Deep Point Correlation Design

Designing point patterns with desired properties can require substantial effort, both in hand-crafting coding and mathematical derivation. Retaining these properties in multiple dimensions or for a substantial number of points can be challenging and computationally expensive. Tackling those two issues, we suggest to automatically generate scalable point patterns from design goals using deep learning. We phrase pattern generation as a deep composition of weighted distance-based unstructured filters. Deep point pattern design means to optimize over the space of all such compositions according to a user-provided point correlation loss, a small program which measures a pattern’s fidelity in respect to its spatial or spectral statistics, linear or non-linear (e. g., radial) projections, or any arbitrary combination thereof. Our analysis shows that we can emulate a large set of existing patterns (blue, green, step, projective, stair, etc.-noise), generalize them to countless new combinations in a systematic way and leverage existing error estimation formulations to generate novel point patterns for a user-provided class of integrand functions. Our point patterns scale favorably to multiple dimensions and numbers of points: we demonstrate nearly 10 k points in 10-D produced in one second on one GPU. All the resources (source code and the pre-trained networks) can be found at https://sampling.mpi-inf.mpg.de/deepsampling.html

Hollow condensates, topological ladders and quasiperiodic chains

This thesis presents three distinct topics pertaining to the intersection of condensed matter and atomic, molecular and optical (AMO) physics. We theoretically address the physics of hollow Bose-Einstein condensates and the behavior of vortices within them then discuss localization-delocalization physics of one-dimensional quasiperiodic models, and end by focusing on the physics of localized edge modes and topological phases in quasi-one-dimensional ladder models. For all three topics we maintain a focus on experimentally accessible, physically realistic systems and explicitly discuss experimental implementations of our work or its implications for future experiments. First, we study shell-shaped Bose-Einstein condensates (BECs). This work is motivated by experiments aboard the International Space Station (ISS) in the Cold Atom Laboratory (CAL) where hollow condensates are being engineered. Additionally, shell-like structures of superfluids form in interiors of neutron stars and with ultracold bosons in three-dimensional optical lattices. Our work serves as a theoretical parallel to CAL studies and a step towards understanding these more complex systems. We model hollow BECs as confined by a trapping potential that allows for transitions between fully-filled and hollow geometries. Our study is the first to consider such a real-space topological transition. We find that collective mode frequencies of spherically symmetric condensates show non-monotonic features at the hollowing-out point. We further determine that for fully hollow spherically symmetric BECs effects of Earth's gravity are very destructive and consequently focus on microgravity environments. Finally, we study quantized vortices on hollow condensate shells and their response to system rotation. Vortex behavior interesting as a building block for studies of more complicated quantum fluid equilibration processes and physics of rotating neutron stars interiors. Condensate shells' closed and hollow geometry constrains possible vortex configurations. We find that those configurations are stable only for high rotation rates. Further, we determine that vortex lines nucleate at lower rotation rates for hollow condensates than those that are fully-filled. Second, we analyze the effects of quasiperiodicity in one-dimensional systems. Distinct from truly disordered systems, these models exhibit delocalization in contrast to well-known facts about Anderson localization. We study the famous Aubry-Andre-Harper (AAH) model, a one-dimensional tight-binding model that localizes only for sufficiently strong quasiperiodic on-site modulation and is equivalent to the Hofstadter problem at its critical point. Generalizations of the AAH modelhave been studied numerically and a generalized self-dual AAH model has been proposed and analytically analyzed by S. Ganeshan, J. Pixley and S. Das Sarma (GPD). For extended and generalized AAH models the appearance of a mobility edge i.e. an energy cut-off dictating which wavefunctions undergo the localization-delocalization transition is expected. For the GPD model this critical energy has been theoretically determined. We employ transfer matrices to study one-dimensional quasiperiodic systems. Transfer matrices characterize localization physics through Lyapunov exponents. The symplectic nature of transfer matrices allows us to represent them as points on a torus. We then obtain information about wavefunctions of the system by studying toroidal curves corresponding to transfer matrix products. Toroidal curves for localized, delocalized and critical wavefunctions are distinct, demonstrating a geometrical characterization of localization physics. Applying the transfer matrix method to AAH-like models, we formulate a geometrical picture that captures the emergence of the mobility edge. Additionally, we connect with experimental findings concerning a realization of the GPD model in an interacting ultracold atomic system. Third, we consider a generalization of the Su-Schrieffer-Heeger (SSH) model. The SSH chain is a one-dimensional tight-binding model that can host localized bound states at its ends. It is celebrated as the simplest model having topological properties captured by invariants calculated from its band-structure. We study two coupled SSH chains i.e. the SSH ladder. The SSH ladder has a complex phase diagram determined by inter-chain and intra-chain couplings. We find three distinct phases: a topological phase hosting localized zero energy modes, a topologically trivial phase having no edge modes and a phase akin to a weak topological insulator where edge modes are not robust. The topological phase of the SSH ladder is analogous to the Kitaev chain, which is known to support localized Majorana fermion end modes. Bound states of the SSH ladder having the same spatial wavefunction profiles as these Majorana end modes are Dirac fermions or bosons. The SSH ladder is consequently more suited for experimental observation than the Kitaev chain. For quasiperiodic variations of the inter-chain coupling, the SSH ladder topological phase diagram reproduces Hofstadter's butterfly pattern. This system is thus a candidate for experimental observation of the famous fractal. We discuss one possible experimental setup for realizing the SSH ladder in its Kitaev chain-like phase in a mechanical meta-material system. This approach could also be used to experimentally study the Hofstadter butterfly in the future. Presented together, these three topics illustrate the richness of the intersection of condensed matter and AMO physics and the many exciting prospects of theoretical work in the realm of the former combining with experimental advances within the latter

Geometry-independent tight-binding method for massless Dirac fermions in two dimensions

The Nielsen-Ninomiya theorem, dubbed `fermion-doubling', poses a problem for the naive discretization of a single (massless) Dirac cone on a two-dimensional surface. The inevitable appearance of an additional, unphysical fermionic mode can, for example, be circumvented by introducing an extra dimension to spatially separate Dirac cones. In this work, we propose a geometry-independent protocol based on a tight-binding model for a three-dimensional topological insulator on a cubic lattice. The low-energy theory, below the bulk gap, corresponds to a Dirac cone on its two-dimensional surface which can have an arbitrary geometry. We introduce a method where only a thin shell of the topological insulator needs to be simulated. Depending on the setup, we propose to gap out the states on the undesired surfaces either by breaking the time-reversal symmetry or by introducing a superconducting pairing. We show that it is enough to have a thickness of the topological-insulator shell of three to nine lattice constants. This leads to an effectively two-dimensional scaling with minimal and fixed shell thickness. We test the idea by comparing the spectrum and probability distribution to analytical results for both a proximitized Dirac mode and a Dirac mode on a sphere which exhibits a nontrivial spin-connection. The protocol yields a tight-binding model on a cubic lattice simulating Dirac cones on arbitrary surfaces with only a small overhead due to the finite thickness of the shell.Comment: 10 pages, 7 figure

Bayesian model based spatiotemporal survey designs and partially observed log Gaussian Cox process

In geostatistics, the spatiotemporal design for data collection is central for accurate prediction and parameter inference. An important class of geostatistical models is log-Gaussian Cox process (LGCP) but there are no formal analyses on spatial or spatiotemporal survey designs for them. In this work, we study traditional balanced and uniform random designs in situations where analyst has prior information on intensity function of LGCP and show that the traditional balanced and random designs are not efficient in such situations. We also propose a new design sampling method, a rejection sampling design, which extends the traditional balanced and random designs by directing survey sites to locations that are a priori expected to provide most information. We compare our proposal to the traditional balanced and uniform random designs using the expected average predictive variance (APV) loss and the expected Kullback-Leibler (KL) divergence between the prior and the posterior for the LGCP intensity function in simulation experiments and in a real world case study. The APV informs about expected accuracy of a survey design in point-wise predictions and the KL-divergence measures the expected gain in information about the joint distribution of the intensity field. The case study concerns planning a survey design for analyzing larval areas of two commercially important fish stocks on Finnish coastal region. Our experiments show that the designs generated by the proposed rejection sampling method clearly outperform the traditional balanced and uniform random survey designs. Moreover, the method is easily applicable to other models in general. (C) 2019 The Author(s). Published by Elsevier B.V.Peer reviewe

Subfactors and Applications

The theory of subfactors connects diverse topics in mathematics and mathematical physics such as tensor categories, vertex operator algebras, quantum groups, quantum topology, free probability, quantum field theory, conformal field theory, statistical mechanics, condensed matter physics and, of course, operator algebras. We invited an international group of researchers from these areas and many fruitful interactions took place during the workshop

Topological Phases of Matter: Classification, Realization and Application.

The recent discovery of topological insulators has led to a tremendous interest in the exploration of topological phases of matter which do not fit into Landau's symmetry breaking paradigm. Numerous exotic topological materials are theoretically predicted. Some of them have been experimentally reported, but many remain not. In this thesis, we explore topological phases of matter from three aspects: their classification, realization and application. We first review some basic classification theories, which provide us a "big picture" and lay the foundation for the rest of the thesis. We then move on to propose a systematic method based on quaternion algebra to construct toy tight-binding Hamiltonians for all the exotic phases in a recently developed periodic table for topological insulators and superconductors. We also introduce two peculiar families of topological phases that are beyond the table---the Hopf and four-dimensional topological insulators without time reversal symmetry. Prototypical Hamiltonians are constructed and their topological properties, such as robust edge states, are numerically studied. Motivated by rapid experimental progress in engineering spin-orbit coupling and artificial gauge field for cold atoms, we continue the thesis by proposing a feasible experimental scheme to realize a three-dimensional chiral topological insulator with cold fermionic atoms in an optical lattice. To unambiguously probe topological phases, we also bring forth systematic and generic methods to measure the characteristic topological invariants, for both free and strongly interacting systems. Moreover, we demonstrate that a kaleidoscope of knot and link structures is encoded in the spin texture of Hopf insulators and show how to observe different knots and links in cold atoms via time-of-flight images. The last part of the thesis is about the application of topological materials. After a demonstration of how to create, braid and detect Majorana fermions with cold atoms, we put forward a proposal to construct a self-test quantum random number generator by using Majorana fermions. Majorana random number generators are able to generate certifiable true random numbers with unconditional security. They offer a new perspective to the utilization of topological materials and may have vital applications in cryptography and related areas.PhDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/111397/1/dldeng_1.pd

Tailoring Plasmon-Enhanced Light-Matter Interaction

Plasmons are the collective oscillation of free electrons in materials. They concentrate light into nanoscale volumes and trigger optical processes in nearby materials. My thesis is devoted to the understanding of optical processes that are mediated by localized surface plasmons. The fundamental excitation of plasmonic modes and the enhancement of optical absorption and Raman scattering in nanoscale materials are studied using experimental and theoretical approaches. I introduce a novel type of plasmonic excitation in layered films of metallic nanoparticles. Because of field retardation, incident light induces antiparallel dipoles in adjacent layers of metallic nanoparticles exciting a dark interlayer plasmon. It benefits from reduced radiative damping and efficient light absorption as I demonstrate with simulations and experiments. The self-assembled nanoparticle films pave the way for large-area coatings with tunable plasmon resonances. An application is the decay of plasmons into hot charge carriers that trigger photocatalytic reactions in molecules. I propose dark interlayer plasmons as ideal excitation channels for hot electrons because of their small radiative damping. Using plasmonic nanostructures for photodetection and sensing requires an understanding of the interaction with adjacent materials. I introduce microscopic theories for the enhancement of optical absorption and Raman scattering by localized surface plasmons. The plasmonic near field of nanoparticle arrays induced non-vertical optical transitions in graphene in dependence of the periodicity of the plasmonic lattice. For plasmon-enhanced Raman scattering I developed a general theoretical framework using perturbation theory. It provides analytic expressions for the enhanced Raman cross section. In a molecular dipole coupled to a plasmonic nanoparticle the enhancement is strongly affected by interference between different scattering channels. Plasmon-enhanced Raman scattering is an ideal tool to study the properties of materials interfaced with plasmonic nanostructures. I analyzed nanoscale strain and doping in graphene on top of a gold nanostructure. I developed a method for separating the contributions from strain and doping in the Raman spectrum of graphene, which is applicable to graphene on arbitrary substrates and in arbitrary strain configurations.Ziel dieser Arbeit ist es, ein besseres Verständnis von optischen Prozessen zu erlangen, die durch lokalisierte Oberflächenplasmonen gesteuert werden. Dafür habe ich grundlegende Anregungsmechanismen von Plasmonmoden, sowie die Verstärkung von optischer Absorption und Ramanstreuung, mit experimentellen und theoretischen Methoden untersucht. Ich stelle eine neuartige plasmonische Anregung in geschichteten Filmen von metallischen Nanopartikeln vor. Dieses dunkle Plasmon besteht aus antiparallelen plasmonischen Dipolen in den Nanopartikeln benachbarter Lagen und kann aufgrund von Feldretardierung direkt mit Licht angeregt werden. Ich zeige mit Experimenten und Simulationen, dass diese Anregung eine reduzierte Strahlungsdämpfung aufweist und zu einer ausgeprägten, durchstimmbaren Lichtabsorption im nahinfraroten Spektralbereich führt. Da die Nanopartikelfilme mittels Selbstorganisation von Nanopartikeln hergestellt werden können, eignen sie sich für die großflächige Beschichtung von Oberflächen. Aufgrund der unterdrückten radiativen Dämpfung sind dunkle Plasmonen in Nanopartikelfilmen ein idealer Anregungskanal für heiße Elektronen, mit Anwendungen in der Fotokatalyse. Mit mikroskopischen Theorien habe ich die Interaktion von plasmonischen Nanostrukturen mit angrenzenden Nanomaterialien untersucht. Ich zeige, dass das plasmonische Nahfeld eines Gitters von Goldnanopartikeln nicht-vertikale optische Übergänge in Graphen anregt. Die Auswahlregeln für diese Übergänge hängen von der Periodizität der plasmonischen Nanostruktur ab. Die mikroskopische Theorie führt zu einem besseren Verständnis der Photostromentstehung in Graphen-basierten optoelektronischen Detektoren. Als Zweites stelle ich ein allgemeines Konzept zur Beschreibung von plasmon-verstärkter Ramanstreuung mit Störungstheorie vor. Die analytischen Ausdrücke aus dieser Theorie eignen sich, um die Abhängigkeit der plasmonischen Verstärkung von der Anregungsenergie zu untersuchen. Mittels einer Implementierung für ein Molekül nahe eines plasmonischen Nanopartikels zeige ich, dass die Verstärkung stark von der Interferenz verschiedener Streuprozesse beeinflusst wird. Plasmon-verstärkte Ramanstreuung ist ideal, um zu untersuchen, wie Materialeigenschaften von einer angrenzenden plasmonischen Nanostruktur beeinflusst werden. Das zeige ich für Materialverspannungen und Dotierung in Graphen durch eine Gold-Nanostruktur. Ich habe dafür eine allgemeine Methodik entwickelt, mit der die Beiträge von Verspannung und Dotierung zum Ramanspektrum von Graphen voneinander getrennt und quantifiziert werden können. Diese eignet sich zur Auswertung von unbekannten Verspannungs-Konfigurationen in Graphen auf verschiedensten Substraten

Basics and Applications in Quantum Optics

Quantum optics has received a lot of attention in recent decades due to the handiness and versatility of optical systems, which have been exploited both to study the foundations of quantum mechanics and for various applications. In this Special Issue, we collect some articles and a review focusing on some research activities that show the potential of quantum optics in the advancement of quantum technologies