Table of contents

In quantum systems that are neither completely integrable nor completely chaotic, energy-level fluctuations are governed by asymptotic scaling laws, analogous to those for the intensity fluctuations of twinkling starlight.

We review clear signatures of "quantum chaos" observable in both open and closed systems realized in GaAs/Al_xGa_1-xAs microstructures. In open ballistic billiards where scattering dynamics determine the characteristics of quantum transport, we find a striking difference in the shape of the negative magneto-resistance peak for transport through chaotic, stadium cavities–Lorentzian line shape, versus non-chaotic, circle cavities–unusual triangular line shape. In a nearly closed system of individual quantum dot single- electron-transistors, the distribution of the height of Coulomb blockade conductance peaks is strongly non-Gaussian and sensitive to the absence or presence of a magnetic field. Even though our findings are in substantial agreement with the prediction of Random Matrix Theory, there exists evidence that deviations can occur due to the effect of electron-electron interaction. Lastly, we briefly summarize recent observations of pairing effect in ultra-small quantum dots associated with the spin degree of freedom.

We investigate experimentally and theoretically the behavior of Coulomb blockade (CB) peaks in a magnetic field that couples principally to the ground-state spin (rather than the orbital moment) of a chaotic quantum dot. In the first part, we discuss numerically observed features in the magnetic field dependence of CB peak and spacings that unambiguously identify changes in spin S of each ground state for successive numbers of electrons on the dot, N. We next evaluate the probability that the ground state of the dot has a particular spin S, as a function of the exchange strength, J, and external magnetic field strength, B. In the second part, we describe recent experiments on gate-defined GaAs quantum dots in which Coulomb peak motion and spacing are measured as a function of in-plane magnetic field, allowing changes in spin between N and N + 1 electron ground states to be inferred.

This paper discusses some of the issues related to quantum dots that were raised during the first session on transport phenomena at the Nobel Symposium.

We utilize a semi-classical approach to calculate conductance and weak localization corrections in a triangular billiard in non-zero magnetic field. Results of the calculations and comparison to numerical quantum mechanical simulations suggest that applicability of the standard semiclassical method for description of the geometry-specific features in the conductance of such systems is not obvious as the unitarity of the semiclassical scattering matrix is violated as well as the symmetry of the conductance/reflectance with respect to the magnetic field and the direction of the current is not satisfied. The reason for this is given. Our findings raise the question to what extent one can rely on numerous predictions for statistical properties of the conductance oscillations of ballistic cavities including the WL lineshapes and fractal conductance which were essentially based on the standard SC approach.

Fractals describe the scaling properties of a spectacular variety of natural objects. In general, fractal studies in natural environments are "passive" in the sense that there is no experimental interaction with the system being observed. In contrast, in this paper we investigate fractals in an artificial environment where controlled changes in the generation process can be used to study how fractals evolve. To do this we construct micron-sized billiards in high quality semiconductor materials where the properties of chaotic electrons can be tuned with precision. By inserting a circle at the centre of a square billiard, we investigate the transition between two distinct forms of fractals observed in the billiard's conductance – from exact to statistical self-affinity.

Semiconductor quantum dots provide an ideal experimental system for studying the transport signatures associated with wavefunction scarring in open billiards. In this report, we present the results of experimental and numerical studies that provide clear evidence for the role of scarred wavefunction states in transport through quantum dots. In contrast to scarring in closed billiards, an important feature of the dots we study is found to be their open nature, which allows us to select the specific scars that are manifest in the transport behavior.

In systems with broken spatial symmetry, directed transport of particles can be observed in the absence of any time-averaged forces or gradients, if the system is kept away from thermal equilibrium. The underlying principle of such "ratchets" has found applications in particle separation and in the modelling of molecular motors in living systems, and has also attracted much interest from a fundamental point of view. In particular, chaos in classical ratchets, and quantum effects such as tunnelling, have each been found to have a strong effect on direction and magnitude of the particle current. Here we are interested in the question how classically chaotic ratchets would behave in the quantum regime. Can chaotic behaviour be observed in quantum ratchets, or, alternatively, could ratchet effects be generated from quantum chaotic behaviour? From the point of view of experimentalists, we discuss these questions with a focus on mesoscopic electron devices, and point to possible topics for theoretical investigations.

Statistics of resonance poles are computed for time-reversal ballistic transport through chaotic and integrable mesoscopic billiards coupled to a pair of single-channel leads in the regime of overlapping resonances. In the case of chaotic open billiard, the width distribution function shows good agreement with the random-matrix-theory prediction in all ranges of the width. In the case of integrable open billiard, however, there exists some deviation and the agreement is perceived only for the tail of the width distribution function. This is understood quantitatively in terms of classical decay-time distributions. On the other hand, the statistics of resonance positions for both chaotic and integrable open billiards show deviations from the Gaussian-orthogonal-ensemble and Poisson predictions. The statistical nature known for eigenvalues of the closed counterparts of the systems is retrieved after eliminating all the broad resonances compared to the mean resonance spacing.

We study the properties of quantum states in the interior and the exterior of magnetic billiards. A weight is associated to each state, providing an objective criterion to distinguish between bulk and edge states. We define a spectral density of edge states, which is then studied statistically and semiclassically. In particular, we observe strong cross-correlations between the interior and exterior edge spectra. They are identified as the quantum signature of a classical duality of periodic orbits.

It might be anticipated that there is statistical universality in the long-time classical dynamics of chaotic systems, corresponding to the universal agreement between their quantum spectral statistics and random matrix theory. It is argued that no such universality exists. Two statistical properties of long period orbits are considered. The distribution of the phase-space density of periodic orbits of fixed length is shown to have a log-normal distribution. Also, a correlation function of periodic-orbit actions is discussed, which has a semiclassical correspondence to the quantum spectral two-point correlation function. It is shown that bifurcations are a mechanism for creating correlations of periodic-orbit actions. They lead to a result which is non-universal, and which in general may not be an analytic function of the action difference.

Quantum dots are small conducting devices containing up to several thousand electrons. We focus here on closed dots whose single-electron dynamics are mostly chaotic. The mesoscopic fluctuations of the conduction properties of such dots reveal the effects of one-body chaos, quantum coherence and electron-electron interactions.

It is shown that within bounds given explicitly, the spectral properties of the k-body embedded unitary and orthogonal ensembles coincide with those of the canonical Gaussian unitary and orthogonal random-matrix ensemble, respectively. We discuss the limits of validity of our approach.

Generic properties of the strength function (local density of states (LDOS)) and chaotic eigenstates are analyzed for isolated systems of interacting particles. Both random matrix models and dynamical systems are considered in the unique approach. Specific attention is paid to the quantum-classical correspondence for the form of the LDOS and eigenstates, and to the localization in the energy shell. A new effect of the non-ergodicity of individual eigenstates in the deep semiclassical limit is briefly discussed.

The quantum numbers of states in ³⁰P have been determined for over 100 states from the ground state to 8 MeV. Previous measurements had provided complete spectroscopy for over 150 states in ²⁶Al in a similar energy region. These N = Z = odd nuclei have the property that states of isospin T = 0 and T = 1 coexist from the ground state. Isospin is approximately conserved in these nuclei. Since the energy E, spin J, parity π, and isospin T are known for every level, these data provide a unique opportunity to test the effect of symmetry breaking on the level statistics and on the transition strength distributions. The level statistics are strongly affected by the small symmetry breaking, in agreement with predictions of Random Matrix Theory (RMT). In ²⁶Al the transition strength distributions differ from the standard Porter-Thomas distribution; deviations at a weaker level also appear in ³⁰P. Until very recently there were no theoretical predictions for the effect of symmetry breaking on the transitions. Recent RMT calculations predict a deviation from the Porter-Thomas distribution, in qualitative agreement with the measurements.

The standard generic quantum computer model is studied analytically and numerically and the border for emergence of quantum chaos, induced by imperfections and residual inter-qubit couplings, is determined. This phenomenon appears in an isolated quantum computer without any external decoherence. The onset of quantum chaos leads to quantum computer hardware melting, strong quantum entropy growth and destruction of computer operability. The time scales for development of quantum chaos and ergodicity are determined. In spite the fact that this phenomenon is rather dangerous for quantum computing it is shown that the quantum chaos border for inter-qubit coupling is exponentially larger than the energy level spacing between quantum computer eigenstates and drops only linearly with the number of qubits n. As a result the ideal multi-qubit structure of the computer remains rather robust against imperfections. This opens a broad parameter region for a possible realization of quantum computer. The obtained results are related to the recent studies of quantum chaos in such many-body systems as nuclei, complex atoms and molecules, finite Fermi systems and quantum spin glass shards which are also reviewed in the paper.

The statistical properties of Quantum Chromodynamics (QCD) show universal features which can be modeled by random matrices. This has been established in detailed analyses of data from lattice gauge calculations. Moreover, systematic deviations were found which link QCD to disordered systems in condensed matter physics. To furnish these empirical findings with analytical arguments, we apply and extend the methods developed in disordered systems to construct a non-linear σ model for the spectral correlations in QCD. Our goal is to derive connections to other low-energy effective theories, such as the Nambu–Jona-Lasinio model, and to chiral perturbation theory.

We consider off-diagonal contributions to double sums over periodic orbits that arise in semiclassical approximations for spectral statistics of classically chaotic quantum systems. We identify pairs of periodic orbits whose actions are strongly correlated. For a class of systems with uniformly hyperbolic dynamics, we demonstrate that these pairs of orbits give rise to a τ² contribution to the spectral form factor K(τ) which agrees with random matrix theory. Most interestingly, this contribution has its origin in a next-to-leading-order behaviour of a classical distribution function for long times.

Special quantum states exist which are quasiclassical quantizations of regions of phase space that are weakly chaotic. In a weakly chaotic region, the orbits are quite regular and remain in the region for some time before escaping and manifesting possible chaotic behavior. Such phase space regions are characterized as being close to periodic orbits of an integrable reference system. The states are often rather striking, and can be concentrated in spatial regions. This leads to possible phenomena. We review some methods we have introduced to characterize such regions and find analytic formulas for the special states and their energies.

Classical periodic orbits responsible for emergence of the superdeformed shell structure of single-particle motion in spheroidal cavities are identified and their relative contributions to the shell structure are evaluated. Fourier transforms of quantum spectra clearly show that three-dimensional periodic orbits born out of bifurcations of planar orbits in the equatorial plane become predominant at large prolate deformations. A new semiclassical method capable of describing the shell structure formation associated with these bifurcations is briefly discussed.

We discuss the localization of wavefunctions along planes containing the shortest periodic orbits in a three-dimensional billiard system with axial symmetry. This model mimics the self-consistent mean field of a heavy nucleus at deformations that occur characteristically during the fission process [M. Brack et al., Rev. Mod. Phys. 44, 320 (1972); S. Bjørnholm et al., Rev. Mod. Phys. 52, 725 (1980)]. Many actinide nuclei become unstable against left-right asymmetric deformations, which results in asymmetric fragment mass distributions. Recently we have shown [M. Brack et al., Phys. Rev. Lett. 79, 1817 (1997); M. Brack et al., in "Similarities and Differences Between Atomic Nuclei and Clusters", (AIP, New York, 1998), p. 17] that the onset of this asymmetry can be explained in the semiclassical periodic orbit theory by a few short periodic orbits lying in planes perpendicular to the symmetry axis. Presently we show that these orbits are surrounded by small islands of stability in an otherwise chaotic phase space, and that the wavefunctions of the diabatic quantum states that are most sensitive to the left-right asymmetry have their extrema in the same planes. An EBK quantization of the classical motion near these planes reproduces the exact eigenenergies of the diabatic quantum states surprisingly well.

Homogeneous neutron matter at subnuclear densities becomes unstable towards the formation of inhomogeneities. Depending on the average value of the neutron density one can observe the appearance of either bubbles, rods, tubes or plates embeded in a neutron gas. We estimate the quantum corrections to the ground state energy (which could be termed either shell correction or Casimir energy) of such phases of neutron matter. The calculations are performed by evaluating the contribution of the shortest periodic orbits in the Gutzwiller trace formula for the density of states. The magnitude of the quantum corrections to the ground state energy of neutron matter are of the same order as the energy differences between various phases.

There is much latitude between the known requirements of Schnirelman's theorem regarding the ergodicity of individual high-energy eigenstates of classically chaotic systems, and the extremes of random matrix theory. Some eigenstate statistics and long-time transport behavior bear nonrandom imprints of the underlying classical dynamics while simultaneously obeying Schnirelman's theorem. Here we review the issues and give evidence for the Sinai billiard having non Random Matrix Theory behavior, even as ℏ → 0.

Over the preceeding twenty years, the role of underlying classical dynamics in quantum mechanical tunneling has received considerable attention. A number of new tunneling phenomena have been uncovered that have been directly linked to the set of dynamical possibilities arising in simple systems that contain at least some chaotic motion. These tunneling phenomena can be identified by their novel ℏ-dependencies and/or statistical behaviors. We summarize a sampling of these phenomena and mention some applications.

We discuss quantum relaxation and fluctuation phenomena in the light of the corresponding classical behaviour. Quantum exponential localization and tunneling effects lead to the universal algebraic decay of the survival probability P(t) ∝ 1/t. This behaviour is observed in particular in the time dependence of the ionization probability of Rydberg atoms driven by a monochromatic field. At a fixed time the quantum survival probability strongly fluctuates as some parameter is varied. These fluctuations have a fractal structure which is quantitatively related to the time dependence of the survival probability.

Systems with two degrees of freedom are comparatively well-understood in contrast to the other, more common, variety which requires more than two degrees of freedom for its description. These multidimensional systems will be the focus of this article, which deals with quantal problems in the Correspondence Principle limit. Our systems of choice are highly excited ("Rydberg") atoms placed in external fields the Hamiltonians of which show soft chaos.

We discuss the final stages of the simultaneous ionization of two or more electrons due to a strong laser pulse. An analysis of the classical dynamics suggests that the dominant pathway for non-sequential escape has the electrons escaping in a symmetric arrangement. Classical trajectory models within and near to this symmetry subspace support the theoretical considerations and give final momentum distributions in close agreement with experiments.

We review two classically-forbidden processes that give rise to prominent structures in atomic photoabsorption spectra. In one, the electron undergoes classically-forbidden reflection above a potential-energy barrier. In the other, absorption occurs into a classically-forbidden region of the electron's motion.

The breakdown of the dissipationless state of the quantum Hall effect at high currents sometimes occurs as a series of regular steps in the dissipative voltage drop measured along the Hall bar. The steps were first seen clearly in two of the Hall bars used to maintain the US Resistance Standard, but have also been reported in other devices. This paper describes a model to account for the origin of the steps. It is proposed that the dissipationless flow of the quantum Hall fluid is unstable at high flow rates due to inter-Landau level tunnelling processes in local microscopic regions of the Hall bar. Electron-hole pairs are generated in the quantum Hall fluid in these regions and the electronic motion can be envisaged as a quantum analogue of the von Karman vortex street which forms when a classical fluid flows past an obstacle.

Within the past five years Albert Einstein's concept of a dilute atomic Bose Condensate has been realized in many experimental laboratories. Temperatures in the nano-Kelvin regime have been achieved using magnetic and optical trapping of laser and evaporatively cooled atoms. At such temperatures the relative de Broglie wavelengths of the gaseous trapped atoms can become long compared to their mean spacing, and through a process of bosonic amplification a "quantum phase transition" takes place involving 10³ to 10¹¹ atoms, most of which end up in identical single particle quantum states, whose length scales are determined by the external trap. Rb, Na, Li and atomic H have been trapped at variable densities of the order of 10¹³/cm³, and in traps of varying geometries. Of these all but Li have an effective repulsive atomic pair interaction, but utilization of molecular Feshbach resonances allows the interactions of other species to be tuned over wide ranges of strengths, including control of the sign of the effective atomic pair interactions. Fully quantum and macroscopic systems at such low densities are a theorist's dream: simple Hartree type mean field theory provides a startlingly accurate description of density profiles, low energy excitation frequencies, and such a description, commonly called the Non-linear Schrödinger Equation (NLSE) or the Gross-Pitaevskii (GP) Equation, will be explored here. The NLSE appropriate for attractive atomic interactions is known to lead to chaotically unstable dynamics, and eventual implosion of the condensate should the local number density exceed a critical value. In this work we illustrate this type of chaotic collapse for attractive condensates, and then explore the types of chaotic dynamics of solitons and vortices, which are the signatures of dynamical non-linearity in the repulsive case. Finally the implications of the symmetry-breaking associated with phase rigidity are explored in model simulations of repulsive condensates: condensates with repulsive atomic interactions break into phase domains when subject to weak shocks and, perhaps surprisingly, break into chaotic "laser-speckle" type patterns as the shock level increases. The fully quantum mechanical NLSE thus displays a full range of chaotic types of motion: from particle-like chaotic collisions of solitons and vortices to fully developed time dependent wave chaos.

Two experiments with super- and normal conducting microwave cavities which are used as an analog to two-dimensional, infinitely deep quantum potentials are presented. The eigenvalues of such a potential can be measured directly through the resonance frequencies of the cavity, while the eigenvectors can be determined by measuring the field distributions inside the cavity. In the case of open systems another information – the imaginary part of the eigenvalues – is observable by measuring the widths of the resonances. As examples for this experimental approach to quantum chaos the semiclassical reconstruction of length spectra of billiards with varying chaoticity based of various trace formulas proposed by Gutzwiller, by Ullmo, Grinberg and Tomsovic and by Berry and Tabor for systems with chaotic, mixed and regular dynamics, respectively, and the first observation of square root singularities – so called exceptional points recently proposed by Heiss – in the energy spectrum of coupled cavities are presented.

We use high resolution acoustic spectroscopy as an analog system to quantum systems. We study transitions between regularity and chaos and specific symmetry breaking. We compare with the predictions of Random Matrix Theory that was developed for the study of the complicated compound nuclear states. The acoustic system is interesting in its own right. It is governed by a vectorial wave-equation in contrast to the scalar Schrödinger equation, and we study the complex interplay of different resonant modes arising from this fact.

We present experimental results for the ultrasound transmission spectra and standing wave patterns of a rectangular plate of fused quartz. We demonstrate that mode conversion for the in-plane modes leads to complicated behavior and the fluctuation statistics are described by the Gaussian orthogonal ensemble (GOE) of random matrix theory (RMT). Studying the distribution of normalized frequency shifts upon a temperature increase of 5 °C, and observing the corresponding measured standing wave patterns, it is revealed that a small number of special resonances are protected from mode conversion and therefore do not take part in the mixing. Such states serve as reference points in our data analysis and allow us to compare our results to a random matrix model for symmetry breaking.

We show that the autocorrelation of quantum spectra of an open chaotic system is well described by the classical Ruelle-Pollicott resonances of the associated chaotic strange repeller. This correspondence is demonstrated utilizing microwave experiments on 2-D n-disk billiard geometries, by determination of the wave-vector autocorrelation C(κ) from the experimental quantum spectra S₂₁(k). The correspondence is also established via "numerical experiments" that simulate S₂₁(k) and C(κ) using periodic orbit calculations of the quantum and classical resonances. Semiclassical arguments that relate quantum and classical correlation functions in terms of fluctuations of the density of states and correlations of particle density are also examined and support the experimental results. The results establish a correspondence between quantum spectral correlations and classical decay modes in an open system.

A personal view is given on the interplay between classical wave experiments and theory.

Deformed cylindrical and spherical dielectric optical resonators and lasers are analyzed from the perspective of non-linear dynamics and quantum chaos theory. In the short-wavelength limit such resonators behave like billiard systems with non-zero escape probability due to refraction. A ray model is introduced to predict the resonance lifetimes and emission patterns from such a cavity. A universal wavelength-independent broadening is predicted and found for large deformations of the cavity. However there are significant wave-chaotic corrections to the model which arise from chaos-assisted tunneling and dynamical localization effects. Highly directional emission from lasers based on these resonators is predicted from chaotic "whispering gallery" modes for index of refraction less than two. The detailed nature of the emission pattern can be understood from the nature of the phase-space flow in the billiard, and a dramatic variation of this pattern with index of refraction is found due to an effect we term "dynamical eclipsing". Semiconductor lasers of this type also show highly directional emission and high output power but from different modes associated with periodic orbits, both stable and unstable. A semiclassical approach to these modes is briefly reviewed. These asymmetric resonant cavities (ARCs) show promise as components in future integrated optical devices, providing perhaps the first application of quantum chaos theory.

Angular momentum ceases to be the preferred basis for identifying dynamical localization in an oval billiard at large eccentricity. We give reasons for this, and comment on the classical phase-space structure that is encoded in the wave functions of "leaky" dielectric resonators with oval cross section.

The objective of this paper is to show that time-reversal invariance can be exploited in acoustics to accurately control wave propagation through random propagating media as well as through chaotic reverberant cavities. To illustrate these concepts, several experiments are presented. They show that chaotic dynamics reduces the number of transducers needed to insure an accurate time-reversal experiment. Multi-pathing in random media and in chaotic cavities enhances resolution in time-reversal acoustics by making the effective size of time reversal mirrors much larger than its physical size. Comparisons with phase conjugated experiments will show that this effect is typical of broadband time-reversed acoustics and is not observed in monochromatic phase conjugation. Self averaging properties of time reversal experiments conducted in chaotic scattering environments will be emphasized.

We investigate the low-frequency dynamics for transmission or reflection of a wave by a cavity with chaotic scattering. We compute the probability distribution of the phase derivative ϕ' = dϕ/dω of the scattered wave amplitude, known as the single-mode delay time. In the case of a cavity connected to two single-mode waveguides we find a marked distinction between detection in transmission and in reflection: The distribution P(ϕ') vanishes for negative ϕ' in the first case but not in the second case.

The Nobel Foundation arranges on a regular basis special Nobel Symposia in scientific disciplines that are related to the Nobel prizes. Since 1965, when the series of symposia was first initiated, more than one hundred have been arranged. A Nobel symposium corresponds to a rather small meeting with leading scientists in their particular field, facilitating informal discussions, in plenum as well as in private, on critical issues and latest results. In addition to the specialists a small number graduate students, young researchers, and researchers from nearby fields are also invited to participate in the symposia.

This time the Nobel Foundation has selected the field of quantum chaos for its 116th symposium. Thus "Quantum Chaos Y2K" was held in June 13-17, 2000, at the historic and beautiful Bäckaskogs Castle in the south of Sweden.

Quantum chaos is becoming a very wide field that ranges from experiments to theoretical physics and purely mathematical issues. Because of this grand span we decided at an early stage of the planning of this symposium to focus on experiments and theory, and to encourage the interplay between them. We also wanted to stress the interdisciplinary character of the subject, and involved physics from a broad range of subjects, including condensed matter physics, nuclear physics, atomic physics, and elementary particle physics. The physics involved in quantum chaos has much in common with acoustics, microwaves, optics, etc. Therefore we choose to include also aspects of wave chaos in this broader sense. The program was structured according to the following headings:

• Manifestations of Classical Chaos in Quantum Systems

• Transport Phenomena

• Description of Quantal Spectra in Terms of Periodic Orbits

• Semiclassical and Random Matrix Approaches

• Quantum Chaos in Interacting Systems

• Chaos and Tunneling

• Wave-Dynamic Chaos

The meeting at Bäckaskog was conducted in a truly interdisciplinary atmosphere with invited talks and poster presentations. There was ample time for discussions in which the chairmen played very important roles in summarising and highlighting the different sub-fields. In addition to invited talks and poster contributions the present proceedings therefore also contain informal contributions from some of the chairmen. In this way we hope the proceedings will convey some of the enthusiasm felt at Bäckaskog.

Finally we wish to thank the Nobel Foundation for selecting the field of quantum chaos to be the topic of the 116th Nobel symposium and for its generous support. We also wish to thank the international program committee for all its help when designing the symposium.

The after-dinner speech given by Martin C Gutzwiller at the Conference Dinner, Friday 16 June 2000.