_R$ and $\chi_q = -\varphi_L + 2\varphi_R$. These trajectories correspond to the constructive interference of two Cooper pairs traversing the three terminals.
2. Quantized Winding Trajectories: The resonances formed a diamond-like closed loop in the $(\varphi_L, \varphi_R)$ phase space, representing quantized winding trajectories on the compact torus of superconducting phase variables. This confirms the topological nature of the multipair transport.
3. Hybridization and Avoided Crossings: A critical finding was the observation of avoided crossings between the two distinct quartet resonance branches. While classical topological arguments might suggest independent trajectories that cross freely, the experimental data showed clear hybridization. Theoretical calculations confirmed that without hybridization, the coexistence of degenerate states would generate partition noise, contradicting the noiseless nature of Josephson currents at zero temperature. The observed avoided crossings indicate the coherent quantum mechanical hybridization of ABS branches belonging to different winding sectors.
4. Robustness at High Bias: The quartet resonance patterns were observed to re-emerge at high bias voltages ($V_t \approx 2.78\Delta$), demonstrating the robustness of these higher-order processes against large bias conditions and the versatility of the device for engineering correlated states.

Significance and Claims
The paper claims to provide the first direct, phase-resolved spectroscopic evidence of higher-order Cooper quartet processes in a multiterminal device. By visualizing the evolution of ABS across a two-dimensional superconducting phase space, the authors demonstrate that multiterminal junctions naturally host two distinct forms of topology:

1. Classical Topology: Defined by integer phase-winding sectors that classify multiplet processes (e.g., the specific slopes of the resonance lines).
2. Quantum Topology: Encoded in the Andreev band structure on the phase torus, where quantum coherence dictates the interaction between modes.

The observation of avoided crossings serves as a direct signature that the Andreev spectrum cannot be understood by classical winding arguments alone; it requires the full quantum mechanical structure of the spectrum. The authors conclude that graphene-based MTJJs serve as a versatile platform for exploring exotic quantum states in synthetic superconducting lattices and pave the way for designing topological band structures based on phase-controlled, higher-order superconducting transport. The work establishes a quantitative benchmark for detecting Cooper quartets and suggests that extending this spectroscopic approach to four-terminal junctions could reveal even higher-order multiplets (such as sextets) and new topological phases.