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We report first experiments with a macroscopic-size superconducting microwave resonator that has the geometric structure of the C60 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.e., of modes with energy values at the Dirac point existent in the band structure due to the hexagonal arrangements of the carbon atoms, and to test the Atiyah-Singer index theorem which relates this number to the topology of the curved carbon lattice. [ABSTRACT FROM AUTHOR]

A novel method for the calculation of the energy dispersion relation and density of states (DOS) in one (ID) and two (2D) dimensions is introduced and applied to linear lattices (ID) and square and hexagonal lattices (2D). The (van Hove) singularities and (Dirac) zeros of the DOS are discussed. Results for the 2D hexagonal lattice (graphenelike materials) are compared with experimental data in microwave photonic crystals. [ABSTRACT FROM AUTHOR]

We determined experimentally the eigenvalues of two rectangular quantum billiards that contain circular scatterers forming a triangular grid, so-called Dirac billiards. For this we performed measurements of unprecedented accuracy using superconducting macroscopic-size microwave billiards that enclose a photonic crystal. The objective was the investigation of the peculiar features of the density of the eigenvalues (DOE), which resemble that of a graphene flake, and of their fluctuation properties. We identified in the measured resonance spectra Dirac points and in their adjacent bands the van Hove singularities (VHSs), that show up as sharp peaks in the DOE. The analysis of the experimental resonance frequencies and of the band structure, which was computed with a tight-binding model, revealed that the VHSs divide the associated band into regions where the system is governed by the nonrelativistic Schrodinger equation of the quantum billiard and the Dirac equation of the graphene billiard of corresponding shape, respectively. Furthermore, we demonstrate that Dirac billiards are most suitable for the modeling of idealized graphene. Indeed, the DOEs of both systems are well described by a finite tight-binding model which includes first-, second-, and third-nearest-neighbor couplings. [ABSTRACT FROM AUTHOR]

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. Together, both cavities will accelerate ions from 300 to 1400 keV/u with only one external quadrupole triplet for transverse focusing in between. Due to the demanding requirements of the APF concept on the voltage distribution along the beam axis on the one hand and the CW operation on the other hand, the optimization of each cavity concerning RF, mechanical, and thermal properties is crucial for the successful operation of the HELIAC injector. [ABSTRACT FROM AUTHOR]

We present experimental results for the density of states (DOS) of a superconducting microwave Dirac billiard which serves as an idealized model for the electronic properties of graphene. The DOS exhibits two sharp peaks which evolve into van Hove singularities with increasing system size. They divide the band structure into regions governed by the relativistic Dirac equation and by the nonrelativistic Schrodinger equation, respectively. We demonstrate that in the thermodynamic limit, a topological transition appears as a neck-disrupting Lifshitz transition in the number susceptibility and as an excited-state transition in the electronic excitations. Furthermore, we recover the finite-size scaling typical for excited-state quantum phase transitions involving logarithmic divergences and identify a quasiorder parameter [ABSTRACT FROM AUTHOR]

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. The predictions are confirmed by calculations based on a conventional numerical method as well as experimental measurements of the spectra and electric field intensity distributions of electromagnetic waveguides. [ABSTRACT FROM AUTHOR]

This paper presents experimental results on properties of waves propagating in an unbounded and a bounded photonic crystal consisting of metallic cylinders that are arranged in a triangular lattice. First, we present transmission measurements of plane waves traversing a photonic crystal. The experiments are performed in the vicinity of a Dirac point, i.e., an isolated conical singularity of the photonic band structure. There, the transmission shows a pseudodiffusive \/L dependence, with L being the thickness of the crystal, a phenomenon also observed in graphene. Second, eigenmode intensity distributions measured in a microwave analog of a relativistic Dirac billiard, a rectangular microwave billiard that contains a photonic crystal, are discussed. Close to the Dirac point, states have been detected that are localized at the straight edge of the photonic crystal corresponding to a zigzag edge in graphene [ABSTRACT FROM AUTHOR]

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. The aim is to determine the effect of the billiard dynamics on their structure. For this the development of a method for the construction of a directed wave packet by means of an array of multiple antennas was crucial. The interference patterns show a sensitive dependence not only on the billiard dynamics but also on the initial position and direction of the wave packet. [ABSTRACT FROM AUTHOR]

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. It consists of two separate plates with milled slits, aligned in the horizontal and vertical direction. The collimation system and alignment measurements are proposed, investigated, and realized. The complete setup of this system and its integration into the existing environment at the GSI High Charge State Injector are presented, as well as the results of the recent reference measurements. [ABSTRACT FROM AUTHOR]

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 TLHe≃4 K. The cavity had the shape of a chaotic Africa billiard. Superconductivity rendered possible the accurate determination of complete sequences of the resonance frequencies and of the widths and strengths of the resonances, an indispensable prerequisite for the unambiguous detection of T invariance violation, especially when it is only partially violated. This allows for the first time the precise specification of the size of T invariance violation from the fluctuation properties of the resonance frequencies and from the strength distribution, which actually depends sensitively on it and thus provides a most suitable measure. For this purpose we derived an analytical expression for the latter which is valid for isolated resonances in the range from no T invariance violation to complete violation. [ABSTRACT FROM AUTHOR]

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. They exhibit two sharp peaks at the van Hove singularities which separate the band structure into regions with a linear and a quadratic dispersion relation, respectively. In the vicinity of the van Hove singularities we observe rapid changes in, e.g., the wave function structure. Accordingly, we question whether the spectral properties are there still determined by the shapes of the DBs. The commonly used statistical measures are no longer applicable; however, we demonstrate in this Letter that the ratio distributions provide suitable measures. [ABSTRACT FROM AUTHOR]

The new heavy ion superconducting continuous wave HElmholtz LInear ACcelerator (HELIAC) is under construction at GSI. A normal conducting injector, comprising an ECR ion source, an RFQ and a DTL, is recently in development. The new Interdigital H-mode DTL, presented in this paper, accelerates the heavy ion beam from 300 to 1400 keV/u, applying an Alternating Phase Focusing (APF) beam dynamics scheme. This APF section, consisting of two separately controlled tanks, has to provide for stable routine operation with assistance of dedicated beam diagnostics devices in the Intertank section. The installed quadrupole lenses and beam steerers installed there ensure full transmission in a wide range of input beam parameters. [Display omitted] • Alternating phase focusing has been applied for low emittance growth heavy ion beam transport. • Realistic 6D coupling for all phase planes has been considered. • Two radio frequency resonators have been designed to provide for continuous wave operation. • For beam dynamics design, a parallel global optimization search strategy has been adopted. [ABSTRACT FROM AUTHOR]

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. The momentum space representations of the latter evidenced that the resonant states are each related to a specific classical torus, leading to the regular structure of the spectrum. Furthermore, they allow for a precise determination of the refractive index. Measurements with different arrangements of the emitting and the receiving antennas were performed and their influence on the symmetry properties of the field distributions was investigated in detail, showing that resonances with specific symmetries can be selected purposefully. In addition, the length spectrum deduced from the measured resonance spectra and the trace formula for the dielectric square resonator are discussed in the framework of the semiclassical model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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). The behavior of the eigenvalues and eigenvectors when encircling the EP in parameter space was studied, including the geometric amplitude that builds up in the case of broken TRI. A one-dimensional subspace of parameters was found where the differences of the eigenvalues are either real or purely imaginary. There, the Hamiltonians were found to be PT invariant under the combined operation of parity (V) and time reversal (T) in a generalized sense. The EP is the point of transition between both regions. There a spontaneous breaking of PT occurs. [ABSTRACT FROM AUTHOR]

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. Based on these observations a semiclassical model was developed. It shows excellent agreement with all but a single class of measured field distributions that will be treated separately. [ABSTRACT FROM AUTHOR]

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. A long-standing problem was the computation of the distribution of the off-diagonal scattering- matrix elements. We report here an exact solution to this problem and present analytical results for systems with preserved and with violated time-reversal invariance. Our derivation is based on a new variant of the supersymmetry method. We also validate our results with scattering data obtained from experiments with microwave billiards. [ABSTRACT FROM AUTHOR]

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. E 78, 056202 (2008)]. We found good qualitative agreement between the experimental data and the predictions of the trace formula. Deviations could be attributed to missing resonances in the measured spectra in accordance with previous experiments [Phys. Rev. E 81, 066215 (2010)]. The investigation of the numerical length spectrum showed good qualitative and reasonable quantitative agreement with the trace formula. It demonstrated, however, the need for higher-order corrections of the trace formula. The application of a curvature correction to the Fresnel reflection coefficients entering the trace formula yielded better agreement, but deviations remained, indicating the necessity of further investigations. [ABSTRACT FROM AUTHOR]

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. Furthermore, small deviations between the experimental length spectrum and the trace formula prediction were attributed to the systematic error of the effective index of refraction approximation. In summary, the methods developed in this article enable the application of the trace formula for two-dimensional dielectric resonators also to realistic, flat three-dimensional dielectric microcavities and -lasers, allowing for the interpretation of their spectra in terms of classical periodic orbits. [ABSTRACT FROM AUTHOR]

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. After applying a purely imaginary diagonal shift to the Hamiltonian, its eigenvalues are either real or complex conjugate on a curve, which passes through the EP. An appropriate basis choice reveals its PT symmetry. Spontaneous symmetry breaking occurs at the EP. [ABSTRACT FROM AUTHOR]

The standalone superconducting continuous wave heavy ion linac HELIAC (HElmholtz LInear ACcelerator) is a common project of GSI Helmholtz Centre for Heavy Ion Research and Helmholtz Institute Mainz (HIM) under key support of Goethe University Frankfurt (IAP) and in collaboration with National Research Nuclear University MEPhI and NRC "Kurchatov Institute" ITEP. In 2017 the first superconducting section of the linac has been successfully commissioned and extensively tested with beam at GSI. The measurements sufficiently present the capability of 216.816 MHz multi-gap Crossbar H-mode (CH) DTL-structures. An acceleration of heavy ions up to the design beam energy and beyond has been reached with the setup. The required acceleration gain of 0.5 MeV/u was achieved with heavy ion beams even above the design mass-to-charge ratio at maximum available beam intensity and full beam transmission. Recently the HELIAC beam dynamics concept foresees twelve superconducting CH-DTL cavities, assembled in four cryomodules. Each module is equipped with three accelerating CH structures, two superconducting solenoids for transverse beam focusing and a Spoke-type buncher for longitudinal beam matching. The linac is used to accelerate ions with a mass-to-charge ratio up to A ∕ z = 6 from the input energy of 1.4 MeV/u up to the smoothly variable output energy between 3.5 MeV/u and 7.3 MeV/u. The manuscript presents the reference beam dynamics layout of the entire HELIAC facility. [ABSTRACT FROM AUTHOR]