Planar Josephson junctions made from atomically thin films exhibit poor transverse screening, causing the magnetic-field dependence of the Josephson current Ic(B) to deviate from the standard Fraunhofer pattern of conventional junctions. Given the weak screening, the junction is highly sensitive to Pearl vortices in the leads. Vortices alter the phase pattern and affect the Josephson current. The importance of thermal and quantum fluctuations can be quantified through the superfluid density ρ_s and the sheet resistance R. The low carrier density and large effective mass associated with flat electronic bands result in a small ρ_s and a large R, which enhance fluctuation effects in both the thermal and quantum regimes. These properties make graphene particularly well suited to study vortex dynamics. Fluctuations can cause vortices to jump in and out of the leads, leading to shifts in the Fraunhofer-like pattern, as observed in recent experiment [1]. Our model quantitatively explains these jumps, whose timescale depends on magnetic field, current, temperature, and superfluid stiffness. At low temperatures Pearl vortices can tunnel through the edge barriers. At elevated temperatures, fast vortex jumps may wash out the Fraunhofer pattern well below Tc. By analyzing the timescale of these jumps, we can determine the superfluid stiffness and the Berezinskii-Kosterlitz-Thouless transition temperature of magic-angle twisted four-layer graphene. These values are in agreement with recent kinetic inductance measurements [2,3|.

1. M. Perego et al. Nature Communications 16, 10259 (2025).

2. Banerjee, A. et al. Nature 638, 93–98 (2025).

3. Tanaka, M. et al. Nature 638, 99–105 (2025).