From graphene to fullerene: experiments with microwave photonic crystals

Abstract

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.

Kac's isospectrality question revisited in neutrino billiards

“…This question is of relevance in the field of relativistic quantum chaos which emerged recently [27][28][29][30][31][32][33][34][35][36][37][38] with the pioneering fabrication of graphene [39][40][41]; see also Refs. [42][43][44] for recent reviews. To obtain an answer, we analysed neutrino billiards (NBs) of corresponding shapes which were introduced in the seminal work Ref.…”

Eigenfunction and eigenmode-spacing statistics in chaotic photonic crystal graphs

“…Recent photonic topological insulator studies present an alternative form of PC waveguide using the interface between two different topological domains [30][31][32][33][34]. Bulk PC systems have also been employed to construct chaotic billiard systems [12,35]. Here, we will utilize the defect waveguide modes to build chaotic graph structures.…”

“…Wave-chaotic phenomena have been studied in various systems, ranging from 1D graphs [1][2][3][4][5], 2D billiards [6][7][8][9][10][11][12] to 3D enclosures [13][14][15][16][17][18]. The statistical properties of many system quantities, such as the closed system eigenvalues and the open system scattering/impedance matrices, exhibit universal characteristics, which only depend on general symmetries (e.g., time-reversal, symplectic) and the degree of system loss.…”

Dirac cones for bi- and trilayer Bernal-stacked graphene in a quantum graph model

“…and we have a Dirac cone, since the valence (the "-" sign above) and the conducting (the "+" sign) bands touch (approximately) linearly. We present a precise definition in Section 4. See [34,14,30,10] for descriptions of interesting experiments with artificial graphene, that is, some synthetic structures that permit the study of photonic crystals with Dirac cone dispersion and topologically protected edge states; the idea is to built systems for which the physics is simpler to explore than graphene itself, and electrons may be replaced with photons, plasmons or microcavity polaritons.…”