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Dietz B, Klaus T, Masi M, Miski-Oglu M, Richter A, Skipa T, Wunderle M. Closed and open superconducting microwave waveguide networks as a model for quantum graphs. Phys Rev E 2024; 109:034201. [PMID: 38632749 DOI: 10.1103/physreve.109.034201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/03/2023] [Accepted: 02/08/2024] [Indexed: 04/19/2024]
Abstract
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.
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Affiliation(s)
- Barbara Dietz
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Korea
- Basic Science Program, Korea University of Science and Technology (UST), Daejeon 34113, Republic of Korea
| | - Tobias Klaus
- Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany
| | - Marco Masi
- Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany
| | - Maksym Miski-Oglu
- GSI Helmholtzzentrum für Schwerionenforschung GmbH, D-64291 Darmstadt, Germany
| | - Achim Richter
- Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany
| | - Tatjana Skipa
- Darmstadt University of Applied Sciences, D-64295 Darmstadt, Germany
| | - Marcus Wunderle
- Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany
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2
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Akhshani A, Białous M, Sirko L. Quantum graphs and microwave networks as narrow-band filters for quantum and microwave devices. Phys Rev E 2023; 108:034219. [PMID: 37849123 DOI: 10.1103/physreve.108.034219] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/01/2023] [Accepted: 09/19/2023] [Indexed: 10/19/2023]
Abstract
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^{'}
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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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3
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Che J, Zhang X, Zhang W, Dietz B, Chai G. Fluctuation properties of the eigenfrequencies and scattering matrix of closed and open unidirectional graphs with chaotic wave dynamics. Phys Rev E 2022; 106:014211. [PMID: 35974604 DOI: 10.1103/physreve.106.014211] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2022] [Accepted: 07/04/2022] [Indexed: 06/15/2023]
Abstract
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.
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Affiliation(s)
- Jiongning Che
- Lanzhou Center for Theoretical Physics and the Gansu Provincial Key Laboratory of Theoretical Physics, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Xiaodong Zhang
- Lanzhou Center for Theoretical Physics and the Gansu Provincial Key Laboratory of Theoretical Physics, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Weihua Zhang
- Lanzhou Center for Theoretical Physics and the Gansu Provincial Key Laboratory of Theoretical Physics, Lanzhou University, Lanzhou, Gansu 730000, China
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Korea
| | - Barbara Dietz
- Lanzhou Center for Theoretical Physics and the Gansu Provincial Key Laboratory of Theoretical Physics, Lanzhou University, Lanzhou, Gansu 730000, China
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Korea
| | - Guozhi Chai
- School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China
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4
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Farooq O, Ławniczak M, Akhshani A, Bauch S, Sirko L. The Generalized Euler Characteristics of the Graphs Split at Vertices. ENTROPY 2022; 24:e24030387. [PMID: 35327898 PMCID: PMC8947457 DOI: 10.3390/e24030387] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 02/02/2022] [Revised: 02/25/2022] [Accepted: 03/08/2022] [Indexed: 12/10/2022]
Abstract
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.
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Abstract
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.
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Seim E, Kohler A, Lukacs R, Brandsrud MA, Marstein ES, Olsen E, Blümel R. Wave chaos enhanced light trapping in optically thin solar cells. CHAOS (WOODBURY, N.Y.) 2021; 31:063136. [PMID: 34241303 DOI: 10.1063/5.0049330] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/2021] [Accepted: 06/02/2021] [Indexed: 06/13/2023]
Abstract
Enhancing the energy output of solar cells increases their competitiveness as a source of energy. Producing thinner solar cells is attractive, but a thin absorbing layer demands excellent light management in order to keep transmission- and reflection-related losses of incident photons at a minimum. We maximize absorption by trapping light rays to make the mean average path length in the absorber as long as possible. In chaotic scattering systems, there are ray trajectories with very long lifetimes. In this paper, we investigate the scattering dynamics of waves in a model system using principles from the field of quantum chaotic scattering. We quantitatively find that the transition from regular to chaotic scattering dynamics correlates with the enhancement of the absorption cross section and propose the use of an autocorrelation function to assess the average path length of rays as a possible way to verify the light-trapping efficiency experimentally.
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Affiliation(s)
- E Seim
- RealTek, Norwegian University of Life Sciences, Ås 1430, Norway
| | - A Kohler
- RealTek, Norwegian University of Life Sciences, Ås 1430, Norway
| | - R Lukacs
- The Norwegian Public Service Pension Fund, Oslo 0212, Norway
| | - M A Brandsrud
- RealTek, Norwegian University of Life Sciences, Ås 1430, Norway
| | - E S Marstein
- Department of Solar Energy, Institute of Energy Technology, Kjeller 2007, Norway
| | - E Olsen
- RealTek, Norwegian University of Life Sciences, Ås 1430, Norway
| | - R Blümel
- Department of Physics, Wesleyan University, Middletown, Connecticut 06459-0155, USA
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Che J, Lu J, Zhang X, Dietz B, Chai G. Missing-level statistics in classically chaotic quantum systems with symplectic symmetry. Phys Rev E 2021; 103:042212. [PMID: 34005854 DOI: 10.1103/physreve.103.042212] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2020] [Accepted: 03/30/2021] [Indexed: 11/07/2022]
Abstract
We present experimental and theoretical results for the fluctuation properties in the incomplete spectra of quantum systems with symplectic symmetry and a chaotic dynamics in the classical limit. To obtain theoretical predictions, we extend the random-matrix theory (RMT) approach introduced in Bohigas and Pato [O. Bohigas and M. P. Pato, Phys. Rev. E 74, 036212 (2006)PLEEE81539-375510.1103/PhysRevE.74.036212] for incomplete spectra of quantum systems with orthogonal symmetry. We validate these RMT predictions by randomly extracting a fraction of levels from complete sequences obtained numerically for quantum graphs and experimentally for microwave networks with symplectic symmetry and then apply them to incomplete experimental spectra to demonstrate their applicability. Independently of their symmetry class, quantum graphs exhibit nongeneric features which originate from nonuniversal contributions. Part of the associated eigenfrequencies can be identified in the level dynamics of parameter-dependent quantum graphs and extracted, thereby yielding spectra with systematically missing eigenfrequencies. We demonstrate that, even though the RMT approach relies on the assumption that levels are missing at random, it is possible to determine the fraction of missing levels and assign the appropriate symmetry class by comparison of their fluctuation properties with the RMT predictions.
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Affiliation(s)
- Jiongning Che
- Lanzhou Center for Theoretical Physics and the Gansu Provincial Key Laboratory of Theoretical Physics, Lanzhou University, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Junjie Lu
- Lanzhou Center for Theoretical Physics and the Gansu Provincial Key Laboratory of Theoretical Physics, Lanzhou University, Lanzhou University, Lanzhou, Gansu 730000, China.,Institut de Physique de Nice, CNRS UMR 7010, Université Côte d'Azur, 06108 Nice, France
| | - Xiaodong Zhang
- Lanzhou Center for Theoretical Physics and the Gansu Provincial Key Laboratory of Theoretical Physics, Lanzhou University, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Barbara Dietz
- Lanzhou Center for Theoretical Physics and the Gansu Provincial Key Laboratory of Theoretical Physics, Lanzhou University, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Guozhi Chai
- School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China
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8
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Ławniczak M, Lipovský J, Białous M, Sirko L. Application of topological resonances in experimental investigation of a Fermi golden rule in microwave networks. Phys Rev E 2021; 103:032208. [PMID: 33862759 DOI: 10.1103/physreve.103.032208] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/29/2020] [Accepted: 03/02/2021] [Indexed: 11/07/2022]
Abstract
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.
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Affiliation(s)
- Michał Ławniczak
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland
| | - Jiří Lipovský
- Department of Physics, Faculty of Science, University of Hradec Králové, Rokitanského 62, 500 03 Hradec Králové, Czechia
| | - 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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9
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Ławniczak M, van Tiggelen B, Sirko L. 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. Phys Rev E 2020; 102:052214. [PMID: 33327135 DOI: 10.1103/physreve.102.052214] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/08/2020] [Accepted: 11/03/2020] [Indexed: 11/07/2022]
Abstract
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.
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Affiliation(s)
- Michał Ławniczak
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland
| | | | - Leszek Sirko
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland
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10
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Lu J, Che J, Zhang X, Dietz B. Experimental and numerical investigation of parametric spectral properties of quantum graphs with unitary or symplectic symmetry. Phys Rev E 2020; 102:022309. [PMID: 32942473 DOI: 10.1103/physreve.102.022309] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/05/2020] [Accepted: 07/29/2020] [Indexed: 11/07/2022]
Abstract
We present experimental and numerical results for the parametric fluctuation properties in the spectra of classically chaotic quantum graphs with unitary or symplectic symmetry. A level dynamics is realized by changing the lengths of a few bonds parametrically. The long-range correlations in the spectra reveal at a fixed parameter value deviations from those expected for generic chaotic systems with corresponding universality class. They originate from modes which are confined to individual bonds or explore only a fraction of the quantum graph. Similarly, discrepancies are observed in the avoided-crossing distribution, velocity correlation function, and the curvature distribution of the level dynamics which also may be attributed to such localized modes. We demonstrate that these may be easily identified by inspecting the level dynamics and consequently their nonuniversal contributions to the parametric spectral properties may be diminished considerably. This is corroborated by numerical studies.
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Affiliation(s)
- Junjie Lu
- School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Jiongning Che
- School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Xiaodong Zhang
- School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Barbara Dietz
- School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China
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Seim E, Kohler A, Lukacs R, Brandsrud MA, Marstein ES, Olsen E, Blümel R. Chaos: A new mechanism for enhancing the optical generation rate in optically thin solar cells. CHAOS (WOODBURY, N.Y.) 2019; 29:093132. [PMID: 31575130 DOI: 10.1063/1.5111042] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2019] [Accepted: 09/06/2019] [Indexed: 06/10/2023]
Abstract
The photogenerated current of solar cells can be enhanced by light management with surface structures. For solar cells with optically thin absorbing layers, it is especially important to take advantage of this fact through light trapping. The general idea behind light trapping is to use structures, either on the front surface or on the back, to scatter light rays to maximize their path length in the absorber. In this paper, we investigate the potential of chaotic scattering for light trapping. It is well known that the trajectories close to the invariant set of a chaotic scatterer spend a very long time inside of the scatterer before they leave. The invariant set, also called the chaotic repeller, contains all rays of infinite length that never enter or leave the region of the scatterer. If chaotic repellers exist in a system, a chaotic dynamics is present in the scatterer. As a model system, we investigate an elliptical dome structure placed on top of an optically thin absorbing film, a system inspired by the chaotic Bunimovich stadium. A classical ray-tracing program has been developed to classify the scattering dynamics and to evaluate the absorption efficiency, modeled with Beer-Lambert's law. We find that there is a strong correlation between the enhancement of absorption efficiency and the onset of chaotic scattering in such systems. The dynamics of the systems was shown to be chaotic by their positive Lyapunov exponents and the noninteger fractal dimension of their scattering fractals.
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Affiliation(s)
- E Seim
- RealTek, Norwegian University of Life Sciences, Ås 1430, Norway
| | - A Kohler
- RealTek, Norwegian University of Life Sciences, Ås 1430, Norway
| | - R Lukacs
- RealTek, Norwegian University of Life Sciences, Ås 1430, Norway
| | - M A Brandsrud
- RealTek, Norwegian University of Life Sciences, Ås 1430, Norway
| | - E S Marstein
- Department of Solar Energy, Institute of Energy Technology, Kjeller 2007, Norway
| | - E Olsen
- RealTek, Norwegian University of Life Sciences, Ås 1430, Norway
| | - R Blümel
- Department of Physics, Wesleyan University, Middletown, CT 06457, USA
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12
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Ławniczak M, Lipovský J, Sirko L. Non-Weyl Microwave Graphs. PHYSICAL REVIEW LETTERS 2019; 122:140503. [PMID: 31050459 DOI: 10.1103/physrevlett.122.140503] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/29/2018] [Indexed: 06/09/2023]
Abstract
One of the most important characteristics of a quantum graph is the average density of resonances, ρ=(L/π), where L denotes the length of the graph. This is a very robust measure. It does not depend on the number of vertices in a graph and holds also for most of the boundary conditions at the vertices. Graphs obeying this characteristic are called Weyl graphs. Using microwave networks that simulate quantum graphs we show that there exist graphs that do not adhere to this characteristic. Such graphs are called non-Weyl graphs. For standard coupling conditions we demonstrate that the transition from a Weyl graph to a non-Weyl graph occurs if we introduce a balanced vertex. A vertex of a graph is called balanced if the numbers of infinite leads and internal edges meeting at a vertex are the same. Our experimental results confirm the theoretical predictions of [E. B. Davies and A. Pushnitski, Analysis and PDE 4, 729 (2011)] and are in excellent agreement with the numerical calculations yielding the resonances of the networks.
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Affiliation(s)
- Michał Ławniczak
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland
| | - Jiří Lipovský
- Department of Physics, Faculty of Science, University of Hradec Králové, Rokitanského 62, 500 03 Hradec Králové, Czechia
| | - Leszek Sirko
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland
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13
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Investigation of the diagonal elements of the Wigner's reaction matrix for networks with violated time reversal invariance. Sci Rep 2019; 9:5630. [PMID: 30948771 PMCID: PMC6449383 DOI: 10.1038/s41598-019-42123-y] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/03/2018] [Accepted: 03/25/2019] [Indexed: 11/11/2022] Open
Abstract
The distributions of the diagonal elements of the Wigner’s reaction \documentclass[12pt]{minimal}
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\begin{document}$$\hat{K}$$\end{document}Kˆ matrix for open systems with violated time reversal T invariance in the case of large absorption are for the first time experimentally studied. The Wigner’s reaction matrix links the properties of chaotic systems with the scattering processes in the asymptotic region. Microwave networks consisting of microwave circulators were used in the experiment to simulate quantum graphs with violated T invariance. The distributions of the diagonal elements of the reaction \documentclass[12pt]{minimal}
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\begin{document}$$\hat{K}$$\end{document}Kˆ matrix were experimentally evaluated by measuring of the two-port scattering matrix \documentclass[12pt]{minimal}
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\begin{document}$$\hat{S}$$\end{document}Sˆ. The violation of T invariance in the networks with large absorption was demonstrated by calculating the enhancement factor W of the matrix \documentclass[12pt]{minimal}
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\begin{document}$$\hat{S}$$\end{document}Sˆ. Our experimental results are in very good agreement with the analytic ones attained for the Gaussian unitary ensemble in the random matrix theory. The obtained results suggest that the distributions P(ʋ) and P(u) of the imaginary and the real parts of the diagonal elements of the Wigner’s reaction \documentclass[12pt]{minimal}
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\begin{document}$$\hat{K}$$\end{document}Kˆ matrix together with the enhancement factor W can be used as a powerful tool for identification of systems with violated T symmetry and quantification of their absorption.
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Dietz B, Yunko V, Białous M, Bauch S, Ławniczak M, Sirko L. Nonuniversality in the spectral properties of time-reversal-invariant microwave networks and quantum graphs. Phys Rev E 2017; 95:052202. [PMID: 28618543 DOI: 10.1103/physreve.95.052202] [Citation(s) in RCA: 35] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2017] [Indexed: 11/06/2022]
Abstract
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.
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Affiliation(s)
- Barbara Dietz
- Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
| | - Vitalii Yunko
- Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
| | - Małgorzata Białous
- Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
| | - Szymon Bauch
- Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
| | - Michał Ławniczak
- Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
| | - Leszek Sirko
- Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland
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15
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Białous M, Yunko V, Bauch S, Ławniczak M, Dietz B, Sirko L. Power Spectrum Analysis and Missing Level Statistics of Microwave Graphs with Violated Time Reversal Invariance. PHYSICAL REVIEW LETTERS 2016; 117:144101. [PMID: 27740838 DOI: 10.1103/physrevlett.117.144101] [Citation(s) in RCA: 35] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2016] [Indexed: 06/06/2023]
Abstract
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.
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Affiliation(s)
- Małgorzata Białous
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland
| | - Vitalii Yunko
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland
| | - Szymon Bauch
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland
| | - Michał Ławniczak
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland
| | - Barbara Dietz
- 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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16
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Vinayak, Kumar S, Pandey A. Parametric number covariance in quantum chaotic spectra. Phys Rev E 2016; 93:032217. [PMID: 27078354 DOI: 10.1103/physreve.93.032217] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2015] [Indexed: 11/07/2022]
Abstract
We study spectral parametric correlations in quantum chaotic systems and introduce the number covariance as a measure of such correlations. We derive analytic results for the classical random matrix ensembles using the binary correlation method and obtain compact expressions for the covariance. We illustrate the universality of this measure by presenting the spectral analysis of the quantum kicked rotors for the time-reversal invariant and time-reversal noninvariant cases. A local version of the parametric number variance introduced earlier is also investigated.
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Affiliation(s)
- Vinayak
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
| | - Sandeep Kumar
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India.,Department of Physics, H. N. B. Government PG College, Naini, Allahabad 211008, India
| | - Akhilesh Pandey
- School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India
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17
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Ławniczak M, Bauch S, Hul O, Sirko L. Experimental investigation of the enhancement factor for microwave irregular networks with preserved and broken time reversal symmetry in the presence of absorption. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:046204. [PMID: 20481804 DOI: 10.1103/physreve.81.046204] [Citation(s) in RCA: 43] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/28/2009] [Indexed: 05/29/2023]
Abstract
We present the results of the experimental study of the two-port scattering matrix S[over ] elastic enhancement factor W{S,beta} for microwave irregular networks simulating quantum graphs with preserved and broken time reversal symmetry in the presence of moderate and strong absorption. In the experiment, quantum graphs with preserved time reversal symmetry were simulated by microwave networks which were built of coaxial cables and attenuators connected by joints. Absorption in the networks was controlled by the length of microwave cables and the use of microwave attenuators. In order to simulate quantum graphs with broken time reversal symmetry we used the microwave networks with microwave circulators. We show that the experimental results obtained for networks with moderate and strong absorption are in good agreement with the ones obtained within the framework of random matrix theory.
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Affiliation(s)
- Michał Ławniczak
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland
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18
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Hul O, Seba P, Sirko L. Departure of some parameter-dependent spectral statistics of irregular quantum graphs from random matrix theory predictions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:066204. [PMID: 19658580 DOI: 10.1103/physreve.79.066204] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/12/2009] [Indexed: 05/28/2023]
Abstract
Parameter-dependent statistical properties of spectra of totally connected irregular quantum graphs with Neumann boundary conditions are studied. The autocorrelation functions of level velocities c(x) and c[over ](omega,x) as well as the distributions of level curvatures and avoided crossing gaps are calculated. The numerical results are compared with the predictions of random matrix theory for Gaussian orthogonal ensemble (GOE) and for coupled GOE matrices. The application of coupled GOE matrices was justified by studying localization phenomena in graphs' wave functions Psi(x) using the inverse participation ratio and the amplitude distribution P(Psi(x)) .
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Affiliation(s)
- Oleh Hul
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland
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19
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Lawniczak M, Hul O, Bauch S, Seba P, Sirko L. Experimental and numerical investigation of the reflection coefficient and the distributions of Wigner's reaction matrix for irregular graphs with absorption. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:056210. [PMID: 18643145 DOI: 10.1103/physreve.77.056210] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/19/2008] [Indexed: 05/26/2023]
Abstract
We present the results of an 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 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 modeled by optical potentials. The distribution of the reflection coefficient P(R) and the distributions of the reaction K matrix were obtained from 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.
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Affiliation(s)
- Michał Lawniczak
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland
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20
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Bhullar AS, Blümel R, Koch PM. Ghost orbit spectroscopy. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:016211. [PMID: 16486262 DOI: 10.1103/physreve.73.016211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/25/2005] [Indexed: 05/06/2023]
Abstract
Direct periodic-orbit expansions of individual spectral eigenvalues is a new direction in quantum mechanics. Using a unitary -matrix theory, we present exact, convergent, integral-free ghost orbit expansions of spectral eigenvalues for a step potential in the tunneling regime. We suggest an experiment to extract ghost orbit information from measured spectra in the tunneling regime (ghost orbit spectroscopy). We contrast our unitary, convergent theory with a recently published nonunitary, divergent theory [Yu. Dabaghian and R. Jensen, Eur. J. Phys. 26, 423 (2005)].
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Affiliation(s)
- A S Bhullar
- Department of Physics, Wesleyan University, Middletown, Connecticut 06459-0155, USA
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21
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Hul O, Savytskyy N, Tymoshchuk O, Bauch S, Sirko L. Investigation of nodal domains in the chaotic microwave ray-splitting rough billiard. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:066212. [PMID: 16486045 DOI: 10.1103/physreve.72.066212] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2005] [Indexed: 05/06/2023]
Abstract
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 of the billiard were measured up to the level number . We show that in the regime of Shnirelman ergodicity 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 on the level number was found. We show that in the limit the least squares fit of the experimental data yields , which is close to the theoretical prediction . We demonstrate that for higher level numbers the variance of the mean number of nodal domains is scattered around the theoretical limit . We also found that the distribution of the areas of nodal domains has power behavior , where the scaling exponent is equal to . This result is in good agreement with the prediction of percolation theory.
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Affiliation(s)
- Oleh Hul
- Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa, Poland
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22
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Vaa C, Koch PM, Blümel R. Weyl formula: experimental test of ray splitting and corner corrections. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:056211. [PMID: 16383731 DOI: 10.1103/physreve.72.056211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2005] [Indexed: 05/05/2023]
Abstract
The number of resonances N(f) of a resonator below frequency f is an essential concept in physics. Smooth approximations N(f) are known as Weyl formulas. An abrupt change in the properties of the wave propagation medium in a resonator was predicted by Prange [Phys. Rev. E 53, 207 (1996)] to produce a universal ray-splitting correction to N(f). We confirm this effect experimentally. Our results with a quasi-two-dimensional dielectric-loaded microwave cavity are directly relevant to the ray-splitting correction in two-dimensional quantal ray-splitting billiards. Our experimental spectra have sufficient accuracy and extent to allow, as far as we are aware, the first experimental determination of the corner correction, which we find to agree with theory. We show that our movable-bar setup enhances non-Newtonian periodic orbits, thereby providing an experimental technique for periodic-orbit spectroscopy. This technique, differential spectroscopy, will facilitate the study of non-Newtonian classical physics.
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Affiliation(s)
- C Vaa
- Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA
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23
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Décanini Y, Folacci A. Weyl formulas for annular ray-splitting billiards. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:046204. [PMID: 14683032 DOI: 10.1103/physreve.68.046204] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2003] [Indexed: 05/24/2023]
Abstract
We consider the distribution of eigenvalues for the wave equation in annular (electromagnetic or acoustic) ray-splitting billiards. These systems are interesting in that the derivation of the associated smoothed spectral counting function can be considered as a canonical problem. This is achieved by extending a formalism developed by Berry and Howls for ordinary (without ray-splitting) billiards [Berry and Howls, Proc. R. Soc. London, Ser. A 447, 527 (1994)]. Our results are confirmed by numerical computations and permit us to infer a set of rules useful in order to obtain Weyl formulas for more general ray-splitting billiards.
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Affiliation(s)
- Yves Décanini
- SPE, UMR CNRS 6134, Equipe Physique Semi-Classique (et) de la Matière Condensée, Faculté des Sciences, Université de Corse, Boîte Postale 52, 20250 Corte, France.
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24
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Smolyarenko IE, Simons BD. Parametric spectral statistics in unitary random matrix ensembles: from distribution functions to intra-level correlations. ACTA ACUST UNITED AC 2003. [DOI: 10.1088/0305-4470/36/12/339] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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25
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Schaadt K, Tufaile APB, Ellegaard C. Chaotic sound waves in a regular billiard. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:026213. [PMID: 12636782 DOI: 10.1103/physreve.67.026213] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/01/2002] [Indexed: 05/24/2023]
Abstract
We present experimental results for the ultrasound transmission spectra and standing wave patterns of a rectangular block of fused quartz. A comparison is made between our data and an approximation of the theoretical staircase function for three-dimensional isotropic elasticity. The main emphasis of our study is on the role of mode conversion in regular ray-splitting billiards. We present the fluctuation statistics and find that these are described by the Gaussian orthogonal ensemble of random matrix theory, despite the fact that the system is not classically chaotic, as demonstrated with numerical simulation. Using temperature perturbation, we find that the vast majority of the resonances are mixtures of transverse and longitudinal wave motion, yet a small number of special resonances remain pure. We further illuminate this by presenting standing wave patterns measured on one face of the block.
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Affiliation(s)
- K Schaadt
- Complexity Lab, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark
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26
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Smolyarenko IE, Simons BD. Parametric statistics of individual energy levels in random Hamiltonians. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:025202. [PMID: 12636733 DOI: 10.1103/physreve.67.025202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2002] [Indexed: 05/24/2023]
Abstract
We establish a general framework to explore parametric statistics of individual energy levels in disordered and chaotic quantum systems of unitary symmetry. The method is applied to the calculation of the universal intralevel parametric velocity correlation function and the distribution of level shifts under the influence of an arbitrary external perturbation.
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Affiliation(s)
- I E Smolyarenko
- Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 0HE, United Kingdom
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