We present the results of experimental and theoretical study of irregular, tetrahedral microwave networks consisting of coaxial cables (annular waveguides) connected by $T$ joints. The spectra of the networks were measured in the frequency range $0.0001–16\text{GHz}$ in order to obtain their statistical properties such as the integrated nearest neighbor spacing distribution and the spectral rigidity ${\Delta }_3\left(L\right)$. The comparison of our experimental and theoretical results shows that microwave networks can simulate quantum graphs with time reversal symmetry. In particular, we use the spectra of the microwave networks to study the periodic orbits of the simulated quantum graphs. We also present experimental study of directional microwave networks consisting of coaxial cables and Faraday isolators for which the time reversal symmetry is broken. In this case our experimental results indicate that spectral statistics of directional microwave networks deviate from predictions of Gaussian orthogonal ensembles in random matrix theory approaching, especially for small eigenfrequency spacing $s$, results for Gaussian unitary ensembles. Experimental results are supported by the theoretical analysis of directional graphs.

We present experimental and theoretical results on highly excited Rydberg atoms passing through a waveguide. The waveguide is excited in a coherent mode with a superimposed component of technically generated noise. In the theoretical part of the paper we derive and solve a master equation for a Rydberg atom driven by a monochromatic coherent microwave field in the presence of noise. We show that a Rydberg atom subjected to a mixture of coherent modes and noise fields exhibits four dynamical regimes: (i) diffusive broadening, (ii) localization, (iii) destruction of coherence and localization, and (iv) relaxation to equilibrium. The four regimes are passed one after the other as a function of irradiation time. They occur on different time scales and are thus temporally well separated from each other. The theory is checked by an experiment on the time dependence of the population distribution of highly excited rubidium Rydberg atoms initially prepared in a unique and well-defined Rydberg state and irradiated by a strong microwave field. The localization regime, characterized by a ‘‘freezing’’ of the width of the wave packet with respect to the Rydberg levels, has been observed. The addition of a small noise component was shown to lead to delocalization after times inversely proportional to the noise power, as predicted by our theory.

We present the results of the experimental study of the two-port scattering matrix S\ifmmode \hat{}\else \^{}\fi{} elastic enhancement factor WS,$\beta${} for microwave irregular networks simulating quantum graphs with preserved and broken time reversal symmetry in the presence of moderate and strong absorption. In the experiment, quantum graphs with preserved time reversal symmetry were simulated by microwave networks which were built of coaxial cables and attenuators connected by joints. Absorption in the networks was controlled by the length of microwave cables and the use of microwave attenuators. In order to simulate quantum graphs with broken time reversal symmetry we used the microwave networks with microwave circulators. We show that the experimental results obtained for networks with moderate and strong absorption are in good agreement with the ones obtained within the framework of random matrix theory.

The famous question of Kac ``can one hear the shape of a drum?'' addressing the unique connection between the shape of a planar region and the spectrum of the corresponding Laplace operator, can be legitimately extended to scattering systems. In the modified version, one asks whether the geometry of a vibrating system can be determined by scattering experiments. We present the first experimental approach to this problem in the case of microwave graphs (networks) simulating quantum graphs. Our experimental results strongly indicate a negative answer. To demonstrate this we consider scattering from a pair of isospectral microwave networks consisting of vertices connected by microwave coaxial cables and extended to scattering systems by connecting leads to infinity to form isoscattering networks. We show that the amplitudes and phases of the determinants of the scattering matrices of such networks are the same within the experimental uncertainties. Furthermore, we demonstrate that the scattering matrices of the networks are conjugated by the so-called transplantation relation.

We measured the transmission spectra of asymmetrically shaped three-dimensional (3D) microwave cavities to determine resonant frequencies. We used the Balian-Bloch formula [R. Balian and C. Bloch, Ann. Phys. (N.Y.) 84, 559 (1974), and 64, 271 (1971), Eq. (I.1)] for electromagnetic waves in a three-dimensional cavity with smooth walls to check that very few resonances were missed up to 14 GHz. After normalizing them with the local mean eigenmode spacing, we unfolded the resonance spectra and found that the distribution of electromagnetic eigenmodes of the irregular 3D microwave cavities displays a statistical behavior characteristic for classically chaotic quantum systems, viz., the Wigner distribution. We found that this result did not depend on the exact irregular shape of the 3D cavity, suggesting that it is universal.

We present experimental studies of the power spectrum and other fluctuation properties in the spectra of microwave networks simulating chaotic quantum graphs with violated time reversal invariance. On the basis of our data sets, we demonstrate that the power spectrum in combination with other long-range and also short-range spectral fluctuations provides a powerful tool for the identification of the symmetries and the determination of the fraction of missing levels. Such a procedure is indispensable for the evaluation of the fluctuation properties in the spectra of real physical systems like, e.g., nuclei or molecules, where one has to deal with the problem of missing levels.