Asymptotically Efficient Quasi-Newton Type Identification with Quantized Observations Under Bounded Persistent Excitations

This paper is concerned with the optimal identification problem of dynamical systems in which only quantized output observations are available under the assumption of fixed thresholds and bounded persistent excitations. Based on a time-varying projection, a weighted Quasi-Newton type projection (WQNP) algorithm is proposed. With some mild conditions on the weight coefficients, the algorithm is proved to be mean square and almost surely convergent, and the convergence rate can be the reciprocal of the number of observations, which is the same order as the optimal estimate under accurate measurements. Furthermore, inspired by the structure of the Cramer-Rao lower bound, an information-based identification (IBID) algorithm is constructed with adaptive design about weight coefficients of the WQNP algorithm, where the weight coefficients are related to the parameter estimates which leads to the essential difficulty of algorithm analysis. Beyond the convergence properties, this paper demonstrates that the IBID algorithm tends asymptotically to the Cramer-Rao lower bound, and hence is asymptotically efficient. Numerical examples are simulated to show the effectiveness of the information-based identification algorithm.Comment: 16 pages, 3 figures, submitted to Automatic

Topological data analysis and geometry in quantum field dynamics

Many non-perturbative phenomena in quantum field theories are driven or accompanied by non-local excitations, whose dynamical effects can be intricate but difficult to study. Amongst others, this includes diverse phases of matter, anomalous chiral behavior, and non-equilibrium phenomena such as non-thermal fixed points and thermalization. Topological data analysis can provide non-local order parameters sensitive to numerous such collective effects, giving access to the topology of a hierarchy of complexes constructed from given data. This dissertation contributes to the study of topological data analysis and geometry in quantum field dynamics. A first part is devoted to far-from-equilibrium time evolutions and the thermalization of quantum many-body systems. We discuss the observation of dynamical condensation and thermalization of an easy-plane ferromagnet in a spinor Bose gas, which goes along with the build-up of long-range order and superfluidity. In real-time simulations of an over-occupied gluonic plasma we show that observables based on persistent homology provide versatile probes for universal dynamics off equilibrium. Related mathematical effects such as a packing relation between the occurring persistent homology scaling exponents are proven in a probabilistic setting. In a second part, non-Abelian features of gauge theories are studied via topological data analysis and geometry. The structure of confining and deconfining phases in non-Abelian lattice gauge theory is investigated using persistent homology, which allows for a comprehensive picture of confinement. More fundamentally, four-dimensional space-time geometries are considered within real projective geometry, to which canonical quantum field theory constructions can be extended. This leads to a derivation of much of the particle content of the Standard Model. The works discussed in this dissertation provide a step towards a geometric understanding of non-perturbative phenomena in quantum field theories, and showcase the promising versatility of topological data analysis for statistical and quantum physics studies

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

Quantum Transport in Semiconductor Nanostructures

I. Introduction (Preface, Nanostructures in Si Inversion Layers, Nanostructures in GaAs-AlGaAs Heterostructures, Basic Properties). II. Diffusive and Quasi-Ballistic Transport (Classical Size Effects, Weak Localization, Conductance Fluctuations, Aharonov-Bohm Effect, Electron-Electron Interactions, Quantum Size Effects, Periodic Potential). III. Ballistic Transport (Conduction as a Transmission Problem, Quantum Point Contacts, Coherent Electron Focusing, Collimation, Junction Scattering, Tunneling). IV. Adiabatic Transport (Edge Channels and the Quantum Hall Effect, Selective Population and Detection of Edge Channels, Fractional Quantum Hall Effect, Aharonov-Bohm Effect in Strong Magnetic Fields, Magnetically Induced Band Structure).Comment: 111 pages including 109 figures; this review from 1991 has retained much of its usefulness, but it was not yet available electronicall

The Study of Exciton-Polariton Phase Transitions Through Spontaneous Vortices and First-Order Correlation

Phase transitions are among the most fascinating phenomena discovered in the last century, with collective molecular dynamics leading to novel observations such as superconductivity and superfluidity. Yet even in the highly-ordered state of such systems, defects could still form and persist. Investigating the behavior of such defects is not just a matter of fundamental interest, but also valuable in applications where macroscopic order is important. Our studies of order and defects make use of an exciton-polariton (polaritons hereafter) system. These are quasi-particles with part-matter and part-light components, with a bosonic nature that allows for observation of macroscopic quantum phenomena typically seen in conventional atomic condensates. Their low effective mass allows for such observations at relatively high temperatures (10K - 300K), and the photonic component allows for experimental access using standard table-top optical techniques. These factors have generated much interest in the field of polaritonics over the past two decades. In a two-dimensional system like the polaritons, defects appear in phase transitions as spontaneously-formed vortices. While such vortices have been observed, their reported behaviors were influenced by sample properties or pump configurations. Formation and behaviors in which phase-transition mechanisms and polariton hydrodynamics are primary contributing factors have yet to be observed. Such observations would allow for comparisons with analogous results in atomic condensates and allow for deeper studies in universality. In this thesis, I present efforts taken towards the realization of such observations. I will propose the design of a Compact Mirroring Mach-Zehnder interferometer (CoMMZI) for the detection of photonic orbital angular momentum (OAM) states. The observation of such states would be an unequivocal indication of spontaneously-formed quantum vortices. I will demonstrate that the proposed interferometer is capable of detecting OAM states with a low number of photons, thereby making it suitable for the detection of moving vortices in a single-shot realization of a polariton condensate. I will then discuss how the interferometer was tested with vortex states formed with a spatial light modulator and continuous-wave lasers. I will show that the interferometer is capable of detecting vortex phases and present techniques to increase the chances of successful detection. I will also show preliminary experiments with a polariton condensate formed with non-resonant pump configurations. OAM states within an optically-induced ring trap have been detected. Finally, I will show spectrometric and temporal first-order correlation functions for polaritons within an optically-induced ring trap, a potential system for the observation of vortices. I will show the presence of three population fractions with coherence times spanning three orders of magnitude and briefly discuss possible implications for vortex-detection experiments. The efforts in this thesis demonstrate the possibilities and challenges associated with the detection of spontaneously-formed vortices within a single instance of a polariton condensate, providing insight for attempts at experimental realizations.PHDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163142/1/lhofai_1.pd

Investigating generalized Kitaev magnets using machine learning

Frustration in Kitaev-Materialien führt zu einem sehr reichhaltigen und komplexen Phasendiagramm, einschließlich der klassischen Spinflüssigkeitsphase. Die Suche nach und das Verständnis von Spinflüssigkeiten und weiteren neuartigen komplexen Phasen der Materie stehen im Mittelpunkt der heutigen Forschung zu kondensierter Materie. Mittels analytischer Methoden Ordnungsparameter zur Charakterisierung dieser Phasen zu finden, ist nahezu unmöglich. Bei niedrigen Temperaturen ordnen sich die meisten klassischen Spinsysteme in komplizierte Strukturen, die große magnetische Elementarzellen belegen, was die Komplexität des Problems noch weiter erhöht und außerhalb des Anwendungsbereichs der meisten herkömmlichen Methoden liegt. In dieser Arbeit untersuchen wir die Hamilton-Operatoren realitätsnaher Kitaev-Materialien mithilfe maschinellen Lernens. Hauptmerkmale des zugrundeliegenden Algorithmus sind unbeaufsichtigtes Lernen, welches ermöglicht die Topologie eines Phasendiagramms ohne jegliche Vorkenntnisse erforschen, und Interpretierbarkeit, welche zur Analyse der Struktur der klassischen Grundzustände notwendig ist. In den ersten drei Kapiteln werden wir den Algorithmus des maschinellen Lernens auf verschiedene Hamilton-Operatoren anwenden, die zur Modellierung von Kitaev-Materialien eingesetzt werden, um zu untersuchen inwieweit die Quantenmodelle und die experimentellen Beobachtungen allein durch deren klassischen Grenzfall erklärt werden können. Darüber hinaus erforschen wir weitere Features dieses Algorithmus, die es uns ermöglichen, verborgene Symmetrien, lokale Einschränkungen der klassischen Spinflüssigkeiten, sowie bisher unbekannte Phasen im hochdimensionalen Phasenraum aufzudecken. In den letzten beiden Kapiteln werden wir uns mit dem Verständnis der Struktur der klassischen Grundzustände befassen, welche durch die Verflechtung mehrerer Helices charakterisiert sind. Wir werden auch versuchen, die Signatur dieser Phasen in Experimenten zu verstehen, indem wir die Dynamik und den Transport durch Kitaev-Magnete untersuchen. Diese Arbeit beweist die Tauglichkeit von maschinellem Lernen, hochkomplexe Phasendiagramme mit wenig bis gar keinem Vorwissen aufzudecken und hochfrustrierten Magnetismus zu erforschen. Die Kombination aus maschinellem und menschlichem Einsatz ebnet den Weg zu neuen und spannenden physikalischen Erkenntnissen.Bond frustration in Kitaev materials leads to a very rich phase diagram with highly intricate phases including the classical spin liquid phase. The search and understanding of spin liquids and novel complex phases of matter is at the heart pf present day condensed matter research. To search and design order parameters to characterize these phases using analytical approaches is a nearly impossible task. At low temperatures, most of the classical spins order into complicated spin structures occupying large magnetic unit cells which further adds to the complication and is out of the realm of most traditional methods. In this thesis we investigate realistic Kitaev material Hamiltonians using a machine learning framework whose key features, of unsupervised learning which helps us study the topology of the phase diagram without prior knowledge and interpretability which helps us analyse the structure of the classical ground states, are exploited. In the first three chapters, we shall use this framework on different Hamiltonians used to model Kitaev materials and understand to what extent the quantum limit and experimental results could be explained just by the classical limit of these models. We in addition explore other features of this framework which lets us uncover hidden symmetries as well as local constraints for the classical spin liquids and hitherto unreported new phases in the high dimensional phase space. In the last two chapters we shall dwell on the understanding the structure of the classical ground states which is quite complicated as it hosts a tangle of multiple helices. We shall also try and understand the signature of these phases on experiments by studying the dynamics and transport through Kitaev magnets thus bridging the gap between experiment and theory. This thesis proves instances of using machine learning to uncover highly complex phase diagrams with little to no previous knowledge and serve as a paradigm to explore highly frustrated magnetism. Through a combination of machine and human effort we are on the way to uncover new and exciting physics

Topological Signatures and Quenches in One Dimensional Fermionic Systems

L'abstract è presente nell'allegato / the abstract is in the attachmen