Fractal energy spectrum of a polariton gas in a Fibonacci quasi-periodic potential

We report on the study of a polariton gas confined in a quasi-periodic one dimensional cavity, described by a Fibonacci sequence. Imaging the polariton modes both in real and reciprocal space, we observe features characteristic of their fractal energy spectrum such as the opening of mini-gaps obeying the gap labeling theorem and log-periodic oscillations of the integrated density of states. These observations are accurately reproduced solving an effective 1D Schr\"{o}dinger equation, illustrating the potential of cavity polaritons as a quantum simulator in complex topological geometries.