Relativistic quantum chaos-An emergent interdisciplinary field

Imaging Quantum Interference in Stadium-Shaped Monolayer and Bilayer Graphene Quantum Dots

Experimental realizations of graphene-based stadium-shaped quantum dots (QDs) have been few and have been incompatible with scanned probe microscopy. Yet, the direct visualization of electronic states within these QDs is crucial for determining the existence of quantum chaos in these systems. We report the fabrication and characterization of electrostatically defined stadium-shaped QDs in heterostructure devices composed of monolayer graphene (MLG) and bilayer graphene (BLG). To realize a stadium-shaped QD, we utilized the tip of a scanning tunneling microscope to charge defects in a supporting hexagonal boron nitride flake. The stadium states visualized are consistent with tight-binding-based simulations but lack clear quantum chaos signatures. The absence of quantum chaos features in MLG-based stadium QDs is attributed to the leaky nature of the confinement potential due to Klein tunneling. In contrast, for BLG-based stadium QDs (which have stronger confinement) quantum chaos is precluded by the smooth confinement potential which reduces interference and mixing between states.

Spectral statistics of light in one-dimensional graphene-based photonic crystals

The optical properties and spectral statistics of light in one-dimensional photonic crystals in the representative classes of (AB)^{N} (composed of dielectric layers) and (AGBG)^{N} (composed of periodic stacking of graphene-dielectric layers) have been investigated using the transfer matrix method and random matrix theory. The proposed method provides new predictions to determine the chaos and regularity of the optical systems. In this analysis, the chaoticity parameter with q=0 for Poisson distribution and q→1 for Wigner distribution is determined based on the random matrix theory. It has been shown that two kinds of chaos and regularity modes can be found with Brody distribution. Also, as a part of this work, we found out the regular pattern in both classes of (AB)^{N} and (AGBG)^{N} when results were fit to a Brody distribution. Moreover, the effects of different parameters such as the number of unit cells, incident angle, state of polarization, and chemical potential of the graphene nanolayers on the structures' regularity are discussed. It is found that the regular patterns are seen in the band gaps. The results show that the structure (AGBG)^{N} has an extra photonic band gap compared to (AB)^{N}, which is tunable by changing the chemical potential of the graphene nanolayers. Therefore, the possibility of external control of the regularity using a gate voltage in the graphene-based photonic crystals is obtained. Finally, comparing of TE and TM waves based on the random matrix theory, which interpolates between regular and chaotic systems, indicates that the Poisson statistics well describes the TE waves.

Semiclassical quantization of neutrino billiards

The impact of the classical dynamic on the fluctuation properties in the eigenvalue spectrum of nonrelativistic quantum billiards (QBs) are now well understood based on the semiclassical approach which provides an approximation for the fluctuating part ρ^{fluc}(k) of the spectral density in terms of a trace formula, that is, a sum over classical periodic orbits of its classical counterpart, abbreviated as CB. This connection between the eigenvalue spectrum of a quantum system and the classical periodic orbits is discernible in the Fourier transform of ρ^{fluc}(k) from eigenwave number k to length, which exhibits peaks at the lengths of the periodic orbits. The uprise of interest in properties of graphene related to their relativistic Dirac spectrum implicated the emergence of intensive studies of relativistic neutrino billiards (NBs), consisting of a spin-1/2 particle governed by the Dirac equation and confined to a bounded planar domain. In distinction to QBs, NBs do not have a well-defined classical limit. Yet comparison of their length spectra showed that for massless spin-1/2 particles those of the NB exhibit peaks at positions corresponding to the lengths of periodic orbits with an even number of reflections at the boundary of the CB associated with the corresponding QB. In order to understand the transition from the relativistic to the nonrelativistic regime, we derive an exact quantization condition for massive NBs and use it to obtain a trace formula. This trace formula provides a direct link between the spectral density of a NB and the classical dynamic of the corresponding QB through the periodic orbits of the associated CB.

Kac's isospectrality question revisited in neutrino billiards

"Can one hear the shape of a drum?" Kac raised this famous question in 1966, referring to the possibility of the existence of nonisometric planar domains with identical Dirichlet eigenvalue spectra of the Laplacian. Pairs of nonisometric isospectral billiards were eventually found by employing the transplantation method which was deduced from Sunada's theorem. Our main focus is the question to what extent isospectrality of nonrelativistic quantum billiards is present in the corresponding relativistic case, i.e., for massless spin-1/2 particles governed by the Dirac equation and confined to a domain of corresponding shape by imposing boundary conditions on the wave function components. We consider those for neutrino billiards [Berry and Mondragon, Proc. R. Soc. London A 412, 53 (1987)2053-916910.1098/rspa.1987.0080] and demonstrate that the transplantation method fails and thus isospectrality is lost when changing from the nonrelativistic to the relativistic case. To confirm this we compute the eigenvalues of pairs of neutrino billiards with the shapes of various billiards which are known to be isospectral in the nonrelativistic limit. Furthermore, we investigate their spectral properties, in particular, to find out whether not only their eigenvalues but also the fluctuations in their spectra and their length spectra differ.