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This book introduces the quantum mechanics of classically chaotic systems, or quantum chaos for short. The author's philosophy has been to keep the discussion simple and to illustrate theory, wherever possible, with experimental or numerical examples. The microwave billiard experiments, initiated by the author and his group, play a major role in this respect. Topics covered include the various types of billiard experiment, random matrix theory, systems with periodic time dependences, the analogy between the dynamics of a one-dimensional gas with a repulsive interaction and spectral level dynamics, where an external parameter takes the role of time, scattering theory distributions and fluctuation, properties of scattering matrix elements, semiclassical quantum mechanics, periodic orbit theory, and the Gutzwiller trace formula. This book will be of great value to anyone working in quantum chaos.

The eigenfrequencies of resonance cavities shaped as stadium or Sinai billiards are determined by microwave absorption. In the applied frequency range 0-18.74 GHz the used cavities can be considered as two dimensional.For this case quantum-mechanical and electromagnetic boundary conditions are equivalent, and the resonance spectrum of the cavity is, if properly normalized, identical with the quantum-mechanical eigenvalue spectrum. Spectra, containing up to a thousand eigenfrequencies, are obtained within minutes. Statistical properties of the spectra as well as their correlation with classical periodic orbits are discussed. PACS numbers: 05.45.+bThe statistical properties of the eigenvalue spectrum of a Hamiltonian of a quantum system change in a characteristic way if the corresponding classical system shows a transition from integrable to nonintegrable behavior.Whereas in integrable systems for the distribution of energy differences of successive eigenenergies Poisson statistics is observed, the spectra of nonintegrable systems obey Wigner statistics. Experimentally, Wigner statistics were already observed many years ago in the spectra of highly excited nuclei, ' a very recent example is given by the optical fluorescence spectra of N02 molecules. Intense theoretical studies exist for the periodically kicked top, " nonlinearly coupled oscillators, 6 and billiards of different shapes. ' Pechukas and Yukawa' showed that the strength of the nonintegrable part of the Hamiltonian may be interpreted as a pseudotime. As a function of this "time, " the eigenenergies move on the energy axis in a similar way as the particles of an interacting one-dimensional gas.An alternative approach to the understanding of the statistics of eigenvalues is opened by the semiclassical approximation.Gutzwiller'' showed that the density of eigenvalues p(E) can be decomposed into a monotonic and an oscillatory part, p(E) =po(E)+gp"cos[(I/h)S"-y"l (I) (see also Ref. 12 for a review). In the case of twodimensional billiards, the monotonic part po(E) is given b 13 po(E)-A 4x L 1 8n JE where A is the area and L is the circumference of the billiard (the units were chosen such that h /2m =1, i.e. , E k, where k is the de Broglie wave number of the particle). The oscillatory part of p(E) in Eq. (1) is a sum over all classical periodic orbits y. 5" is the classical action for the orbit. The prefactor p"and the phase p" can be calculated within the frame of the semiclassical approximation. If the action S" is proportional to a power of E, then the contributions from the different periodic orbits to p(E) can be obtained by a Fourier transformation of the spectrum as was demonstrated by Wintgen. ' The classical periodic orbits show further up in so-called "scars, " regions of extra high amplitudes of some eigenfunctions in the neighborhood of periodic or-15, 16Calculations of eigenvalues of nonintegrable Hamiltonians are extremely time consuming even on modern computers. Probably for this reason in all publications mentioned above the total number of e...

The coupling of a quantum mechanical system to open decay channels has been theoretically studied in numerous works, mainly in the context of nuclear physics but also in atomic, molecular and mesoscopic physics. Theory predicts that with increasing coupling strength to the channels the resonance widths of all states should first increase but finally decrease again for most of the states. In this letter, the first direct experimental verification of this effect, known as resonance trapping, is presented. In the experiment a microwave Sinai cavity with an attached waveguide with variable slit width was used.PACS numbers: 03.65.Nk, 84.40.Az, 85.30.Vw Since more than ten years, interference phenomena in open quantum systems have been studied theoretically in the framework of different models. Common to all these studies is the appearance of different time scales as soon as the resonance states start to overlap see [1] and the recent papers [2] with references therein). Some of the states align with the decay channels and become short-lived while the remaining ones decouple to a great deal from the continuum and become long-lived (trapped). Due to this phenomenon, the number of relevant states will, in the short-time scale, be reduced while the system as a whole becomes dynamically stabilized. The phenomenologically introduced doorway states in nuclear physics provide an example for the alignment of the short-lived states with the channels [3]. Calculations for microwave resonators showed that the trapped resonance states can be identified in the time-delay function and that short-lived collective modes are formed at large openings of the resonator [4]. Resonance narrowing is inherent also in the Fano formalism [5]. Similar effects have been found in the linewidths in a semiconductor microcavity with variable strength of normal-mode coupling [6]. In spite of the many theoretical studies, the effect of resonance trapping has not yet been verified unambigously in an experiment. A theoretical study of neutron resonances in nuclei as a function of the interaction of a doorway state with narrow resonances [7] allowed only to draw the conclusion that resonance trapping is not in contradiction with experimental data. For a clear experimental demonstration of the trapping effect, the coupling strength to the decay channels should be tunable, which was not possible in all above mentioned experiments.The mechanism of resonance trapping can be illustrated best on the basis of a schematical model. In an open quantum system the resonance states are allowed to decay, i. e. their energies are complex,The Hamilton operator is non-hermitian,Here H 0 describes the N discrete states of the closed quantum system coupled to K decay channels by the N × K matrix V . H 0 and V V † are hermitian and α is a real parameter for the total coupling strength between the closed system and the channels. The complex eigenvalues of H give the energy positions E R and widths Γ R of the resonance states. Studies on the basis of this model were perfo...

Microwave transport experiments have been performed in a quasi-two-dimensional resonator with randomly distributed conical scatterers. At high frequencies, the flow shows branching structures similar to those observed in stationary imaging of electron flow. Semiclassical simulations confirm that caustics in the ray dynamics are responsible for these structures. At lower frequencies, large deviations from Rayleigh's law for the wave height distribution are observed, which can only partially be described by existing multiple-scattering theories. In particular, there are "hot spots" with intensities far beyond those expected in a random wave field. The results are analogous to flow patterns observed in the ocean in the presence of spatially varying currents or depth variations in the sea floor, where branches and hot spots lead to an enhanced frequency of freak or rogue wave formation.

We study the fundamental question of dynamical tunneling in generic two-dimensional Hamiltonian systems by considering regular-to-chaotic tunneling rates. Experimentally, we use microwave spectra to investigate a mushroom billiard with adjustable foot height. Numerically, we obtain tunneling rates from high precision eigenvalues using the improved method of particular solutions. Analytically, a prediction is given by extending an approach using a fictitious integrable system to billiards. In contrast to previous approaches for billiards, we find agreement with experimental and numerical data without any free parameter. Typical Hamiltonian systems have a mixed phase space in which regular and chaotic motion coexist. While classically these regions are separated, quantum mechanically they are coupled by tunneling. This process has been called "dynamical tunneling" [1] as it occurs across a dynamically generated barrier in phase space. Tunneling has been studied between symmetry related regular regions (chaos-assisted tunneling) [2,3,4,5,6,7] and from a single regular region to the chaotic sea [8,9,10,11,12]. In contrast to the well understood 1D tunneling through a barrier, the quantitative prediction of dynamical tunneling is a major challenge. Results have been found for specific systems or system classes only, e.g. recently for 2D quantum maps with an approach using a fictitious integrable system [12]. However, a precise knowledge of tunneling rates is of great importance. Recent examples are spectral statistics in systems with a mixed phase space [13], eigenstates affected by flooding of regular islands [14], and emission properties of optical micro-cavities [15].Billiards are an important class of Hamiltonian systems. Classically, a point particle moves along straight lines inside a domain with elastic reflections at its boundary. Quantum-mechanical approaches for dynamical tunneling rates have so far escaped a full quantitative prediction as they required fitting by a factor of 6 for the annular billiard [3] and by a factor of 100 (see below) for the mushroom billiard [16].In this paper we present a combined experimental, theoretical, and numerical investigation of dynamical tunneling rates in mushroom billiards [17], which are of great current interest [13,16,18,19,20] due to their sharply divided phase space. Experiments are performed using a microwave cavity. Extending the approach using a fictitious integrable system [12] to billiards, we find quantitative agreement in the experimentally accessible regime, see Fig. 1, without a free parameter. In addition, numerical computations verify the predictions over 18 orders of magnitude with errors typically smaller than a factor of 2, see Fig. 4. The theoretical approach thus provides unprecedented agreement for tunneling rates in billiards.We consider the desymmetrized mushroom billiard, i.e. the 2D autonomous system H(p, q) = p 2 /2M + V (q) shown in Fig. 2b, characterized by the radius of the quarter circle R, the foot width a and the foot height l. The pote...

Wave functions of a stadium billiard are determined in a microwave analog experiment. It is shown that Gutzwiller's semiclassical representation of the Green's function in terms of classical trajectories can account not only for eigenvalue spectra but also for eigenfunction patterns.PACS numbers: 05.45.+bThe perhaps most impressive way to demonstrate the qualitative difference between integrable and nonintegrable systems is the presentation of eigenfunction patterns of billiards. Whereas for a rectangular or circular billiard the nodal lines form a regular grid with a large number of crossings, for nonintegrable billiards the node lines perform meandric walks while avoiding any crossing (in the presence of degeneracies the crossings can be destroyed by suitable superpositions of eigenfunctions also in integrable billiards; for details see Chap. of Ref. [l]). This was first demonstrated by McDonald and Kaufman[2] who calculated wave functions in a stadium billiard (a more detailed account of this work can be found in Ref.[3]). A surprise was then the discovery by Heller that many wave functions are not distributed more or less uniformly over the whole billiard but form so-called scars, regions of extra high amplitudes near classical periodic orbits [4,5]. An approach to explain these scars was made by Bogomolny [6]. He used the fact that the Green's function,can be expressed semiclassically as a sum over classically allowed trajectories [7], G(q A ,qB,E)Him) 1/2 XlA, I 1/2 exp -rS r (q A ,q B ,E) -im rn l(2)In Eq. (1) E n and 3(^,<7*,£)/9?!|, M-U2,where the p A are the two components of the momentum at the starting point, written as functions of q A ,q B ,E y and ql are the two components of the end position. Classically, A r is proportional to the density of trajectories at point qs starting isotropically from point q A (a very readable account of these questions can be found in Ref. [8]). Taking q A =q B =<7 in Eq.(2) and integrating over q, one gets the Gutzwiller trace formula establishing a correspondence between periodic orbits and the quantum-mechanical spectrum. This correspondence could be demonstrated experimentally by the present authors for electromagnetic eigenfrequency spectra of billiard-shaped microwave resonators (Ref.[9], hereafter...

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.