arXiv subjects

Szymon Bauch

Publications and source records attributed to Szymon Bauch.

17 recordsLinked to original sources

arXiv2025

Families of isospectral and isoscattering quantum graphs

A concept of germ graphs and the M-function formalism are employed to construct large families of isospectral and isoscattering graphs. This approach represents a complete departure from the original approach pioneered by Sunada, where isospectral graphs are obtained as quotients of a certain large symmetric graph. Using the M-function formalism and the symmetries of the graph itself we construct isospectral and isoscattering pairs. In our novel approach isospectral pairs do not need to be embedded into a larger symmetric graph as in Sunada's approach. We demonstrate that the introduced formalism can also be extended to graphs with dissipation. The theoretical predictions are validated experimentally using microwave networks emulating open quantum graphs with dissipation.

quant-ph↗

arXiv2022

Distributions of the Wigner reaction matrix for microwave networks with symplectic symmetry in the presence of absorption

We report on experimental studies of the distribution of the reflection coefficients, and the imaginary and real parts of Wigner's reaction (K) matrix employing open microwave networks with symplectic symmetry and varying size of absorption. The results are compared to analytical predictions derived for the single-channel scattering case within the framework of random matrix theory (RMT). Furthermore, we performed Monte Carlo simulations based on the Heidelberg approach for the scattering (S) and K matrix of open quantum-chaotic systems and the two-point correlation function of the S-matrix elements. The analytical results and the Monte Carlo simulations depend on the size of absorption. To verify them, we performed experiments with microwave networks for various absorption strengths. We show that deviations from RMT predictions observed in the spectral properties of the corresponding closed quantum graph, and attributed to the presence of nonuniversal short periodic orbits, does not have any visible effects on the distributions of the reflection coefficients and the K and S matrices associated with the corresponding open quantum graph.

quant-ph↗

arXiv2020

Hearing Euler characteristic of graphs

The Euler characteristic $χ=|V|-|E|$ and the total length $\mathcal{L}$ are the most important topological and geometrical characteristics of a metric graph. Here, $|V|$ and $|E|$ denote the number of vertices and edges of a graph. The Euler characteristic determines the number $β$ of independent cycles in a graph while the total length determines the asymptotic behavior of the energy eigenvalues via the Weyl's law. We show theoretically and confirm experimentally that the Euler characteristic can be determined (heard) from a finite sequence of the lowest eigenenergies $λ_1, \ldots, λ_N$ of a simple quantum graph, without any need to inspect the system visually. In the experiment quantum graphs are simulated by microwave networks. We demonstrate that the sequence of the lowest resonances of microwave networks with $β\leq 3$ can be directly used in determining whether a network is planar, i.e., can be embedded in the plane. Moreover, we show that the measured Euler characteristic $χ$ can be used as a sensitive revealer of the fully connected graphs.

math-ph↗

arXiv2019

Missing level statistics and the power spectrum of level fluctuations analysis of three-dimensional chaotic microwave cavities

We present an experimental study of missing level statistics of three-dimensional chaotic microwave cavities. The investigation is reinforced by the power spectrum of level fluctuations analysis which also takes into account the missing levels. On the basis of our data sets we demonstrate that the power spectrum of level fluctuations in combination with short- and long-range spectral fluctuations provides a powerful tool for the determination of the fraction of randomly missing levels in systems that display wave chaos such as the three-dimensional chaotic microwave cavities. The experimental results are in good agreement with the analytical expressions that explicitly take into account the fraction of observed levels. We also show that in the case of incomplete spectra with many unresolved states the above procedures may fail. In such a case the random matrix theory calculations can be useful for the determination of missing levels.

cond-mat.mes-hall↗

arXiv2018

Experimental and numerical study of spectral properties of three-dimensional chaotic microwave cavities: The case of missing levels

We present an experimental and numerical study of missing-level statistics of chaotic three-dimensional microwave cavities. The nearest-neighbor spacing distribution, the spectral rigidity, and the power spectrum of level fluctuations were investigated. We show that the theoretical approach to a problem of incomplete spectra does not work well when the incompleteness of the spectra is caused by unresolved resonances. In such a case the fraction of missing levels can be evaluated by calculations based on random matrix theory.

physics.class-ph↗

arXiv2017

Non-universality 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 prob- lems 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 corrobo- rate 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.

nlin.CD↗

arXiv2016

Long-range correlations in rectangular cavities containing point-like 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 point-like perturbations. We demonstrate that their spectral properties exhibit the features typical for singular statistics. Hitherto, only the nearest-neighbor spacing distribution had been studied. We, in addition 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, 12302 (2010)].

nlin.CD↗

arXiv2016

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 in- variance. 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 pro- cedure 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.

nlin.CD↗

arXiv2015

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 the integrable one due to presence of antennas acting as scatterers is characterised by the parameter of chaoticity k = 2.8. The experimental results for the rectangular cavity are compared with the ones obtained for a microwave rough cavity simulating a chaotic quantum billiard. The experimental results were obtained for the frequency range v = 16 - 18.5 GHz and moderate absorption strength y = 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 the one obtained for the rough cavity. The results obtained for the microwave rough cavity are smaller than the ones obtained within the framework of Random Matrix Theory and lie between them and the ones predicted within a recently introduced model of the two-channel coupling (V. Sokolov and O. Zhirov, arXiv:1411.6211v2[nucl-th], 12 Dec 2014).

nlin.CD↗

arXiv2014

Resonances and poles in isoscattering microwave networks and graphs

Can one hear the shape of a graph? This is a modification of the famous question of Mark Kac "Can one hear the shape of a drum?" which can be asked in the case of scattering systems such as quantum graphs and microwave networks. It addresses an important mathematical problem whether scattering properties of such systems are uniquely connected to their shapes? Recent experimental results based on a characteristics of graphs such as the cumulative phase of the determinant of the scattering matrices indicate a negative answer to this question (O. Hul, M. Lawniczak, S. Bauch, A. Sawicki, M. Kus, L. Sirko, Phys. Rev. Lett 109, 040402 (2012).). In this paper we consider important local characteristics of graphs such as structures of resonances and poles of the determinant of the scattering matrices. Using these characteristics we study experimentally and theoretically properties of graphs and directly confirm that the pair of graphs considered in the cited paper is isoscattering. The experimental results are compared to the theoretical ones for a broad frequency range from 0.01 to 3 GHz. In the numerical calculations of the resonances of the graphs absorption present in the experimental networks is taken into account.

physics.optics↗

arXiv2012

Are Scattering Properties of Graphs Uniquely Connected to Their Shapes?

The famous question of Mark 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.

quant-ph↗

arXiv2009

Experimental and numerical investigation of the reflection coefficient and the distributions of Wigner's reaction matrix for irregular graphs with absorption

We present the results of experimental and numerical study of the distribution of the reflection coefficient P(R) and the distributions of the imaginary P(v) and the real P(u) parts of the Wigner's reaction K matrix for irregular fully connected hexagon networks (graphs) in the presence of strong absorption. In the experiment we used microwave networks, which were built of coaxial cables and attenuators connected by joints. In the numerical calculations experimental networks were described by quantum fully connected hexagon graphs. The presence of absorption introduced by attenuators was modelled by optical potentials. The distribution of the reflection coefficient P(R) and the distributions of the reaction K matrix were obtained from the measurements and numerical calculations of the scattering matrix S of the networks and graphs, respectively. We show that the experimental and numerical results are in good agreement with the exact analytic ones obtained within the framework of random matrix theory (RMT).

nlin.CD↗

arXiv2009

Investigation of nodal domains in a chaotic three-dimensional microwave rough billiard with the translational symmetry

We show that using the concept of the two-dimensional level number N_{|} one can experimentally study of the nodal domains in a three-dimensional (3D) microwave chaotic rough billiard with the translational symmetry. Nodal domains are regions where a wave function has a definite sign. We found the dependence of the number of nodal domains aleph_{N_{|}} lying on the cross-sectional planes of the cavity on the two-dimensional level number N_{|}. We demonstrate that in the limit N_{|} -> infinity the least squares fit of the experimental data reveals the asymptotic ratio aleph_{N_{|}}/N_{|} = 0.059 +- 0.029 that is close to the theoretical prediction aleph_{N_{|}}/N_{|} = 0.062. This result is in good agreement with the predictions of percolation theory.

nlin.CD↗

arXiv2009

Experimental investigation of electric field distributions in a chaotic 3D microwave rough billiard

We present the first experimental study of the electric field distributions E_N of a three-dimensional (3D) microwave chaotic rough billiard with the translational symmetry. The translational symmetry means that the cross-section of the billiard is invariant under translation along z direction. The 3D electric field distributions were measured up to the level number N = 489. In this way the experimental spatial correlation functions C_{N,p}(x,s) ~ < E_{N,p}(x+1/2*s) E_{N,p}^(*)(x - 1/2*s)> were found and compared with the theoretical ones. The experimental results for higher two-dimensional level number N_{|} appeared to be in good agreement with the theoretical predictions.

nlin.CD↗

arXiv2009

Experimental investigation of Wigner's reaction matrix for irregular graphs with absorption

We use tetrahedral microwave networks consisting of coaxial cables and attenuators connected by T-joints to make an experimental study of Wigner's reaction K matrix for irregular graphs in the presence of absorption. From measurements of the scattering matrix S for each realization of the microwave network we obtain distributions of the imaginary and real parts of K. Our experimental results are in good agreement with theoretical predictions.

nlin.CD↗

arXiv2009

Investigation of nodal domains in the chaotic microwave ray-splitting rough billiard

We study experimentally nodal domains of wave functions (electric field distributions) lying in the regime of Shnirelman ergodicity in the chaotic microwave half-circular ray-splitting rough billiard. For this aim the wave functions Psi_N of the billiard were measured up to the level number N=415. We show that in the regime of Shnirelman ergodicity (N>208) wave functions of the chaotic half-circular microwave ray-splitting rough billiard are extended over the whole energy surface and the amplitude distributions are Gaussian. For such ergodic wave functions the dependence of the number of nodal domains aleph_N on the level number N was found. We show that in the limit N->infty the least squares fit of the experimental data yields aleph_N/N = 0.063 +- 0.023 that is close to the theoretical prediction aleph_N/N = 0.062. We demonstrate that for higher level numbers N = 215-415 the variance of the mean number of nodal domains sigma^2_N/ N is scattered around the theoretical limit sigma^2_N /N = 0.05. We also found that the distribution of the areas s of nodal domains has power behavior n_s ~ s^{-tau}, where the scaling exponent is equal to tau = 2.14 +- 0.12. This result is in a good agreement with the prediction of percolation theory.

nlin.CD↗

arXiv2004

Experimental simulation of quantum graphs by microwave networks

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 GHz in order to obtain their statistical properties such as the integrated nearest neighbor spacing distribution and the spectral rigidity. 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 (GOE) in random matrix theory approaching, especially for small eigenfrequency spacing s, results for Gaussian unitary ensembles (GUE). Experimental results are supported by the theoretical analysis of directional graphs.

nlin.CD↗