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

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. 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

Analysis of Missing Level Statistics for Microwave Networks Simulating Quantum Chaotic Graphs Without Time Reversal Symmetry - the Case of Randomly Lost Resonances

“…The NNSD, the spectral rigidity and the average power spectrum are displayed in the panels (a), (b), and (c), respectively. One should point out that the power spectrum obtained for the incomplete experimental spectra with randomly removed additional resonance frequencies are presented in [47]. The results obtained for the NNSD after removing certain number of resonances from the experimental spectra to achieve φ = 0.925, 0.865, 0.810, respectively, are presented by histograms (red in color) and by empty circles (red in color) in the other panels.…”

“…Recently, a new procedure to obtain information on the degree of chaoticity of a classical system from the spectral properties of the corresponding quantum system was developed for incomplete sequences of levels [47,48]. Incomplete spectra pose major problems in real physical systems like, e.g., nuclei and molecules [49][50][51][52], which have to be overcome, so such procedures are indispensable for their analysis [53,54].…”

Experimental Study of Quantum Graphs with Simple Microwave Networks: Non-Universal Features

“…In [22,23], experiments with microwave networks are carried out and showed non-universal behavior for long-range fluctuating properties. According to theory, the universal behavior is obtained only in the limit of infinitely intricate graphs with infinitely many bonds and nodes [21].…”

Multi‐scale response of runoff to climate fluctuation in the headwater region of the Kaidu River in Xinjiang of China

“…Humid or arid weather conditions occurred in some previous period may impact now and future, accordingly neighboring observed values in the time series are correlated to some extent. Some traditional approaches such as power spectrum analysis, correlation analysis and statistical analysis are suitable for determining the correlation characteristics of stationary signals (Białous et al ., ; Fong et al ., ; Li et al ., ; Rysak et al ., ). However, the runoff time series usually is affected by noise or nonstationary signals, the mean, standard deviation, high order values and correlation function change as time goes on.…”