From graphene to fullerene: experiments with microwave photonic crystals

Ultracold quantum gases serve as ideal models for the characterization of universal properties of a variety of phenomena in quantum systems. In a formal analogy to such 'quantum simulators' we demonstrate in this brief review that photonic crystals embedded into flat microwave billiards can also serve as ideal model systems for the experimental study of non-relativistic and relativistic phenomena in flat and curved structures like Graphene and Fullerene, respectively. We determined in high precision experiments with superconducting microwave billiards modeling Graphene billiards (flakes) a few thousands of eigenvalues and thereby were able to study thoroughly fluctuation properties in their spectra. Using a microwave cavity modeling a Fullerene billiard with the geometrical structure of a C-60 molecule it became possible to determine in high resolution experiments for the first time the number of zero-energy modes, i.e. of modes with energy values at the Dirac point existing due to the honeycomb structure in the carbon lattice. We were thereby able to test the so-called Atiyah-Singer index theorem which relates this number to the topology of the curved carbon lattice.