Spatial correlation in quantum chaotic systems with time-reversal symmetry: theory and experiment

Transmission phase of a quantum dot and statistical fluctuations of partial-width amplitudes

Experimentally, the phase of the amplitude for electron transmission through a quantum dot (transmission phase) shows the same pattern between consecutive resonances. Such universal behavior, found for long sequences of resonances, is caused by correlations of the signs of the partial-width amplitudes of the resonances. We investigate the stability of these correlations in terms of a statistical model. For a classically chaotic dot, the resonance eigenfunctions are assumed to be Gaussian distributed. Under this hypothesis, statistical fluctuations are found to reduce the tendency towards universal phase evolution. Long sequences of resonances with universal behavior only persist in the semiclassical limit of very large electron numbers in the dot and for specific energy intervals. Numerical calculations qualitatively agree with the statistical model but quantitatively are closer to universality.

Universal impedance fluctuations in wave chaotic systems

We experimentally investigate theoretical predictions of universal impedance fluctuations in wave chaotic systems using a microwave analog of a quantum chaotic infinite square well potential. We emphasize the use of the radiation impedance to remove the nonuniversal effects of the particular coupling between the outside world and the scatterer. Specific predictions that we test include the probability density functions (PDFs) of the real and imaginary parts of the universal impedance, the equality of the variances of these PDFs, and the dependence of these PDFs on a single loss parameter.

Fermi-edge singularities in the mesoscopic x-ray edge problem

We study the x-ray edge problem for a chaotic quantum dot or nanoparticle displaying mesoscopic fluctuations. In the bulk, x-ray physics is known to produce Fermi-edge singularities-deviations from the naively expected photoabsorption cross section in the form of a peaked or rounded edge. For a coherent system with chaotic dynamics, we find substantial changes; in particular, a photoabsorption cross section showing a rounded edge in the bulk will change to a slightly peaked edge on average as the system size is reduced to a mesoscopic (coherent) scale.

Generalized Berry conjecture and mode correlations in chaotic plates

We consider a modification of the Berry conjecture for eigenmode statistics in wave-bearing systems. The eigenmode correlator is conjectured to be proportional to the imaginary part of the Green's function. The generalization is applicable not only to scalar waves in the interior of homogeneous isotropic systems where the correlator is a Bessel function, but to arbitrary points of heterogeneous systems as well. In view of recent experimental measurements, expressions for the intensity correlator in chaotic plates are derived.

Semiclassical construction of random wave functions for confined systems

We develop a statistical description of chaotic wave functions in closed systems obeying arbitrary boundary conditions by combining a semiclassical expression for the spatial two-point correlation function with a treatment of eigenfunctions as Gaussian random fields. Thereby we generalize Berry's isotropic random wave model by incorporating confinement effects through classical paths reflected at the boundaries. Our approach allows one to explicitly calculate highly nontrivial statistics, such as intensity distributions, in terms of usually few short orbits, depending on the energy window considered. We compare with numerical quantum results for the Africa billiard and derive nonisotropic random wave models for other prominent confinement geometries.

Numerical study of eigenvector statistics for random banded matrices

The statistics of eigenvector amplitudes near the band center in random-banded-matrix ensembles is studied numerically. The nonlinear sigma model provides a rigorous description of the statistics in these ensembles. We are interested in the extension of the predictions of the sigma model approach to complex quantum systems. We study the validity range of the perturbation theory beginning from the well-known formulas in random matrix theory.

Semiclassical spatial correlations in chaotic wave functions

We study the spatial autocorrelation of energy eigenfunctions psi(n)(q) corresponding to classically chaotic systems in the semiclassical regime. Our analysis is based on the Weyl-Wigner formalism for the spectral average C(epsilon)(q(+),q(-),E) of psi(n)(q(+))psi(*)(n)(q(-)), defined as the average over eigenstates within an energy window epsilon centered at E. In this framework C(epsilon) is the Fourier transform in the momentum space of the spectral Wigner function W(x,E;epsilon). Our study reveals the chord structure that C(epsilon) inherits from the spectral Wigner function showing the interplay between the size of the spectral average window, and the spatial separation scale. We discuss under which conditions is it possible to define a local system independent regime for C(epsilon). In doing so, we derive an expression that bridges the existing formulas in the literature and find expressions for C(epsilon)(q(+),q(-),E) valid for any separation size /q(+)-q(-)/.

Measurement of wave chaotic eigenfunctions in the time-reversal symmetry-breaking crossover regime

We present experimental results on eigenfunctions of a wave chaotic system in the continuous crossover regime between time-reversal symmetric and time-reversal symmetry-broken states. The statistical properties of the eigenfunctions of a two-dimensional microwave resonator are analyzed as a function of an experimentally determined time-reversal symmetry-breaking parameter. We test four theories of one-point eigenfunction statistics and introduce a new theory relating the one-point and two-point statistical properties in the crossover regime. We also find a universal correlation between the one-point and two-point statistical parameters for the crossover eigenfunctions.

Correlations due to localization in quantum eigenfunctions of disordered microwave cavities

Statistical properties of experimental eigenfunctions of quantum chaotic and disordered microwave cavities are shown to demonstrate nonuniversal correlations due to localization. Varying energy E and mean free path l enable us to experimentally tune from localized to delocalized states. Large level-to-level inverse participation ratio ( I2) fluctuations are observed for the disordered billiards, whose distribution is strongly asymmetric about <I2>. The spatial density autocorrelations of eigenfunctions are shown to spatially decay exponentially and the decay lengths are experimentally determined. All the results are quantitatively consistent with calculations based upon nonlinear sigma models.

Distribution of Husimi zeros in polygonal billiards

The zeros of the Husimi function provide a minimal description of individual quantum eigenstates and their distribution is of considerable interest. We provide here a numerical study for pseudointegrable billiards which suggests that the zeros tend to diffuse over phase space in a manner reminiscent of chaotic systems but nevertheless contain a subtle signature of pseudointegrability. We also find that the zeros depend sensitively on the position and momentum uncertainties (Delta q and Delta p, respectively) with the classical correspondence best when Delta q = Delta p = square root of [Planck's constant/2]. Finally, short-range correlations seem to be well described by the Ginibre ensemble of complex matrices.