Title: Josephson effect in a fractal geometry

Abstract: The Josephson effect is a hallmark signature of the superconducting state, which, however, has been sparsely explored in non-crystalline superconducting materials. Motivated by this, we consider a Josephson junction consisting of two superconductors with a fractal metallic interlayer, which is patterned as a Sierpi\'nski carpet by removing atomic sites in a self-similar and scale-invariant manner. We here show that the fractal geometry has direct observable consequences on the Josephson effect. In particular, we demonstrate that the form of the supercurrent-magnetic field relation as the fractal generation number increases can be directly related to the self-similar fractal geometry of the normal metallic layer. Furthermore, the maxima of the corresponding diffraction pattern directly encode the self-repeating fractal structure in the course of fractal generation, implying that the corresponding magnetic length directly probes the shortest length scale in the given fractal generation. Our results should motivate future experimental efforts to verify these predictions in designer quantum materials.