Closed and open superconducting microwave waveguide networks as a model for quantum graphs.

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.

Experimental study of closed and open microwave waveguide graphs with preserved and partially violated time-reversal invariance.

We report on experiments that were performed with microwave waveguide systems and demonstrate that in the frequency range of a single transversal mode they may serve as a model for closed and open quantum graphs. These consist of bonds that are connected at vertices. On the bonds, they are governed by the one-dimensional Schrödinger equation with boundary conditions imposed at the vertices.

Continuous wave interdigital H-mode cavities for alternating phase focusing heavy ion acceleration.

In the future, a new superconducting (SC) continuous wave (CW) high intensity heavy ion HElmholtz LInear ACcelerator (HELIAC) should provide ion beams with maximum beam energy above the Coulomb barrier for the Super Heavy Element program at GSI (Gesellschaft für Schwerionenforschung, in Engl.: Association for Heavy Ion Research). The HELIAC consists of a SC main accelerator supplied by a normal conducting injector, which comprises an electron cyclotron resonance ion source, a radio-frequency quadrupole, and two separate interdigital H-mode drift-tube linear accelerator cavities, based on an Alternating Phase Focusing (APF) scheme.

A dynamic collimation and alignment system for the Helmholtz linear accelerator.

The upcoming commissioning of the superconducting (SC) continuous wave Helmholtz linear accelerators first of series cryomodule is going to demand precise alignment of the four internal SC cavities and two SC solenoids. For optimal results, a beam-based alignment method is used to reduce the misalignment of the whole cryomodule, as well as its individual components. A symmetric beam of low transverse emittance is required for this method, which is to be formed by a collimation system.

Partial Time-Reversal Invariance Violation in a Flat, Superconducting Microwave Cavity with the Shape of a Chaotic Africa Billiard.

We report on the experimental realization of a flat, superconducting microwave resonator, a microwave billiard, with partially violated time-reversal (T) invariance, induced by inserting a ferrite into the cavity and magnetizing it with an external magnetic field perpendicular to the resonator plane. In order to prevent its expulsion caused by the Meissner-Ochsenfeld effect, we used a cavity of which the top and bottom plate were made from niobium, a superconductor of type II, and cooled it down to liquid-helium temperature T_{LHe}≃4  K. The cavity had the shape of a chaotic Africa billiard.

Spectral Properties of Dirac Billiards at the van Hove Singularities.

We study distributions of the ratios of level spacings of rectangular and Africa-shaped superconducting microwave resonators containing circular scatterers on a triangular grid, so-called Dirac billiards (DBs). The high-precision measurements allowed the determination of, respectively, all 1651 and 1823 eigenfrequencies in the first two bands. The resonance densities are similar to that of graphene.

Fullerene C(60) Simulated with a Superconducting Microwave Resonator and Test of the Atiyah-Singer Index Theorem.

We report first experiments with a macroscopic-size superconducting microwave resonator that has the geometric structure of the C(60) fullerene molecule. Our high-resolution measurements reveal the exceptional spectral properties that stem from the icosahedral symmetry of its carbon lattice. In particular, they allow us to determine the number of zero-energy modes, i.

Dielectric square resonator investigated with microwave experiments.

We present a detailed experimental study of the symmetry properties and the momentum space representation of the field distributions of a dielectric square resonator as well as the comparison with a semiclassical model. The experiments have been performed with a flat ceramic microwave resonator. Both the resonance spectra and the field distributions were measured.

Spectral properties and dynamical tunneling in constant-width billiards.

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.

Scattering experiments with microwave billiards at an exceptional point under broken time-reversal invariance.

Scattering experiments with microwave cavities were performed and the effects of broken time-reversal invariance (TRI), induced by means of a magnetized ferrite placed inside the cavity, on an isolated doublet of nearly degenerate resonances were investigated. All elements of the effective Hamiltonian of this two-level system were extracted. As a function of two experimental parameters, the doublet and the associated eigenvectors could be tuned to coalesce at a so-called exceptional point (EP).

Experimental observation of localized modes in a dielectric square resonator.

We investigated the frequency spectra and field distributions of a dielectric square resonator in a microwave experiment. Since such systems cannot be treated analytically, the experimental studies of their properties are indispensable. The momentum representation of the measured field distributions shows that all resonant modes are localized on specific classical tori of the square billiard.

Distribution of scattering matrix elements in quantum chaotic scattering.

Scattering is an important phenomenon which is observed in systems ranging from the micro- to macroscale. In the context of nuclear reaction theory, the Heidelberg approach was proposed and later demonstrated to be applicable to many chaotic scattering systems. To model the universal properties, stochasticity is introduced to the scattering matrix on the level of the Hamiltonian by using random matrices.

Bound states in sharply bent waveguides: analytical and experimental approach.

Quantum wires and electromagnetic waveguides possess common features since their physics is described by the same wave equation. We exploit this analogy to investigate experimentally with microwave waveguides and theoretically with the help of an effective potential approach the occurrence of bound states in sharply bent quantum wires. In particular, we compute the bound states, study the features of the transition from a bound to an unbound state caused by the variation of the bending angle, and determine the critical bending angles at which such a transition takes place.

Trace formula for chaotic dielectric resonators tested with microwave experiments.

We measured the resonance spectra of two stadium-shaped dielectric microwave resonators and tested a semiclassical trace formula for chaotic dielectric resonators proposed by Bogomolny et al. [Phys. Rev.

Application of a trace formula to the spectra of flat three-dimensional dielectric resonators.

The length spectra of flat three-dimensional dielectric resonators of circular shape were determined from a microwave experiment. They were compared to a semiclassical trace formula obtained within a two-dimensional model based on the effective index of refraction approximation and a good agreement was found. It was necessary to take into account the dispersion of the effective index of refraction for the two-dimensional approximation.

PT symmetry and spontaneous symmetry breaking in a microwave billiard.

We demonstrate the presence of parity-time (PT) symmetry for the non-Hermitian two-state Hamiltonian of a dissipative microwave billiard in the vicinity of an exceptional point (EP). The shape of the billiard depends on two parameters. The Hamiltonian is determined from the measured resonance spectrum on a fine grid in the parameter plane.

Double-slit experiments with microwave billiards.

Single and double-slit experiments are performed with two microwave billiards with the shapes of a rectangle and a quarter stadium, respectively. The classical dynamics of the former is regular, whereas that of the latter is chaotic. Microwaves can leave the billiards via slits in the boundary, forming interference patterns on a screen.

Exceptional points in a microwave billiard with time-reversal invariance violation.

We report on the experimental study of an exceptional point (EP) in a dissipative microwave billiard with induced time-reversal invariance (T) violation. The associated two-state Hamiltonian is non-Hermitian and nonsymmetric. It is determined experimentally on a narrow grid in a parameter plane around the EP.

Experimental test of a trace formula for two-dimensional dielectric resonators.

Resonance spectra of two-dimensional dielectric microwave resonators of circular and square shapes have been measured. The deduced length spectra of periodic orbits were analyzed and a trace formula for dielectric resonators recently proposed by Bogomolny [Phys. Rev.

Quantum chaotic scattering in microwave resonators.

In a frequency range where a microwave resonator simulates a chaotic quantum billiard, we have measured moduli and phases of reflection and transmission amplitudes in the regimes of both isolated and of weakly overlapping resonances and for resonators with and without time-reversal invariance. Statistical measures for S -matrix fluctuations were determined from the data and compared with extant and/or newly derived theoretical results obtained from the random-matrix approach to quantum chaotic scattering. The latter contained a small number of fit parameters.

Friedel oscillations in microwave billiards.

Friedel oscillations of electron densities near step edges have an analog in microwave billiards. A random plane-wave model, normally only appropriate for the eigenfunctions of a purely chaotic system, can be applied and is tested for non-purely-chaotic dynamical systems with measurements on pseudointegrable and mixed dynamics geometries. It is found that the oscillations in the pseudointegrable microwave cavity match the random plane-wave modeling.

Nonperiodic echoes from quantum mushroom-billiard hats.

Nonperiodic tunable quantum echoes have been observed in experiments with an open microwave billiard whose geometry under certain conditions provides Fibonacci-like sequences of classical delay times. These sequences combined with the reflection at the opening induced by the wave character of the experiment and the size of the opening allow to shape quantum pulses. The pulses are obtained by response of an integrable scattering system.

Induced violation of time-reversal invariance in the regime of weakly overlapping resonances.

We measure the complex scattering amplitudes of a flat microwave cavity (a "chaotic billiard"). Time-reversal (T) invariance is partially broken by a magnetized ferrite placed within the cavity. We extend the random-matrix approach to T violation in scattering, determine the parameters from some properties of the scattering amplitudes, and successfully predict others.

Chaotic scattering in the regime of weakly overlapping resonances.

We measure the transmission and reflection amplitudes of microwaves in a resonator coupled to two antennas at room temperature in the regime of weakly overlapping resonances and in a frequency range of 3-16GHz . Below 10.1GHz the resonator simulates a chaotic quantum system.

Properties of nodal domains in a pseudointegrable barrier billiard.

We investigate experimentally the scaling properties of the number of nodal domains of the wave functions and the distribution of the nodal domain areas in a pseudointegrable barrier billiard. The number of nodal domains is smaller than expected for billiards whose dynamics is chaotic or regular. This reduction is explained by the appearance of superscars in the barrier billiard, i.