Long-range phase coherence and phase patterns in hybrid Josephson junction arrays

The coherence of superconductivity and its suppression near a quantum phase transition is governed by the interplay between local pairing and macroscopic phase coherence. Using scanning SQUID, we image the local susceptibility in a hybrid Josephson junction array. On a square lattice of narrow islands, we simultaneously access both the amplitude and spatial phase structure of sensitive superconducting states. We observe periodic phase patterns at commensurate magnetic fillings. At zero field the long-range phase coherence is strongest. At a finite field, smaller than one percent of flux quantum per unit cell, the system fragments into large regions of constant superconducting phase, as a function of the applied field. Our results provide the first direct measurement of long-range phase coherence in a Josephson junction array.

💡 Research Summary

In this work the authors employ a scanning superconducting quantum interference device (SQUID) microscope to directly image the local magnetic susceptibility of a hybrid Josephson junction array (JJA), thereby accessing both the amplitude and spatial phase structure of the superconducting state. The study proceeds in two stages. First, a continuous Nb wire‑grid is examined to benchmark the SQUID’s sensitivity to phase coherence. By imaging the static flux landscape together with the AC susceptibility, the authors demonstrate that regions where weak links form closed loops exhibit a uniform diamagnetic response, whereas open‑circuit regions show a dramatically reduced signal. This confirms that the SQUID can distinguish between phase‑coherent clusters and disconnected islands even when the superfluid stiffness is very weak.

The second stage focuses on a deliberately engineered Al/InAs hybrid JJA. The array consists of cross‑shaped Al islands on a square lattice with a 3 µm pitch and 0.3 µm line width, each island coupled to its four nearest neighbours via Josephson junctions. With gate voltages set to zero, the charging energy (E_C) and Josephson energy (E_J) are fixed, allowing the magnetic field to serve as the sole tuning parameter. By applying a perpendicular field the authors vary the frustration f = Φ/Φ₀ from zero to several integer and fractional values while maintaining a base temperature of 20 mK.

Transport measurements reveal Little‑Parks‑type oscillations in the array resistance, with pronounced minima at integer fillings and at fractional fillings such as ½ and ⅓, indicating enhanced global phase coherence at these commensurate points. Simultaneously, scanning SQUID susceptibility maps show that at f = 0 the response is spatially homogeneous and strong, reflecting a fully phase‑locked state across the entire array. As f is increased to integer values (±1, ±2, …) the peak amplitude of the susceptibility diminishes and the underlying grid pattern becomes visible, signalling a progressive loss of phase stiffness. The reduction of the peaks with higher integer filling is attributed to the need for larger phase twists around each plaquette to accommodate multiple vortices, which weakens the Josephson currents.

At fractional fillings the susceptibility maps reveal richer structures. At f = −½ a checkerboard pattern emerges, consistent with theoretical predictions of alternating 0‑π phase domains (vortex‑antivortex arrangements) that minimize the Josephson energy on a frustrated lattice. Fast Fourier Transform analysis confirms that the periodicity matches the lattice spacing. In contrast, at incommensurate fillings such as f = −1.06 the susceptibility displays the bare lattice, and the inferred phase pattern consists of sparse, mobile vortices that generate local phase fluctuations and suppress global coherence.

A striking observation is that even a minute magnetic field—less than 1 % of a flux quantum per plaquette—causes the array to fragment into large, phase‑coherent domains separated by regions where the phase is no longer locked. Within each domain the Josephson currents form closed loops that screen the local AC field, while across domain boundaries the screening collapses. This “phase‑coherent clustering” demonstrates that long‑range phase order can survive locally even when the global stiffness is vanishingly small.

Overall, the paper provides (i) a clear experimental demonstration that scanning SQUID susceptometry can directly probe superconducting phase textures, (ii) quantitative evidence of how integer and fractional magnetic frustration control the emergence of ordered phase patterns and the strength of phase stiffness in a clean JJA, and (iii) the discovery of field‑induced spatial fragmentation of phase coherence. These findings have immediate relevance for studies of quantum phase transitions in low‑dimensional superconductors, for the design of superconducting qubits that rely on controllable Josephson coupling, and for the engineering of artificial superconducting metamaterials with tailored phase landscapes.