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We compute the real-space profile of the superconducting order parameter (OP)in a hybrid ring that consists of a 1D superconductor connected to a Fibonaccichain using a self-consistent approach. In our study, the strength of thepenetration, as measured by the order parameter at the center of thequasicrystal, depends on the structural parameter $\phi$, or phason angle, thatcharacterizes different realizations of the Fibonacci chains of a given length.We show that the penetration strength dependence on $\phi$ reflects propertiesof the topological edge states of the Fibonacci chain. We show that the inducedsuperconducting order parameter averaged over all chains has a power law decayas a function of distance from the S-N interface. More interestingly, we showthat there are large OP fluctuations for individual chains and that thepenetration strength in a finite Fibonacci chain can be significantly $larger$than in a normal periodic conductor for special values of $\phi$.

The Proximity Effect in a Superconductor-Quasicrystal Hybrid Ring