Effects of Repulsive Coupling in Ensembles of Excitable Elements

Die vorliegende Arbeit behandelt die kollektive Dynamik identischer Klasse-I-anregbarer Elemente. Diese können im Rahmen der nichtlinearen Dynamik als Systeme nahe einer Sattel-Knoten-Bifurkation auf einem invarianten Kreis beschrieben werden. Der Fokus der Arbeit liegt auf dem Studium aktiver Rotatoren als Prototypen solcher Elemente. In Teil eins der Arbeit besprechen wir das klassische Modell abstoßend gekoppelter aktiver Rotatoren von Shinomoto und Kuramoto und generalisieren es indem wir höhere Fourier-Moden in der internen Dynamik der Rotatoren berücksichtigen. Wir besprechen außerdem die mathematischen Methoden die wir zur Untersuchung des Aktive-Rotatoren-Modells verwenden. In Teil zwei untersuchen wir Existenz und Stabilität periodischer Zwei-Cluster-Lösungen für generalisierte aktive Rotatoren und beweisen anschließend die Existenz eines Kontinuums periodischer Lösungen für eine Klasse Watanabe-Strogatz-integrabler Systeme zu denen insbesondere das klassische Aktive-Rotatoren-Modell gehört und zeigen dass (i) das Kontinuum eine normal-anziehende invariante Mannigfaltigkeit bildet und (ii) eine der auftretenden periodischen Lösungen Splay-State-Dynamik besitzt. Danach entwickeln wir mit Hilfe der Averaging-Methode eine Störungstheorie für solche Systeme. Mit dieser können wir Rückschlüsse auf die asymptotische Dynamik des generalisierten Aktive-Rotatoren-Modells ziehen. Als Hauptergebnis stellen wir fest dass sowohl periodische Zwei-Cluster-Lösungen als auch Splay States robuste Lösungen für das Aktive-Rotatoren-Modell darstellen. Wir untersuchen außerdem einen "Stabilitätstransfer" zwischen diesen Lösungen durch sogenannte Broken-Symmetry States. In Teil drei untersuchen wir Ensembles gekoppelter Morris-Lecar-Neuronen und stellen fest, dass deren asymptotische Dynamik der der aktiven Rotatoren vergleichbar ist was nahelegt dass die Ergebnisse aus Teil zwei ein qualitatives Bild für solch kompliziertere und realistischere Neuronenmodelle liefern.We study the collective dynamics of class I excitable elements, which can be described within the theory of nonlinear dynamics as systems close to a saddle-node bifurcation on an invariant circle. The focus of the thesis lies on the study of active rotators as a prototype for such elements. In part one of the thesis, we motivate the classic model of repulsively coupled active rotators by Shinomoto and Kuramoto and generalize it by considering higher-order Fourier modes in the on-site dynamics of the rotators. We also discuss the mathematical methods which our work relies on, in particular the concept of Watanabe-Strogatz (WS) integrability which allows to describe systems of identical angular variables in terms of Möbius transformations. In part two, we investigate the existence and stability of periodic two-cluster states for generalized active rotators and prove the existence of a continuum of periodic orbits for a class of WS-integrable systems which includes, in particular, the classic active rotator model. We show that (i) this continuum constitutes a normally attracting invariant manifold and that (ii) one of the solutions yields splay state dynamics. We then develop a perturbation theory for such systems, based on the averaging method. By this approach, we can deduce the asymptotic dynamics of the generalized active rotator model. As a main result, we find that periodic two-cluster states and splay states are robust periodic solutions for systems of identical active rotators. We also investigate a 'transfer of stability' between these solutions by means of so-called broken-symmetry states. In part three, we study ensembles of higher-dimensional class I excitable elements in the form of Morris-Lecar neurons and find the asymptotic dynamics of such systems to be similar to those of active rotators, which suggests that our results from part two yield a suitable qualitative description for more complicated and realistic neural models

Molecular Dynamics and Stochastic Simulations of Surface Diffusion

Despite numerous advances in experimental methodologies capable of addressing the various phenomenon occurring on metal surfaces, atomic scale resolution of the microscopic dynamics remains elusive for most systems. Computational models of the processes may serve as an alternative tool to fill this void. To this end, parallel molecular dynamics simulations of self-diffusion on metal surfaces have been developed and employed to address microscopic details of the system. However these simulations are not without their limitations and prove to be computationally impractical for a variety of chemically relevant systems, particularly for diffusive events occurring in the low temperature regime. To circumvent this difficulty, a corresponding coarse-grained representation of the surface is also developed resulting in a reduction of the required computational effort by several orders of magnitude, and this description becomes all the more advantageous with increasing system size and complexity. This representation provides a convenient framework to address fundamental aspects of diffusion in nonequilibrium environments and an interesting mechanism for directing diffusive motion along the surface is explored. In the ensuing discussion, additional topics including transition state theory in noisy systems and the construction of a checking function for protein structure validation are outlined. For decades the former has served as a cornerstone for estimates of chemical reaction rates. However, in complex environments transition state theory most always provides only an upper bound for the true rate. An alternative approach is described that may alleviate some of the difficulties associated with this problem. Finally, one of the grand challenges facing the computational sciences is to develop methods capable of reconstructing protein structure based solely on readily-available sequence information. Herein a checking function is developed that may prove useful for addressing whether a particular proposed structure is a viable possibility.Ph.D.Committee Chair: Hernandez, Rigoberto; Committee Member: Bredas, Jean-Luc; Committee Member: Ludovice, Peter; Committee Member: Orlando, Thomas; Committee Member: Sherrill, C. Davi

Quantum State Estimation and Tracking for Superconducting Processors Using Machine Learning

Quantum technology has been rapidly growing; in particular, the experiments that have been performed with superconducting qubits and circuit QED have allowed us to explore the light-matter interaction at its most fundamental level. The study of coherent dynamics between two-level systems and resonator modes can provide insight into fundamental aspects of quantum physics, such as how the state of a system evolves while being continuously observed. To study such an evolving quantum system, experimenters need to verify the accuracy of state preparation and control since quantum systems are very fragile and sensitive to environmental disturbance. In this thesis, I look at these continuous monitoring and state estimation problems from a modern point of view. With the help of machine learning techniques, it has become possible to explore regimes that are not accessible with traditional methods: for example, tracking the state of a superconducting transmon qubit continuously with dynamics fast compared with the detector bandwidth. These results open up a new area of quantum state tracking, enabling us to potentially diagnose errors that occur during quantum gates. In addition, I investigate the use of supervised machine learning, in the form of a modified denoising autoencoder, to simultaneously remove experimental noise while encoding one and two-qubit quantum state estimates into a minimum number of nodes within the latent layer of a neural network. I automate the decoding of these latent representations into positive density matrices and compare them to similar estimates obtained via linear inversion and maximum likelihood estimation. Using a superconducting multiqubit chip, I experimentally verify that the neural network estimates the quantum state with greater fidelity than either traditional method. Furthermore, the network can be trained using only product states and still achieve high fidelity for entangled states. This simplification of the training overhead permits the network to aid experimental calibration, such as the diagnosis of multi-qubit crosstalk. As quantum processors increase in size and complexity, I expect automated methods such as those presented in this thesis to become increasingly attractive

On the dynamics of topological solitons

This thesis investigates the dynamics of lump-like objects in non-integrable field theories, whose stability is due to topological considerations. The work concerns three different low dimensional ((1 + 1)- and (2 + l)-dimensional) systems and addresses the questions of how the topology and metric structure of physical space, the quantum mechanics of the basic field quanta and intersoliton interactions affect soliton dynamics. In chapter 2 a sine-Gordon system in discrete space, but with continuous time, is presented. This has some novel features, namely a topological lower bound on the energy of a kink and an explicit static kink which saturates this bound. Kink dynamics in this model is studied using a geodesic approximation which, on comparison with numerical simulations, is found to work well for moderately low kink speeds. At higher speeds the dynamics becomes significantly dissipative, and the approximation fails. Some of the dissipative phenomena observed are explained by means of a dispersion relation for phonons on the spatial lattice. Chapter 3 goes on to quantize the kink sector of this model. A quantum induced potential called the kink Casimir energy is computed numerically in the weak coupling approximation by quantizing the lattice phonons. The effect of this potential on classical kink dynamics is discussed. Chapter 4 presents a study of the low-energy dynamics of a CP(^1) lump on the two-sphere in the geodesic approximation. By considering the isometry group inherited from globalsymmetries of the model, the structure of the induced metric on the unit-charge moduli space is so restricted that the metric can be calculated explicitly. Some totally geodesic submanifolds are found, and the qualitative features of motion on these described. The moduli space is found to be geodesically incomplete. Finally, chapter 5 contains an analysis of long range intervortex forces in the abelian Higgs model, a massive field theory, extending a point source. approximation previously only used in massless theories. The static intervortex potential is rederived from a new viewpoint and used to model type II vortex scattering. Velocity dependent forces are then calculated, providing a model of critical vortex scattering, and leading to a conjecture for the analytic asymptotic form of the metric on the two-vortex moduli space

Algebraic quantum theory

The main objective consists in endowing the elementary particles with an algebraic space-time structure in the perspective of unifying quantum field theory and general relativity: this is realized in the frame of the Langlands global program based on the infinite dimensional representations of algebraic groups over adele rings. In this context, algebraic quanta, strings and fields of particles are introduced.Comment: 137 p