Session Q8: Focus Session: Wave Chaos: Theory and Applications

Caustics Formation and Sharp Focusing in PT-symmetric Waveguide Arrays

We investigate focusing effects and curved optical beam trajectories in waveguide arrays consisting of coupled dimers with local Parity-Time symmetry i.e. one element of the dimer has loss and the other an equal amount of gain. When the intra-dimer coupling is stronger than the inter-dimer coupling the propagation constants of this array are real (exact PT-symmetric phase). We find that, under such conditions, appropriate tailoring of the phases and amplitudes of the initial beam can lead to reconfigurable caustic phenomena and curved beam propagation,as well as focusing of the initial beam at paraxial distances controlled by the degree of gain and loss involved in these PT-symmetric structures.   

Many-Body scattering through mesoscopic chaotic cavities: Universal effects of indistinguishability and interaction

We consider the mesoscopic scattering of identical particles, and study the interplay between three physical effects: universality of single-particle transport, many-body correlations due to quantum indistinguishability, similar to the Hong-Ou-Mandel effect in quantum optics, and the presence of interparticle interactions. Starting from a rigorous construction of the many-body scattering amplitudes, the well-known universality of chaotic wave transport is encoded in the statistical correlations between single-particle scattering matrices and ultimately between classical single-particle paths joining incoming and outgoing channels. For non-interacting systems, very non-trivial combinations of scattering matrices arise due to the symmetrization postulate and a mesoscopic version of the Hong-Ou-Mandel profile is obtained[1]. In a further step, a universal Hamiltonian representing interactions in the base of scattering single-particle states is constructed that allows us to study how interparticle interactions affect the Hong-Ou-Mandel correlations in the regime of mesoscopic chaotic transport.\\[4pt] [1] J. D. Urbina, J. Kuipers, Q. Hummel, K. Richter, arXiv:1409.1558v1   

Luukko Scars - a New Mechanism for Wavefunction Scar Formation

A new type of scarring phenomenon in bound quantum systems is reported that creates classical orbit localization as a result of localization in classical action. Perturbation of quasi-degenerate manifolds of unperturbed quantum states by a weak random potential is involved. The localization may be related to Anderson localization but this is still under investigation.   

Focusing Waves at Arbitrary Locations in a Ray-Chaotic Enclosure Using Time-Reversed Synthetic Sonas

Time reversal mirrors have been widely used to achieve wave focusing in wave-chaotic acoustic and electromagnetic enclosures. A typical time reversal experiment requires that a transmitter be initially present at the target focusing point, which limits the application of this technique. In this presentation, we propose a method to focus waves at an arbitrary location inside a complex (wave chaotic) enclosure using a numerically calculated wave signal. We use a semi-classical ray algorithm to calculate the signal that would be received at a transceiver port resulting from the injection of a short pulse at the desired target location. This pre-calculated signal is then time-reversed and sent into the enclosure by the transceiver, resulting in a time reversed short pulse at the focusing point. Since a physical wave source is not required at the target point, one can focus a signal at any desired location given knowledge of the ray propagation paths. Three parameters are used to quantify reconstruction quality, the peak-to-peak voltage, focus ratio, and energy transfer ratio. It is shown that the values of these quality metrics can be predicted by the statistics of the scattering-parameter \textbar S\textunderscore 21\textbar \textasciicircum 2 between the transceiver and target points in the enclosure. We experimentally demonstrate the method using a flat microwave billiard and quantify the reconstruction quality as a function of enclosure loss, port coupling and other considerations.   

Eigenfunction scarring in distorted quantum wells

Conventional scarring refers to pronounced localization of eigenfunctions along unstable classical periodic orbits [1]. Here we apply a highly efficient eigenvalue solver of arbitrary two-dimensional (2D) systems [2] to study scarring phenomena in generic situations. In particular, we report unexpectedly strong scarring in 2D quantum wells perturbed by random potential bumps of variable characteristics [3]. The scars resemble classical periodic orbits of the unperturbed system (no bumps), but there appears to be no clear connection to the periodic orbits of the perturbed system, as would be the case for conventional scarring. The scars are also robust to increasing distortion amplitude, and show a tendency to pin to the potential bumps. We have used a variety of tools to analyze the origin of the scarring, in particular its relation with Anderson localization. [1] E. J. Heller, Phys. Rev. Lett. 53, 1515 (1984); [2] P. J. J. Luukko and E. Rasanen, Comp. Phys. Comm. 184, 769 (2013); [3] P. J. J. Luukko, A. Klales, B. Drury, L. Kaplan, E. J. Heller, and E. Rasanen (2015).   

Semiclassical propagation of correlation functions in closed electromagnetic environments

Field-field correlation functions can be propagated efficiently within confined systems through the Wigner-Weyl formalism. A semiclassical Frobenius-Perron operator is derived for the propagation of Wigner functions as a solution of the associated boundary integral equation. This idea is used to study the effect of non-integrable (chaotic) dynamics on the propagation of classical noisy fields. Model systems for quantum mechanics are used to mimic the radiation into closed spaces. A realistic model of statistical sources into a semi-open polygonal billiard is also presented. We find that the simplest description in terms of the classical Frobenius-Perron operator provides a description of the frequency-averaged correlation function but that wave-resonant and interference effects can also be accounted for. Applications of the theory focus on the prediction of energy distribution through electromagnetic environments in electromagnetic compatibility, wireless communication systems, and imaging optics.   

Massive simulation of complex electromagnetic cavities

The analysis of the chaotic behavior of complex electromagnetic cavities takes benefit from the availability of a large amount of data on field samples. The application of a code running on a supercomputer is able to return a precise electromagnetic simulation of electrically large structures. The simulations of mode-stirred reverberation chamber (RC) were performed using an in-house parallel finite-difference time-domain (FDTD) code. The code is divided into three modules that are managed by a unique, single-step job: the electromagnetic solver based on the FDTD method; a fast Fourier transform (FFT) to obtain the frequency domain behavior; a statistical tool to obtain the RC properties. A unique run produces statistical results for all the investigated stirrer angles, without the burden of saving intermediate data. The code implements a hybrid parallelization as function of stirrer angle and cavity volume. Specifically, such a computation is known to be ``embarrassingly parallel'' with respect to the stirrer angle. The excitation is a Gaussian pulse modulated sinusoid at 1.1 GHz: the 95\% bandwidth is 0.2 and 2 GHz. After the FDTD simulation is completed, the FFT module gives the frequency behavior of the fields in each point with a resolution of about 50 kHz.   

Nonlinear elastic waves in solids: Deriving simplicity from complexity

The introduction of nonlinearities to the dynamics of a homogeneous elastic medium alters the underlying wave dispersion characteristics. In this work, we present an exact formulation for the treatment of geometric nonlinearity in one-dimensional elastic wave propagation in a rod, considering both a thin rod where the thickness is small compared to the wavelength and a thick rod where lateral inertia is accounted for. Our derivation starts with the implementation of Hamilton's principle and terminates with an expression for the finite-strain dispersion relation in closed form. We explore the effect of wave amplitude on the derived dispersion relation and compare with results obtained by direct time-domain simulations followed by Fourier transformations. While often dispersion is attributed to only linear mechanisms, here we show that an otherwise linearly nondispersive elastic solid may exhibit dispersion solely due to the presence of a nonlinearity. This work provides insights into the fundamentals of nonlinear wave propagation in solids, which represents one of the agents of wave chaos in complex systems.   

Geometry and topology of tangled vortices in wave chaos

Linear waves in three-dimensional chaotic systems can contain complex vortex tangles that are difficult to describe analytically, even in non-dynamic systems such as solutions of the time-independent Schr\"odinger equation in which the vortices are zeros of a complex scalar wavefunction. Despite the linear nature of such eigenfunctions, the local geometry of their vortices leads to random conformations for which certain properties appear universal on large scales, even when compared to vortices in different physical systems such as superfluid turbulence or models of cosmic strings [1]. We numerically track vortex tangle in the random wave model of chaotic eigenfunctions in different systems with the same limiting behaviour at high energy [2]. While many quantities reveal only a common statistical scaling on the large scale, the topology---particularly the occurrence of knotted loops---discriminates between tangles arising from different systems. In fact, knotting seems to depend on the nature of the chaotic system, and can be surprisingly rare when compared to standard random walk models.\\[4pt] [1] A J Taylor and M R Dennis, \emph{J Phys A} \textbf{47}, 465101 (2014)\\[0pt] [2] M V Berry and M R Dennis, \emph{Proc R Soc A} \textbf{456}, 2059-79 (2000)   

Non-Harmonic Pressure Fluctuations by the Self-Excited Oscillations in a Reactor-Column

Self-excited non-harmonic pressure oscillations that result from non-linearity in the system are generated in an air-flow in a reactor column. The uniform steady flow is converted spontaneously into an oscillatory flow, under the especial experimental conditions in a reactor column that includes a thin layer of dissipative porous medium. The resulting large-amplitude non-harmonic pressure fluctuations in the air-flow are similar to the bifurcation in chaotic systems; where two or more energy states can occur simultaneously, with the system oscillating between them. Experimental results will be presented to demonstrate this abrupt change in flow-regime, from steady-flow to chaotic turbulent vibrations. Our results show that a low-pressure shock-wave-front is established in the column and precedes the self-excitation oscillations in the system. Results show that there exists a threshold for flow-rate, beyond which the transition from steady-flow to pulsating-flow occurs. A numerical model is developed to express this behavior in terms of system variables, such as; dominant frequencies, obtained from fast-Fourier-transforms of time-domain pressure signals, flow-rate, dimensionless aerodynamic characteristic numbers, relaxation-time, and energy dissipation in the system.   

A spectral force based version of the Wigner-Liouville equation

Traditionally, a direct numerical solution of the Wigner-Liouville (WL) equation has been plagued with high computational burden and instability inherent to the integration of the highly oscillatory Wigner potential kernel. We have developed a method based on the spectral decomposition of the force which recasts the WL equation into a manageable form. By removing one integral, this new form is computationally less demanding. Furthermore a damping term naturally appears which reduces the instability caused by the oscillatory terms. Finally, the new form is local in position as opposed to the original WL equation which is non-local in both position and momentum. The spectral force WL equation is interpreted as representing two processes; a classical evolution with a constant force, and a local quantum generation term with positive and negative contributions mediated by the spectral components of the force. This interpretation allows for a straightforward implementation using a finite difference scheme for the classical evolution coupled with direct evaluation of the discretized generation terms. We observe a good match between results obtained using our method and theoretical results.