We determine with unprecedented accuracy the lowest 900 eigenvalues of two quantum constant-width billiards from resonance spectra measured with flat, superconducting microwave resonators. While the classical dynamics of the constant-width billiards is unidirectional, a change of the direction of motion is possible in the corresponding quantum system via dynamical tunneling. This becomes manifest in a splitting of the vast majority of resonances into doublets of nearly degenerate ones. The fluctuation properties of the two respective spectra are demonstrated to coincide with those of a random-matrix model for systems with violated time-reversal invariance and a mixed dynamics. Furthermore, we investigate tunneling in terms of the splittings of the doublet partners. On the basis of the random-matrix model we derive an analytical expression for the splitting distribution which is generally applicable to systems exhibiting dynamical tunneling between two regions with (predominantly) chaotic dynamics.
We determine with unprecedented accuracy the lowest 900 eigenvalues of two quantum constant-width billiards from resonance spectra measured with flat, superconducting microwave resonators. While the classical dynamics of the constant-width billiards is unidirectional, a change of the direction of motion is possible in the corresponding quantum system via dynamical tunneling. This becomes manifest in a splitting of the vast majority of resonances into doublets of nearly degenerate ones. The fluctuation properties of the two respective spectra are demonstrated to coincide with those of a random-matrix model for systems with violated time-reversal invariance and a mixed dynamics. Furthermore, we investigate tunneling in terms of the splittings of the doublet partners. On the basis of the random-matrix model we derive an analytical expression for the splitting distribution which is generally applicable to systems exhibiting dynamical tunneling between two regions with (predominantly) chaotic dynamics.
Two experiments with super- and normal conducting microwave cavities which are used as an analog to two-dimensional, infinitely deep quantum potentials are presented. The eigenvalues of such a potential can be measured directly through the resonance frequencies of the cavity, while the eigenvectors can be determined by measuring the field distributions inside the cavity. In the case of open systems another information – the imaginary part of the eigenvalues – is observable by measuring the widths of the resonances. As examples for this experimental approach to quantum chaos the semiclassical reconstruction of length spectra of billiards with varying chaoticity based of various trace formulas proposed by Gutzwiller, by Ullmo, Grinberg and Tomsovic and by Berry and Tabor for systems with chaotic, mixed and regular dynamics, respectively, and the first observation of square root singularities – so called exceptional points recently proposed by Heiss – in the energy spectrum of coupled cavities are presented.
Abstract We investigate the spectral properties and Husimi functions of relativistic quantum billiards (QBs) consisting of a spin-1/2 particle governed by the Dirac equation and confined to a planar domain of constant-width (CW) by imposing boundary conditions (BCs) on the spinor components. We consider those of neutrino billiards (NBs) proposed in (Berry and Mondragon 1987 Proc. R. Soc. A 412 53). The classical dynamics of billiards of corresponding shape is predominantly chaotic. CW billiards attracted particular attention because they exhibit unusual properties. Their classical dynamics features unidirectionality, whereas in the corresponding nonrelativistic QB a change of the rotational direction of motion is possible via dynamical tunneling, and the spectral properties coincide with those of typical quantum systems with violated time-reversal invariance. Unidirectionality of the quantum dynamics would arise in the structure of the Husimi functions. We analyze them for two realizations of CW NBs in the ultra-relativistic, i.e. the massless case and for massive cases and come to the result, that the modes can be separated into clockwise and counterclockwise modes and dynamical tunneling is absent. This is attributed to the BCs and the unidrectionality of the local current arising from them.
We report on the experimental study of the spectral properties of quantum systems consisting of two quantum billiards (QBs), one with chaotic, the other one with integrable classical dynamics, that are coupled to each other via an opening in a common wall. They are compared to those of a special case of the Rosenzweig-Porter model with random matrices composed of two diagonal blocks modeling the spectral properties of the QBs, that are coupled with a tunable parameter. We demonstrate that this model is suitable for the description of the experimental data and thus may be employed to determine the strength of the coupling. It results from the increasing overlap of eigenmodes in the QBs penetrating through the opening into the other one, leading to a mixing of their eigenstates, and the breaking of the symmetry present in the QB with integrable dynamics. This implicates deviations of the spectral properties from those of typical quantum systems with integrable and chaotic dynamics, respectively, and approaches those of a fully chaotic system for sufficiently large coupling strength. In contrast in previous studies the transition from integrable to chaotic dynamics was induced by introducing a random potential of increasing strength into such a QB and applicability of a variant of the Rosenzweig-Porter model was tested.
The present doctoral thesis is concerned, on the one hand, with the investigation of the decay probability in a quantum billiard, and, on the other, the realization of the double-slit experiment in microwave billiards. For both purposes transmission elements of the scattering matrix are experimentally measured in microwave cavities with regular and chaotic dynamics. One or more pieces of the billiards' boundary are removed and the leaked outward eld is investigated. The general aim is to distinguish the eect of the billiard dynamics on the temporal decay behaviour and on the interference patterns. The decay probability in an open quantum billiard can be explored by means of the scattering matrix formalismus derived in the framework of nuclear physics. In a circular billiard, archetype of regular systems, the opening breaks the spatial symmetry causing a splitting of the degenerate states observable at low frequencies. One state of the doublet remains unaected contributing to the long-time behavior inside the cavity (survival probability), while the short-lived one is strongly influenced by the opening and decays much faster. The case of chaotic dynamics is studied by means of a tilted stadium billiard. For sufficiently long times the decay probability shows an essentially nonexponential behaviour. Agreement with theoretical results derived by Harney, Dittes and Muller is obtained for a few of the open channels and intermediate times. For both, the circle and the stadium,the short time decay resembles the classical behaviour. Furthermore, a double-slit experiment is performed with a wave packet starting inside a microwave billiard domain, both for regular and chaotic dynamics. Stationary and temporal patterns formed by the leaking of the wave packet are experimentally determined. The evolution of a directed initial state is constructed by means of the electromagnetic propagator. Also single-slit experiments are carried out and compared with the corresponding double-slit experiments. The present work sheds light on the fundamental question of the decay of open quantum systems and provides plentiful information about the structure of the diracted eld from microwave billards with a double-slit on the boundary connecting classical properties with the corresponding quantum features in both cases.
The present doctoral thesis describes experimentally measured properties of the resonance spectra of flat microwave billiards with partially broken timereversal invariance induced by an embedded magnetized ferrite. A vector network analyzer determines the complex scattering matrix elements. The data is interpreted in terms of the scattering formalism developed in nuclear physics. At low excitation frequencies the scattering matrix displays isolated resonances. At these the effect of the ferrite on isolated resonances (singlets) and pairs of nearly degenerate resonances (doublets) is investigated. The hallmark of time-reversal symmetry breaking is the violation of reciprocity, i.e. of the symmetry of the scattering matrix. One finds that reciprocity holds in singlets; it is violated in doublets. This is modeled by an effective Hamiltonian of the resonator. A comparison of the model to the data yields time-reversal symmetry breaking matrix elements in the order of the level spacing. Their dependence on the magnetization of the ferrite is understood in terms of its magnetic properties. At higher excitation frequencies the resonances overlap and the scattering matrix elements fluctuate irregularly (Ericson fluctuations). They are analyzed in terms of correlation functions. The data are compared to three models based on random matrix theory. The model by Verbaarschot, Weidenmüller and Zirnbauer describes time-reversal invariant scattering processes. The one by Fyodorov, Savin and Sommers achieves the same for systems with complete time-reversal symmetry breaking. An extended model has been developed that accounts for partial breaking of time-reversal invariance. This extended model is in general agreement with the data, while the applicability of the other two models is limited. The cross-correlation function between forward and backward reactions determines the time-reversal symmetry breaking matrix elements of the Hamiltonian to up to 0.3 mean level spacings. Finally the sensitivity of the elastic enhancement factor to time-reversal symmetry breaking is studied. Based on the data elastic enhancement factors below 2 are found which is consistent with breaking of time-reversal invariance in the regime of overlapping resonances. The present work provides the framework to probe for broken time-reversal invariance in any scattering data by a multitude of methods in the whole range between isolated and overlapping resonances.
Since the early days of quantum mechanics, tunneling has been recognized as one of the hallmarks of the wave character of microscopic physics. The possibility of a quantum particle to penetrate an energetic barrier represents certainly one of the most spectacular implications of quantum theory and has led to various applications in nuclear, atomic, and molecular physics as well as in mesoscopic science. Typical scenarios in which tunneling manifests are the escape route of a quantum particle from a quasibounded region, the transition between two or more symmetry-related, but classically disconnected wells (which we shall focus on in the following), as well as scattering or transport through potential barriers. The spectrum of scenarios becomes even richer when the concept of tunneling is generalized to any kind of classically forbidden transitions in phase space, that is, to transitions that are not necessarily inhibited by static potential barriers but by some other constraints of the underlying classical dynamics (such as integrals of motion). Such “dynamical tunneling” processes arise frequently in molecular systems [1] and were realized with cold atoms propagating in periodically modulated optical lattices [2-4]. Moreover, the electromagnetic analog of dynamical tunneling was also obtained with microwaves in billiards [5].