Scaling Theory of the Anderson Transition in Random Graphs: Ergodicity and Universality.

We study the Anderson transition on a generic model of random graphs with a tunable branching parameter 1

Multifractality of quantum wave functions in the presence of perturbations.

We present a comprehensive study of the destruction of quantum multifractality in the presence of perturbations. We study diverse representative models displaying multifractality, including a pseudointegrable system, the Anderson model, and a random matrix model. We apply several types of natural perturbations which can be relevant for experimental implementations. We construct an analytical theory for certain cases and perform extensive large-scale numerical simulations in other cases. The data are analyzed through refined methods including double scaling analysis. Our results confirm the recent conjecture that multifractality breaks down following two scenarios. In the first one, multifractality is preserved unchanged below a certain characteristic length which decreases with perturbation strength. In the second one, multifractality is affected at all scales and disappears uniformly for a strong-enough perturbation. Our refined analysis shows that subtle variants of these scenarios can be present in certain cases. This study could guide experimental implementations in order to observe quantum multifractality in real systems.

Two scenarios for quantum multifractality breakdown.

We expose two scenarios for the breakdown of quantum multifractality under the effect of perturbations. In the first scenario, multifractality survives below a certain scale of the quantum fluctuations. In the other one, the fluctuations of the wave functions are changed at every scale and each multifractal dimension smoothly goes to the ergodic value. We use as generic examples a one-dimensional dynamical system and the three-dimensional Anderson model at the metal-insulator transition. Based on our results, we conjecture that the sensitivity of quantum multifractality to perturbation is universal in the sense that it follows one of these two scenarios depending on the perturbation. We also discuss the experimental implications.

Trace formula for chaotic dielectric resonators tested with microwave experiments.

We measured the resonance spectra of two stadium-shaped dielectric microwave resonators and tested a semiclassical trace formula for chaotic dielectric resonators proposed by Bogomolny et al. [Phys. Rev. E 78, 056202 (2008)]. We found good qualitative agreement between the experimental data and the predictions of the trace formula. Deviations could be attributed to missing resonances in the measured spectra in accordance with previous experiments [Phys. Rev. E 81, 066215 (2010)]. The investigation of the numerical length spectrum showed good qualitative and reasonable quantitative agreement with the trace formula. It demonstrated, however, the need for higher-order corrections of the trace formula. The application of a curvature correction to the Fresnel reflection coefficients entering the trace formula yielded better agreement, but deviations remained, indicating the necessity of further investigations.

Trace formula for dielectric cavities. III. TE modes.

The construction of the semiclassical trace formula for resonances with transverse electric polarization for two-dimensional dielectric cavities is discussed. Special attention is given to the derivation of the two first terms of Weyl's series for the average number of such resonances. The formulas obtained agree well with numerical calculations for dielectric cavities of different shapes.

Trace formula for dielectric cavities. II. Regular, pseudointegrable, and chaotic examples.

Dielectric resonators are open systems particularly interesting due to their wide range of applications in optics and photonics. In a recent paper [Phys. Rev. E 78, 056202 (2008)] the trace formula for both the smooth and the oscillating parts of the resonance density was proposed and checked for the circular cavity. The present paper deals with numerous shapes which would be integrable (square, rectangle, and ellipse), pseudointegrable (pentagon), and chaotic (stadium), if the cavities were closed (billiard case). A good agreement is found between the theoretical predictions, the numerical simulations, and experiments based on organic microlasers.

Trace formula for dielectric cavities: general properties.

The construction of the trace formula for open dielectric cavities is examined in detail. Using the Krein formula it is shown that the sum over cavity resonances can be written as a sum over classical periodic orbits for the motion inside the cavity. The contribution of each periodic orbit is the product of the two factors. The first is the same as in the standard trace formula and the second is connected with the product of reflection coefficients for all points of reflection with the cavity boundary. Two asymptotic terms of the smooth resonance counting function related with the area and the perimeter of a convex cavity are derived. The coefficient of the perimeter term differs from the one for closed cavities due to unusual high-energy asymptotics of the S matrix for the scattering on the cavity. Corrections to the leading semi-classical formula are briefly discussed. Obtained formulas agree well with numerical calculations for circular dielectric cavities.