G. A. Luna-Acosta

Classical versus quantum structure of the scattering probability matrix: Chaotic waveguides

The purely classical counterpart of the scattering probability matrix (SPM)/S(n,m)/(2) of the quantum scattering matrix S is defined for two-dimensional quantum waveguides for an arbitrary number of propagating modes M. We compare the quantum and classical structures of /S(n,m)/(2) for a waveguide with generic Hamiltonian chaos. It is shown that even for a moderate number of channels, knowledge of the classical structure of the SPM allows us to predict the global structure of the quantum one and, hence, understand important quantum transport properties of waveguides in terms of purely classical dynamics. It is also shown that the SPM, being an intensity measure, can give additional dynamical information to that obtained by the Poincaré maps.

Periodic chaotic billiards: quantum-classical correspondence in energy space.

We investigate the properties of eigenstates and local density of states (LDOS) for a periodic two-dimensional rippled billiard, focusing on their quantum-classical correspondence in energy representation. To construct the classical counterparts of LDOS and the structure of eigenstates (SES), the effects of the boundary are first incorporated (via a canonical transformation) into an effective potential, rendering the one-particle motion in the 2D rippled billiard equivalent to that of two interacting particles in 1D geometry. We show that classical counterparts of SES and LDOS in the case of strong chaotic motion reveal quite a good correspondence with the quantum quantities. We also show that the main features of the SES and LDOS can be explained in terms of the underlying classical dynamics, in particular, of certain periodic orbits. On the other hand, statistical properties of eigenstates and LDOS turn out to be different from those prescribed by random matrix theory. We discuss the quantum effects responsible for the nonergodic character of the eigenstates and individual LDOS that seem to be generic for this type of billiards with a large number of transverse channels.

Chaotic classical scattering and dynamics in oscillating 1-D potential wells

Abstract We study the motion of a classical particle interacting with one, two, and finally an infinite chain of 1-D square wells with oscillating depth. For a single well we find complicated scattering behavior even though there is no topological chaos due to the absence of hyperbolic periodic orbits. In contrast, for two coupled square wells there is chaotic scattering. The infinite oscillating chain yields the generic transition to chaos, with diffusion in energy and in space, as the separation between wells is increased. We briefly discuss the relevance of our results to solid state physics.

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.

Understanding quantum scattering properties in terms of purely classical dynamics: two-dimensional open chaotic billiards.

We study classical and quantum scattering properties of particles in the ballistic regime in two-dimensional chaotic billiards that are models of electron- or micro-waveguides. To this end we construct the purely classical counterparts of the scattering probability (SP) matrix |S(n,m)|(2) and Husimi distributions specializing to the case of mixed chaotic motion (incomplete horseshoe). Comparison between classical and quantum quantities allows us to discover the purely classical dynamical origin of certain general as well as particular features that appear in the quantum description of the system. On the other hand, at certain values of energy the tunneling of the wave function into classically forbidden regions produces striking differences between the classical and quantum quantities. A potential application of this phenomenon in the field of microlasers is discussed briefly. We also see the manifestation of whispering gallery orbits as a self-similar structure in the transmission part of the classical SP matrix.

Quantum-classical correspondence for local density of states and eigenfunctions of a chaotic periodic billiard

Abstract Classical-quantum correspondence for conservative chaotic Hamiltonians is investigated in terms of the structure of the eigenfunctions and the local density of states, using as a model a 2D rippled billiard in the regime of global chaos. The influence of the observed localized and sparsed states in the quantum-classical correspondence is discussed.

Quantum weak chaos in a degenerate system

Quantum weak chaos is studied in a perturbed degenerate system: a charged particle interacting with a monochromatic wave in a transverse magnetic field. The evolution operator for an arbitrary number of periods of the external field is built and its structure is explored in terms of the quasienergy eigenstates under resonance conditions (when the wave frequency equals the cyclotron frequency) in the regime of weak classical chaos. The new phenomenon of diffusion via the quantum separatrices and the influence of chaos on diffusion are investigated and, in the quasiclassical limit, compared with its classical dynamics. We determine the crossover from purely quantum diffusion to a diffusion that is the quantum manifestation of classical diffusion along the stochastic web. This crossover results from the nonmonotonic dependence of the characteristic localization length of the quasienergy states on the wave amplitude. The width of the quantum separatrices was computed and compared with the width of the classical stochastic web. We give the physical parameters that can be realized experimentally to show the manifestation of quantum chaos in a nonlinear acoustic resonance.

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.

Chaotic waveguide-based resonators for microlasers

We propose the construction of highly directional emission microlasers using two-dimensional high-index semiconductor waveguides as open resonators. The prototype waveguide is formed by two collinear leads connected to a cavity of certain shape. The proposed lasing mechanism requires that the shape of the cavity yield mixed chaotic ray dynamics so as to have the appropiate (phase space) resonance islands. These islands allow, via Heisenberg’s uncertainty principle, the appearance of quasi bound states (QBS) which, in turn, propitiate the lasing mechanism. The energy values of the QBS are found through the solution of the Helmholtz equation. We use classical ray dynamics to predict the direction and intensity of the lasing produced by such open resonators for typical values of the index of refraction.

Nearest level spacing statistics in open chaotic systems: generalization of the Wigner surmise.

We investigate the nearest level spacing statistics of open chaotic wave systems. To this end we derive the spacing distributions for the three Wigner ensembles in the one-channel case. The theoretical results give a clear physical meaning of the modifications on the spacing distributions produced by the coupling to the environment. Based on the analytical expressions obtained, we then propose general expressions of the spacing distributions for any number of channels, valid from weak to strong coupling. The latter expressions contain one free parameter. The surmise is successfully compared with numerical simulations of non-Hermitian random matrices and with experimental data obtained with a lossy electromagnetic chaotic cavity.