Non-Weyl Microwave Graphs

Closed and open superconducting microwave waveguide networks as a model for quantum graphs

We report on high-precision measurements that were performed with superconducting waveguide networks with the geometry of a tetrahedral and a honeycomb graph. They consist of junctions of valency three that connect straight rectangular waveguides of equal width but incommensurable lengths. The experiments were performed in the frequency range of a single transversal mode, where the associated Helmholtz equation is effectively one-dimensional and waveguide networks may serve as models of quantum graphs with the joints and waveguides corresponding to the vertices and bonds. The tetrahedral network comprises T junctions, while the honeycomb network exclusively consists of Y junctions, that join waveguides with relative angles 90^{∘} and 120^{∘}, respectively. We demonstrate that the vertex scattering matrix, which describes the propagation of the modes through the junctions, strongly depends on frequency and is nonsymmetric at a T junction and thus differs from that of a quantum graph with Neumann boundary conditions at the vertices. On the other hand, at a Y junction, similarity can be achieved in a certain frequency range. We investigate the spectral properties of closed waveguide networks and fluctuation properties of the scattering matrix of open ones and find good agreement with random matrix theory predictions for the honeycomb waveguide graph.

Quantum graphs and microwave networks as narrow-band filters for quantum and microwave devices

We investigate properties of the transmission amplitude of quantum graphs and microwave networks composed of regular polygons such as triangles and squares. We show that for the graphs composed of regular polygons, with the edges of the length l, the transmission amplitude displays a band of transmission suppression with some narrow peaks of full transmission. The peaks are distributed symmetrically with respect to the symmetry axis kl=π, where k is the wave vector. For microwave networks the transmission peak amplitudes are reduced and their symmetry is broken due to the influence of internal absorption. We demonstrate that for the graphs composed of the same polygons but separated by the edges of length l^{'} Collapse

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Affiliation(s)

Afshin Akhshani Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland
Małgorzata Białous Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland
Leszek Sirko Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland

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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.

Enhancement factor in the regime of semi-Poisson statistics in a singular microwave cavity

We investigated properties of a singular billiard, that is, a quantum billiard which contains a pointlike (zero-range) perturbation. A singular billiard was simulated experimentally by a rectangular microwave flat resonator coupled to microwave power via wire antennas which act as singular scatterers. The departure from regularity was quantitatively estimated by the short-range plasma model in which the parameter η takes the values 1 and 2 for the Poisson and semi-Poisson statistics, respectively. We show that in the regime of semi-Poisson statistics the experimental power spectrum and the second nearest-neighbor-spacing distribution P(2,s) are in good agreement with their theoretical predictions. Furthermore, the measurement of the two-port scattering matrix allowed us to evaluate experimentally the enhancement factor F(γ^{tot}) in the regime of the semi-Poisson statistics as a function of the total absorption factor γ^{tot}. The experimental results were compared with the analytical formula for F(γ^{tot}) evaluated in this article. The agreement between the experiment and theory is good.

Experimental study of closed and open microwave waveguide graphs with preserved and partially violated time-reversal invariance

We report on experiments that were performed with microwave waveguide systems and demonstrate that in the frequency range of a single transversal mode they may serve as a model for closed and open quantum graphs. These consist of bonds that are connected at vertices. On the bonds, they are governed by the one-dimensional Schrödinger equation with boundary conditions imposed at the vertices. The resulting transport properties through the vertices may be expressed in terms of a vertex scattering matrix. Quantum graphs with incommensurate bond lengths attracted interest within the field of quantum chaos because, depending on the characteristics of the vertex scattering matrix, its wave dynamic may exhibit features of a typical quantum system with chaotic counterpart. In distinction to microwave networks, which serve as an experimental model of quantum graphs with Neumann boundary conditions, the vertex scattering matrices associated with a waveguide system depend on the wave number and the wave functions can be determined experimentally. We analyze the spectral properties of microwave waveguide systems with preserved and partially violated time-reversal invariance, and the properties of the associated wave functions. Furthermore, we study properties of the scattering matrix describing the measurement process within the framework of random matrix theory for quantum chaotic scattering systems.

Fluctuation properties of the eigenfrequencies and scattering matrix of closed and open unidirectional graphs with chaotic wave dynamics

We present experimental and numerical results for the fluctuation properties in the eigenfrequency spectra and of the scattering matrix of closed and open unidirectional quantum graphs, respectively. Unidirectional quantum graphs, that are composed of bonds connected by reflectionless vertices, were introduced by Akila and Gutkin [Akila and Gutkin, J. Phys. A: Math. Theor. 48, 345101 (2015)1751-811310.1088/1751-8113/48/34/345101]. The nearest-neighbor spacing distribution of their eigenvalues was shown to comply with random-matrix theory predictions for typical chaotic systems with completely violated time-reversal invariance. The occurrence of short periodic orbits confined to a fraction of the system, that lead in conventional quantum graphs to deviations of the long-range spectral correlations from the behavior expected for typical chaotic systems, is suppressed in unidirectional ones. Therefore, we pose the question whether such graphs may serve as a more appropriate model for closed and open chaotic systems with violated time-reversal invariance than conventional ones. We compare the fluctuation properties of their eigenvalues and scattering matrix elements and observe especially in the long-range correlations larger deviations from random-matrix theory predictions for the unidirectional graphs. These are attributed to a loss of complexity of the underlying dynamic, induced by the unidirectionality.

The Generalized Euler Characteristics of the Graphs Split at Vertices

We show that there is a relationship between the generalized Euler characteristic Eo(|VDo|) of the original graph that was split at vertices into two disconnected subgraphs i=1,2 and their generalized Euler characteristics Ei(|VDi|). Here, |VDo| and |VDi| denote the numbers of vertices with the Dirichlet boundary conditions in the graphs. The theoretical results are experimentally verified using microwave networks that simulate quantum graphs. We demonstrate that the evaluation of the generalized Euler characteristics Eo(|VDo|) and Ei(|VDi|) allow us to determine the number of vertices where the two subgraphs were initially connected.

Spectral duality in graphs and microwave networks

Quantum graphs and their experimental counterparts, microwave networks, are ideally suited to study the spectral statistics of chaotic systems. The graph spectrum is obtained from the zeros of a secular determinant derived from energy and charge conservation. Depending on the boundary conditions at the vertices, there are Neumann and Dirichlet graphs. The first ones are realized in experiments, since the standard junctions connecting the bonds obey Neumann boundary conditions due to current conservation. On average, the corresponding Neumann and Dirichlet eigenvalues alternate as a function of the wave number, with the consequence that the Neumann spectrum is described by random matrix theory only locally, but adopts features of the interlacing Dirichlet spectrum for long-range correlations. Another spectral interlacing is found for the Green's function, which in contrast to the secular determinant is experimentally accessible. This is illustrated by microwave studies and numerics.

A new spectral invariant for quantum graphs

The Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic [Formula: see text], with [Formula: see text] denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic [Formula: see text] of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristic [Formula: see text] can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic [Formula: see text] a new powerful tool for studying of physical systems modeled by differential equations on metric graphs including isoscattering and neural networks where both Neumann and Dirichlet boundary conditions occur.

Missing-level statistics in a dissipative microwave resonator with partially violated time-reversal invariance

We report on the experimental investigation of the fluctuation properties in the resonance frequency spectra of a flat resonator simulating a dissipative quantum billiard subject to partial time-reversal-invariance violation (TIV) which is induced by two magnetized ferrites. The cavity has the shape of a quarter bowtie billiard of which the corresponding classical dynamics is chaotic. Due to dissipation it is impossible to identify a complete list of resonance frequencies. Based on a random-matrix theory approach we derive analytical expressions for statistical measures of short- and long-range correlations in such incomplete spectra interpolating between the cases of preserved time-reversal invariance and complete TIV and demonstrate their applicability to the experimental spectra.

Application of topological resonances in experimental investigation of a Fermi golden rule in microwave networks

We investigate experimentally a Fermi golden rule in two-edge and five-edge microwave networks with preserved time reversal invariance. A Fermi golden rule gives rates of decay of states obtained by perturbing embedded eigenvalues of graphs and networks. We show that the embedded eigenvalues are connected with the topological resonances of the analyzed systems and we find the trajectories of the topological resonances on the complex plane.

Isoscattering strings of concatenating graphs and networks

We identify and investigate isoscattering strings of concatenating quantum graphs possessing n units and 2n infinite external leads. We give an insight into the principles of designing large graphs and networks for which the isoscattering properties are preserved for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \rightarrow \infty $$\end{document}n→∞. The theoretical predictions are confirmed experimentally using \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=2$$\end{document}n=2 units, four-leads microwave networks. In an experimental and mathematical approach our work goes beyond prior results by demonstrating that using a trace function one can address the unsettled until now problem of whether scattering properties of open complex graphs and networks with many external leads are uniquely connected to their shapes. The application of the trace function reduces the number of required entries to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2n \times 2n $$\end{document}2n×2n scattering matrices \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{S}}$$\end{document}S^ of the systems to 2n diagonal elements, while the old measures of isoscattering require all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2n)^2$$\end{document}(2n)2 entries. The studied problem generalizes a famous question of Mark Kac “Can one hear the shape of a drum?”, originally posed in the case of isospectral dissipationless systems, to the case of infinite strings of open graphs and networks.

Experimental investigation of distributions of the off-diagonal elements of the scattering matrix and Wigner's K[over ̂] matrix for networks with broken time reversal invariance

We present an extensive experimental study of the distributions of the real and imaginary parts of the off-diagonal elements of the scattering matrix S[over ̂] and the Wigner's reaction K[over ̂] matrix for open microwave networks with broken time (T) reversal invariance. Microwave Faraday circulators were applied in order to break T invariance. The experimental distributions of the real and imaginary parts of the off-diagonal entries of the scattering matrix S[over ̂] are compared with the theoretical predictions from the supersymmetry random matrix theory [A. Nock, S. Kumar, H.-J. Sommers, and T. Guhr, Ann. Phys. (NY) 342, 103 (2014)10.1016/j.aop.2013.11.006]. Furthermore, we show that the experimental results are in very good agreement with the recent predictions for the distributions of the real and imaginary parts of the off-diagonal elements of the Wigner's reaction K[over ̂] matrix obtained within the framework of the Gaussian unitary ensemble of random matrix theory [S. B. Fedeli and Y. V. Fyodorov, J. Phys. A: Math. Theor. 53, 165701 (2020)1751-811310.1088/1751-8121/ab73ab]. Both theories include losses as tunable parameters and are therefore well adapted to the experimental verification.

How time-reversal-invariance violation leads to enhanced backscattering with increasing openness of a wave-chaotic system

We report on the experimental investigation of the dependence of the elastic enhancement, i.e., enhancement of scattering in the backward direction over scattering in other directions, of a wave-chaotic system with partially violated time-reversal (T) invariance on its openness. The elastic enhancement factor is a characteristic of quantum chaotic scattering which is of particular importance in experiments, like compound-nuclear reactions, where only cross sections, i.e., the moduli of the associated scattering matrix elements, are accessible. In the experiment a quantum billiard with the shape of a quarter bow tie, which generates a chaotic dynamics, is emulated by a flat microwave cavity. Partial T-invariance violation of varying strength 0≤ξ≲1 is induced by two magnetized ferrites. The openness is controlled by increasing the number M of open channels, 2≤M≤9, while keeping the internal absorption unchanged. We investigate the elastic enhancement as function of ξ and find that for a fixed M it decreases with increasing T-invariance violation, whereas it increases with increasing openness beyond a certain value of ξ≳0.2. The latter result is surprising because it is opposite to that observed in systems with preserved Tinvariance (ξ=0). We come to the conclusion that the effect of T-invariance violation on the elastic enhancement then dominates over the openness, which is crucial for experiments which rely on enhanced backscattering, since, generally, a decrease of the openness is unfeasible. Motivated by these experimental results, we performed theoretical investigations based on random matrix theory which confirm our findings.

Edge switch transformation in microwave networks

We investigated the spectra of resonances of four-vertex microwave networks simulating both quantum graphs with preserved and with partially violated time-reversal invariance before and after an edge switch operation. We show experimentally that under the edge switch operation, the spectra of the microwave networks with preserved time-reversal symmetry are level-1 interlaced, i.e., ν_{n-r}≤ν[over ̃]_{n}≤ν_{n+r}, where r=1, in agreement with the recent theoretical predictions of Aizenman et al. [M. Aizenman, H. Schanz, U. Smilansky, and S. Warzel, Acta Phys. Pol. A 132, 1699 (2017)ATPLB60587-424610.12693/APhysPolA.132.1699]. Here, we denote by {ν_{n}}_{n=1}^{∞} and {ν[over ̃]_{n}}_{n=1}^{∞} the spectra of microwave networks before and after the edge switch transformation. We demonstrate that the experimental distribution P(ΔN) of the spectral shift ΔN is close to the theoretical one. Furthermore, we show experimentally that in the case of the four-vertex networks with partially violated time-reversal symmetry, the spectra are level-1 interlaced. Our experimental results are supplemented by the numerical calculations performed for quantum graphs with violated time-reversal symmetry. In this case, the edge switch transformation also leads to the spectra which are level-1 interlaced. Moreover, we demonstrate that for microwave networks simulating graphs with violated time-reversal symmetry, the experimental distribution P(ΔN) of the spectral shift ΔN agrees, within the experimental uncertainty, with the numerical one.

Hearing Euler characteristic of graphs

The Euler characteristic χ=|V|-|E| and the total length 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 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},...,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≤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.