Topological Photonics

Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.Comment: 87 pages, 30 figures, published versio

Traveling wave modeling of nonlinear dynamics in multisection semiconductor lasers

A hierarchy of 1 (time) + 1 (space) dimensional first-order partial differential equation (traveling wave) models is used for a description of dynamics in individual semiconductor lasers, various multisection semiconductor lasers, and coupled laser systems. Consequent modifications of the basic traveling wave model allow for taking into account different physical effects such as the gain dispersion, the thermal detuning, the spatial hole burning of carriers, the nonlinear gain saturation, or various carrier exchange processes in quantum dot lasers. For illustration, the model was applied for simulations of dynamics in complex ring laser with four branches of filtered feedback. Finally, several advanced techniques for model analysis such as calculation of instantaneous optical modes, finding of steady states, and numerical continuation and bifurcation analysis of the model equations were discussed and illustrated by example simulations

A preliminary investigation into the effects of nonlinear response modification within coupled oscillators

This thesis provides an account of an investigation into possible dynamic interactions between two coupled nonlinear sub-systems, each possessing opposing nonlinear overhang characteristics in the frequency domain in terms of positive and negative cubic stiffnesses. This system is a two degree-of-freedom Duffing oscillator coupled in series in which certain nonlinear effects can be advantageously neutralised under specific conditions. This theoretical vehicle has been used as a preliminary methodology for understanding the interactive behaviour within typical industrial ultrasonic cutting components. Ultrasonic energy is generated within a piezoelectric exciter, which is inherently nonlinear, and which is coupled to a bar-horn or block-horn to one, or more, material cutting blades, for example. The horn/blade configurations are also nonlinear, and within the whole system there are response features which are strongly reminiscent of positive and negative cubic stiffness effects. The two degree-of-freedom model is analysed and it is shown that a practically useful mitigating effect on the overall nonlinear response of the system can be created under certain conditions when one of the cubic stiffnesses is varied. It has also bfeen shown experimentally that coupling of ultrasonic components with different nonlinear characteristics can strongly influence the performance of the system and that the general behaviour of the hypothetical theoretical model is indeed borne out in practice

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”

Periodically driven many-body quantum systems : Quantum Ratchets, Topological States and the Floquet-Boltzmann Equation

Controlling and manipulating complex many-body quantum systems will be a key ingredient for the development of next-generation technologies. While the realisation of a universal quantum machine is still out of reach, in recent years experimental systems of ultracold atoms have already evolved into a vivid field of research for quantum simulation. Crucially, such systems even allow for the successful quantum engineering of targeted many-body systems by means of coherent periodic driving. The essential properties of these Floquet systems encompass two main aspects: fast driving facilitates the simulation of effective static systems, and interactions lead to unique heating effects as energy is only conserved modulo the driving frequency. Within this thesis we theoretically study both of these aspects in respective model systems. In part I of this thesis, we investigate the dynamics of excitations of a bosonic Mott insulator in a designed one-dimensional Floquet system. Here, periodic driving in combination with breaking all mirror symmetries of the system can induce directed motion of particles. In the limit of small excitation densities, the effectively non-interacting quantum ratchet determines the motion of holes and doublons in the Mott insulator and can in fact be used to manipulate the dynamics of such. This little quantum machine can also be used to drive particles against an external force, where transport is possible but requires the fulfilment of a commensurability condition for long times. In part II, we discuss the role of interactions for periodically driven systems by means of a Floquet version of the Boltzmann equation. Starting from the Keldysh approach, we develop this semiclassical formalism based on a clear separation of time scales. The result is a description of the dynamics and the scattering of Floquet quasiparticles in such systems. Here, the property of discrete energy violation is naturally encoded in our formalism predicting the heating of interacting Floquet systems to infinite temperatures in the long-time limit. As a first application of this approach, we investigate a cold atom setup realising the Haldane model by means of periodic shaking. While homogeneous systems heat up globally, a confining potential evokes thermoelectric transport effects resulting from spatially dependent heating characteristics. Moreover, we show that the interplay of intrinsic heating, macroscopic diffusion and non-trivial topological properties of the Haldane model lead to an anomalous Floquet-Nernst effect, which describes anomalous particle transport as the result of developing temperature gradients. In part III, we elaborate on the quantum simulator aspect of ultracold atoms by providing a theoretical framework for a possible simulation of a topological edge state in a one-dimensional optical lattice. In this case, the one-dimensional Dirac equation with spatially varying mass is important, which captures the topological properties of a corresponding system of the BDI symmetry class. We analytically discuss such system and investigate the role of mean-field interaction effects. We also identify the emergence of dynamical instabilities in a realisation with bosonic atoms

GAUGE FIELDS AND GEOMETRIC PHASES IN PERIODIC SYSTEMS

Ph.DDOCTOR OF PHILOSOPH

Nonlinear Dynamics

This volume covers a diverse collection of topics dealing with some of the fundamental concepts and applications embodied in the study of nonlinear dynamics. Each of the 15 chapters contained in this compendium generally fit into one of five topical areas: physics applications, nonlinear oscillators, electrical and mechanical systems, biological and behavioral applications or random processes. The authors of these chapters have contributed a stimulating cross section of new results, which provide a fertile spectrum of ideas that will inspire both seasoned researches and students