Cooper Quartet Resonances in Graphene Devices

• Cooper quartet resonances are correlated multipair tunneling phenomena in multiterminal Josephson junctions, especially in graphene-based devices.
• They arise from the coherent transfer of two Cooper pairs across independent superconducting phases, resulting in distinctive topological and spectroscopic signatures.
• Tunneling spectroscopy and phase tomography techniques map quantized ABS spectra and reveal topologically protected winding invariants in these systems.

Cooper quartet resonances are a distinctive manifestation of correlated multipair tunneling in multiterminal Josephson junctions, notably in graphene-based three-terminal devices. Unlike conventional supercurrent carried by paired electrons (Cooper pairs) between two superconductors, quartet resonances involve the coherent transfer of two Cooper pairs—i.e., four electrons—across multiple superconducting terminals, leading to spectroscopically and topologically rich transport phenomena. These processes fundamentally rely on the additional degrees of freedom provided by multiple independent superconducting phase differences, resulting in emergent topological structures and higher-order Josephson effects (Rashid et al., 25 Jan 2026, Jung et al., 2024, Arnault et al., 2022).

1. Device Architectures for Multiterminal Josephson Physics

Graphene-based three-terminal Josephson junctions (3TJJs) serve as the prototypical platform for observing Cooper quartet resonances due to their tunable ballistic transport, transparent interfaces, and flexible device geometries. A typical 3TJJ consists of a monolayer graphene flake encapsulated in hBN, contacted by three superconducting electrodes (e.g., MoRe or Al), which are often situated at 120° relative angles in a “Y”, “T,” or star geometry. The independent superconducting phases of the electrodes constitute the phase space for the junction.

A canonical device for Cooper quartet studies includes:

• Independent flux loops for at least two leads, enabling control over the two non-redundant phase differences, often denoted as φL and φR, with the remaining terminal (e.g., B) as the reference (φB = 0).
• A separate tunnel probe (e.g., an Al terminal weakly coupled to the central graphene region) for direct tunneling spectroscopy of Andreev bound states (ABS), allowing spatial and phase-resolved conductance mapping (Rashid et al., 25 Jan 2026, Jung et al., 2024).

These microstructures support phase dispersions and quantum interference effects inaccessible in two-terminal geometries, facilitating observation of higher-order resonances such as quartets.

2. Theoretical Framework: Andreev Bound States and Quartet Formation

The physics of Cooper quartet resonances arises from coherent Andreev processes involving all three superconducting terminals. The Josephson energy of a multiterminal junction is determined by the ABS spectrum, which depends explicitly on the independent superconducting phase differences.

The prototypical quantization condition for ABS in the short-junction limit is derived using the scattering-matrix formalism: $\det[1 - a(E)^2 S_A(\phi)] = 0,$ where $S_A(\phi)$ incorporates Andreev reflections at the contacts, and $a(E)$ captures the energy/dynamical phase accumulated by the quasiparticles (Jung et al., 2024).

Quartet resonances correspond to energy minima or level crossings in the ABS spectrum that persist along specific lines in the (φL, φR) phase space, defined for the primary resonances by the constraints: $$2\phi_L - \phi_R = \text{const} \,\qquad -\phi_L + 2\phi_R = \text{const}$$ These lines represent correlated four-electron tunneling trajectories, each characterized by unique winding numbers (e.g., (2,–1) or (–1,2)), corresponding to their topological class on the phase torus (Rashid et al., 25 Jan 2026). The coherent mixing (hybridization) of multiple ABS branches at intersections of these lines is a quantum hallmark of the system, leading to avoided crossings and robust spectroscopic signatures.

3. Experimental Observation: Tunneling Spectroscopy and Phase Tomography

Phase-resolved tunneling spectroscopy provides direct access to the ABS spectrum underlying quartet resonances. This is accomplished by:

• Using on-chip flux lines to sweep φL and φR over the entire phase torus.
• Employing a superconducting tunnel probe to measure differential conductance (dI/dV) as a function of bias and phase coordinates.

Experiments reveal sharp minima in the conductance spectra that track quantized trajectories corresponding to quartet resonance conditions (Rashid et al., 25 Jan 2026). Mapping dI/dV over (φL, φR) produces two families of resonance lines—the QL and QR quartet branches—characterized by quantized slopes (e.g., 2 and ½). The observation of avoided crossings at their intersections signals quantum hybridization of topologically distinct quartet modes.

By reconstructing the full two-dimensional ABS spectrum (i.e., spectral tomography), experiments recover both gapped and gapless phases, with transitions between them controlled by the evolution of ABS nodal lines corresponding to quartet conditions (Jung et al., 2024). The location and structure of nodal lines are consistent with theoretical predictions derived from the minimal scattering-matrix model.

4. Topological Structure of Quartet Resonances

The space of independent phase differences (φL, φR) forms a compact torus T², analogous to a Brillouin zone for Bloch electrons. Quartet resonance lines correspond to topologically protected winding sectors in this space, characterized by their integer directional vectors.

Topological phenomena manifested in these systems include:

• Nodal-line structures: Zero-energy crossings (ABS band nodes) form closed loops in T², each separating different vorticity sectors for the ground state (Jung et al., 2024).
• Winding invariants: Each gapped “island” on the torus is labeled by a pair of vorticity indices (νL, νR), which jump by ±1 upon crossing a nodal line, signifying a topological transition.
• Quantum hybridization: At intersections of distinguished winding lines, quantum coupling leads to spectrally resolvable avoided crossings, a clear signature of coherent mixing between topologically distinct quartet modes (Rashid et al., 25 Jan 2026).
• Berry curvature and Chern numbers: While topological invariants such as Berry curvature can be computed for the ABS bands, time-reversal symmetry ensures that Chern numbers vanish unless symmetry is broken (e.g., by breaking time-reversal or introducing magnetic flux) (Jung et al., 2024).

5. Dynamical and Nonequilibrium Aspects

Dynamical stabilization of multiplet supercurrents (including quartets) can occur due to phase-locking—when a combination of Josephson voltages satisfies resonance conditions of the form $nV_1 + mV_2 = 0$. The resulting dynamical state supports robust supercurrent even out of equilibrium (Arnault et al., 2022). Theoretical analysis (using the Kapitza inverted pendulum analogy) shows that these states are stabilized by a time-averaged effective potential with higher periodicities (e.g., cos 2φ for quartets), supporting multiplet supercurrent branches with zero differential resistance about the resonance line.

Fractional Shapiro steps and correlated switching phenomena provide additional confirmation of strongly coupled phase dynamics underlying quartets (Arnault et al., 2020). These effects arise because the multidimensional Josephson washboard potential supports periodic orbits (including half-integer cycles) set by the multiplet resonance conditions.

6. Implications, Applications, and Outlook

Cooper quartet resonances in multiterminal Josephson junctions open pathways toward artificial topological band structure engineering, higher-order Josephson devices, and quantum information platforms:

• Synthetic Andreev crystals: Control of multiple phase differences enables the realization of multi-dimensional Andreev minibands with engineered topological features—nodal lines, Weyl points, and quantized transconductance (Jung et al., 2024, Rashid et al., 25 Jan 2026, Draelos et al., 2018).
• Noise-protected Josephson elements: Quartet dynamics yield robust cos 2φ energy terms, foundational for double-well Josephson circuits and noise-resistant qubit implementations (Arnault et al., 2022).
• Nonreciprocal transport and diodes: Multiterminal geometry allows realization of superconducting diode effects—nonreciprocal supercurrents without applied magnetic field—by engineering device asymmetry (Zhang et al., 2023, Chiles et al., 2022).
• Topological and quantum logic devices: The topological winding structure and hybridization phenomena constitute essential components for multi-phase qubits, quantum simulators, and novel device concepts harnessing higher-order coherent dynamics (Rashid et al., 25 Jan 2026, Jung et al., 2024).

In summary, Cooper quartet resonances constitute a direct experimental and theoretical signature of multipair correlated transport in multiterminal Josephson networks, with graphene-based 3TJJs as the leading platform for their exploration. These resonances anchor a rapidly expanding research field in phase-engineered superconducting devices, enabling access to new topological regimes and robust quantum-coherent phenomena (Rashid et al., 25 Jan 2026, Jung et al., 2024, Arnault et al., 2022, Arnault et al., 2020).