No abstract

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.

Using an energy-independent non-Hermitian Hamiltonian approach to open systems, we fully describe transport through a sequence of potential barriers as external barriers are varied. Analyzing the complex eigenvalues of the non-Hermitian Hamiltonian model, a transition to a superradiant regime is shown to occur. Transport properties undergo a strong change at the superradiance transition, where the transmission is maximized and a drastic change in the structure of resonances is demonstrated. Finally, we analyze the effect of the superradiance transition in the Anderson localized regime.

We examine the effect of short unstable periodic orbits on wave function statistics in a classically chaotic system, and find that the tail of the wave function intensity distribution in phase space is dominated by scarring associated with the least unstable periodic orbits. In an ensemble average over systems with classical orbits of different instabilities, a power-law tail is found, in sharp contrast to the exponential prediction of random matrix theory. The calculations are compared with numerical data, and quantitative agreement is obtained. [S0031-9007 (98)05543-4] PACS numbers: 05.45. + b, 03.65.SqQuantum eigenstates of classically chaotic systems generically exhibit a phenomenon known as scarring, the enhancement of intensity along short unstable periodic orbits for some fraction of the wave functions. Scarring is a fascinating example of the influence of identifiable classical structures on stationary quantum properties and on long-time quantum transport in a classically ergodic system. The occurrence of scars is in some sense paradoxical, because classically, all such short-time information is destroyed at long times, and a classical probability distribution after being evolved for a sufficiently long time retains no memory of its initial state. Scarring is one of the most dramatic examples of a departure of quantum chaotic systems from the predictions of random matrix theory (RMT), according to which wave functions must be evenly distributed over phase space, up to quantum fluctuations. Scarring has now been observed experimentally in a variety of systems, including microwave cavities [1,2], tunnel junctions [3], and the hydrogen atom in a uniform magnetic field [4,5].Examples of scarring were observed numerically in [6], and a theory based on the semiclassical evolution of Husimi states near a periodic orbit was provided. Later work by Bogomolny [7] and Berry [8] involved calculations in coordinate space and Wigner phase space, respectively. All these works were based on the linearized dynamics around the unstable periodic orbit, and were thus, by construction, theories of the short-time behavior only. Yet to get a true understanding of the properties of individual eigenstates it is essential to understand the longtime quantum dynamics, including returns of amplitude to the original periodic orbit after undergoing excursions into other areas of phase space. In a recent paper [9], a formalism was developed for dealing with these nonlinear contributions to scarring, providing quantitative agreement of the theory with numerical results. This work used a measure of scarring based on Husimi intensities. (Recently Fishman, Agam, and others have provided interesting new perspectives on the problem of scarring, and have offered a measure of scarring related to, but somewhat different from ours [10]. A number of other authors have also made significant contributions in this area; we cannot list them all but a few recent references are provided in [11].) In this paper, we will apply a result previously obtained...

We consider quantum mechanics on constrained surfaces which have non-Euclidean metrics and variable Gaussian curvature. The old controversy about the ambiguities involving terms in the Hamiltonian of order ប 2 multiplying the Gaussian curvature is addressed. We set out to clarify the matter by considering constraints to be the limits of large restoring forces as the constraint coordinates deviate from their constrained values. We find additional ambiguous terms of order ប 2 involving freedom in the constraining potentials, demonstrating that the classical constrained Hamiltonian or Lagrangian cannot uniquely specify the quantization: the ambiguity of directly quantizing a constrained system is inherently unresolvable. However, there is never any problem with a physical quantum system, which cannot have infinite constraint forces and always fluctuates around the mean constraint values. The issue is addressed from the perspectives of adiabatic approximations in quantum mechanics and Feynman path integrals, and semiclassically in terms of adiabatic actions.

The phenomenon of periodic orbit scarring of eigenstates of classically chaotic systems is attracting increasing attention. Scarring is one of the most important "corrections" to the ideal random eigenstates suggested by random matrix theory. This paper discusses measures of scars and in so doing also tries to clarify the concepts and effects of eigenfunction scarring. We propose a universal scar measure which takes into account an entire periodic orbit and the linearized dynamics in its vicinity. This measure is tuned to pick out those structures which are induced in quantum eigenstates by unstable periodic orbits and their manifolds. It gives enhanced scarring strength as measured by eigenstate overlaps and inverse participation ratios, especially for longer orbits. We also discuss off-resonance scars which appear naturally on either side of an unstable periodic orbit.

Using a non-Hermitian Hamiltonian approach to open systems, we study the interplay of disorder and superradiance in a one-dimensional Anderson model. Analyzing the complex eigenvalues of the non-Hermitian Hamiltonian, a transition to a superradiant regime is shown to occur. As an effect of openness the structure of eigenstates undergoes a strong change in the superradiant regime: we show that the sensitivity to disorder of the superradiant and the subradiant subspaces is very different; superradiant states remain delocalized as disorder increases, while subradiant states are sensitive to the degree of disorder.Comment: 7 pages, submitted to the special issue on "Physics with non-Hermitian operators: Theory and Experiment" of the journal "Fortschritte der Physik - Progress of Physics

[1] Refraction of a Longuet-Higgins Gaussian sea by random ocean currents creates persistent local variations (in the form of lumps or streaks) in average energy and wave action distributions. These variations explicitly survive averaging over wavelength and wave propagation direction. The lumps and streaks in average local action mean that the uniform sampling assumed in the venerable Longuet-Higgins theory does not apply. Proper handling of the nonuniform sampling results in greatly increased probability of freak wave formation. The present theory represents a synthesis of Longuet-Higgins Gaussian seas and the refraction model of White and Fornberg, which used a nonGaussian nonstatistical plane wave incident seaway. Using the linearized equations for deep ocean waves, we obtain quantitative predictions for the increased probability of freak wave formation when the refractive effects are taken into account. The wave height distribution depends primarily on the ''freak index,'' g, which measures the strength of refraction relative to the angular spread of the incoming sea. Dramatic effects are obtained in the tail of this distribution even for the modest values of the freak index that are expected to occur commonly in nature. Extensive comparisons are made between the analytical description and numerical simulations.