Superscarred wave functions as models for doorway states
Barrier billiards have the peculiar property that a considerable part of the (nontrivial) wave functions are superscarred, that is, localized around families of neutrally stable classical periodic orbits. Examples of measured superscars are shown in the figure. In contrast to chaotic systems the scar structure does not disappear in the limit of high resonance frequencies. The superscarring wave functions themselves form families which live inside an infinitely long periodic orbit channel and can be approximated by a constructed superscar state. We have observed experimentally that the superscarring wave functions in the respective periodic orbit channels have a certain overlap with the neighbouring chaotic states, i.e. they “leak out” of the channel. Moreover, the distribution of the overlap of a constructed superscar state with the measured nonscarring wave functions follows a Breit-Wigner distribution. These properties are reminiscent of the so-called doorway state phenomenon in nuclear physics. A typical example is a nuclear giant resonance spread out into underlying complicated particle-hole states. Thus, superscar states provide an ideal model for a thorough investigation of the doorway mechanism. The results are published in Phys. Rev. Lett. 100, 204101 (2008).
Nodal domain and nodal line statistics
Several hundreds of wave functions were measured in a pseudointegrable barrier billiard proposed by Bogomolny and Schmit. Based on these large data sets we studied properties of the nodal lines, where a wave function vanishes, and nodal domains, where it has a definite sign. It has been conjectured a few years ago that the distribution of the number of nodal domains is universal and different for regular and chaotic systems. There were no theoretical or experimental results on properties of the nodal domains of systems with mixed dynamics or pseudointegrable ones. The aim of the experiments with the barrier billiard was to fill this gap. For this a method to find nodal domains had to be developed which was not an easy task especially in regions where the measured intensity and hence the squared wave functions are small. The scaling properties of the number of nodal domains of the wave functions and the distribution of the sizes of the nodal domains were evaluated. Deviations of the former from the prediction for chaotic systems were observed, whereas the latter is well described by that predicted for chaotic systems based on a percolation model. The results and their interpretation are published in Phys. Rev. E 78, 045201 (2008).
Symmetry breaking for a system with two symmetry classes of unequal sizes
The effect of the mixing of one isospin T>=9/2 into a sequence of 124 isospin T<=7/2 states in the compound nucleus 93Tc on spectral properties and the strength distribution was recently studied [Phys. Lett. B 598, 42 (2004)]. Regarding the spectral properties deviations from RMT were attributed to missing levels, and for the strength distribution they were interpreted as an effect of isospin mixing. A major drawback of these statistical evaluations is the low number and the scarcity of data points. We thus performed experiments where the isospin mixing is modelled by a small and a large chaotic microwave billiard which are coupled with a variable strength. In the framework of a diploma thesis the resonance frequencies of such a billiard system were determined and their statistical properties were evaluated with a RMT model for symmetry breaking, which also had been applied to the 93Tc data. These promising studies have not yet been completed, since the statistical significance of the data needs to be improved. Moreover, the coupling mechanism has been further developed in the meantime such that the coupling strength can be varied with considerably higher precision.
Observation of a Dirac point in microwave experiments with a photonic crystal
We recently started with a new series of experiments concerning the modelling of properties of graphene with microwave photonic crystals. Due to its peculiar electronic properties graphene, a monoatomic layer of carbon atoms arranged on a honeycomb lattice, attracted a lot of attention over the last years which culminated in the Nobel Prize only recently. The conductance and the valence band form conically shaped valleys that touch each other at the six corners of the hexagonal Brillouin zone. Close to these, the electron energy depends linearly on the quasi momentum. This linear or conical dispersion relation implies an energy independent velocity, that is, near these points the electrons and holes behave like massless relativistic particles described by the Dirac equation for spin-½ particles. This results in a number of peculiar electronic properties which have an analogue in relativistic quantum mechanics. In the literature the touching points are referred to as Dirac points. Interestingly, the band structure of photonic crystals possesses similar properties. In our laboratory a photonic crystal composed of a total of 874 metallic cylinders, which are arranged on a triangular lattice and squeezed between two metal plates, was constructed. Its band structure is similar to that of graphene. In the experiments transmission and reflection spectra were measured. At the frequency of the Dirac point, the spectra reveal a clear cusp structure which is directly related to the local density of states which tends to zero linearly with the excitation frequency in the vicinity of the Dirac point (see figure). The results on the detection of a Dirac point in a microwave photonic crystal are published in Phys. Rev. B 82, 014301 (2010).
Quantum chaotic scattering in the regimes of weakly overlapping resonances and Ericson fluctuations
The theory of chaotic scattering has been largely developed in the framework of the statistical theory of compound-nucleus reactions. Predictions for the fluctuation properties of the associated S-matrix were obtained based on RMT and the supersymmetry method. To be more precise, analytic expressions for S-matrix correlation functions and the distribution of the S-matrix elements have been derived which are valid from the regime of isolated resonances with Γ/d << 1 to that of strongly overlapping ones with Г/d >> 1. Here, Г denotes the resonance width and d the mean resonance spacing. For T-invariant systems the fluctuation properties of nuclear cross-sections have been investigated experimentally in the regime of isolated resonances and the Ericson regime of strongly overlapping ones. Upper bounds on the strength of T-invariance violation were deduced from studies of nuclear spectra and compound-nucleus reactions in the Ericson regime. However, there was a lack of experimental tests of the RMT predictions on chaotic scattering in the regime of weakly overlapping resonances. We demonstrated in numerous experiments that microwave billiards indeed provide ideal systems for tests of RMT predictions derived in the context of nuclear reaction theory. First, we have access to modulus and phase of the S-matrix elements and thus obtain valuable additional information not accessible in nuclear reaction experiments. Second, the scattering data sets are considerably larger than those collected in nuclear physics. Experiments were performed with T-invariant microwave billiards and with partial T violation. For the evaluation of the latter we extended predictions derived for T-invariant systems by Verbaarschot, Weidenmüller and Zirnbauer for certain autocorrelation functions of pairs of complex conjugate scattering matrix elements to systems with partial T violation. The scattering data allows tests with unprecedented accuracy of predictions on S-matrix correlation functions, the elastic enhancement factor and the distribution of the S-matrix elements in the regimes of isolated and weakly overlapping resonances. Due to the large sets of scattering data measured as function of the excitation frequency we are also in a position to study higher order correlation functions, as e.g. the cross-section autocorrelation function to test predictions on cross-section fluctuations. Results on S-matrix properties, which corroborate the validity of the RMT approach, were published in a series of papers [Phys. Rev. E 78, 055204(R) (2008), Phys. Rev. Lett. 103, 064101 (2009), Phys. Rev. E 81, 036205 (2010) and Phys. Lett. B 685, 263 (2010)]. In a recent publication [Phys. Lett. B 693, 316 (2010)] we furthermore addressed experiments planned within ELI, the “Extreme Light Infrastructure”, an ambitious european project to generate laser beams of extremely high intensity. These include photonuclear reactions induced by an intense laser pulse allowing the direct measurement of the decay in time of nuclei. We proposed a tool for the evaluation of such experiments in terms of a time-decay function which we derived based on RMT results for the autocorrelation function of the associated S-matrix.
Double slit experiments with microwave billiards
A few years ago, the famous double slit experiment was analysed by Casati and Prosen numerically for a situation where an initial wave packet is prepared inside a billiard, which has two slits as openings in a straight part of its boundary. They came to the conclusion that, if the classical dynamics inside the billiard is regular, the interference fringes observed on a screen outside of the billiard have the same structure as in a double slit experiment with plane waves. In contrast, for a chaotic billiard these fringes disappear and the intensity distribution observed on the screen equals the sum of those obtained from two independent one slit experiments. Motivated by these numerical considerations we performed experiments with a regular rectangular microwave billiard and a chaotic one with the shape of a tilted stadium. A method was developed for the construction of a directed wave packet by means of an array of emitting antennas. We observed a sensitive dependence of the interference pattern, measured with a moving antenna at a large number of positions in the vicinity of the slits in the exterior (see figure), on the initial propagation direction of the wave packet and on the dynamics of the billiard. The temporal evolution of the electromagnetic waves leaving the billiard was obtained by measuring complete frequency spectra at a large number of locations of the receiving antenna in the exterior and by computing their Fourier transforms. The results are published in Phys. Rev. E 84, 016221 (2011).
Avoided level-crossing statistics in open chaotic billiards
During the first period of the SFB 634 resonance frequencies of a chaotic microwave billiard whose shape depends on a parameter were measured [Phys. Rev. E 73, 035201(R) (2006)]. Deviations of the parametric spectral properties from RMT predictions could be explained by treating the microwave billiard as an open scattering system with the antennas acting as single scattering channels. Since the experiments were performed at superconducting conditions no fictitious scattering channels needed to be introduced. Due to a lack of analytic expressions concerning the parametric spectral properties the experimental distributions were compared with numerical simulations based on an effective Hamiltonian including the parameter-dependent Hamiltonian of the closed system and a term for the coupling of the resonator modes to the exterior via the antennas. Within a simple two-level random matrix ensemble we derived in collaboration with C. Poli, O. Legrand and F. Mortessagne an analytic expression for the distribution of avoided crossings of the resonances of chaotic open quantum systems, which describes the experimental distribution very well. This corroborates our interpretation of the deviations as an effect of the openness of the microwave billiard. The results are published in Phys. Rev. E 80, 035204(R) (2010).
Friedel oscillations in microwave billards
Friedel oscillations of the electronic density of states e.g., in metals near the Fermi level about impurity atoms or near step edges, which have received much attention recently, have an analogue in microwave billiards. We evaluated the measured field intensities of the pseudointegrable Barrier billiard and the mushroom billiard with mixed classical dynamics both at boundaries with Dirichlet conditions, where the wave function vanishes, and with Neumann conditions, where the normal derivative of the wave function vanishes (see figure). The aim was an experimental test of predictions derived on the basis of a random-plane wave model for Friedel oscillations. While for the barrier billiard the experimental results reproduce the predicted oscillations very well, an appropriate phase-space projection had to be incorporated in the random-plane wave model to obtain agreement for the mushroom billiard. The results are published in Phys. Rev. E 80, 066210 (2009).
Nonperiodic echoes from quantum mushroom-billiard hats
Numerical studies on properties of classical mushroom-billiard hats with an opening obtained by removing the stem showed that for a fixed angular momentum the number of bounces that a particle sent into the hat experiences before leaving it is selective. In fact, three different numbers, i.e. three delay times, are possible. To test how far this selectivity influences the pulse structure of the corresponding open quantum billiard, experiments with open microwave billiards with the shape of a quarter circle were performed. Microwave power was emitted into the resonator and received by an antenna positioned outside the resonator in front of the opening. A Fourier transform of the so obtained reflection spectra yields the response to a short pulse in the time domain. Its modulus shows an aperiodic sequence of peaks; an example is shown in Fig. 6. An analysis of the peak positions in terms of classical delay times showed that the pulse structure is indeed determined by the possible classical escape times and by diffraction at the opening. The results are published in Phys. Rev. E 80, 036212 (2009).
Strength distributions in mixed systems
The motivation for this project comes from results obtained from the nuclear data related to the electric pygmy dipole resonance in four different isotones with neutron number N=82. Their spectral properties indicate that they are governed by a mixed dynamics. However, due to the large number of missing levels no final conclusions could be drawn. In our laboratory complete spectra of several 100 resonance frequencies and resonance strengths of diverse microwave billiards with mixed classical dynamics have been measured. Based on a superstatistical approach a prediction was derived for the nearest neighbour-spacing distribution of the eigenvalues and for the resonance strength distribution of quantum systems with mixed classical dynamics. The comparison with corresponding experimental distributions all in all yielded a better agreement than that with the other well-established distributions. The related analysis and the results are published in Phys. Rev. E 77, 046202 (2008).
Bouncing ball orbits and symmetry-breaking effects in a three-dimensional chaotic billiard
We have studied in collaboration with U. Reif and B. Mößner from the Department of Applied Mathematics at the TU Darmstadt the classical and quantum mechanics of a three-dimensional stadium billiard [see Phys. Rev. E 77, 046221 (2008)]. It consists of two quarter cylinders that are rotated with respect to each other by 90° and its classical dynamics is chaotic. The billiard exhibits only a few families of nongeneric periodic orbits. We introduced an analytic method for the determination of their contribution to the spectral density of the corresponding quantum billiard in Phys. Rev. Lett. 89, 064101 (2002), where the spectral properties of a three-dimensional microwave resonator with this shape (see photo) were investigated experimentally. Note, that even though there is no analogy between quantum billiards and microwave resonators in three dimensions, in both cases a semiclassical approximation for the spectral density, a so-called trace formula, is given in terms of a sum over all periodic orbits of the classical billiard. The length spectrum, which is computed from the spectral density, could be understood in terms of nongeneric and unstable periodic orbits. For unequal radii of the quarter cylinders the level statistics agrees well with predictions from RMT. For equal radii the billiard exhibits an additional symmetry. We investigated the effects of symmetry breaking on spectral properties. Moreover, for equal radii, we observed a small deviation of the level statistics from random matrix theory. This led to the discovery of stable and marginally stable orbits, which are absent for unequal radii.
Time-reversal invariance violation at an exceptional point
Recently, experiments on properties of the eigenvalues and the eigenfunctions of a dissipative system with broken T invariance in the vicinity of an exceptional point (EP) were completed. The experimental setup is similar to that used in the experiments for T-invariant systems, except for a magnetized ferrite inside the resonator to induce T violation. Close to an EP the system is described by a complex two-state Hamiltonian which is neither Hermitian nor symmetric and depends on two parameters. At an EP its eigenvalues and also the associated eigenvectors coalesce. In contrast to previous experiments concerning EPs, we now have the possibility to determine the two-state Hamiltonian on a very narrow grid in the parameter plane. For this we apply the S-matrix formalism successfully tested in Phys. Rev. Lett. 98, 074103 (2007) for isolated single resonances and for pairs of nearly degenerate ones. This yields an unprecedented set of data, allowing the very precise determination of the location of the EP, the size of the T violation, and the geometric phases and amplitudes gathered by the eigenvectors along arbitrary contours around the EP. The results are well described by an analytic two-state model developed in the course of the data analysis and have been published in Phys. Rev. Lett. 106, 150403 (2011).
Spectral properties of dielectric microwave resonators
Dielectric microlasers and microcavities have received much attention over the last years. They typically consist of a flat cylindrical dielectric resonator whose cross section shape determines the emission properties such as the directionality of radiation or the lifetimes of the resonances. Generally, even for the simplest geometries, as e.g., a flat dielectric disk with a height much smaller than its planar extension, the Helmholtz equations cannot be solved analytically. The aim of a series of microwave experiments performed in our laboratory with flat cylindrical dielectric resonators was to test an approximation which reduces the associated three-dimensional Helmholtz equation to a two-dimensional one by introducing a so-called effective index of refraction. Therefore, the resonances were identified using the measured intensity distributions (see figure). Even though the dimensions of the microwave resonators are considerably larger than those of a typical microlaser the results are applicable to the latter due to the scaling invariance of the Helmholtz equation, the only relevant quantity being their ratio to the wavelength. These studies provided the first rigorous test of the accuracy of this approximation, which revealed that it is limited [see Phys. Rev. A 80, 023825 (2009)].
Furthermore, over the last years the study of classical ray dynamics and of the role played by the periodic orbits gained considerable attention, especially in the context of the emission properties, such as the directions of maximal emission. Therefore, in another series of experiments a trace formula in terms of the periodic orbits recently proposed for the fluctuating part of the resonance density of two-dimensional dielectric resonators was tested. To obtain a two-dimensional dielectric resonator, it was squeezed between two metallic plates. As mentioned above all modes in a dielectric resonator are only quasi bound. Consequently, a large part of the expected resonances is not observed in the measured resonance spectra. Accordingly, our main concern was to test whether the trace formula is applicable to systems with such a large number of missing levels as is the typical situation in nuclear physics. We came to the result that the lengths of the classical periodic orbits can be obtained from the experimental resonance density even if only 10% of the expected number of resonances is found [see Phys. Rev. E. 81, 066215 (2010)].