Denis Ullmo

Manifestations of classical phase space structures in quantum mechanics

Abstract Using two coupled quartic oscillators for illustration, the quantum mechanics of simple systems whose classical analogs have varying degrees of non-integrability is investigated. By taking advantage of discrete symmetries and dynamical quasidegeneracies it is shown that Percivals semiclassical classification scheme, i.e., eigenstates may be separated into a regular and an irregular group, basically works. This allows us to probe deeply into the workings of semiclassical quantization in mixed phase space systems. Some observations of intermediate status states are made. The standard modeling of quantum fluctuation properties exhibited by the irregular states and levels by random matrix ensembles is then put on a physical footing. Generalized ensembles are constructed incorporating such classical information as fluxes crossing partial barriers and relative fractions of phase space volume occupied by interesting subregions. The ensembles apply equally well to both spectral and eigenstate properties. They typically show non-universal, but nevertheless characteristic level fluctuations. In addition, they predict “semiclassical localization” of eigenfunctions and “quantum suppression of chaos” which are quantitatively borne out in the quantum systems.

Wireless propagation in buildings: a statistical scattering approach

A new approach to the modeling of wireless propagation in buildings is introduced. We treat the scattering by walls and local clutter probabilistically through either a relaxation-time approximation in a Boltzmann equation or by using a diffusion equation. The result is a range of models in which one can vary the tradeoff between the complexity of the building description and the accuracy of the prediction. The two limits of this range are ray tracing at the most accurate end and a simple decay law at the most simple. By comparing results for two of these new models with measurements, we conclude that a reasonably accurate description of propagation can be obtained with a relatively simple model. The most effective way to use the models is by combining them with a few measurements through a sampling technique.

ORBITAL MAGNETISM IN THE BALLISTIC REGIME: GEOMETRICAL EFFECTS

Abstract We present a general semiclassical theory of the orbital magnetic response of noninteracting electrons confined in two-dimensional potentials. We calculate the magnetic susceptibility of singly-connected and the persistent currents of multiply connected geometries. We concentrate on the geometric effects by studying confinement by perfect (disorder free) potentials stressing the importance of the underlying classical dynamics. We demonstrate that in a constrained geometry the standard Landau diamagnetic response is always present, but is dominated by finite-size corrections of a quasi-random sign which may be orders of magnitude larger. These corrections are very sensitive to the nature of the classical dynamics. Systems which are integrable at zero magnetic field exhibit larger magnetic response than those which are chaotic. This difference arises from the large oscillations of the density of states in integrable systems due to the existence of families of periodic orbits. The connection between quantum and classical behavior naturally arises from the use of semiclassical expansions. This key tool becomes particularly simple and insightful at finite temperature, where only short classical trajectories need to be kept in the expansion. In addition to the general theory for integrable systems, we analyze in detail a few typical examples of experimental relevance: circles, rings and square billiards. In the latter, extensive numerical calculations are used as a check for the success of the semiclassical analysis. We study the weak-field regime where classical trajectories remain essentially unaffected, the intermediate field regime where we identify new oscillations characteristic for ballistic mesoscopic structures, and the high-field regime where the typical de Haas-van Alphen oscillations exhibit finite-size corrections. We address the comparison with experimental data obtained in high-mobility semiconductor microstructures discussing the differences between individual and ensemble measurements, and the applicability of the present model.

Interactions and interference in quantum dots: Kinks in Coulomb-blockade peak positions

We investigate the spin of the ground state of a geometrically confined many-electron system. For atoms, shell structure simplifies this problem—the spin is prescribed by the well-known Hund’s rule. In contrast, quantum dots provide a controllable setting for studying the interplay of quantum interference and electronelectron interactions in general cases. In a generic confining potential, the shell-structure argument suggests a singlet ground state for an even number of electrons. The interaction among the electrons produces, however, accidental occurrences of spin-triplet ground states, even for weak interaction, a limit which we analyze explicitly. Variation of an external parameter causes sudden switching between these states and hence a kink in the conductance. Experimental study of these kinks would yield the exchange energy for the ‘‘chaotic electron gas.’’ @S0163-1829~99!51448-3# The evolution of the properties of a system as a continuous change is made to it is a ubiquitous topic in quantum physics. The classic example is the evolution of energy lev

Correlation-induced inhomogeneity in circular quantum dots

Properties of the ‘electron gas’—in which conduction electrons interact by means of Coulomb forces but ionic potentials are neglected—change dramatically depending on the balance between kinetic energy and Coulomb repulsion. The limits are well understood1. For very weak interactions (high density), the system behaves as a Fermi liquid, with delocalized electrons. In contrast, in the strongly interacting limit (low density), the electrons localize and order into a Wigner crystal phase. The physics at intermediate densities, however, remains a subject of fundamental research2,3,4,5,6,7,8. Here, we study the intermediate-density electron gas confined to a circular disc, where the degree of confinement can be tuned to control the density. Using accurate quantum Monte Carlo techniques9, we show that the electron–electron correlation induced by an increase of the interaction first smoothly causes rings, and then angular modulation, without any signature of a sharp transition in this density range. This suggests that inhomogeneities in a confined system, which exist even without interactions, are significantly enhanced by correlations.

THE LEVEL SPLITTING DISTRIBUTION IN CHAOS-ASSISTED TUNNELLING

A compound tunneling mechanism from one integrable region to another mediated by a delocalized state in an intermediate chaotic region of phase space was recently introduced to explain peculiar features of tunneling in certain two-dimensional systems. This mechanism is known as chaos-assisted tunneling. We study its consequences for the distribution of the level splittings and obtain a general analytical form for this distribution under the assumption that chaos assisted tunneling is the only operative mechanism. We have checked that the analytical form we obtain agrees with splitting distributions calculated numerically for a model system in which chaos-assisted tunneling is known to be the dominant mechanism. The distribution depends on two parameters: The first gives the scale of the splittings and is related to the magnitude of the classically forbidden processes, the second gives a measure of the efficiency of possible barriers to classical transportwhich may exist in the chaotic region. If these are weak, this latter parameter is irrelevant; otherwise it sets an energy scale at which the splitting distribution crosses over from one type of behavior to another. The detailed form of the crossover is also obtained and found to be in good agreement with numerical results for models for chaos-assisted tunneling.

Many-body physics and quantum chaos

Experimental progresses in the miniaturisation of electronic devices have made routinely available in the laboratory small electronic systems, on the micron or sub-micron scale, which at low temperature are sufficiently well isolated from their environment to be considered as fully coherent. Some of their most important properties are dominated by the interaction between electrons. Understanding their behaviour therefore requires a description of the interplay between interference effects and interactions. The goal of this review is to address this relatively broad issue, and more specifically to address it from the perspective of the quantum chaos community. I will therefore present some of the concepts developed in the field of quantum chaos which have some application to study many-body effects in mesoscopic and nanoscopic systems. Their implementation is illustrated on a few examples of experimental relevance such as persistent currents, mesoscopic fluctuations of Kondo properties or Coulomb blockade. I will furthermore try to bring out, from the various physical illustrations, some of the specific advantages on more general grounds of the quantum chaos based approach.

Interactions in chaotic nanoparticles: Fluctuations in Coulomb blockade peak spacings

We use random matrix models and a Fermi-liquid approach to investigate the ground state energy of electrons confined to a nanoparticle. Our expression for the energy includes the charging effect, the single-particle energies, and the residual screened interactions treated in Hartree-Fock. This model is applicable to chaotic quantum dots or nanoparticles\char22{}in these systems the single-particle statistics follows random matrix theory at energy scales less than the Thouless energy. We find the distribution of Coulomb blockade peak spacings first for a large dot in which the residual interactions can be taken constant: the spacing fluctuations are of order the mean level separation

Semiclassical magnetotransport in graphene n-p junctions

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Berry phase in graphene: a semiclassical perspective

Corrections to this limit are studied using the small parameter