Geometry and Topology of Escape I: Epistrophes

We consider a dynamical system given by an area-preserving map on a two-dimensional phase plane and consider a one-dimensional line of initial conditions within this plane. We record the number of iterates it takes a trajectory to escape from a bounded region of the plane as a function along the line of initial conditions, forming an ``escape-time plot''. For a chaotic system, this plot is in general not a smooth function, but rather has many singularities at which the escape time is infinite; these singularities form a complicated fractal set. In this article we prove the existence of regular repeated sequences, called ``epistrophes'', which occur at all levels of resolution within the escape-time plot. (The word ``epistrophe'' comes from rhetoric and means ``a repeated ending following a variable beginning''.) The epistrophes give the escape-time plot a certain self-similarity, called ``epistrophic'' self-similarity, which need not imply either strict or asymptotic self-similarity.Comment: 15 pages, 9 figures, to appear in Chaos, first of two paper

Multi-utility Learning: Structured-output Learning with Multiple Annotation-specific Loss Functions

Structured-output learning is a challenging problem; particularly so because of the difficulty in obtaining large datasets of fully labelled instances for training. In this paper we try to overcome this difficulty by presenting a multi-utility learning framework for structured prediction that can learn from training instances with different forms of supervision. We propose a unified technique for inferring the loss functions most suitable for quantifying the consistency of solutions with the given weak annotation. We demonstrate the effectiveness of our framework on the challenging semantic image segmentation problem for which a wide variety of annotations can be used. For instance, the popular training datasets for semantic segmentation are composed of images with hard-to-generate full pixel labellings, as well as images with easy-to-obtain weak annotations, such as bounding boxes around objects, or image-level labels that specify which object categories are present in an image. Experimental evaluation shows that the use of annotation-specific loss functions dramatically improves segmentation accuracy compared to the baseline system where only one type of weak annotation is used

Analysis of Chaos-Induced Pulse Trains in the Ionization of Hydrogen

We examine excitation (by a short laser pulse) of a hydrogen atom in parallel electric and magnetic fields, from an initial tightly bound state to a state above the classical ionization threshold. We predict that the atom ionizes by emitting a train of electron pulses. This prediction is based on the classical dynamics of electron escape. In particular, the pulse train is due to classical chaos, which occurs for nonvanishing magnetic field. We connect the structure of the pulse train to fractal structure in the escape dynamics, and discuss several issues of experimental interest, with a particular emphasis on understanding the resolution of individual pulses. A brief account of this work appeared previously as a Letter [Phys. Rev. Lett. 92, 073001 (2004)]

Chaos-Induced Pulse Trains in the Ionization of Hydrogen

We predict that a hydrogen atom in parallel electric and magnetic fields, excited by a short laser pulse to an energy above the classical saddle, ionizes via a train of electron pulses. These pulses are a consequence of classical chaos induced by the magnetic field. We connect the structure of this pulse train (e.g., pulse size and spacing) to fractal structure in the classical dynamics. This structure displays a weak self-similarity, which we call “epistrophic self-similarity.” We demonstrate how this self-similarity is reflected in the pulse train

Elastic Wave Transmission at an Abrupt Junction in a Thin Plate, with Application to Heat Transport and Vibrations in Mesoscopic Systems

The transmission coefficient for vibrational waves crossing an abrupt junction between two thin elastic plates of different widths is calculated. These calculations are relevant to ballistic phonon thermal transport at low temperatures in mesoscopic systems and the Q for vibrations in mesoscopic oscillators. Complete results are calculated in a simple scalar model of the elastic waves, and results for long wavelength modes are calculated using the full elasticity theory calculation. We suggest that thin plate elasticty theory provide a useful and tractable approximation to the full three dimensional geometry.Comment: 35 pages, including 12 figure

Dynamics of a Josephson Array in a Resonant Cavity

We derive dynamical equations for a Josephson array coupled to a resonant cavity by applying the Heisenberg equations of motion to a model Hamiltonian described by us earlier [Phys. Rev. B {\bf 63}, 144522 (2001); Phys. Rev. B {\bf 64}, 179902 (E)]. By means of a canonical transformation, we also show that, in the absence of an applied current and dissipation, our model reduces to one described by Shnirman {\it et al} [Phys. Rev. Lett. {\bf 79}, 2371 (1997)] for coupled qubits, and that it corresponds to a capacitive coupling between the array and the cavity mode. From extensive numerical solutions of the model in one dimension, we find that the array locks into a coherent, periodic state above a critical number of active junctions, that the current-voltage characteristics of the array have self-induced resonant steps (SIRS's), that when $N_a$ active junctions are synchronized on a SIRS, the energy emitted into the resonant cavity is quadratic in $N_a$, and that when a fixed number of junctions is biased on a SIRS, the energy is linear in the input power. All these results are in agreement with recent experiments. By choosing the initial conditions carefully, we can drive the array into any of a variety of different integer SIRS's. We tentatively identify terms in the equations of motion which give rise to both the SIRS's and the coherence threshold. We also find higher-order integer SIRS's and fractional SIRS's in some simulations. We conclude that a resonant cavity can produce threshold behavior and SIRS's even in a one-dimensional array with appropriate experimental parameters, and that the experimental data, including the coherent emission, can be understood from classical equations of motion.Comment: 15 pages, 10 eps figures, submitted to Phys. Rev.

Quantum interference and Coulomb interaction in arrays of tunnel junctions

We study the electronic properties of an array of small metallic grains connected by tunnel junctions. Such an array serves as a model for a granular metal. Previous theoretical studies of junction arrays were based on models of quantum dissipation which did not take into account the diffusive motion of electrons within the grains. We demonstrate that these models break down at sufficiently low temperatures: for a correct description of the screening properties of a granular metal at low energies the diffusive nature of the electronic motion within the grains is crucial. We present both a diagrammatic and a functional integral approach to analyse the properties of junction arrays. In particular, a new effective action is obtained which enables us to describe the array at arbitrary temperature. In the low temperature limit, our theory yields the correct, dynamically screened Coulomb interaction of a normal metal, whereas at high temperatures the standard description in terms of quantum dissipation is recovered.Comment: 14 pages, 7 figure

Parity Effect and Charge Binding Transition in Submicron Josephson Junction Arrays

We reconsider the issue of Berezinskii-Kosterlitz-Thouless (BKT) transition into an insulating state in the Coulomb-dominated Josephson junction arrays. We show that previously predicted picture of the Cooper-pair BKT transtion at T = T_2 is valid only under the condition that T_2 is considerably below the parity-effect temperature (which is usually almost 10 times below the value of superconductive transition temperature), and even in this case it is not a rigorous phase transition but only a crossover, whereas the real phase transition takes place at T_1 = T_2/4. Our theory is in agreement with available experimental data on Coulomb-dominated Josephson arrays and also sheds some light on the origin of unusual reentrant temperature dependence of resistivity in the array with nearly-criticial ratio of Coulomb to Josephson energies.Comment: 4 pages, Revtex, to be published in JETP Letters, April 9

Theory of Two-Dimensional Josephson Arrays in a Resonant Cavity

We consider the dynamics of a two-dimensional array of underdamped Josephson junctions placed in a single-mode resonant cavity. Starting from a well-defined model Hamiltonian, which includes the effects of driving current and dissipative coupling to a heat bath, we write down the Heisenberg equations of motion for the variables of the Josephson junction and the cavity mode, extending our previous one-dimensional model. In the limit of large numbers of photons, these equations can be expressed as coupled differential equations and can be solved numerically. The numerical results show many features similar to experiment. These include (i) self-induced resonant steps (SIRS's) at voltages V = (n hbar Omega)/(2e), where Omega is the cavity frequency, and n is generally an integer; (ii) a threshold number N_c of active rows of junctions above which the array is coherent; and (iii) a time-averaged cavity energy which is quadratic in the number of active junctions, when the array is above threshold. Some differences between the observed and calculated threshold behavior are also observed in the simulations and discussed. In two dimensions, we find a conspicuous polarization effect: if the cavity mode is polarized perpendicular to the direction of current injection in a square array, it does not couple to the array and there is no power radiated into the cavity. We speculate that the perpendicular polarization would couple to the array, in the presence of magnetic-field-induced frustration. Finally, when the array is biased on a SIRS, then, for given junction parameters, the power radiated into the array is found to vary as the square of the number of active junctions, consistent with expectations for a coherent radiation.Comment: 11 pages, 8 eps figures, submitted to Phys. Rev

Full capacitance-matrix effects in driven Josephson-junction arrays

We study the dynamic response to external currents of periodic arrays of Josephson junctions, in a resistively capacitively shunted junction (RCSJ) model, including full capacitance-matrix effects}. We define and study three different models of the capacitance matrix $C_{\vec{r},\vec{r}'}$: Model A includes only mutual capacitances; Model B includes mutual and self capacitances, leading to exponential screening of the electrostatic fields; Model C includes a dense matrix $C_{\vec{r},\vec{r}'}$ that is constructed approximately from superposition of an exact analytic solution for the capacitance between two disks of finite radius and thickness. In the latter case the electrostatic fields decay algebraically. For comparison, we have also evaluated the full capacitance matrix using the MIT fastcap algorithm, good for small lattices, as well as a corresponding continuum effective-medium analytic evaluation of a finite voltage disk inside a zero-potential plane. In all cases the effective $C_{\vec{r},\vec{r}'}$ decays algebraically with distance, with different powers. We have then calculated current voltage characteristics for DC+AC currents for all models. We find that there are novel giant capacitive fractional steps in the I-V's for Models B and C, strongly dependent on the amount of screening involved. We find that these fractional steps are quantized in units inversely proportional to the lattice sizes and depend on the properties of $C_{\vec{r},\vec{r}'}$. We also show that the capacitive steps are not related to vortex oscillations but to localized screened phase-locking of a few rows in the lattice. The possible experimental relevance of these results is also discussed.Comment: 12 pages 18 Postscript figures, REVTEX style. Paper to appear in July 1, Vol. 58, Phys. Rev. B 1998 All PS figures include