Experimental simulation of quantum graphs by microwave networks

Hul, Oleh; Bauch, Szymon; Pakoński, Prot; Savytskyy, Nazar; Życzkowski, Karol; Sirko, Leszek

doi:10.1103/physreve.69.056205

Cited by 183 publications

(233 citation statements)

References 39 publications

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“…Accordingly, the fluctuation properties in the spectra of classically chaotic quantum graphs with and without T invariance are expected to coincide with those of random matrices from the GOE and the GUE, respectively. This was confirmed experimentally [10,11] for the nearest-neighbor spacing distribution using microwave networks [22][23][24][25][26].…”

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confidence: 48%

“…However, there are nongeneric features in the spectra of real physical systems that are not yet fully understood. Such problems are best tackled experimentally with the help of model systems like microwave billiards [8,9] and microwave graphs [10,11]. In the experiments with microwave billiards the analogy between the scalar Helmholtz equation and the Schrödinger equation of the corresponding quantum billiard is exploited.…”

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confidence: 99%

“…In the experiments with microwave billiards the analogy between the scalar Helmholtz equation and the Schrödinger equation of the corresponding quantum billiard is exploited. Microwave graphs [10,11] simulate the spectral properties of quantum graphs [12][13][14], networks of onedimensional wires joined at vertices. They provide an extremely rich system for the experimental and the theoretical study of quantum systems, that exhibit a chaotic dynamics in the classical limit.…”

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confidence: 99%

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Power Spectrum Analysis and Missing Level Statistics of Microwave Graphs with Violated Time Reversal Invariance

et al. 2016

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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. Introduction.-In the last decades the concept of quantum chaos, that is, the understanding of the features of the classical dynamics in terms of the spectral properties of the corresponding quantum system, like nuclei, atoms, molecules, quantum wires and dots or other complex systems [1][2][3], has been elaborated extensively. It has been established by now that the spectral properties of generic quantum systems with classically regular dynamics agree with those of Poissonian random numbers [4] while they coincide with those of the eigenvalues of random matrices [6] from the Gaussian orthogonal ensemble (GOE) and the Gaussian unitary ensemble (GUE) for classically chaotic systems with and without timereversal (T ) invariance [5], respectively, in accordance with the Bohigas-Giannoni-Schmit (BGS) conjecture [7].A multitude of studies with focus on problems from the field of quantum chaos have been performed by now theoretically and numerically. However, there are nongeneric features in the spectra of real physical systems that are not yet fully understood. Such problems are best tackled experimentally with the help of model systems like microwave billiards [8,9] and microwave graphs [10,11]. In the experiments with microwave billiards the analogy between the scalar Helmholtz equation and the Schrödinger equation of the corresponding quantum billiard is exploited. Microwave graphs [10,11] simulate the spectral properties of quantum graphs [12][13][14], networks of onedimensional wires joined at vertices. They provide an extremely rich system for the experimental and the theoretical study of quantum systems, that exhibit a chaotic dynamics in the classical limit.The idea of quantum graphs was introduced by Linus Pauling to model organic molecules [15] and they are also used to simulate, e.g., quantum wires [16], optical waveguides [17] and mesoscopic quantum systems [18,19]. The validity of the BGS conjecture was proven rigourously for graphs with incommensurable bond lengths in Refs. [20,21]. Accordingly, the fluctuation properties in the spectra of classically chaotic quantum graphs with and without T invariance are expected

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Power Spectrum Analysis and Missing Level Statistics of Microwave Graphs with Violated Time Reversal Invariance

et al. 2016

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“…-Length swapping can be easily implemented in experimental simulations of quantum graphs by networks of RF wave-guides [19].…”

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confidence: 99%

Edge Switching Transformations of Quantum Graphs

et al. 2017

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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 { En} ∞ n=1 correspondingly, are level-2 interlaced, so that En−2 ≤ En ≤ En+2. The proofs are guided by considerations of the quantum graphs' discrete analogs. [2,3], and they are defined and discussed at great length and detail in review articles and books (see e.g., [4][5][6][7]). In this note, we consider a general compact quantum graph, G = (V, E, L), whose edges e ∈ E are metrized and of finite lengths belonging to the list, may include external potential V and possibly also a magnetic potential A:The operator definition is incomplete without specifying also the boundary conditions (bc) at vertices. They are assumed here to be local, i.e., expressed in terms of linear relations on the limiting values of the function and of its derivatives along the deg(v) edges which meet at each vertex v ∈ V. The possible choices which ensure self-adjointness are reviewed in detail in, e.g., [5,6]. Under mild conditions on V and A the spectrum of the Schrödinger operator {E n } ∞ n=1 is discrete and bounded below, but not above. It suffices to assume V, A are integrable, but for a more transparent presentation we focus on the case these are piecewise continuous [6].This spectrum of H is unaffected by the operation of edge splitting, through the insertion of a vertex of degree 2 with Kirchhoff boundary conditions. These require Ψ to be continuous at v ∈ V having there directional derivatives satisfying:(To avoid confusion: these boundary conditions are not assumed here for the other quantum graph vertices.) Our purpose here is to discuss the effects on the spectrum of another basic graph transformation, that of edge switching. It is defined in the following extension of a notion which is used in combinatorial graph theory [8]. Definition 1.1 For a quantum graph G with a selfadjoint Schrödinger operator H, an edge switch is a transformation in which a pair of edges in E exchange the graph designations of one of their end points. Preserved under this exchange are the edge lengths, the local action of H along the corresponding edges, and the vertex boundary conditions -up to the corresponding transposition of the functions whose limiting behavior at the two vertices enter the local boundary conditions.Under an edge switch the collection of the edge lengths remains unchanged. However the spectra will in the generic case be affected. For example, for quantum graph Laplacians it is shown in [9] that the topology of the quantum graph with rationally independent edge length L "can be heard" in the sense of Marc Kac [10]. For clarity let us add that the graph topology may be affected by a switch, but it does not have to. However, regardless of that, and even in case the swi...

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“…In this case the disorder is introduced either by choosing random bond lengths [77,79,80] (which is a parameter not present in our network model) or by randomizing the vertex-scattering matrices [78] (somehow equivalent to consider random connection strengths). Some of the RMT properties of quantum graphs have already been tested experimentally by the use of small ensembles of small microwave networks with fixed connectivity [81]. Furthermore, complex networks having specific topological properties (such as small-world and scale-free networks, where randomness is applied only to the connectivity) show signatures of RMT behavior in their spectral and eigenfunction properties [34-36, 38, 48].…”

Section: Other Erdős-rényi Random Networkmentioning

confidence: 99%

Universality in the spectral and eigenfunction properties of random networks

et al. 2015

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By the use of extensive numerical simulations we show that the nearest-neighbor energy level spacing distribution P (s) and the entropic eigenfunction localization length of the adjacency matrices of Erdős-Rényi (ER) fully random networks are universal for fixed average degree ξ ≡ αN (α and N being the average network connectivity and the network size, respectively). We also demonstrate that Brody distribution characterizes well P (s) in the transition from α = 0, when the vertices in the network are isolated, to α = 1, when the network is fully connected. Moreover, we explore the validity of our findings when relaxing the randomness of our network model and show that, in contrast to standard ER networks, ER networks with diagonal disorder also show universality. Finally, we also discuss the spectral and eigenfunction properties of small-world networks.

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Experimental simulation of quantum graphs by microwave networks

Cited by 183 publications

References 39 publications

Power Spectrum Analysis and Missing Level Statistics of Microwave Graphs with Violated Time Reversal Invariance

Power Spectrum Analysis and Missing Level Statistics of Microwave Graphs with Violated Time Reversal Invariance

Edge Switching Transformations of Quantum Graphs

Universality in the spectral and eigenfunction properties of random networks

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