Semiclassical construction of random wave functions for confined systems

Numerical realization of diffuse sound pressure fields using prolate spheroidal wave functions

A diffuse sound field is conventionally defined as a zero-mean circularly symmetric complex Gaussian random field. A more recent, generalized definition is that of a sound field having mode shapes that are diffuse in the conventional sense, and eigenfrequencies that conform to the Gaussian orthogonal ensemble. Such a generalized diffuse sound field can represent a random ensemble of sound fields that share gross features, such as modal density and total absorption, but otherwise have any possible arrangement of local wave scattering features. The problem of generating realizations or Monte Carlo samples of a conventional diffuse sound field or, equivalently, of the mode shapes of a generalized diffuse sound field, is addressed here. Such realizations can be obtained from an eigenvalue decomposition of the spatial correlation function. A discrete decomposition is numerically expensive when the sound pressures at many locations are of interest, so a fast analytical decomposition based on prolate spheroidal wave functions is developed. The approach is numerically validated by comparison with a detailed room model, where random wave scatterers are explicitly modeled as acoustic point masses with random positions, and good correspondence is observed. Furthermore, applications involving correlated sound sources and sound-structure interaction are presented.

Characterization of random features of chaotic eigenfunctions in unperturbed basis

In this paper we study random features manifested in components of energy eigenfunctions of quantum chaotic systems, given in the basis of unperturbed, integrable systems. Based on semiclassical analysis, particularly on Berry's conjecture, it is shown that the components in classically allowed regions can be regarded as Gaussian random numbers in a certain sense, when appropriately rescaled with respect to the average shape of the eigenfunctions. This suggests that when a perturbed system changes from integrable to chaotic, deviation of the distribution of rescaled components in classically allowed regions from the Gaussian distribution may be employed as a measure for the "distance" to quantum chaos. Numerical simulations performed in the Lipkin-Meshkov-Glick model and the Dicke model show that this deviation coincides with the deviation of the nearest-level-spacing distribution from the prediction of random-matrix theory. Similar numerical results are also obtained in two models without classical counterpart.

Correlation between peak-height modulation and phase lapses in transport through quantum dots

We show that two intriguing features of mesoscopic transport, namely, the modulation of Coulomb blockade peak heights and the transmission phase lapses occurring between subsequent peaks, are closely related. Our analytic arguments are corroborated by numerical simulations for chaotic ballistic quantum dots. The correlations between the two properties are experimentally testable. The statistical distribution of the partial-width amplitude, at the heart of the previous relationship, is determined, and its characteristic parameters are estimated from simple models.

Correlations in eigenfunctions of quantum chaotic systems with sparse Hamiltonian matrices

In most realistic models for quantum chaotic systems, the Hamiltonian matrices in unperturbed bases have a sparse structure. We study correlations in eigenfunctions of such systems and derive explicit expressions for some of the correlation functions with respect to energy. The analytical results are tested in several models by numerical simulations. Some applications are discussed for a relation between transition probabilities and for expectation values of some local observables.

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 spatial correlations in random spinor fields

We identify universal spatial fluctuations in systems with nontrivial spin dynamics. To this end we calculate by exact numerical diagonalization a variety of experimentally relevant correlations between spinor amplitudes, spin polarizations, and spin currents, both in the bulk and near the boundary of a confined two-dimensional clean electron gas in the presence of spin-orbit interaction. We support our claim of universality with the excellent agreement between the numerical results and system-independent spatial correlations of a random field defined on both the spatial and spin degrees of freedom. A rigorous identity relating our universal predictions with response functions provides a direct physical interpretation of our results in the framework of linear response theory.

Scattering phase of quantum dots: emergence of universal behavior

We investigate scattering through chaotic ballistic quantum dots in the Coulomb-blockade regime. Focusing on the scattering phase, we show that large universal sequences emerge in the short wavelength limit, where phase lapses of π systematically occur between two consecutive resonances. Our results are corroborated by numerics and are in qualitative agreement with existing experiments.

A hybrid approach for predicting the distribution of vibro-acoustic energy in complex built-up structures

Finding the distribution of vibro-acoustic energy in complex built-up structures in the mid-to-high frequency regime is a difficult task. In particular, structures with large variation of local wavelengths and/or characteristic scales pose a challenge referred to as the mid-frequency problem. Standard numerical methods such as the finite element method (FEM) scale with the local wavelength and quickly become too large even for modern computer architectures. High frequency techniques, such as statistical energy analysis (SEA), often miss important information such as dominant resonance behavior due to stiff or small scale parts of the structure. Hybrid methods circumvent this problem by coupling FEM/BEM and SEA models in a given built-up structure. In the approach adopted here, the whole system is split into a number of subsystems that are treated by either FEM or SEA depending on the local wavelength. Subsystems with relative long wavelengths are modeled using FEM. Making a diffuse field assumption for the wave fields in the short wave length components, the coupling between subsystems can be reduced to a weighted random field correlation function. The approach presented results in an SEA-like set of linear equations that can be solved for the mean energies in the short wavelength subsystems.

Experimental examination of the effect of short ray trajectories in two-port wave-chaotic scattering systems

Predicting the statistics of realistic wave-chaotic scattering systems requires, in addition to random matrix theory, introduction of system-specific information. This paper investigates experimentally one aspect of system-specific behavior, namely, the effects of short ray trajectories in wave-chaotic systems open to outside scattering channels. In particular, we consider ray trajectories of limited length that enter a scattering region through a channel (port) and subsequently exit through a channel (port). We show that a suitably averaged value of the impedance can be computed from these trajectories and that this can improve the ability to describe the statistical properties of the scattering systems. We illustrate and test these points through experiments on a realistic two-port microwave scattering billiard.

Universal and nonuniversal properties of wave-chaotic scattering systems

Prediction of the statistics of scattering in typical wave-chaotic systems requires combining system-specific information with universal aspects of chaotic scattering as described by random matrix theory. This Rapid Communication shows that the average impedance matrix, which characterizes such system-specific properties, can be semiclassically calculated in terms of ray trajectories between ports. Theoretical predictions are compared with experimental results for a microwave billiard, demonstrating that the theory successfully uncovered universal statistics of wave-chaotic scattering systems.

Density and correlation functions of vortex and saddle points in open billiard systems

We present microwave measurements for the density and spatial correlation of current critical points in an open billiard system and compare them with new and previous predictions of the random-wave model (RWM). In particular, due to an improvement of the experimental setup, we determine experimentally the spatial correlation of saddle points of the current field. An asymptotic expression for the vortex-saddle and saddle-saddle correlation functions based on the RWM is derived, with experiment and theory agreeing well. We also derive an expression for the density of saddle points in the presence of a straight boundary with general mixed boundary conditions in the RWM and compare with experimental measurements of the vortex and saddle density in the vicinity of a straight wall satisfying Dirichlet conditions.

Residual Coulomb interaction fluctuations in chaotic systems: the boundary, random plane waves, and semiclassical theory

New fluctuation properties arise in problems where both spatial integration and energy summation are necessary ingredients. The quintessential example is given by the short-range approximation to the first order ground state contribution of the residual Coulomb interaction. The dominant features come from the region near the boundary where there is an interplay between Friedel oscillations and fluctuations in the eigenstates. Quite naturally, the fluctuation scale is significantly enhanced for Neumann boundary conditions as compared to Dirichlet. Elements missing from random plane wave modeling of chaotic eigenstates lead surprisingly to significant errors, which can be corrected within a purely semiclassical approach.

Wave-vector resonance in a nonlinear multiwavespeed chaotic billiard

Nonlinear coupling between eigenmodes of a system leads to spectral energy redistribution. For multiwavespeed chaotic billiards, the average coupling strength can exhibit sharp discontinuities as a function of frequency related to wave-vector coincidences between constituent waves of different wavespeeds. The phenomenon is investigated numerically for an ensemble of two-dimensional square two-wavespeed billiards with rough boundaries and quadratic nonlinearity representative of elastodynamic waves. Results of direct numerical simulations are compared with theoretical predictions.

Phase-space correlations of chaotic eigenstates

It is shown that the Husimi representations of chaotic eigenstates are strongly correlated along classical trajectories. These correlations extend across the whole system size and, unlike the corresponding eigenfunction correlations in configuration space, they persist in the semiclassical limit. A quantitative theory is developed on the basis of Gaussian wave packet dynamics and random-matrix arguments. The role of symmetries is discussed for the example of time-reversal invariance.