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1
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Quantum Graphology

100%
Kottos T.
Acta Physica Polonica A
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2006
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vol. 109
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issue 1
7-21
EN
We review quantum chaos on graphs. We construct a unitary operator which represents the quantum evolution on the graph and study its spectral and wave function statistics. This operator is the analogue of the classical evolution operator on the graph. It allows us to establish a connection between the corresponding periodic orbits and the statistical properties of eigenvalues and eigenfunctions. Specifically, for the energy-averaged spectral form factor we derived an exact combinatorial expression which illustrate the role of correlations between families of isometric orbits. We also show that enhanced wave function localization due to the presence of short unstable periodic orbits and strong scarring can rely on completely different mechanisms
2
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Nodal Domains in Chaotic Microwave Rough Billiards with and without Ray-Splitting Properties

100%
Hul O., Savytskyy N., Tymoshchuk O., Bauch S., Sirko L.
Acta Physica Polonica A
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2006
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vol. 109
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issue 1
73-87
EN
We study experimentally nodal domains of wave functions (electric field distributions) lying in the regime of Breit-Wigner ergodicity in the chaotic microwave half-circular ray-splitting rough billiard. Using the rough billiard without ray-splitting properties we also study the wave functions lying in the regime of Shnirelman ergodicity. The wave functionsΨ_N of the ray-splitting billiard were measured up to the level number N=204. In the case of the rough billiard without ray-splitting properties, the wave functions were measured up to N=435. We show that in the regime of Breit-Wigner ergodicity most of wave functions are delocalized in the n, l basis. In the regime of Shnirelman ergodicity wave functions are homogeneously distributed over the whole energy surface. For such wave functions, lying both in the regimes of Breit-Wigner and Shnirelman ergodicity, the dependence of the number of nodal domainsƝ_N on the level number N was found. We show that in the regimes of Breit-Wigner and Shnirelman ergodicity least squares fits of the experimental data reveal the numbers of nodal domains that in the asymptotic limit N→∞ coincide within the error limits with the theoretical predictionƝ_N/N≃ 0.062. Finally, we demonstrate that the signed area distributionΣ_A can be used as a useful criterion of quantum chaos.
3
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Numerical and Experimental Studies of the Elastic Enhancement Factor for 2D Open Systems

100%
Ławniczak M., Białous M., Yunko V., Bauch S., Sirko L.
Acta Physica Polonica A
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2015
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vol. 128
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issue 6
974-978
EN
We present the results of numerical and experimental studies of the elastic enhancement factor W for microwave rough and rectangular cavities simulating two-dimensional chaotic and partially chaotic quantum billiards in the presence of moderate absorption. We show that for the frequency range ν=15.0-18.5 GHz, in which the coupling between antennas and the system is strong enough, the values of W for the microwave rough cavity lie below the predictions of random matrix theory and on average above the theoretical results of V. Sokolov and O. Zhirov, Phys. Rev. E 91, 052917 (2015). We also show that for the partially chaotic rectangular billiard the enhancement factor W calculated by applying the Potter-Rosenzweig model with κ=2.8±0.5 is close to the experimental one.
4
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Experimental Determination of the Autocorrelation Function of Level Velocities for Microwave Networks Simulating Quantum Graphs

100%
Ławniczak M., Borkowska A., Hul O., Bauch S., Sirko L.
Acta Physica Polonica A
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2011
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vol. 120
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issue 6A
A-185-A-190
EN
The autocorrelation function c(x) of level velocities is studied experimentally. The measurements were performed for microwave networks simulating quantum graphs. One and two ports measurements of the scattering matrix Ŝ necessary for determining c(x) were realized for the networks possessing 5 and 6 vertices, respectively. The network with six vertices was fully connected. In the case of the networks with five vertices, additionally to the fully connected configuration, we measured the networks without the bond connecting input/output vertices. The obtained experimental results besides the autocorrelation function of level velocities, also the nearest-neighbor spacing distribution and parametric velocities distribution are compared to the predictions of random matrix theory and numerical results.
5
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Manifold Approach for a Many-Body Wannier-Stark System: Localization and Chaos in Energy Space

100%
Parra-Murillo C., Wimberger S.
Acta Physica Polonica A
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2013
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vol. 124
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issue 6
1091-1097
EN
We study the resonant tunneling effect in a many-body Wannier-Stark system, realized by ultracold bosonic atoms in an optical lattice subjected to an external Stark force. The properties of the many-body system are effectively described in terms of upper-band excitation manifolds, which allow for the study of the transition between regular and quantum chaotic spectral statistics. We show that our system makes it possible to control the spectral statistics locally in energy space by the competition of the force and the interparticle interaction. By a time-dependent sweep of the Stark force the dynamics is reduced to a Landau-Zener problem in the single-particle setting.
6
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Experimental and Numerical Studies of One-Dimensional and Three-Dimensional Chaotic Open Systems

100%
Ławniczak M., Hul O., Bauch S., Sirko L.
Acta Physica Polonica A
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2009
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vol. 116
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issue 5
749-755
EN
We present the results of experimental studies of microwave irregular networks and a three-dimensional microwave rough cavity in the presence of absorption. Microwave networks are also analyzed numerically. Microwave networks simulate one-dimensional quantum graphs. The networks consist of coaxial cables connected by joints and attenuators to control absorption. Three-dimensional microwave rough cavities have no formal analog in quantum 3D systems. However, some statistical properties of their spectra such as the level spacing distribution confirms that they belong to the wave-chaotic systems. Absorption of the cavity was changed by using a foam microwave absorber.
7
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Scattering Properties of Chaotic Microwave Billiards

100%
Stöckmann H.
Acta Physica Polonica A
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2009
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vol. 116
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issue 5
783-789
EN
Reflection and transmission measurements in microwave billiards with attached antennas allow the determination of all components of the scattering matrix including their phases. This is an extraordinary situation, since in usual scattering experiments in nuclear or atomic physics only reduced information such as the scattering cross-section can be obtained where the phase information is completely lost. This allows experimental tests of theoretical predictions of scattering theory inaccessible by any other method. As an example the distribution of reflection coefficients and of the line widths in a chaotic microwave billiard are discussed.
8
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Edge Switching Transformations of Quantum Graphs

100%
Aizenman M., Schanz H., Smilansky U., Warzel S.
Acta Physica Polonica A
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2017
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vol. 132
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issue 6
1699-1703
EN
Discussed here are the effects of basics graph transformations on the spectra of associated quantum graphs. In particular it is shown that under an edge switch the spectrum of the transformed Schrödinger operator is interlaced with that of the original one. By implication, under edge swap the spectra before and after the transformation, denoted by {Eₙ}^{∞}ₙ₌₁ and {Ẽₙ}^{∞}ₙ₌₁ correspondingly, are level-2 interlaced, so that Eₙ-₂ ≤ Ẽₙ ≤ Eₙ₊₂. The proofs are guided by considerations of the quantum graphs' discrete analogs.
9
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Experimental Investigation of Reflection Coefficient and Wigner's Reaction Matrix for Microwave Graphs

100%
Hul O., Bauch S., Ławniczak M., Sirko L.
Acta Physica Polonica A
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2007
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vol. 112
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issue 4
655-664
EN
We present the results of experimental study of the distribution P(R) of the reflection coefficient R and the distributions of Wigner's reaction K matrix for irregular, tetrahedral microwave graphs (networks) in the presence of absorption. Our experimental results are in good agreement with the exact theoretical predictions of Savin et al.
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Anderson Localization Phenomenon in One-Dimensional Elastic Systems

80%
Méndez-Sánchez R., Gutiérrez L., Morales A., Flores J., Díaz-de-Anda A., Monsivais G.
Acta Physica Polonica A
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2013
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vol. 124
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issue 6
1063-1068
EN
The phenomenon of the Anderson localization of waves in elastic systems is studied. We analyze this phenomenon in two different sets of systems: disordered linear chains of harmonic oscillators and disordered rods which oscillate with torsional waves. The first set is analyzed numerically whereas the second one is studied both experimentally and theoretically. In particular, we discuss the localization properties of the waves as a function of the frequency. In doing that we have used the inverse participation ratio, which is related to the localization length. We find that the normal modes localize exponentially according to the Anderson theory. In the elastic systems, the localization length decreases with frequency. This behavior is in contrast with what happens in analogous quantum mechanical systems, for which the localization length grows with energy. This difference is explained by means of the properties of the reflection coefficient of a single scatterer in each case.
11
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Electrical Resonance Circuits as Analogs to Quantum Mechanical Billiards

80%
Berggren K., Larsson J., Bengtsson O.
Acta Physica Polonica A
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2006
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vol. 109
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issue 1
33-42
EN
We propose that a two-dimensional electric network may be used for fundamental studies of wave function properties, transport, and related statistics. Using Kirchhoff's current law and the jω-method we find that the network is analogous to a discretized Schrödinger equation for quantum billiards and dots. Thus the complex electric potentials play the role of quantum mechanical wave functions.
12
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Probing Eigenfunction Nonorthogonality by Parametric Shifts of Resonance Widths

80%
Savin D., De Vaulx J.
Acta Physica Polonica A
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2013
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vol. 124
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issue 6
1074-1077
EN
Recently, it has been shown that the change of resonance widths in an open system under a perturbation of its interior is a sensitive indicator of the nonorthogonality of resonance states. We apply this measure to quantify parametric motion of the resonances. In particular, a strong redistribution of the widths is linked with the maximal degree of nonorthogonality. Then for weakly open chaotic systems we discuss the effect of spectral rigidity on the statistical properties of the parametric width shifts, and derive the distribution of the latter in a picket-fence model with equidistant spectrum.
13
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Asymmetry Induced Localization

80%
Kohler H.
Acta Physica Polonica A
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2013
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vol. 124
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issue 6
1053-1059
EN
We consider a two-level system, which couples via non-commuting operators to two independent oscillator baths. When the coupling is symmetric, the renormalized hopping matrix element is finite even for infinitely strong coupling strength. The two-level system is in a delocalized phase. For finite coupling strength a localization transition occurs for a critical asymmetry angle, which separates the localized from the delocalized phase. Using the method of flow equations we are able to monitor real time dynamics.
14
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Impedance and Scattering Variance Ratios of Complicated Wave Scattering Systems in the Low Loss Regime

80%
Yeh J., Drikas Z., Gil Gil J., Hong S., Taddese B., Ott E., Antonsen T., Andreadis T., Anlage S.
Acta Physica Polonica A
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2013
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vol. 124
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issue 6
1045-1052
EN
Random matrix theory successfully predicts universal statistical properties of complicated wave scattering systems in the semiclassical limit, while the random coupling model offers a complete statistical model with a simple additive formula in terms of impedance to combine the predictions of random matrix theory and nonuniversal system-specific features. The statistics of measured wave properties generally have nonuniversal features. However, ratios of the variances of elements of the impedance matrix are predicted to be independent of such nonuniversal features and thus should be universal functions of the overall system loss. In contrast with impedance variance ratios, scattering variance ratios depend on nonuniversal features unless the system is in the high loss regime. In this paper, we present numerical tests of the predicted universal impedance variance ratios and show that an insufficient sample size can lead to apparent deviation from the theory, particularly in the low loss regime. Experimental tests are carried out in three two-port microwave cavities with varied loss parameters, including a novel experimental system with a superconducting microwave billiard, to test the variance-ratio predictions in the low loss time-reversal-invariant regime. It is found that the experimental results agree with the theoretical predictions to the extent permitted by the finite sample size.
15
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Scattering from a Ring Graph - A~Simple Model for the Study of Resonances

80%
Waltner D., Smilansky U.
Acta Physica Polonica A
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2013
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vol. 124
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issue 6
1087-1090
EN
Scattering from the very simple ring graph is shown to display several basic features which underlie the complex (chaotic) phenomena observed in scattering from more complex graphs. In particular we demonstrate the appearance of arbitrarily narrow resonances - the "topological resonances" which are directly linked to the existence of cycles. We use the ring graph to study the response of such resonances to perturbations induced by a time-dependent random noise.
16
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Correlation Functions of Impedance and Scattering Matrix Elements in Chaotic Absorbing Cavities

80%
Savin D., Fyodorov Y., Sommers H.
Acta Physica Polonica A
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2006
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vol. 109
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issue 1
53-64
EN
Wave scattering in chaotic systems with a uniform energy loss (absorption) is considered. Within the random matrix approach we calculate exactly the energy correlation functions of different matrix elements of impedance or scattering matrices for systems with preserved or broken time-reversal symmetry. The obtained results are valid at any number of arbitrary open scattering channels and arbitrary absorption. Elastic enhancement factors (defined through the ratio of the corresponding variance in reflection to that in transmission) are also discussed.
17
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Elastic Enhancement Factor: Mesoscopic Systems versus Macroscopic 2D Electromagnetic Analogue Devices

80%
Sokolov V., Zhirov O.
Acta Physica Polonica A
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2015
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vol. 128
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issue 6
990-993
EN
Excess of probabilities of the elastic processes over the inelastic ones is a common feature of the resonance phenomena described in the framework of the random matrix theory. This effect is quantitatively characterized by the elastic enhancement factor F^{(β)} that is the typical ratio of elastic and inelastic cross-sections. Being measured experimentally, this quantity can supply us with information on specific features of the dynamics of the intermediate complicated open system. We discuss properties of the enhancement factor in a wide scope from mesoscopic systems to macroscopic analogue electromagnetic resonators and demonstrate essential qualitative distinction between the elastic enhancement factor's peculiarities in these two cases. Complete analytical solution is found for the case of systems without time-reversal symmetry and only a few open equivalent scattering channels.
18
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Aspects of Integrability in a Classical Model for Non-Interacting Fermionic Fields

80%
Grosse-Holz S., Engl T., Richter K., Urbina J.
Acta Physica Polonica A
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2015
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vol. 128
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issue 6
994-1001
EN
In this work we investigate the issue of integrability in a classical model for non-interacting fermionic fields. This model is constructed via classical-quantum correspondence obtained from the semiclassical treatment of the quantum system. Our main finding is that the classical system, contrary to the quantum system, is not integrable in general. Regarding this contrast it is clear that in general classical models for fermionic quantum systems have to be handled with care. Further numerical investigation of the system showed that there may be islands of stability in the phase space. We also investigated a similar model that is used in theoretical chemistry and found this one to be most probably integrable, although also here the integrability is not assured by the quantum-classical correspondence principle.
19
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Chaotic Scattering: Exact Results and Microwave Experiments

80%
Guhr T.
Acta Physica Polonica A
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2015
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vol. 128
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issue 6
963-967
EN
Scattering experiments are indispensable for the study of classical and quantum systems. In the Heidelberg approach, universal features are addressed by assuming that the reaction zone is fully quantum chaotic. Although it stems from nuclear physics, it later on turned out to be applicable to a large variety of systems on different scales, including classical wave systems. For a long time, the distribution of the off-diagonal scattering-matrix elements resisted analytical treatment. I review two recent studies in which my collaborators and I fully solved this problem. We also carried out a comparison with data from microwave experiments.
20
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Fading Statistics in Communications - a Random Matrix Approach

80%
Yeh J., Ott E., Antonsen T., Anlage S.
Acta Physica Polonica A
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2011
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vol. 120
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issue 6A
A-85-A-88
EN
Fading is the time-dependent variations in signal strength measured at a receiver, due to temporally evolving multipath scattering and interference. In our previous work we introduced a statistical fading model for the time-reversal invariant case by combining the predictions of random matrix theory with the random coupling model that includes system-specific properties such as the radiation impedance of the ports and short-orbit effects. In the high-loss limit this random matrix theory model reduced to the most common fading models in the wireless communication field. In this paper we discuss the theoretical model in more detail and extend it to the case of broken time-reversal invariance.
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