We theoretically investigate the Josephson effect between two proximized Fibonacci quasicrystals. A quasiperiodic modulation of the chemical potential on a superconducting substrate induces topological gaps and edge modes with energies above the superconducting gap. We reveal that these supragap edge modes, which we term Fibonacci-Andreev bound states, acquire a dispersion with the phase difference between superconductors when they localize at the Josephson junction interface. Shifting the quasiperiodic modulation controls the localization of these states, and their contribution to the supercurrent in short junctions can completely dominate over the conventional one from subgap Andreev bound states. Topological edge modes in quasiperiodic systems can thus strongly modify the Josephson effect, offering a route for exploring exotic physics in quasicrystals.

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• Andreev bound states
• Josephson effect
• Proximity effect
• Superconductivity
• Superconductors
• Topological materials