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Register a new userSpectral properties of microwave graphs with local absorption
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The influence of absorption on the spectra of microwave graphs has been studied experimentally. The microwave networks were made up of coaxial cables and T junctions. First, absorption was introduced by attaching a 50 Ohm load to an additional vertex for graphs with and without time-reversal symmetry. The resulting level-spacing distributions were compared with a generalization of the Wigner surmise in the presence of open channels proposed recently by Poli et al. [Phys. Rev. Lett. 108, 174101 (2012)]. Good agreement was found using an effective coupling parameter. Second, absorption was introduced along one individual bond via a variable microwave attenuator, and the influence of absorption on the length spectrum was studied. The peak heights in the length spectra corresponding to orbits avoiding the absorber were found to be independent of the attenuation, whereas, the heights of the peaks belonging to orbits passing the absorber once or twice showed the expected decrease with increasing attenuation.
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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.
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.
Following an idea by Joyner et al. [EPL, 107 (2014) 50004] a microwave graph with antiunitary symmetry T obeying T^2=-1 has been realized. The Kramers doublets expected for such systems have been clearly identified and could be lifted by a perturbation which breaks the antiunitary symmetry. The observed spectral level spacings distribution of the Kramers doublets is in agreement with the predictions from the Gaussian symplectic ensemble (GSE), expected for chaotic systems with such a symmetry. In addition results on the two-point correlation function, the spectral form factor, the number variance and the spectral rigidity are presented, as well as on the transition from GSE to GOE statistics by continuously changing T from T^2=-1 to T^2=1.
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 Wigners 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).
We introduce a new model for investigating spectral properties of quantum graphs, a quantum circulant graph. Circulant graphs are the Cayley graphs of cyclic groups. Quantum circulant graphs with standard vertex conditions maintain important features of the prototypical quantum star graph model. In particular, we show the spectrum is encoded in a secular equation with similar features. The secular equation of a quantum circulant graph takes two forms depending on whether the edge lengths respect the cyclic symmetry of the graph. When all the edge lengths are incommensurate, the spectral statistics correspond to those of random matrices from the Gaussian Orthogonal Ensemble according to the conjecture of Bohigas, Giannoni and Schmit. When the edge lengths respect the cyclic symmetry the spectrum decomposes into subspectra whose corresponding eigenfunctions transform according to irreducible representations of the cyclic group. We show that the subspectra exhibit intermediate spectral statistics and analyze the small and large parameter asymptotics of the two-point correlation function, applying techniques developed from star graphs. The particular form of the intermediate statistics differs from that seen for star graphs or Dirac rose graphs. As a further application, we show how the secular equations can be used to obtain spectral zeta functions using a contour integral technique. Results for the spectral determinant and vacuum energy of circulant graphs are obtained from the zeta functions.
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