Michał Ławniczak

Power Spectrum Analysis and Missing Level Statistics of Microwave Graphs with Violated Time Reversal Invariance

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.

Experimental investigation of properties of hexagon networks with and without time reversal symmetry

We present the results of experimental study of the distribution P(R) of the reflection coefficient R and the distributions of Wigners reaction K matrix for irregular hexagon fully connected microwave networks simulating quantum graphs with time reversal symmetry (TRS) in the presence of absorption. The distributions of the reflection coefficient R and the real and imaginary parts of K matrix were obtained from the measurements of the scattering matrix S of the microwave networks. Furthermore, we show that the microwave networks with microwave circulators can be used to study systems with broken TRS. Properties of such networks are investigated using the cross-correlation function c12(ν) and the integrated nearest-neighbor spacing distribution I(s).

Experimental investigation of microwave networks simulating quantum chaotic systems: the role of direct processes

The influence of direct processes on the cross-correlation function c12(ν) and the elastic enhancement factor WS, β was studied experimentally using microwave networks in the presence of absorption. Microwave networks simulate quantum graphs with and without time-reversal symmetry. Direct processes exist because, for example, of non-perfect coupling between a studied system and a measuring device. We show that direct processes strongly influence the cross-correlation function c12(ν) in contrast to the enhancement factor WS, β, which is not sensitive to these processes. It makes the enhancement factor WS, β an important measure of the chaoticity of a quantum system.

Experimental investigation of the enhancement factor and the cross-correlation function for graphs with and without time-reversal symmetry: the open system case

We present the results of an experimental study of the enhancement factor WS,? of the two-port scattering matrix and the cross-correlation function c12(?) for microwave irregular networks simulating quantum graphs with time-reversal symmetry (TRS), symmetry index ?=1, and without TRS, ?=2, in the presence of absorption. We show that even for systems with notable losses the elastic enhancement factor WS,? remains a statistical measure that allows the determination of a symmetry class of the chaotic quantum systems under investigation. Therefore, the enhancement factor WS,? may be contrasted with many others statistical measures such as, for example, the nearest neighbor spacing distribution or the spectral rigidity, which are not directly applicable in the presence of strong absorption. The quantum graphs with TRS were simulated by the microwave networks built of coaxial cables connected by joints and attenuators, which allowed us to control absorption. The microwave circulators were added to the networks to simulate graphs with broken TRS. The experimental results are compared to the ones obtained within the framework of random matrix theory.

Nonuniversality in the spectral properties of time-reversal-invariant microwave networks and quantum graphs

We present experimental and numerical results for the long-range fluctuation properties in the spectra of quantum graphs with chaotic classical dynamics and preserved time-reversal invariance. Such systems are generally believed to provide an ideal basis for the experimental study of problems originating from the field of quantum chaos and random matrix theory. Our objective is to demonstrate that this is true only for short-range fluctuation properties in the spectra, whereas the observation of deviations in the long-range fluctuations is typical for quantum graphs. This may be attributed to the unavoidable occurrence of short periodic orbits, which explore only the individual bonds forming a graph and thus do not sense the chaoticity of its dynamics. In order to corroborate our supposition, we performed numerous experimental and corresponding numerical studies of long-range fluctuations in terms of the number variance and the power spectrum. Furthermore, we evaluated length spectra and compared them to semiclassical ones obtained from the exact trace formula for quantum graphs.

Experimental investigation of the elastic enhancement factor in a transient region between regular and chaotic dynamics

We present the results of an experimental study of the elastic enhancement factor W for a microwave rectangular cavity simulating a two-dimensional quantum billiard in a transient region between regular and chaotic dynamics. The cavity was coupled to a vector network analyzer via two microwave antennas. The departure of the system from an integrable one due to the presence of antennas acting as scatterers is characterized by the parameter of chaoticity κ=2.8. The experimental results for the rectangular cavity are compared with those obtained for a microwave rough cavity simulating a chaotic quantum billiard. The experimental results were obtained for the frequency range ν=16-18.5 GHz and moderate absorption strength γ=5.2-7.4. We show that the elastic enhancement factor for the rectangular cavity lies below the theoretical value W=3 predicted for integrable systems, and it is significantly higher than that obtained for the rough cavity. The results obtained for the microwave rough cavity are smaller than those obtained within the framework of random matrix theory, and they lie between them and those predicted within a recently introduced model of the two-channel coupling [V. V. Sokolov and O. V. Zhirov, arXiv:1411.6211 [nucl-th]].

Experimental investigation of the scattering fidelity in microwave networks simulating quantum graphs

We present the results of the evaluation of the scattering fidelity in fully connected five vertices microwave networks with absorption. The scattering fidelity was calculated on the basis of the measurements of the one- and two-port scattering matrices S of the networks performed in a frequency range 8–9 GHz. We show that the scattering fidelity depends on the length of the bonds and openness of the networks.

Experimental and numerical determination of the correlation function of level velocities for microwave networks simulating quantum graphs

The parameter-dependent correlation function of level velocities is studied experimentally and numerically. The measurements were made for microwave networks simulating quantum graphs. One- and two-port measurements of the scattering matrix necessary for determining the correlation function were realized for the five vertices networks. For the fully connected six vertices network, one-port measurements were made. The obtained experimental and numerical results are compared with the predictions of random matrix theory.

Long-range correlations in rectangular cavities containing pointlike perturbations.

We investigated experimentally the short- and long-range correlations in the fluctuations of the resonance frequencies of flat, rectangular microwave cavities that contained antennas acting as pointlike perturbations. We demonstrate that their spectral properties exhibit the features typical for singular statistics. Hitherto, only the nearest-neighbor spacing distribution had been studied. In addition we considered statistical measures for the long-range correlations and analyzed power spectra. Thereby we could corroborate that the spectral properties change to semi-Poisson statistic with increasing microwave frequency. Furthermore, the experimental results are shown to be well described by a model applicable to billiards containing a zero-range perturbation [T. Tudorovskiy et al., New J. Phys. 12, 123021 (2010)NJOPFM1367-263010.1088/1367-2630/12/12/123021].