H-J Stöckmann

One dimensional Kronig-Penney model with positional disorder: Theory versus experiment

We study the effects of random positional disorder in the transmission of waves in the one-dimensional Kronig-Penny model formed by two alternating dielectric slabs. Numerical simulations and experimental data revealed that the so-called resonance bands survive even for relatively strong disorder and large number of cells, while the nonresonance bands disappear already for weak disorder. For weak disorder we derive an analytical expression for the localization length and relate it to the transmission coefficient for finite samples. The obtained results describe very well the experimental frequency dependence of the transmission in a microwave realization of the model. Our results can be applied both to photonic crystals and semiconductor superlattices.

Correlation functions of scattering matrix elements in microwave cavities with strong absorption

The scattering matrix was measured for microwave cavities with two antennae. It was analysed in the regime of overlapping resonances. The theoretical description in terms of a statistical scattering matrix and the rescaled Breit–Wigner approximation has been applied to this regime. The experimental results for the auto-correlation function show that the absorption in the cavity walls yields an exponential decay. This behaviour can only be modelled using a large number of weakly coupled channels. In comparison to the auto-correlation functions, the cross-correlation functions of the diagonal S-matrix elements display a more pronounced difference between regular and chaotic systems.

Impurity effects on the band structure of one-dimensional photonic crystals: experiment and theory

We study the effects of single impurities on the transmission in microwave realizations of the photonic Kronig–Penney model, consisting of arrays of Teflon pieces alternating with air spacings in a microwave guide. As only the first propagating mode is considered, the system is essentially one-dimensional (1D) obeying the Helmholtz equation. We derive analytical closed form expressions from which the band structure, frequency of defect modes and band profiles can be determined. These agree very well with experimental data for all types of single defects considered (e.g. interstitial and substitutional) and show that our experimental set-up serves to explore some of the phenomena occurring in more sophisticated experiments. Conversely, based on the understanding provided by our formulae, information about the unknown impurity can be determined by simply observing certain features in the experimental data for the transmission. Further, our results are directly applicable to the closely related quantum 1D Kronig–Penney model.

Recovery of the fidelity amplitude for the Gaussian ensembles

Using supersymmetry techniques analytical expressions for the average of the fidelity amplitude f(τ) = ψ(0)|exp(2πiHτ) exp(−2πiH0τ)|ψ(0) are obtained, where , and H0 and H are taken from the Gaussian unitary ensemble (GUE) or the Gaussian orthogonal ensemble (GOE), respectively. As long as the perturbation strength is small compared to the mean level spacing, a Gaussian decay of the fidelity amplitude is observed, whereas for stronger perturbations a change to a single-exponential decay takes place, in accordance with results from the literature. Close to the Heisenberg time τ = 1, however, a partial revival of the fidelity is found, which hitherto remained unnoticed. Random matrix simulations have been performed for the three Gaussian ensembles. For the case of the GOE and the GUE they are in perfect agreement with the analytical results.

Algebraic Fidelity Decay for Local Perturbations

From a reflection measurement in a rectangular microwave billiard with randomly distributed scatterers the scattering and the ordinary fidelity was studied. The position of one of the scatterers is the perturbation parameter. Such perturbations can be considered as local since wave functions are influenced only locally, in contrast to, e.g., the situation where the fidelity decay is caused by the shift of one billiard wall. Using the random-plane-wave conjecture, an analytic expression for the fidelity decay due to the shift of one scatterer has been obtained, yielding an algebraic 1/t decay for long times. A perfect agreement between experiment and theory has been found, including a predicted scaling behavior concerning the dependence of the fidelity decay on the shift distance. The only free parameter has been determined independently from the variance of the level velocities.

Directed emission from a dielectric microwave billiard with quadrupolar shape

We present microwave measurements on dielectric billiards with mixed phase space. Transmission spectra were measured from a fixed antenna inside the billiard to a moveable antenna, thus scanning both the inside and outside region of the billiard. A Fourier transform of the transmission spectra yields the pulse propagation, which is used to study in particular the long-time dynamics of the system. The Poynting vector, which describes the energy flow of the microwaves, is obtained from the measurement for each time step of the pulse propagation. Close to the boundary of the billiard it reveals a characteristic directionality of the microwave emission, which is in accordance with measurements of the far-field intensities of micro-disc lasers of the same shape. To achieve a direct comparison to the internal dynamics of the classical system, the Husimi distributions of the pulse propagation are calculated. Averaging the Husimi distributions for the long-time dynamics provides a very clear picture of the internal dynamics in phase space. The results are in agreement with classical simulations.

Trends in quantum chaotic scattering

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.

Microwave realization of a periodically driven system

A realization of a periodically driven microwave system is presented. The principal element of the scheme is a variable capacity, i.e. a varicap, introduced as an element of the resonant circuit. Sideband structures, corresponding to different driving signals, have been measured experimentally. In the linear regime we observed sideband structures with specific shapes. The main peculiarities of these shapes can be explained within a semiclassical approximation. A good agreement between experimental data and theoretical expectations has been found.

A supersymmetry approach to billiards with randomly distributed scatterers

The density of states for a chaotic billiard with randomly distributed point-like scatterers is calculated, doubly averaged over the positions of the impurities and shape of the billiard. Truncating the billiard Hamiltonian to an N ? N matrix, an explicit analytic expression is obtained for the case of broken time-reversal symmetry, depending on the rank N of the matrix, number L of scatterers and strength of the scattering potential. In the strong coupling limit a discontinuous change is observed in the density of states as soon as L exceeds N.