Table of contents

Quantum scattering of waves in classically chaotic systems has been the subject of rather intensive research activity for more than two decades, both theoretically and experimentally. This interest was motivated by phenomena discovered in various areas, ranging from nuclear, atomic and molecular physics, to mesoscopics and theory of electron transport, quantum chaos, and classical wave scattering. Recently, the interest in this subject was renewed due to technological developments in quantum optics (in particular the ability to construct microlasers with chaotic resonators which produce high-power directional emission) as well as the experimental realizations of the so-called random lasers where the feedback is due to multiple scattering within the medium.

The articles collected in this special issue, which contains both review-style contributions and regular research papers, should give an up-to-date, fairly representative (but definitely not complete) picture of current activity related to the various facets of chaotic wave scattering, and its diverse manifestations. We hope that this will serve as a good basis for boosting the research in this fascinating area to a new level of understanding.

We review recent research on the transport properties of classical waves through chaotic systems with special emphasis on microwaves and sound waves. Inasmuch as these experiments use antennas or transducers to couple waves into or out of the systems, scattering theory has to be applied for a quantitative interpretation of the measurements. Most experiments concentrate on tests of predictions from random matrix theory and the random plane wave approximation. In all studied examples a quantitative agreement between experiment and theory is achieved. To this end it is necessary, however, to take absorption and imperfect coupling into account, concepts that were ignored in most previous theoretical investigations. Classical phase space signatures of scattering are being examined in a small number of experiments.

Signatures of photon localization are observed in a constellation of transport phenomena which reflect the transition from diffusive to localized waves. The dimensionless conductance, g, and the ratio of the typical spectral width and spacing of quasimodes, δ, are key indicators of electronic and classical wave localization when inelastic processes are absent. However, these can no longer serve as localization parameters in absorbing samples since the effect of absorption depends upon the length of the trajectories of partial waves traversing the sample, which are superposed to create the scattered field. A robust determination of localization in the presence of absorption is found, however, in steady-state measurements of the statistics of radiation transmitted through random samples. This is captured in a single parameter, the variance of the total transmission normalized to its ensemble average value, which is equal to the degree of intensity correlation of the transmitted wave, κ. The intertwined effects of localization and absorption can also be disentangled in the time domain since all waves emerging from the sample at a fixed time delay from an exciting pulse, t, are suppressed equally by absorption. As a result, the relative weights of partial waves emerging from the sample, and hence the statistics of intensity fluctuations and correlation, and the suppression of propagation by weak localization are not changed by absorption, and manifest the growing impact of weak localization with t.

We use tetrahedral microwave networks consisting of coaxial cables and attenuators connected by T-joints to make an experimental study of Wigner's reaction K matrix for irregular graphs in the presence of absorption. From measurements of the scattering matrix S for each realization of the microwave network, we obtain distributions of the imaginary and real parts of K. Our experimental results are in good agreement with theoretical predictions.

The random laser represents a non-conventional laser whose feedback is mediated by random fluctuation of the dielectric constant in space. Depending on whether the feedback is intensity or field feedback, random lasers are classified into two categories: a random laser with incoherent and non-resonant feedback, and a random laser with coherent and resonant feedback. This paper reviews some of the latest developments in the latter, including experiments and theories on the 'classical' and 'quantum' type of random lasers with coherent feedback, the photon localization lasers. The quantum theories of random lasers are briefly introduced, followed by a discussion of the recent studies on the interplay between light localization and coherent amplification.

We present a semiclassical laser theory for multimode lasing in optical resonators with overlapping modes. Nonlinear saturation and mode competition are characterized in terms of left and right eigenmodes of a non-Hermitian operator. The theory is applied to wave-chaotic cavities and weakly disordered random media. In the limit of sufficiently strong pumping, we find that the mean number of laser peaks increases with the 1/3 power of the pump strength.

We calculate the field–field correlation function in a disordered optically active medium. It is shown that this function exhibits characteristic spatial oscillations, with a period related to the chirality parameter of the medium. Similar oscillations show up in correlations of the polarization-resolved intensity. In contrast, correlations in the total intensity, i.e. summed over all polarizations, are not sensitive to chirality.

We examine the effect of the initial atomic momentum distribution on the dynamics of the atom-optical realization of the quantum kicked rotor. The atoms are kicked by a pulsed optical lattice, the periodicity of which implies that quasi-momentum is conserved in the transport problem. We study and compare experimentally and theoretically two resonant limits of the kicked rotor: in the vicinity of the quantum resonances and in the semiclassical limit of the vanishing kicking period. It is found that for the same experimental distribution of quasi-momenta, significant deviations from the kicked rotor model are induced close to quantum resonance, while close to the classical resonance (i.e. for a small kicking period) the effect of the quasi-momentum vanishes.

We present a discussion of the charge response and the charge fluctuations of mesoscopic chaotic cavities in terms of a generalized Wigner–Smith matrix. The Wigner–Smith matrix is well known in investigations of time-delay of quantum scattering. It is expressed in terms of the scattering matrix and its derivatives with energy. We consider a similar matrix but instead of an energy derivative, we investigate the derivative with regard to the electric potential. The resulting matrix is then the operator of charge. If this charge operator is combined with a self-consistent treatment of Coulomb interaction, the charge operator determines the capacitance of the system, the non-dissipative ac-linear response, the RC-time with a novel charge relaxation resistance, and in the presence of transport a resistance that governs the displacement currents induced into a nearby conductor. In particular, these capacitances and resistances determine the relaxation rate and dephasing rate of a nearby qubit (a double quantum dot). We discuss the role of screening of mesoscopic chaotic detectors. Coulomb interaction effects in quantum pumping and in photon assisted electron–hole shot noise are treated similarly. For the latter, we present novel results for chaotic cavities with non-ideal leads.

We review the random matrix description of electron transport through open quantum dots, subject to time-dependent perturbations. All characteristics of the current linear in the bias can be expressed in terms of the scattering matrix, calculated for a time-dependent Hamiltonian. Assuming that the Hamiltonian belongs to a Gaussian ensemble of random matrices, we investigate various statistical properties of the direct current in the ensemble. Particularly, even at zero bias the time-dependent perturbation induces current, called photovoltaic current. We discuss dependence of the photovoltaic current and its noise on the frequency and the strength of the perturbation. We also describe the effect of time-dependent perturbation on the weak localization correction to the conductance and on conductance fluctuations.

We review recent results on the anomalous transport in one-dimensional and quasi-one-dimensional systems with bulk and surface disorder. Principal attention is paid to the role of long-range correlations in random potentials for the bulk scattering and in corrugated profiles for the surface scattering. It is shown that with the proper type of correlations one can construct such a disorder that results in a selective transport with given properties. Of particular interest is the possibility to arrange windows of a complete transparency (or reflection) with dependence on the wave number of incoming classical waves or electrons.

As an alternative to Büttiker's dephasing lead model, we examine a dephasing stub. Both models are phenomenological ways to introduce decoherence in chaotic scattering by a quantum dot. The difference is that the dephasing lead opens up the quantum dot by connecting it to an electron reservoir, while the dephasing stub is closed at one end. Voltage fluctuations in the stub take over the dephasing role from the reservoir. Because the quantum dot with dephasing lead is an open system, only expectation values of the current can be forced to vanish at low frequencies, while the outcome of an individual measurement is not so constrained. The quantum dot with dephasing stub, in contrast, remains a closed system with a vanishing low-frequency current at each and every measurement. This difference is a crucial one in the context of quantum algorithms, which are based on the outcome of individual measurements rather than on expectation values. We demonstrate that the dephasing stub model has a parameter range in which the voltage fluctuations are sufficiently strong to suppress quantum interference effects, while still being sufficiently weak that classical current fluctuations can be neglected relative to the nonequilibrium shot noise.

We consider a system that consists of two single-quantum billiards (QBs) coupled by a waveguide and study the transmission through this system as a function of length and width of the waveguide. To interpret the numerical results for the transmission, we explore a simple model with a small number of states which allows us to consider the problem analytically. The transmission is described in the S-matrix formalism by using the non-Hermitian effective Hamilton operator for the open system. The coupling of the single QBs to the internal waveguide characterizes the 'internal' coupling strength u of the states of the system while that of the system as a whole to the attached leads determines the 'external' coupling strength v of the resonance states via the continuum (waves in the leads). The transmission is resonant for all values of v/u in relation to the effective Hamiltonian. It depends strongly on the ratio v/u via the eigenvalues and eigenfunctions of the effective Hamiltonian. The results obtained are compared qualitatively with those from simulation calculations for larger systems. Most interesting is the existence of resonance states with vanishing widths that may appear at all values of v/u. They cause zeros in the transmission through the double QB due to trapping of the particle in the waveguide.

We review properties of open chaotic mesoscopic systems with a finite Ehrenfest time τ_E. The Ehrenfest time separates a short-time regime of the quantum dynamics, where wave packets closely follow the deterministic classical motion, from a long-time regime of fully-developed wave chaos. For a vanishing Ehrenfest time the quantum systems display a degree of universality which is well described by random-matrix theory. In the semiclassical limit, τ_E becomes parametrically larger than the scattering time off the boundaries and the dwell time in the system. This results in the emergence of an increasing number of deterministic transport and escape modes, which induce strong deviations from random-matrix universality. We discuss these deviations for a variety of physical phenomena, including shot noise, conductance fluctuations, decay of quasi-bound states and the mesoscopic proximity effect in Andreev billiards.

We analyse simple models of quantum chaotic scattering, namely quantized open baker's maps. We numerically compute the density of quantum resonances in the semiclassical regime. This density satisfies a fractal Weyl law, where the exponent is governed by the (fractal) dimension of the set of trapped trajectories. This type of behaviour is also expected in the (physically more relevant) case of Hamiltonian chaotic scattering. Within a simplified model, we are able to rigorously prove this Weyl law and compute quantities related to the 'coherent transport' through the system, namely the conductance and 'shot noise'. The latter is close to the prediction of random matrix theory.

In quasi-classical studies of closed systems, e.g. a billiard, resurgence means that the contribution of long periodic orbits to the spectral determinant can be expressed in terms of composites of short orbits, and the resulting expression for the determinant is manifestly real. The question has thus long been posed whether something like resurgence applies to a scattering system with its resonances. We find here a resurgent expression for Wigner's R-matrix (which gives the S-matrix by a Cayley transform) in which long scattering pseudo-orbits are expressed in terms of composites of short pseudo-orbits, both scattering and periodic, and the result is manifestly Hermitian, giving a unitary expression for the S-matrix. This is particularly useful in the case that the resonance width is comparable with the resonance spacing. The pseudo-orbits are defined in terms of a fictitious and to some extent arbitrary closed reference system. We relate the results to other formulations. We give a simple but non-trivial approximation for a particular example which illustrates the phenomenon of 'resonance trapping'.

This paper deals with the classical and quantum dynamics on convex co-compact surfaces. We review the recent developments of the theory and compare the asymptotic behaviour of both classical and quantum observables. We show rigorously that the classical decay rate is larger than the quantum decay rate. This is well known in the physics literature on chaotic scattering but has never been verified mathematically.

We review recent progress in analysing wave scattering in systems with both intrinsic chaos and/or disorder and internal losses, when the scattering matrix is no longer unitary. By mapping the problem onto a nonlinear supersymmetric σ-model, we are able to derive closed-form analytic expressions for the distribution of reflection probability in a generic disordered system. One of the most important properties resulting from such an analysis is statistical independence between the phase and the modulus of the reflection amplitude in every perfectly open channel. The developed theory has far-reaching consequences for many quantities of interest, including local Green functions and time delays. In particular, we point out the role played by absorption as a sensitive indicator of mechanisms behind the Anderson localization transition. We also provide a random-matrix-based analysis of S-matrix and impedance correlations for various symmetry classes as well as the distribution of transmitted power for systems with broken time-reversal invariance, completing previous works on the subject. The results can be applied, in particular, to the experimentally accessible impedance and reflection in a microwave or an ultrasonic cavity attached to a system of antennas.

We review recent developments in quantum scattering from mesoscopic systems. Various spatial geometries whose closed analogues show diffusive, localized or critical behaviour are considered. These are the features that cannot be described by the universal random matrix theory results. Instead, one has to go beyond this approximation and incorporate them in a non-perturbative way. Here, we pay particular attention to the traces of these non-universal characteristics, in the distribution of the Wigner delay times and resonance widths. The former quantity captures time-dependent aspects of quantum scattering while the latter is associated with the poles of the scattering matrix.

We discuss signatures of quantum chaos in open chaotic billiards. Solutions for such a system are given by complex scattering wavefunctions ψ = u + iv when a steady current flows through the billiard. For slightly opened chaotic billiards the current distributions are simple and universal. It is remarkable that the resonant transmission through integrable billiards also gives the universal current distribution. Currents induced by the Rashba spin–orbit interaction can flow even in closed billiards. Wavefunction and current distributions for a chaotic billiard with weak and strong spin–orbit interactions have been derived and compared with numerics. Similarities with classical waves are considered. In particular we propose that the networks of electric resonance RLC circuits may be used to study wave chaos. However, being different from quantum billiards, there is a resistance from the inductors which gives rise to heat power and decoherence.

This paper describes a statistical model for decaying quantum systems (e.g. photo-dissociation or -ionization). It takes the interference between direct and indirect decay processes explicitly into account. The resulting expressions for the partial decay amplitudes and the corresponding cross sections may be considered a many-channel many-resonance generalization of Fano's original work on resonance lineshapes (Fano 1961 Phys. Rev.124 1866). A statistical (random matrix) model is then introduced. It allows to describe chaotic scattering systems with tunable couplings to the decay channels. We focus on the autocorrelation function of the total (photo) cross section, and we find that it depends on the same combination of parameters, as the Fano-parameter distribution. These combinations are statistical variants of the one-channel Fano parameter. It is thus possible to study the Fano interference (i.e. the interference between direct and indirect decay paths) on the basis of the autocorrelation function, and thereby in the regime of overlapping resonances. It allows us to study the Fano interference in the limit of strongly overlapping resonances, where we find a persisting effect on the level of the weak localization correction.

We report on numerical calculations of the Fano parameters characteristic of non-symmetric resonance profiles in the electron transport through a waveguide attached to an irregular cavity. The distribution of Fano parameters is calculated for this chaotic scattering system with preserved time-reversal symmetry. We note the role played by the parametrization of the background conductance in comparing random matrix theory predictions for the Fano parameters with numerical or experimental data. Our calculated distribution agrees well with random matrix theory predictions.

The lower boundary of Artin's billiard on the Poincaré half-plane is continuously deformed to generate a class of billiards with classical dynamics varying from fully integrable to completely chaotic. The quantum scattering problem in these open billiards is described and the statistics of both real and imaginary parts of the resonant momenta are investigated. The evolution of the resonance positions is followed as the boundary is varied which leads to large changes in their distribution. The transition to arithmetic chaos in Artin's billiard, which is responsible for the Poissonian level-spacing statistics of the bound states in the continuum (cusp forms) at the same time as the formation of a set of resonances all with width and real parts determined by the zeros of Riemann's zeta function, is closely examined. Regimes are found which obey the universal predictions of random matrix theory (RMT) as well as exhibiting non-universal long-range correlations. The Brody parameter is used to describe the transitions between different regimes.

We focus on the problem of an impurity-free billiard with a random position-dependent boundary coupling to the environment. The response functions of such an open system can be obtained non-perturbatively from a supersymmetric generating functional. The derivation of this functional is based on averaging over the escape rates and results in a nonlinear ballistic σ-model, characterized by system-specific parameters. Particular emphasis is placed on the 'whispering gallery modes' as the origin of surface diffusion modes in the limit of large dimensionless conductance.

The use of multi-antenna arrays in wireless communications through disordered media promises huge increases in the information transmission rate. It is therefore important to analyse the information capacity of such systems in realistic situations of microwave transmission, where the statistics of the transmission amplitudes (channel) may be coloured. Here, we present an approach that provides analytic expressions for the statistics, i.e. the moments of the distribution, of the mutual information for general Gaussian channel statistics. The mathematical method applies tools developed originally in the context of coherent wave propagation in disordered media, such as random matrix theory and replicas. Although it is valid formally for large antenna numbers, this approach produces extremely accurate results even for arrays with as few as two antennas. We also develop a method to analytically optimize over the input signal distribution, which enables us to calculate analytic capacities when the transmitter has knowledge of the statistics of the channel. The emphasis of this paper is on elucidating the novel mathematical methods used. We do this by analysing a specific case when the channel matrix is a complex Gaussian with arbitrary mean and unit covariance, which is usually called the Ricean channel.

We propose an interpolation formula for the distribution of the reflection coefficient in the presence of time reversal symmetry for chaotic cavities with absorption. This is done assuming a similar functional form as when time reversal invariance is absent. The interpolation formula reduces to the analytical expressions for the strong and weak absorption limits. Our proposal is compared to the quite complicated exact result existing in the literature.