Experimental investigations of chaos-assisted tunneling in a microwave annular billiard.

We present detailed investigations of the experimental signatures of chaos-assisted tunneling in the two-dimensional annular billiard, as already summarized in Phys. Rev. Lett. 84, 867 (2000). We have performed analog experiments with two-dimensional, electromagnetic resonators allowing for a direct simulation of the corresponding quantum system. Spectra from a superconducting cavity with a high-frequency resolution are combined with electromagnetic intensity distributions of high spatial resolution experimentally determined using a normal conducting twin cavity. Thereby all eigenmodes were obtained with properly identified quantum numbers. Besides distributions of quasi-doublet splittings, which serve as fundamental observables for the tunneling between whispering gallery types of modes, we also focus on the distributions of resonance widths of the doublets. These directly reflect the role of lifetime of certain modes in the tunneling process. Here, as theoretically expected, the class of so-called beach modes is found to play a particular role in mediating between regular and chaotic states to enhance the tunneling strength. This behavior is found in the spectrum and also in the structure of the wave functions.

Distribution of resonance strengths in microwave billiards of mixed and chaotic dynamics.

A new measure for statistical properties of the wave function components of quantum systems, the distribution of the product of two partial widths, is introduced. It is tested with data obtained in analog experiments with microwave billiards, where the product of two partial widths equals the resonance strengths in the microwave spectra. The billiards are from the family of the Limaçons, one with chaotic and two with mixed classical dynamics. For completely chaotic systems the partial widths generically obey a Porter-Thomas distribution. We show that in this case the distribution of their product equals a K0 distribution. While we find deviations of the experimental strength distribution from the K0 distribution for the billiards with mixed dynamics, the distributions agree perfectly for the chaotic billiard, when taking into account the experimental threshold of detection in the theoretical description. Hence, the strength distribution provides another stringent test for the connection between statistical properties of systems with classical chaotic dynamics and random matrix theory.

First experimental evidence for quantum echoes in scattering systems.

A self-pulsing effect termed quantum echoes has been observed in experiments with an open superconducting and a normal conducting microwave billiard whose geometry provides soft chaos, i.e., a mixed phase space portrait with a large stable island. For such systems a periodic response to an incoming pulse has been predicted. Its period has been associated with the degree of development of a horseshoe describing the topology of the classical dynamics. The experiments confirm this picture and reveal the topological information.

Encircling an exceptional point.

We calculate analytically the geometric phases that the eigenvectors of a parametric dissipative two-state system described by a complex symmetric Hamiltonian pick up when an exceptional point (EP) is encircled. An EP is a parameter setting where the two eigenvalues and the corresponding eigenvectors of the Hamiltonian coalesce. We show that it can be encircled on a path along which the eigenvectors remain approximately real and discuss a microwave cavity experiment, where such an encircling of an EP was realized. Since the wave functions remain approximately real, they could be reconstructed from the nodal lines of the recorded spatial intensity distributions of the electric fields inside the resonator. We measured the geometric phases that occur when an EP is encircled four times and thus confirmed that for our system an EP is a branch point of fourth order.

Observation of a chiral state in a microwave cavity.

A microwave experiment has been realized to measure the phase difference of the oscillating electric field at two points inside the cavity. The technique has been applied to a dissipative resonator which exhibits a singularity-called exceptional point-in its eigenvalue and eigenvector spectrum. At the singularity, two modes coalesce with a phase difference of pi/2. We conclude that the state excited at the singularity has a definitive chirality.

Phase shift experiments identifying Kramers doublets in a chaotic superconducting microwave billiard of threefold symmetry.

The spectral properties of a two-dimensional microwave billiard showing threefold symmetry have been studied with a new experimental technique. This method is based on the behavior of the eigenmodes under variation of a phase shift between two input channels, which strongly depends on the symmetries of the eigenfunctions. Thereby a complete set of 108 Kramers doublets has been identified by a simple and purely experimental method. This set clearly shows Gaussian unitary ensemble statistics, although the system is time-reversal invariant.

R-matrix theory of driven electromagnetic cavities.

The resonances of cylindrical symmetric microwave cavities are analyzed in R-matrix theory, which transforms the input channel conditions to the output channels. Single and interfering double resonances are studied and compared with experimental results obtained with superconducting microwave cavities. Because of the equivalence of the two-dimensional Helmholtz and the stationary Schrödinger equations, the results give insight into the resonance structure of regular and chaotic quantum billiards.

Abnormalities of the enteric nervous system in heterozygous endothelin B receptor deficient (spotting lethal) rats resembling intestinal neuronal dysplasia.

BACKGROUND: A homozygous mutation of the endothelin B receptor (EDNRB) gene in spotting lethal (sl/sl) rats leads to Hirschsprung's disease (HSCR) with long segmented aganglionosis. However, the effects on the development of the enteric nervous system (ENS) promoted by a heterozygous mutation of the EDNRB gene are not known. The present study aimed to describe and morphometrically assess the phenotypic abnormalities of the ENS in heterozygous (+/sl) EDNRB deficient rats in comparison with homozygous (sl/sl) EDNRB deficient and wild-type (+/+) rats. METHODS: The distal small intestine, caecum, and colon were obtained from sl/sl, +/sl, and +/+ rats. To demonstrate the three dimensional organisation of the ENS, the intestinal wall was microdissected into wholemounts and incubated against the pan-neuronal marker protein gene product 9.5. Assessment of the ENS included morphometric quantification of ganglionic size and density, the number of nerve cells per ganglia, and the diameter of nerve fibre strands within both the myenteric and submucous plexus. RESULTS: Sl/sl rats were characterised by complete aganglionosis resembling the same histopathological features observed in patients with HSCR. +/sl rats revealed more subtle abnormalities of the ENS: the submucous plexus was characterised by a significantly increased ganglionic size and density, and the presence of hypertrophied nerve fibre strands. Morphometric evaluation of the myenteric plexus did not show statistically significant differences between +/sl and +/+ rats. CONCLUSIONS: In contrast with sl/sl rats, +/sl rats display non-aganglionated malformations of the ENS. Interestingly, these innervational abnormalities resemble the histopathological criteria for intestinal neuronal dysplasia (IND). Although IND has been described in several intestinal motility disorders, the concept of a clearly defined clinical-histopathological entity is still controversially discussed. The present findings support the concept of IND based on clearly defined morphological criteria suggesting a genetic link, and thus may provide a model for human IND. Furthermore, the data underline the critical role of the "gene dose" for the phenotypic effects promoted by the EDNRB/EDN3 system and confirm that the development of the ENS is not an "all or none" phenomenon.

Experimental test of a trace formula for a chaotic three-dimensional microwave cavity.

We have measured resonance spectra in a superconducting microwave cavity with the shape of a three-dimensional generalized Bunimovich stadium billiard and analyzed their spectral fluctuation properties. The experimental length spectrum exhibits contributions from periodic orbits of nongeneric modes and from unstable periodic orbits of the underlying classical system. It is well reproduced by our theoretical calculations based on the trace formula derived by Balian and Duplantier for chaotic electromagnetic cavities.

Bayesian analysis of level-spacing distributions for chaotic systems with broken symmetry.

Bayesian inference is applied to the nearest-neighbor and next-nearest-neighbor spacing distributions of levels of coupled superconducting microwave billiards. The weakly coupled resonators are equivalent to a quantum system with a partially broken symmetry. The coupling parameters are obtained with help from Bayes's theorem. This procedure does not require the introduction of a set of bins. The results are more accurate than those obtained from other bin-independent procedures.

First experimental test of a trace formula for billiard systems showing mixed dynamics.

In general, trace formulas relate the density of states for a given quantum mechanical system to the properties of the periodic orbits of its classical counterpart. Here we report for the first time on a semiclassical description of microwave spectra taken from superconducting billiards of the Limaçon family showing mixed dynamics in terms of a generalized trace formula derived by Ullmo et al. [Phys. Rev. E 54, 136 (1996)]. This expression not only describes mixed-typed behavior but also the limiting cases of fully regular and fully chaotic systems and thus presents a continuous interpolation between the Berry-Tabor and Gutzwiller formulas.

Impaired sodium excretion, decreased glomerular filtration rate and elevated blood pressure in endothelin receptor type B deficient rats.

The renal endothelin (ET) system, particularly the ET type B receptor, has been implicated in the regulation of sodium excretion and glomerular filtration rate (GFR). We analyzed kidney morphology and function in a rat strain characterized by complete absence of a functional ETB receptor. Due to Hirschsprung's disease limiting lifetime in these rats, studies were performed in 23-day-old rats. Kidney size and morphology (glomerular and interstitial matrix content, glomerular size and cell density and intrarenal vascular morphology) were normal in ETB-deficient rats. There were also no evidence of altered kidney cell cycle regulation in these rats. GFR was significantly lower, by 72% (P<0.001), in homozygous ETB-deficient rats than in wild-type rats. Fractional sodium excretion was likewise markedly reduced by 84% in homozygous ETB-deficient rats (P<0.001 versus wild-type rats). Treatment with the specific epithelial sodium channel blocker amiloride led to a much higher increase in fractional sodium excretion in ETB-deficient rats (934.2+/-73% in ETB-deficient rats versus 297+/-20% in wild-type rats, expressed as percentage of corresponding placebo treated control; P<0.001). Mean arterial blood pressure was elevated by 7.9 mmHg in homozygous ETB-deficient rats (P<0.05 versus wild-type rats). Our study demonstrates that ETB-deficiency causes early onset kidney dysfunction characterized by a markedly reduced sodium excretion, decreased GFR, and slightly elevated blood pressure. The complete absence of the ETB receptor causes in the kidney--in contrast to the colon--a functional rather than a developmental, neural crest cell dependent disease, since kidney morphology was normal in ETB-deficient rats. The much higher increase in the fractional sodium excretion in ETB-deficient rats after pharmacological blockade of the epithelial sodium channel indicates that the decreased fractional sodium excretion in ETB-deficient rats is most probably due to a lack of the inhibitory property of the ETB receptor on the epithelial sodium channel activity.

Experimental observation of the topological structure of exceptional points.

We report on a microwave cavity experiment where exceptional points (EPs), which are square root singularities of the eigenvalues as function of a complex interaction parameter, are encircled in the laboratory. The real and imaginary parts of an eigenvalue are given by the frequency and width of a resonance and the eigenvectors by the field distributions. Repulsion of eigenvalues--always associated with EPs--implies frequency anticrossing (crossing) whenever width crossing (anticrossing) is present. The eigenvalues and eigenvectors are interchanged while encircling an EP, but one of the eigenvectors undergoes a sign change which can be discerned in the field patterns.

Gaussian unitary ensemble statistics in a time-reversal invariant microwave triangular billiard

The spectrum of a chaotic two-dimensional quantum billiard with threefold symmetry has been studied in an experiment with a superconducting microwave cavity. In total 622 eigenvalues were identified experimentally and compared with numerical calculations. The statistical analysis of the data shows that Gaussian unitary ensemble statistics can be observed for a spectrum of a time-reversal invariant system.

First experimental evidence for chaos-assisted tunneling in a microwave annular billiard

We report on first experimental signatures for chaos-assisted tunneling in a two-dimensional annular billiard. Measurements of microwave spectra from a superconducting cavity with high frequency resolution are combined with electromagnetic field distributions experimentally determined from a normal conducting twin cavity with high spatial resolution to resolve eigenmodes with properly identified quantum numbers. Distributions of quasidoublet splittings serve as basic observables for the tunneling between whispering gallery-type modes localized to congruent, but distinct tori which are coupled weakly to irregular eigenstates associated with the chaotic region in phase space.

Phenotype, intestinal morphology, and survival of homozygous and heterozygous endothelin B receptor--deficient (spotting lethal) rats.

BACKGROUND/PURPOSE: Spotting lethal (sl) rats, a model for Hirschsprung's disease, recently have been found to carry a deletion in the endothelin B (ET(B)) gene, causing functional lack of ET(B) receptors. The ET(B) receptor mediates, together with and in counterbalance to the ET(A) receptor, endothelin actions on vessels, cell proliferation, and migration. The authors investigated the effect of homozygosity (sI/sI) or heterozygosity (+/sl) on phenotype, intestinal morphology, and survival. METHODS: Weight, circumference, and serum albumin were measured. Histological tests of major organs and immunoperoxidase reaction for Peripherin, glial fibrillary acid protein (GFAP), and S-100 in small and large intestine were performed. Peripherin-immunostained sections of colon and jejunum were analyzed morphometrically. Screening for sepsis included search for enterocolitis, bacterial infection, endotoxin, and iNOS mRNA. RESULTS: Sl/sl rats died within 4 weeks of life, showing an early and a later death group. Serum albumin levels were decreased in sl/sl rats, whereas signs of sepsis were rare. Immunostaining uncovered alterations in nerve and glial cells in the whole gut of sl/sl rats, and to a subtle degree also in +/sl rats, which appear clinically normal. Morphometric quantification yielded statistically significant alterations in sl/sl rats only. No obvious abnormalities were found in other organs. CONCLUSIONS: Sl/sl rats die from malnutrition rather than sepsis, too early for ischemic complications to occur. Rats of the later death group are a suitable model for studying the ET8 receptor in vivo. Subtle abnormalities in the enteric nervous system of heterozygous rats underline the critical role of the "gene dose" for functional compensation.

Endothelin b receptor deficiency is associated with an increased rate of neuronal apoptosis in the dentate gyrus.

The dentate gyrus retains neuronal proliferative potential throughout life. Using immature endothelin B receptor-deficient (sl/sl) rats, a rabbit model of pneumococcal meningitis and autopsy brains from humans who died from pneumococcal meningitis, we explored the role of endothelin B receptors in physiological and pathological neuronal apoptosis in the dentate gyrus. At postnatal days 3-4, the rate of apoptosis in the dentate gyrus was high in all rats, declining to low levels in wild-type rats (+/+) on days 14 and 22, but remaining high in both homozygous (sl/sl) and heterozygous (sl/+) endothelin B receptor-deficient rats. Increased apoptosis was not significantly compensated for by neuronal proliferation. Hippocampal neuronal cultures also exhibited genotype-dependent apoptosis with the highest rate in neurons from homozygous endothelin B receptor-deficient (sl/sl) rats. In rabbit and human pneumococcal meningitis, increased apoptosis in the dentate gyrus was associated with loss of neuronal endothelin B receptor immunoreactivity. In conclusion, endothelin B receptors appear to act as neuronal survival factors in the dentate gyrus in rodents and man, both during postnatal development and under pathological conditions.

Endothelin B receptor-deficient rats as a subtraction model to study the cerebral endothelin system.

Endothelins, due to their potent vasoactivity and mitogenicity, appear to play an important role in the brain, where all components of the endothelin system, peptides, receptors and converting enzyme, are expressed. To further elucidate the role of the cerebral endothelin system, astrocytes and cerebral vessels from sl/sl rats, devoid of functional endothelin B receptors, have been employed. Astrocytes from sl/sl rats display the following abnormalities as compared to wild-type (+/+) cells: (i) elevated basal extracellular endothelin-1 levels; (ii) exclusive presence of functional endothelin A receptors; (iii) increased extracellular endothelin-1 levels upon endothelin A receptor blockade; (iv) augmented basal endothelin-converting enzyme activity; (v) altered calcium response to endothelin-1. The basilar artery of sl/sl rats shows an enhanced constricting response to endothelin-1 and fails to dilate in response to endothelin-3, shifting the endothelin vasomotor balance to constriction. In conclusion, endothelin B receptors may be essential for restricting extracellular endothelin-1 levels in the brain, as well as for a balanced cerebral vasomotor action of endothelins.

Experimental versus numerical eigenvalues of a Bunimovich stadium billiard: a comparison.

We compare the statistical properties of eigenvalue sequences for a gamma=1 Bunimovich stadium billiard. The eigenvalues have been obtained in two ways: one set results from a measurement of the eigenfrequencies of a superconducting microwave resonator (real system), and the other set is calculated numerically (ideal system). We show influence of mechanical imperfections of the real system in the analysis of the spectral fluctuations and in the length spectra compared to the exact data of the ideal system. We also discuss the influence of a family of marginally stable orbits, the bouncing ball orbits, in two microwave stadium billiards with different geometrical dimensions.