Multi-Terminal Josephson Junctions

• Multi-terminal Josephson junctions are superconducting devices with three or more terminals whose independent phases create multidimensional supercurrent manifolds and engineered topological states.
• They leverage phase-dependent Andreev bound states to form synthetic band structures with measurable topological invariants, such as quantized transconductance and Weyl singularities.
• These junctions enable innovative applications like nonreciprocal superconductivity, parity switches without magnetic fields, and tunable platforms for topological quantum computing.

A multi-terminal Josephson junction is a superconducting nanostructure in which three or more superconducting terminals, each characterized by its own phase, are coupled through a common normal region or a quantum dot. In contrast to the standard two-terminal Josephson effect, the presence of additional terminals introduces fundamentally new physics: the phases become independent variables, enabling (N–1)-dimensional supercurrent manifolds, phase-controlled symmetry breaking, the realization of engineered topological states, novel nonreciprocal effects, and a rich landscape of many-body and dynamical phenomena. These systems serve both as versatile platforms for condensed matter and quantum information research and as testbeds for engineering topological superconductivity, Majorana physics, unconventional supercurrent manipulation, and nontrivial quantum circuit behavior.

1. Phase Dependence and Ground-State Structure

In N-terminal Josephson junctions (N > 2), each superconducting terminal is defined by a phase φₖ. The supercurrent in a given terminal Iₖ becomes a function of all (N–1) independent phase differences, $I_k = I_k(\{\varphi_j - \varphi_i\})$, producing a multidimensional manifold for the allowed supercurrent vectors.

Scattering theory for mesoscopic superconductivity (Pankratova et al., 2018) formalizes the phase dependence by associating the Andreev bound states (ABS) with the determinant equation:

$$\det\big[1 - \alpha(E/\Delta) r S^* r^* S\big] = 0,$$

where $S$ is the normal-state scattering matrix and $r$ encodes the diagonal matrix of superconducting phases via elements $r_{jj} = e^{i\phi_j}$. The ground state energy and the supercurrents are derived as $$I_j = (2e/\hbar) \frac{\partial E_g}{\partial \phi_j}$$.

The critical current in an N-terminal device is no longer a single number but the boundary of an (N–1)-dimensional region: in experiments, zero-voltage regions map out simply-connected manifolds in the space of terminal currents (Pankratova et al., 2018). The form and geometry of these manifolds are acutely sensitive to the symmetry class (orthogonal/unitary) of $S$, with transitions observable upon breaking time-reversal symmetry by, e.g., applying a magnetic field.

2. Andreev Bound States, Topology, and Weyl Singularities

The phase dependences of the ABS in multi-terminal junctions endow them with synthetic band structures in an (N–1)-dimensional “Brillouin zone” of phase space (Gavensky et al., 2022). Topological invariants—Chern numbers—may be defined through the Berry curvature associated with the phase-dependent ABS:

$$C_{\mu\nu}^{(n)} = \frac{1}{2\pi} \int_0^{2\pi}\!\int_0^{2\pi} B_{\mu\nu}^{(n)}\, d\varphi_\mu\,d\varphi_\nu,$$

where $$B_{\mu\nu}^{(n)} = \partial_\mu A_\nu^{(n)} - \partial_\nu A_\mu^{(n)}$$ and $A_\mu^{(n)} = i\langle n|\partial_\mu|n\rangle$ (Ram et al., 21 Jan 2025). Band touchings (gap closings) at isolated points in phase space are realized as Weyl singularities carrying topological charges, analogous to those in Weyl semimetals (Gavensky et al., 2022, Ram et al., 21 Jan 2025).

The Chern number manifests experimentally as quantized transconductance:

$$\bar{G}_{\mu\nu} = -\frac{4e^2}{h} \sum_{\alpha|\epsilon_\alpha<0} C_{\mu\nu}^{(\alpha)}.$$

Accessing quantized conductance and Weyl points requires few-mode operation in the junction, achievable through local quantum point contacts (Graziano et al., 2022).

3. Parity Switches, Discrete Vortices, and Time-Reversal Symmetry Breaking

A multi-terminal geometry allows superconducting phase differences to robustly lift Kramers degeneracy without invoking magnetic fields (Heck et al., 2014). The spectrum can host zero-energy ABS crossings when a “discrete vortex” condition is satisfied—formally, when the superconducting phases wind by $2\pi$ around the junction:

$\sum_{i=1}^N (\varphi_{i+1}-\varphi_i) = 2\pi.$

At these points, ground state fermion parity can flip, protected zero-energy crossings emerge, and time-reversal symmetry is broken locally by the phase configuration, not by external fields (Heck et al., 2014). This enables control over single-fermion states, direct manipulation of spinful ABS, and preparation of parity switches essential for topological superconductivity.

4. Multiplet Supercurrents and Nonlocal Correlations

Multiplet supercurrents—currents resulting from coherent transfer of more than two electrons (such as quartets)—arise in multi-terminal circuits under specific biasing conditions: for example, in a three-terminal “star” circuit, the quartet branch corresponds to $V_1+V_2=0$, leading to phase locking of $\langle d(\varphi_1 + \varphi_2)/dt \rangle=0$ (Arnault et al., 15 Aug 2024, Arnault et al., 2022). These multiplet resonances are dynamically stabilized through synchronization of phase dynamics, analogous to the Kapitza pendulum effect, and can be detected via quantized fractional Shapiro steps under microwave irradiation (Arnault et al., 15 Aug 2024, Arnault et al., 2020).

The circuit effects are robust to elevated temperatures and are not restricted to genuine multiterminal junctions, but can be engineered with two-terminal junctions arranged in networks. Careful modeling, including coupled RCSJ equations, is required to differentiate macroscopic circuit dynamics from truly nonlocal quantum-coherent processes (Zhang et al., 2022).

5. Nonreciprocal and Anomalous Supercurrents

Multi-terminal Josephson junctions provide a platform for intrinsic nonreciprocal superconducting transport without magnetic fields. In particular, breaking spatial mirror symmetry (e.g., by asymmetric channel design in graphene) leads to a magnetic-field-free diode effect, where the critical current from terminal $i$ to $j$ differs from that from $j$ to $i$ (Zhang et al., 2023). Quantitative indicators such as the nonreciprocity efficiency

$$\eta_{jk}^\theta = \frac{|\mathcal{I}_{c,jk}^\theta| - |\mathcal{I}_{c,kj}^\theta|}{|\mathcal{I}_{c,jk}^\theta| + |\mathcal{I}_{c,kj}^\theta|}$$

show efficiencies up to $\sim30\%$ depending on asymmetry.

In four-terminal devices with spin–orbit coupling and in-plane Zeeman fields, both longitudinal and transverse supercurrents can display anomalous Josephson effect (AJE, finite current at zero phase bias) and the Josephson diode effect (JDE, direction-dependent critical current), as well as unidirectional transverse supercurrent and 4π periodicity in current-phase relations (Sahoo et al., 25 Mar 2025). These phenomena are controlled by breaking $k_x\leftrightarrow -k_x$ and $k_y\leftrightarrow -k_y$ symmetries.

6. Implementation Platforms, Control, and Detection

Multi-terminal Josephson junctions have been realized in a variety of materials, including epitaxial InAs/Al and PbTe nanowires, proximitized graphene, and networks with InGaAs and Ge-based heterostructures (Pankratova et al., 2018, Graziano et al., 2022, Gupta et al., 2023, Escribano et al., 24 Jan 2025). Gate-tunable quantum point contact geometries allow exquisite control over the number of conduction modes—which is crucial for accessing few-mode topological phases and resolving detailed Andreev spectroscopy (Graziano et al., 2022).

Distinctive features are probed using differential resistance maps, tunneling spectroscopy, and microwave irradiation (to detect Shapiro steps). Theoretical modeling employs scattering-matrix approaches, tight-binding Bogoliubov–de Gennes simulations, boundary Green’s functions, and minimal Kitaev chain reductions (Ram et al., 21 Jan 2025, Escribano et al., 24 Jan 2025).

Detection of Majorana and "poor man’s Majorana" modes can now be performed without magnetic fields, simply by phase-tuning and local gating, which enables the use of materials with strong spin–orbit coupling but small $g$-factors (such as Ge) (Escribano et al., 24 Jan 2025).

7. Applications and Outlook

Multi-terminal Josephson junctions provide an experimental platform for:

• Engineering topological bands and synthetic quantum matter (Weyl points, Chern insulators) in phase space (Gavensky et al., 2022, Ram et al., 21 Jan 2025).
• Realization of protected qubits based on higher harmonics in energy landscapes, e.g., cos 2φ qubits (Arnault et al., 2022).
• Directional superconducting diodes, on-chip heat circulators, and nonreciprocal superconducting circuits (Hwang et al., 2018, Zhang et al., 2023, Sahoo et al., 25 Mar 2025).
• All-electrical creation and manipulation of Majorana modes for topological quantum computing, without magnetic fields (Heck et al., 2014, Escribano et al., 24 Jan 2025).
• Switchable topological transistors and tunable devices for metrology, based on quantized conductance and Chern number pumping (Gavensky et al., 2022).

The ability to modify topological invariants and induce phase transitions by gate control and phase manipulation opens pathways for reconfigurable quantum electronics and protected information processing.

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This synthesis presents a detailed overview of the state-of-the-art and key principles underlying multi-terminal Josephson junctions, as revealed in recent experimental and theoretical research.