Classical wave experiments on chaotic scattering

Intensity statistics inside an open wave-chaotic cavity with broken time-reversal invariance

Using the supersymmetric method of random matrix theory within the Heidelberg approach framework we provide statistical description of stationary intensity sampled in locations inside an open wave-chaotic cavity, assuming that the time-reversal invariance inside the cavity is fully broken. In particular, we show that when incoming waves are fed via a finite number M of open channels the probability density P(I) for the single-point intensity I decays as a power law for large intensities: P(I)∼I^{-(M+2)}, provided there is no internal losses. This behavior is in marked difference with the Rayleigh law P(I)∼exp(-I/I[over ¯]), which turns out to be valid only in the limit M→∞. We also find the joint probability density of intensities I_{1},...,I_{L} in L>1 observation points, and then we extract the corresponding statistics for the maximal intensity in the observation pattern. For L→∞ the resulting limiting extreme value statistics (EVS) turns out to be different from the classical EVS distributions.

Implementing nonuniversal features with a random matrix theory approach: Application to space-to-configuration multiplexing

We consider the efficiency of multiplexing spatially encoded information across random configurations of a metasurface-programmable chaotic cavity in the microwave domain. The distribution of the effective rank of the channel matrix is studied to quantify the channel diversity and to assess a specific system's performance. System-specific features such as unstirred field components give rise to nontrivial interchannel correlations and need to be properly accounted for in modeling based on random matrix theory. To address this challenge, we propose a two-step hybrid approach. Based on an ensemble of experimentally measured scattering matrices for different random metasurface configurations, we first learn a system-specific pair of coupling matrix and unstirred contribution to the Hamiltonian, and then add an appropriately weighted stirred contribution. We verify that our method is capable of reproducing the experimentally found distribution of the effective rank with good accuracy. The approach can also be applied to other wave phenomena in complex media.

Experimental Microwave Scattering in Polygonal Billiards

Fluctuations in the one-port scattering and normalized impedance matrices in three polygonal and one chaotic time-reversal invariant microwave billiards are experimentally investigated, in several levels of coupling and absorption, at room temperature and at 77 K. The observed distributions of reflection coefficient, phase of the scattering matrix, resistance and reactance exhibit no fingerprint of a given geometry. At low frequencies, the results are consistent with earlier theoretical models by López, Mello and Seligman and by Zheng, Antonsen and Ott, who independently predicted that the scattering fluctuations might be the same for the Wigner and Poisson level spacing distributions in the lossless cavity. The uniqueness of the observed scattering statistics at higher absorption levels is discussed with respect to inherent limitations posed by the experimental technique.

Precise Localization of Multiple Noncooperative Objects in a Disordered Cavity by Wave Front Shaping

Complicated multipath trajectories of waves in disordered cavities cause object localization to be very challenging with traditional ray-tracing approaches. Yet it is known that information about the object position is encoded in the Green's function. After a calibration step, traditional time-reversal approaches retrieve a source's location from a broadband impulse response measurement. Here, we show that a nonemitting object's scattering contribution to a reverberant medium suffices to localize the object. We demonstrate our finding in the microwave domain. Then, we further simplify the scheme by replacing the temporal degrees of freedom (d.o.f.) of the broadband measurement with spatial d.o.f. obtained from wave front shaping. A simple electronically reconfigurable reflectarray inside the cavity dynamically modulates parts of the cavity boundaries, thereby providing spatial d.o.f. The demonstrated ability to localize multiple noncooperative objects with a single-frequency scheme may have important applications for sensors in smart homes.

Envelope and phase distribution of a resonance transmission through a complex environment

A transmission amplitude is considered for quantum or wave transport mediated by a single resonance coupled to the background of many chaotic states. Such a model provides a useful approach to quantify fluctuations in an established signal induced by a complex environment. Applying random matrix theory to the problem, we derive an exact result for the joint distribution of the transmission intensity (envelope) and the transmission phase at arbitrary coupling to the background with finite absorption. The intensity and phase are distributed within a certain region, revealing essential correlations even at strong absorption. In the latter limit, we obtain a simple asymptotic expression that provides a uniformly good approximation of the exact distribution within its whole support, thus going beyond the Rician distribution often used for such purposes. Exact results are also derived for the marginal distribution of the phase, including its limiting forms at weak and strong absorption.

Dynamic Metasurface Aperture as Smart Around-the-Corner Motion Detector

Detecting and analysing motion is a key feature of Smart Homes and the connected sensor vision they embrace. At present, most motion sensors operate in line-of-sight Doppler shift schemes. Here, we propose an alternative approach suitable for indoor environments, which effectively constitute disordered cavities for radio frequency (RF) waves; we exploit the fundamental sensitivity of modes of such cavities to perturbations, caused here by moving objects. We establish experimentally three key features of our proposed system: (i) ability to capture the temporal variations of motion and discern information such as periodicity (“smart”), (ii) non line-of-sight motion detection, and (iii) single-frequency operation. Moreover, we explain theoretically and demonstrate experimentally that the use of dynamic metasurface apertures can substantially enhance the performance of RF motion detection. Potential applications include accurately detecting human presence and monitoring inhabitants’ vital signs.

Distribution of Off-Diagonal Cross Sections in Quantum Chaotic Scattering: Exact Results and Data Comparison

The recently derived distributions for the scattering-matrix elements in quantum chaotic systems are not accessible in the majority of experiments, whereas the cross sections are. We analytically compute distributions for the off-diagonal cross sections in the Heidelberg approach, which is applicable to a wide range of quantum chaotic systems. Thus, eventually, we fully solve a problem that already arose more than half a century ago in compound-nucleus scattering. We compare our results with data from microwave and compound-nucleus experiments, particularly addressing the transition from isolated resonances towards the Ericson regime of strongly overlapping ones.

Fluctuations in an established transmission in the presence of a complex environment

In various situations where wave transport is preeminent, like in wireless communication, a strong established transmission is present in a complex scattering environment. We develop a nonperturbative approach to describe emerging fluctuations which combines a transmitting channel and a chaotic background in a unified effective Hamiltonian. Modeling such a background by random matrix theory, we derive exact results for both transmission and reflection distributions at arbitrary absorption that is typically present in real systems. Remarkably, in such a complex scattering situation, the transport is governed by only two parameters: an absorption rate and the ratio of the so-called spreading width to the natural width of the transmission line. In particular, we find that the established transmission disappears sharply when this ratio exceeds unity. The approach exemplifies the role of the chaotic background in dephasing the deterministic scattering.

Lossy chaotic electromagnetic reverberation chambers: Universal statistical behavior of the vectorial field

The effective Hamiltonian formalism is extended to vectorial electromagnetic waves in order to describe statistical properties of the field in reverberation chambers. The latter are commonly used in electromagnetic compatibility tests. As a first step, the distribution of wave intensities in chaotic systems with varying opening in the weak coupling limit for scalar quantum waves is derived by means of random matrix theory. In this limit the only parameters are the modal overlap and the number of open channels. Using the extended effective Hamiltonian, we describe the intensity statistics of the vectorial electromagnetic eigenmodes of lossy reverberation chambers. Finally, the typical quantity of interest in such chambers, namely, the distribution of the electromagnetic response, is discussed. By determining the distribution of the phase rigidity, describing the coupling to the environment, using random matrix numerical data, we find good agreement between the theoretical prediction and numerical calculations of the response.

Wave-Field Shaping in Cavities: Waves Trapped in a Box with Controllable Boundaries

Electromagnetic cavities are used in numerous domains of applied and fundamental physics, from microwave ovens and electromagnetic compatibility to masers, quantum electrodynamics (QED), and quantum chaos. The wave fields established in cavities are statically fixed by their geometry, which are usually modified by using mechanical parts like mode stirrers in reverberation chambers or screws in masers and QED. Nevertheless, thanks to integral theorems, tailoring the cavity boundaries theoretically permits us to design at will the wave fields they support. Here, we show in the microwave domain that it is achievable dynamically simply by using electronically tunable metasurfaces that locally modify the boundaries, switching them in real time from Dirichlet to Neumann conditions. We prove that at a high modal density, counterintuitively, it permits us to create wave patterns presenting hot spots of intense energy. We explain and model the physical mechanism underlying the concept, which allows us to find a criterion ensuring that modifying parts of a cavity's boundaries turn it into a completely different one. We finally prove that this approach even permits us, in the limiting case where the cavity supports only well-separated resonances, to choose the frequencies at which the latter occur.

Fractal dynamics in chaotic quantum transport

Despite several experiments on chaotic quantum transport in two-dimensional systems such as semiconductor quantum dots, corresponding quantum simulations within a real-space model have been out of reach so far. Here we carry out quantum transport calculations in real space and real time for a two-dimensional stadium cavity that shows chaotic dynamics. By applying a large set of magnetic fields we obtain a complete picture of magnetoconductance that indicates fractal scaling. In the calculations of the fractality we use detrended fluctuation analysis-a widely used method in time-series analysis-and show its usefulness in the interpretation of the conductance curves. Comparison with a standard method to extract the fractal dimension leads to consistent results that in turn qualitatively agree with the previous experimental data.

Distribution of scattering matrix elements in quantum chaotic scattering

Scattering is an important phenomenon which is observed in systems ranging from the micro- to macroscale. In the context of nuclear reaction theory, the Heidelberg approach was proposed and later demonstrated to be applicable to many chaotic scattering systems. To model the universal properties, stochasticity is introduced to the scattering matrix on the level of the Hamiltonian by using random matrices. A long-standing problem was the computation of the distribution of the off-diagonal scattering-matrix elements. We report here an exact solution to this problem and present analytical results for systems with preserved and with violated time-reversal invariance. Our derivation is based on a new variant of the supersymmetry method. We also validate our results with scattering data obtained from experiments with microwave billiards.

Bound states in sharply bent waveguides: analytical and experimental approach

Quantum wires and electromagnetic waveguides possess common features since their physics is described by the same wave equation. We exploit this analogy to investigate experimentally with microwave waveguides and theoretically with the help of an effective potential approach the occurrence of bound states in sharply bent quantum wires. In particular, we compute the bound states, study the features of the transition from a bound to an unbound state caused by the variation of the bending angle, and determine the critical bending angles at which such a transition takes place. The predictions are confirmed by calculations based on a conventional numerical method as well as experimental measurements of the spectra and electric field intensity distributions of electromagnetic waveguides.

Wave systems with direct processes and localized losses or gains: the nonunitary Poisson kernel

We study the scattering of waves in systems with losses or gains simulated by imaginary potentials. This is done for a complex delta potential that corresponds to a spatially localized absorption or amplification. In the Argand plane the scattering matrix moves on a circle C centered on the real axis, but not at the origin, that is tangent to the unit circle. From the numerical simulations it is concluded that the distribution of the scattering matrix, when measured from the center of the circle C, agrees with the nonunitary Poisson kernel. This result is also obtained analytically by extending the analyticity condition, of unitary scattering matrices, to the no-unitary ones. We use this nonunitary Poisson kernel to obtain the distribution of nonunitary scattering matrices when measured from the origin of the Argand plane. The obtained marginal distributions have excellent agreement with the numerical results.

Effect of noise in open chaotic billiards

We investigate the effect of white-noise perturbations on chaotic trajectories in open billiards. We focus on the temporal decay of the survival probability for generic mixed-phase-space billiards. The survival probability has a total of five different decay regimes that prevail for different intermediate times. We combine new calculations and recent results on noise perturbed Hamiltonian systems to characterize the origin of these regimes and to compute how the parameters scale with noise intensity and billiard openness. Numerical simulations in the annular billiard support and illustrate our results.

Nearest level spacing statistics in open chaotic systems: generalization of the Wigner surmise

We investigate the nearest level spacing statistics of open chaotic wave systems. To this end we derive the spacing distributions for the three Wigner ensembles in the one-channel case. The theoretical results give a clear physical meaning of the modifications on the spacing distributions produced by the coupling to the environment. Based on the analytical expressions obtained, we then propose general expressions of the spacing distributions for any number of channels, valid from weak to strong coupling. The latter expressions contain one free parameter. The surmise is successfully compared with numerical simulations of non-Hermitian random matrices and with experimental data obtained with a lossy electromagnetic chaotic cavity.

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.

Statistics of resonance states in a weakly open chaotic cavity with continuously distributed losses

In this Rapid Communication, we demonstrate that a non-Hermitian random matrix description can account for both spectral and spatial statistics of resonance states in a weakly open chaotic wave system with continuously distributed losses. More specifically, the statistics of resonance states in an open two-dimensional chaotic microwave cavity are investigated by solving the Maxwell equations with lossy boundaries subject to Ohmic dissipation. We successfully compare the statistics of its complex-valued resonance states and associated widths with analytical predictions based on a non-Hermitian effective Hamiltonian model defined by a finite number of fictitious open channels.

Statistics of resonance states in open chaotic systems: a perturbative approach

We investigate the statistical properties of the complexness parameter which characterizes uniquely complexness (nonorthogonality) of resonance eigenstates of open chaotic systems. Specifying to the regime of weakly overlapping resonances, we apply the random matrix theory to the effective Hamiltonian formalism and derive analytically the probability distribution of the complexness parameter for two statistical ensembles describing the systems invariant under time reversal. For those with rigid spectra, we consider a Hamiltonian characterized by a picket-fence spectrum without spectral fluctuations. Then, in the more realistic case of a Hamiltonian described by the Gaussian orthogonal ensemble, we reveal and discuss the role of spectral fluctuations.

Quasimodes of a chaotic elastic cavity with increasing local losses

We report noninvasive measurements of the complex field of elastic quasimodes of a silicon wafer with chaotic shape. The amplitude and phase spatial distribution of the flexural modes are directly obtained by Fourier transform of time measurements. We investigate the crossover from real mode to complex-valued quasimode, when absorption is progressively increased on one edge of the wafer. The complexness parameter, which characterizes the degree to which a resonance state is complex valued, is measured for nonoverlapping resonances, and is found to be proportional to the nonhomogeneous contribution to the line broadening of the resonance. A simple two-level model based on the effective Hamiltonian formalism supports our experimental results.

Poincaré recurrences and transient chaos in systems with leaks

In order to simulate observational and experimental situations, we consider a leak in the phase space of a chaotic dynamical system. We obtain an expression for the escape rate of the survival probability by applying the theory of transient chaos. This expression improves previous estimates based on the properties of the closed system and explains dependencies on the position and size of the leak and on the initial ensemble. With a subtle choice of the initial ensemble, we obtain an equivalence to the classical problem of Poincaré recurrences in closed systems, which is treated in the same framework. Finally, we show how our results apply to weakly chaotic systems and justify a split of the invariant saddle into hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential and asymptotic power-law decays of the survival probability.

Scattering fidelity in elastodynamics. II. Further experimental results

The agreement of past measurements of elastodynamic scattering fidelity with predictions based on random matrix theory, even in bodies with ostensibly regular ray dynamics, has motivated further measurements with a precision capable of detecting fine deviations. Two aluminum blocks are studied, one rectangular, one irregular. As previously, it is found that fidelity decays more slowly in the irregular object. It is further shown that the time dependence of that decay corresponds well with predictions based on random matrix theory and the Gaussian orthogonal ensemble and that predictions based on Poissonian model statistics do not correspond. In some cases deviations from theory are seen that are consistent with partially broken reflection symmetries and inhomogeneous dissipation.

Semiclassical mechanism for the quantum decay in open chaotic systems

We address the decay in open chaotic quantum systems and calculate semiclassical corrections to the classical exponential decay. We confirm random matrix predictions and, going beyond, calculate Ehrenfest time effects. To support our results we perform extensive numerical simulations. Within our approach we show that certain (previously unnoticed) pairs of interfering, correlated classical trajectories are of vital importance. They also provide the dynamical mechanism for related phenomena such as photoionization and photodissociation, for which we compute cross-section correlations. Moreover, these orbits allow us to establish a semiclassical version of the continuity equation.

Resonance distribution in open quantum chaotic systems

In order to study the resonance spectra of chaotic cavities subject to some damping (which can be due to absorption or partial reflection at the boundaries), we use a model of damped quantum maps. In the high-frequency limit, the distribution of (quantum) decay rates is shown to cluster near a "typical" value, which is larger than the classical decay rate of the corresponding damped ray dynamics. The speed of this clustering may be quite slow, which could explain why it has not been detected in previous numerical data.

Absorption strength in absorbing chaotic cavities

We derive an exact formula to calculate the absorption strength in absorbing chaotic systems such as microwave cavities or acoustic resonators. The formula allows us to estimate the absorption strength as a function of the averaged reflection coefficient and the real coupling parameter. We also define the weak and strong absorption regimes in terms of the coupling parameter and the absorption strength.

Resonance widths in open microwave cavities studied by harmonic inversion

From the measurement of a reflection spectrum of an open microwave cavity, the poles of the scattering matrix in the complex plane have been determined. The resonances have been extracted by means of the harmonic inversion method. By this, it became possible to resolve the resonances in a regime where the linewidths exceed the mean level spacing up to a factor of 10, a value inaccessible in experiments up to now. The obtained experimental distributions of linewidths were found to be in perfect agreement with predictions from random matrix theory when wall absorption and fluctuations caused by couplings to additional channels are considered.

Unitarization of the classical statistical s matrix for systems with localization

It has recently been observed that a large reverberant cavity admits a classically motivated random s matrix that is not unitary but that can be made so in a minimally invasive manner. A random process with an envelope <s2(t)> approximately exp(-t/TH) representing reflection from a structure having no internal time scales other than Heisenberg time T_{H} was shown to lead to a unitary S matrix exhibiting mesoscopic behaviors not present in the classically inspired original <s2(t)>. These included enhanced backscatter, quantum echo, power law tails, and level repulsion. Here the procedure is extended to two systems having, in addition to Heisenberg times, internal time scales corresponding to conduction and diffusion. The repaired S matrices for coupled rooms and one-dimensional random structures with multiple scattering are found to correspond to Wigner K matrices with signatures of localization.

Short wavelength approximation of a boundary integral operator for homogeneous and isotropic elastic bodies

A short wavelength approximation of a boundary integral operator for two-dimensional isotropic and homogeneous elastic bodies is derived from first principles starting from the Navier-Cauchy equation. Trace formulas for elastodynamics are deduced connecting the eigenfrequency spectrum of an elastic body to the set of periodic rays where mode conversion enters as a dynamical feature.

Nodal domains in open microwave systems

Nodal domains are studied both for real psiR and imaginary part psiI of the wave functions of an open microwave cavity and found to show the same behavior as wave functions in closed billiards. In addition we investigate the variation of the number of nodal domains and the signed area correlation by changing the global phase phig according to psiR+ipsiI=eiphig(psiR'+ipsiI'). This variation can be qualitatively, and the correlation quantitatively explained in terms of the phase rigidity characterizing the openness of the billiard.

Observation of coherent precursors in pulsed mode-stirred reverberation fields

We report on measured statistics of pulsed radio-frequency fields inside a dynamic (mode-stirred) reverberant cavity. Unlike for time-harmonic excitation, the early transient received power during the buildup of energy is quasiperiodic with respect to the stir angle and exhibits unusually long correlations. These correlations decay at the same exponential rate as the increase of energy. By contrast, the probability distribution of the power reaches its asymptotic form in a more rapid but oscillatory manner.

Extended quasimodes within nominally localized random waveguides

We have measured the spatial and spectral dependence of the microwave field inside an open absorbing waveguide filled with randomly juxtaposed dielectric slabs in the spectral region in which the average level spacing exceeds the typical level width. Whenever lines overlap in the spectrum, the field exhibits multiple peaks within the sample. Only then is substantial energy found beyond the first half of the sample. When the spectrum throughout the sample is decomposed into a sum of Lorentzian lines plus a broad background, their central frequencies and widths are found to be essentially independent of position. Thus, this decomposition provides the electromagnetic quasimodes underlying the extended field in nominally localized samples. These quasimodes may exhibit multiple peaks in space when they overlap spectrally.