A relation between billiard geometry and the temperature of its eigenvalue gas

Stöckmann, H.‐J.; Stoffregen, U.; Kollmann, M.

doi:10.1088/0305-4470/30/1/010

Cited by 6 publications

(4 citation statements)

References 26 publications

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“…It is interesting to note that similar results were obtained in [23], where they were attributed to an insufficient experimental resolution in the parameter variation. This was also our first suspicion and we tested it numerically with a random matrix model.…”

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confidence: 74%

Spectral statistics in an open parametric billiard system

et al. 2006

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We present experimental results on the eigenfrequency statistics of a superconducting, chaotic microwave billiard containing a rotatable obstacle. Deviations of the spectral fluctuations from predictions based on Gaussian orthogonal ensembles of random matrices are found. They are explained by treating the billiard as an open scattering system in which microwave power is coupled in and out via antennas. To study the interaction of the quantum (or wave) system with its environment a highly sensitive parametric correlator is used.PACS numbers: 05.45. Mt, 03.65.Nk, 42.25.Bs Classical chaos manifests itself in universal spectral quantum fluctuations that can be described by random matrix theory (RMT) [1]. While the earliest investigations of spectral correlations were confined to nuclear physics [2], during the last twenty years the universality has been tested in other areas, like optical experiments [3], quantum dots [4], and acoustic setups [5]. The (local) spectral statistics depend generically only on the underlying symmetries of the system. In particular, they are described by the Gaussian orthogonal ensemble (GOE) of real symmetric random matrices for spinless systems with time reversal symmetry, and by the Gaussian unitary ensemble (GUE) of complex Hermitian random matrices in the absence of time reversal invariance [6]. The sensitivity of the quantum (or wave) statistical properties to fundamental symmetries is obviously of great interest. For instance, it has been utilized to derive an upper bound for the magnitude of the time or parity violating component in nuclear interactions [7,8].We investigate here spectral properties of a superconducting microwave resonator where currents are induced by the measurement process. Although we study a specific wave system, the results are expected to be of general validity in the physics of complex quantum systems (atoms, molecules, nuclei, quantum dots, ...). The influence of the flux of microwave power flowing from the feeding to the receiving antenna on the spectral properties of the system is so weak that it can only be detected through a highly sensitive diagnosis tool, in our case a parametric statistical measure. In a previous experiment, the wave system was realized by a normal conducting microwave resonator attached to a large number of antennas [9]. There, the distribution of wave functions showed significant deviations from the GOE predictions, which were attributed to the transformation of the standing waves inside the closed microwave billiard into waves propagating from an emitting antenna into a large number of exit channels [10]. The aim of the present paper is to go further into the investigation of this mechanism. We will show that deviations from GOE behaviour are already observed in a resonator with only three (or less) attached antennas, when studying spectral properties as a function of a parameter.The experiment discussed here has been performed with a superconducting microwave resonator, whose high-quality factor is typically Q = 10 5 or larg...

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confidence: 74%

Spectral statistics in an open parametric billiard system

et al. 2006

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“…In microwave billiards the parameter typically is the length of one side or the position of a scatterer. There are detailed microwave studies on eigenvalue level velocities, curvatures, and avoided crossings (Stöckmann et al 1997, Barth et al 1999, Stöckmann 1999. In figure 11(a) the dynamics of the complex eigenvalues for a local perturbation in a rectangular billiard with additional small scatterers is shown, where an additional scatterer is moved along one line as indicated in the inset of figure 11(b).…”

Section: Width Distribution and Width Dynamicsmentioning

confidence: 99%

Microwave studies of the spectral statistics in chaotic systems

Stöckmann¹,

Kuhl²

2022

J. Phys. A: Math. Theor.

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An overview over the microwave studies of chaotic systems is presented, performed by the authors and their coworkers in Marburg and Nice. In an historical overview the impact of Fritz Haake in particular in the beginning of the experiments is recognized. In the following sections two subjects are presented he was particularly interested in. One of them is the Bohigas-Giannoni-Schmit conjecture stating that the universal features of the spectra of chaotic systems are well described by random matrix theory. Microwave realizations of seven of the ten universal ensembles have been achieved, starting with the Gaussian orthogonal ensemble in the very first experiment, and ending with the chiral ensembles in a recent work. To do the measurements the systems have to be opened by attaching antennas to excite the microwaves. Antennas are theoretically taken into account in terms of a non-Hermitian effective Hamiltonian with an imaginary part taking care of the coupling to the environment. Results on level spacing and widths distribution in open systems are presented as well as on resonance trapping observed when changing the coupling to the environment.

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“…As typical for quantum chaos, those experiments were not carried out on eigenvalues of the Schrödinger equation, but rather on related models of quasi-2D microwave cavities or propagation of acoustic waves. We provide an incomplete list of references to beautiful experiments [ 18 , 19 , 20 , 21 , 22 , 23 , 24 ] stressing the work of the Stöckmann group [ 18 ] where a rather complete comparison of different measures with experimental microwave resonance data was carried out.…”

Section: Introductionmentioning

confidence: 99%

Quantum Chaos and Level Dynamics

Zakrzewski

2023

Entropy

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We review the application of level dynamics to spectra of quantally chaotic systems. We show that the statistical mechanics approach gives us predictions about level statistics intermediate between integrable and chaotic dynamics. Then we discuss in detail different statistical measures involving level dynamics, such as level avoided-crossing distributions, level slope distributions, or level curvature distributions. We show both the aspects of universality in these distributions and their limitations. We concentrate in some detail on measures imported from the quantum information approach such as the fidelity susceptibility, and more generally, geometric tensor matrix elements. The possible open problems are suggested.

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A relation between billiard geometry and the temperature of its eigenvalue gas

Cited by 6 publications

References 26 publications

Spectral statistics in an open parametric billiard system

Spectral statistics in an open parametric billiard system

Microwave studies of the spectral statistics in chaotic systems

Quantum Chaos and Level Dynamics

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