Degenerate mixtures of rubidium and ytterbium for engineering open quantum systems

In the last two decades, experimental progress in controlling cold atoms and ions now allows us to manipulate fragile quantum systems with an unprecedented degree of precision. This has been made possible by the ability to isolate small ensembles of atoms and ions from noisy environments, creating truly closed quantum systems which decouple from dissipative channels. However in recent years, several proposals have considered the possibility of harnessing dissipation in open systems, not only to cool degenerate gases to currently unattainable temperatures, but also to engineer a variety of interesting many-body states. This thesis will describe progress made towards building a degenerate gas apparatus that will soon be capable of realizing these proposals. An ultracold gas of ytterbium atoms, trapped by a species-selective lattice will be immersed into a Bose-Einstein condensate (BEC) of rubidium atoms which will act as a bath. Here we describe the challenges encountered in making a degenerate mixture of rubidium and ytterbium atoms and present two experiments performed on the path to creating a controllable open quantum system. The first experiment will describe the measurement of a tune-out wavelength where the light shift of $\Rb{87}$ vanishes. This wavelength was used to create a species-selective trap for ytterbium atoms. Furthermore, the measurement of this wavelength allowed us to extract the dipole matrix element of the $5s \rightarrow 6p$ transition in $\Rb{87}$ with an extraordinary degree of precision. Our method to extract matrix elements has found use in atomic clocks where precise knowledge of transition strengths is necessary to account for minute blackbody radiation shifts. The second experiment will present the first realization of a degenerate Bose-Fermi mixture of rubidium and ytterbium atoms. Using a three-color optical dipole trap (ODT), we were able to create a highly-tunable, species-selective potential for rubidium and ytterbium atoms which allowed us to use $\Rb{87}$ to sympathetically cool $\Yb{171}$ to degeneracy with minimal loss. This mixture is the first milestone creating the lattice-bath system and will soon be used to implement novel cooling schemes and explore the rich physics of dissipation.

Coupling of magnetic, strain and electric polarization fields in the structure of multiferroic material

The comprehensive study on the coupling of magnetism, electrical polarization and the crystalline lattice with the off-stoichiometric effects in self-doped multiferroic hexagonal h-LuMnxO3±δ (0.92≤x≤1.12) ceramic oxides was carried out for the PhD work. There is a complex coupling of the three ferroic degrees. The cancelation of the magnetic moments of ions in the antiferromagnetic order, electric polarization with specific vortex/antivortex topology and lattice properties have pushed researchers to find out ways to disclose the underlying physics and chemistry of magneto-electric and magneto-elastic couplings of h-RMnO3 multiferroic materials. In this research work, self-doping of Lu-sites or Mn-sites of h-LuMnxO3±δ ceramics prepared via solid state route was done to pave a way for deeper understanding of the antiferromagnetic transition, the weak ferromagnetism often reported in the same crystalline lattices and the ferroelectric properties coupled to the imposed lattice changes. Accordingly to the aim of the PhD thesis, the objectives set for the sintering study in the first chapter on experimental results were two. First, study of sintering off-stoichiometric samples within conditions reported in the bibliography and also extracted from the phase diagrams of the LuMnxO3±δ, with a multiple firings ending with a last high temperature step at 1300ºC for 24 hours. Second, explore longer annealing times of up to 240 hours at the fixed temperature of 1300 ºC in a search for improving the properties of the solid solution under study. All series of LuMnxO3±δ ceramics for each annealing time were characterized to tentatively build a framework enabling comparison of measured properties with results of others available in literature. XRD and Rietveld refinement of data give the evolution the lattice parameters as a function to x. Shrinkage of the lattice parameters with increasing x values was observed, the stability limit of the solid solution being determined by analysis of lattice parameters. The evolution of grain size and presence of secondary phases have been investigated by means of TEM, SEM, EDS and EBSD techniques. The dependencies of grain growth and regression of secondary phases on composition x and time were further characterized. Magnetic susceptibility of samples and magnetic irreversibility were extensively examined in the present work. The dependency of magnetic susceptibility, Neel ordering transition and important magnetic parameters are determined and compared to observation in other multiferroics in the following chapter of the thesis. As a tool of high sensitivity to detect minor traces of the secondary phase hausmannite, magnetic measurements are suggested for cross-checking of phase diagrams. Difficulty of previous studies on interpreting the magnetic anomaly below 43 K in h-RMnO3 oxides was discussed and assigned to the Mn3O4 phase, with supported of the electron microscopy. Magneto-electric coupling where AFM ordering is coupled to dielectric polarization is investigated as a function of x and of sintering condition via frequency and temperature dependent complex dielectric constant measurements in the final chapter of the thesis. Within the limits of solid solubility, the crystalline lattice of off-stoichiometric ceramics was shown to preserve the magneto-electric coupling at TN. It represents the first research work on magneto-electric coupling modified by vacancy doping to author’s knowledge. Studied lattices would reveal distortions at the atomic scale imposed by local changes of x dependent on sintering conditions which were widely inspected by using TEM/STEM methods, complemented with EDS and EELS spectroscopy all together to provide comprehensive information on cross coupling of distortions, inhomogeneity and electronic structure assembled and discussed in a specific chapter. Internal interfaces inside crystalline grains were examined. Qualitative explanations of the measured magnetic and ferroelectric properties were established in relation to observed nanoscale features of h-LuMnxO3±δ ceramics. Ferroelectric domains and topological defects are displayed both in TEM and AFM/PFM images, the later technique being used to look at size, distribution and switching of ferroelectric domains influenced by vacancy doping at the micron scale bridging to complementary TEM studies on the atomic structure of ferroelectric domains. In support to experimental study, DFT simulations using Wien2K code have been carried out in order to interpret the results of EELS spectra of O K-edge and to obtain information on the cation hybridization to oxygen ions. The L3,2 edges of Mn is used to access the oxidation state of the Mn ions inside crystalline grains. In addition, rehybridization driven ferroelectricity is also evaluated by comparing the partial density of states of the orbitals of all ions of the samples, also the polarization was calculated and correlated to the off-stoichiometric effect.

Decoherence models for discrete-time quantum walks and their application to neutral atom experiments

We discuss decoherence in discrete-time quantum walks in terms of a phenomenological model that distinguishes spin and spatial decoherence. We identify the dominating mechanisms that affect quantum-walk experiments realized with neutral atoms walking in an optical lattice. From the measured spatial distributions, we determine with good precision the amount of decoherence per step, which provides a quantitative indication of the quality of our quantum walks. In particular, we find that spin decoherence is the main mechanism responsible for the loss of coherence in our experiment. We also find that the sole observation of ballistic-instead of diffusive-expansion in position space is not a good indicator of the range of coherent delocalization. We provide further physical insight by distinguishing the effects of short- and long-time spin dephasing mechanisms. We introduce the concept of coherence length in the discrete-time quantum walk, which quantifies the range of spatial coherences. Unexpectedly, we find that quasi-stationary dephasing does not modify the local properties of the quantum walk, but instead affects spatial coherences. For a visual representation of decoherence phenomena in phase space, we have developed a formalism based on a discrete analogue of the Wigner function. We show that the effects of spin and spatial decoherence differ dramatically in momentum space.

Alternating surgeries

This thesis is concerned with the question of when the double branched cover of an alternating knot can arise by Dehn surgery on a knot in S^3. We approach this problem using a surgery obstruction, first developed by Greene, which combines Donaldson's Diagonalization Theorem with the $d$-invariants of Ozsvath and Szabo's Heegaard Floer homology. This obstruction shows that if the double branched cover of an alternating knot or link L arises by surgery on S^3, then for any alternating diagram the lattice associated to the Goeritz matrix takes the form of a changemaker lattice. By analyzing the structure of changemaker lattices, we show that the double branched cover of L arises by non-integer surgery on S^3 if and only if L has an alternating diagram which can be obtained by rational tangle replacement on an almost-alternating diagram of the unknot. When one considers half-integer surgery the resulting tangle replacement is simply a crossing change. This allows us to show that an alternating knot has unknotting number one if and only if it has an unknotting crossing in every alternating diagram. These techniques also produce several other interesting results: they have applications to characterizing slopes of torus knots; they produce a new proof for a theorem of Tsukamoto on the structure of almost-alternating diagrams of the unknot; and they provide several bounds on surgeries producing the double branched covers of alternating knots which are direct generalizations of results previously known for lens space surgeries. Here, a rational number p/q is said to be characterizing slope for K in S^3 if the oriented homeomorphism type of the manifold obtained by p/q-surgery on K determines K uniquely. The thesis begins with an exposition of the changemaker surgery obstruction, giving an amalgamation of results due to Gibbons, Greene and the author. It then gives background material on alternating knots and changemaker lattices. The latter part of the thesis is then taken up with the applications of this theory.

Evolution of mylonitic microfabrics (EMM), a computer application for educational purposes

`Evolution of mylonitic microfabrics' (EMM) is an interactive Filemaker Pro 3.0 application that documents a series of see-through deformation experiments on polycrystalline norcamphor. The application comprises computer animations, graphics and text explanations designed to give students and researchers insight into the interaction and dynamic nature of small-scale, mylonitic processes like intracrystalline glide, dynamic recrystallization and strain localization (microshearing). EMM shows how mylonitic steady state is achieved at different strain rates and temperatures. First, rotational mechanisms like glide-induced vorticity, subgrain rotation recrystallization and rigid-body rotation bring grains' crystal lattices into orientations that are favorable for intracrystalline glide. In a second stage, selective elimination of grains whose lattices are poorly oriented for glide involves grain boundary migration. This strengthens the texture. Temperature and strain rate affect both the relative activity of different strain accommodation mechanisms and the rate of microfabric change. Steady-state microfabrics are characterized by stable texture, grain size and shape-preferred orientations of grains and domains. This involves the cyclical generation and elimination of dynamically recrystallized grains and microshear zones.

Superlattice structures in twisted bilayers of folded graphene

The electronic properties of bilayer graphene strongly depend on relative orientation of the two atomic lattices. Whereas Bernal-stacked graphene is most commonly studied, a rotational mismatch between layers opens up a whole new field of rich physics, especially at small interlayer twist. Here we report on magnetotransport measurements on twisted graphene bilayers, prepared by folding of single layers. These reveal a strong dependence on the twist angle, which can be estimated by means of sample geometry. At small rotation, superlattices with a wavelength in the order of 10 nm arise and are observed by friction atomic force microscopy. Magnetotransport measurements in this small-angle regime show the formation of satellite Landau fans. These are attributed to additional Dirac singularities in the band structure and discussed with respect to the wide range of interlayer coupling models.

Solid-liquid interactions in microscale structures and devices

Liquid-solid interactions become important as dimensions approach mciro/nano-scale. This dissertation focuses on liquid-solid interactions in two distinct applications: capillary driven self-assembly of thin foils into 3D structures, and droplet wetting of hydrophobic micropatterned surfaces. The phenomenon of self-assembly of complex structures is common in biological systems. Examples include self-assembly of proteins into macromolecular structures and self-assembly of lipid bilayer membranes. The principles governing this phenomenon have been applied to induce self-assembly of millimeter scale Si thin films into spherical and other 3D structures, which are then integrated into light-trapping photovoltaic (PV) devices. Motivated by this application, we present a generalized analytical study of the self-folding of thin plates into deterministic 3D shapes, through fluid-solid interactions, to be used as PV devices. This study consists of developing a model using beam theory, which incorporates the two competing components — a capillary force that promotes folding and the bending rigidity of the foil that resists folding into a 3D structure. Through an equivalence argument of thin foils of different geometry, an effective folding parameter, which uniquely characterizes the driving force for folding, has been identified. A criterion for spontaneous folding of an arbitrarily shaped 2D foil, based on the effective folding parameter, is thus established. Measurements from experiments using different materials and predictions from the model match well, validating the assumptions used in the analysis. As an alternative to the mechanics model approach, the minimization of the total free energy is employed to investigate the interactions between a fluid droplet and a flexible thin film. A 2D energy functional is proposed, comprising the surface energy of the fluid, bending energy of the thin film and gravitational energy of the fluid. Through simulations with Surface Evolver, the shapes of the droplet and the thin film at equilibrium are obtained. A critical thin film length necessary for complete enclosure of the fluid droplet, and hence successful self-assembly into a PV device, is determined and compared with the experimental results and mechanics model predictions. The results from the modeling and energy approaches and the experiments are all consistent. Superhydrophobic surfaces, which have unique properties including self-cleaning and water repelling are desired in many applications. One excellent example in nature is the lotus leaf. To fabricate these surfaces, well designed micro/nano- surface structures are often employed. In this research, we fabricate superhydrophobic micropatterned Polydimethylsiloxane (PDMS) surfaces composed of micropillars of various sizes and arrangements by means of soft lithography. Both anisotropic surfaces, consisting of parallel grooves and cylindrical pillars in rectangular lattices, and isotropic surfaces, consisting of cylindrical pillars in square and hexagonal lattices, are considered. A novel technique is proposed to image the contact line (CL) of the droplet on the hydrophobic surface. This technique provides a new approach to distinguish between partial and complete wetting. The contact area between droplet and microtextured surface is then measured for a droplet in the Cassie state, which is a state of partial wetting. The results show that although the droplet is in the Cassie state, the contact area does not necessarily follow Cassie model predictions. Moreover, the CL is not circular, and is affected by the micropatterns, in both isotropic and anisotropic cases. Thus, it is suggested that along with the contact angle — the typical parameter reported in literature quantifying wetting, the size and shape of the contact area should also be presented. This technique is employed to investigate the evolution of the CL on a hydrophobic micropatterned surface in the cases of: a single droplet impacting the micropatterned surface, two droplets coalescing on micropillars, and a receding droplet resting on the micropatterned surface. Another parameter which quantifies hydrophobicity is the contact angle hysteresis (CAH), which indicates the resistance of the surface to the sliding of a droplet with a given volume. The conventional methods of using advancing and receding angles or tilting stage to measure the resistance of the micropatterned surface are indirect, without mentioning the inaccuracy due to the discrete and stepwise motion of the CL on micropillars. A micronewton force sensor is utilized to directly measure the resisting force by dragging a droplet on a microtextured surface. Together with the proposed imaging technique, the evolution of the CL during sliding is also explored. It is found that, at the onset of sliding, the CL behaves as a linear elastic solid with a constant stiffness. Afterwards, the force first increases and then decreases and reaches a steady state, accompanied with periodic oscillations due to regular pinning and depinning of the CL. Both the maximum and steady state forces are primarily dependent on area fractions of the micropatterned surfaces in our experiment. The resisting force is found to be proportional to the number of pillars which pin the CL at the trailing edge, validating the assumption that the resistance mainly arises from the CL pinning at the trailing edge. In each pinning-and-depinning cycle during the steady state, the CL also shows linear elastic behavior but with a lower stiffness. The force variation and energy dissipation involved can also be determined. This novel method of measuring the resistance of the micropatterned surface elucidates the dependence on CL pinning and provides more insight into the mechanisms of CAH.

Equilibration of quantum gases

Finding equilibration times is a major unsolved problem in physics with few analytical results. Here we look at equilibration times for quantum gases of bosons and fermions in the regime of negligibly weak interactions, a setting which not only includes paradigmatic systems such as gases confined to boxes, but also Luttinger liquids and the free superfluid Hubbard model. To do this, we focus on two classes of measurements: (i) coarse-grained observables, such as the number of particles in a region of space, and (ii) few-mode measurements, such as phase correlators.Weshow that, in this setting, equilibration occurs quite generally despite the fact that the particles are not interacting. Furthermore, for coarse-grained measurements the timescale is generally at most polynomial in the number of particles N, which is much faster than previous general upper bounds, which were exponential in N. For local measurements on lattice systems, the timescale is typically linear in the number of lattice sites. In fact, for one-dimensional lattices, the scaling is generally linear in the length of the lattice, which is optimal. Additionally, we look at a few specific examples, one of which consists ofNfermions initially confined on one side of a partition in a box. The partition is removed and the fermions equilibrate extremely quickly in time O(1 N).

Decomposable approximations and coloured isomorphisms for C*-algebras

In this thesis we introduce nuclear dimension and compare it with a stronger form of the completely positive approximation property. We show that the approximations forming this stronger characterisation of the completely positive approximation property witness finite nuclear dimension if and only if the underlying C*-algebra is approximately finite dimensional. We also extend this result to nuclear dimension at most omega. We review interactions between separably acting injective von Neumann algebras and separable nuclear C*-algebras. In particular, we discuss aspects of Connes' work and how some of his strategies have been used by C^*-algebraist to estimate the nuclear dimension of certain classes of C*-algebras. We introduce a notion of coloured isomorphisms between separable unital C*-algebras. Under these coloured isomorphisms ideal lattices, trace spaces, commutativity, nuclearity, finite nuclear dimension and weakly pure infiniteness are preserved. We show that these coloured isomorphisms induce isomorphisms on the classes of finite dimensional and commutative C*-algebras. We prove that any pair of Kirchberg algebras are 2-coloured isomorphic and any pair of separable, simple, unital, finite, nuclear and Z-stable C*-algebras with unique trace which satisfy the UCT are also 2-coloured isomorphic.

Représentations et fusion des algèbres de Temperley-Lieb originale et diluée

Les algèbres de Temperley-Lieb originales, aussi dites régulières, apparaissent dans de nombreux modèles statistiques sur réseau en deux dimensions: les modèles d'Ising, de Potts, des dimères, celui de Fortuin-Kasteleyn, etc. L'espace d'Hilbert de l'hamiltonien quantique correspondant à chacun de ces modèles est un module pour cette algèbre et la théorie de ses représentations peut être utilisée afin de faciliter la décomposition de l'espace en blocs; la diagonalisation de l'hamiltonien s'en trouve alors grandement simplifiée. L'algèbre de Temperley-Lieb diluée joue un rôle similaire pour des modèles statistiques dilués, par exemple un modèle sur réseau où certains sites peuvent être vides; ses représentations peuvent alors être utilisées pour simplifier l'analyse du modèle comme pour le cas original. Or ceci requiert une connaissance des modules de cette algèbre et de leur structure; un premier article donne une liste complète des modules projectifs indécomposables de l'algèbre diluée et un second les utilise afin de construire une liste complète de tous les modules indécomposables des algèbres originale et diluée. La structure des modules est décrite en termes de facteurs de composition et par leurs groupes d'homomorphismes. Le produit de fusion sur l'algèbre de Temperley-Lieb originale permet de «multiplier» ensemble deux modules sur cette algèbre pour en obtenir un autre. Il a été montré que ce produit pouvait servir dans la diagonalisation d'hamiltoniens et, selon certaines conjectures, il pourrait également être utilisé pour étudier le comportement de modèles sur réseaux dans la limite continue. Un troisième article construit une généralisation du produit de fusion pour les algèbres diluées, puis présente une méthode pour le calculer. Le produit de fusion est alors calculé pour les classes de modules indécomposables les plus communes pour les deux familles, originale et diluée, ce qui vient ajouter à la liste incomplète des produits de fusion déjà calculés par d'autres chercheurs pour la famille originale. Finalement, il s'avère que les algèbres de Temperley-Lieb peuvent être associées à une catégorie monoïdale tressée, dont la structure est compatible avec le produit de fusion décrit ci-dessus. Le quatrième article calcule explicitement ce tressage, d'abord sur la catégorie des algèbres, puis sur la catégorie des modules sur ces algèbres. Il montre également comment ce tressage permet d'obtenir des solutions aux équations de Yang-Baxter, qui peuvent alors être utilisées afin de construire des modèles intégrables sur réseaux.

Discrete breathers in a two-dimensional hexagonal Fermi-Pasta-Ulam lattice

We consider a two-dimensional Fermi-Pasta-Ulam (FPU) lattice with hexagonal symmetry. Using asymptotic methods based on small amplitude ansatz, at third order we obtain a eduction to a cubic nonlinear Schr{\"o}dinger equation (NLS) for the breather envelope. However, this does not support stable soliton solutions, so we pursue a higher-order analysis yielding a generalised NLS, which includes known stabilising terms. We present numerical results which suggest that long-lived stationary and moving breathers are supported by the lattice. We find breather solutions which move in an arbitrary direction, an ellipticity criterion for the wavenumbers of the carrier wave, symptotic estimates for the breather energy, and a minimum threshold energy below which breathers cannot be found. This energy threshold is maximised for stationary breathers, and becomes vanishingly small near the boundary of the elliptic domain where breathers attain a maximum speed. Several of the results obtained are similar to those obtained for the square FPU lattice (Butt \& Wattis, {\em J Phys A}, {\bf 39}, 4955, (2006)), though we find that the square and hexagonal lattices exhibit different properties in regard to the generation of harmonics, and the isotropy of the generalised NLS equation.

MULTI-BAND BOSE HUBBARD MODELS AND EFFECTIVE THREE-BODY INTERACTIONS

Experiments with ultracold atoms in optical lattice have become a versatile testing ground to study diverse quantum many-body Hamiltonians. A single-band Bose-Hubbard (BH) Hamiltonian was first proposed to describe these systems in 1998 and its associated quantum phase-transition was subsequently observed in 2002. Over the years, there has been a rapid progress in experimental realizations of more complex lattice geometries, leading to more exotic BH Hamiltonians with contributions from excited bands, and modified tunneling and interaction energies. There has also been interesting theoretical insights and experimental studies on “un- conventional” Bose-Einstein condensates in optical lattices and predictions of rich orbital physics in higher bands. In this thesis, I present our results on several multi- band BH models and emergent quantum phenomena. In particular, I study optical lattices with two local minima per unit cell and show that the low energy states of a multi-band BH Hamiltonian with only pairwise interactions is equivalent to an effec- tive single-band Hamiltonian with strong three-body interactions. I also propose a second method to create three-body interactions in ultracold gases of bosonic atoms in a optical lattice. In this case, this is achieved by a careful cancellation of two contributions in the pair-wise interaction between the atoms, one proportional to the zero-energy scattering length and a second proportional to the effective range. I subsequently study the physics of Bose-Einstein condensation in the second band of a double-well 2D lattice and show that the collision aided decay rate of the con- densate to the ground band is smaller than the tunneling rate between neighboring unit cells. Finally, I propose a numerical method using the discrete variable repre- sentation for constructing real-valued Wannier functions localized in a unit cell for optical lattices. The developed numerical method is general and can be applied to a wide array of optical lattice geometries in one, two or three dimensions.

CAISL: Simplification Logic for Conditional Attribute Implications

In this work, we present a sound and complete axiomatic system for conditional attribute implications (CAIs) in Triadic Concept Analysis (TCA). Our approach is strongly based on the Simplification paradigm which offers a more suitable way for automated reasoning than the one based on Armstrong’s Axioms. We also present an automated method to prove the derivability of a CAI from a set of CAI s.

Single-Polarization Double Refraction in Plasmonic Crystals: Considerations on Energy Flow

We examined the optical properties of nanolayered metal-dielectric lattices. At subwavelength regimes, the periodic array of metallic nanofilms demonstrates nonlocality-induced double refraction, conventional positive and as well as negative. In particular, we report on energy-flow considerations concerning both refractive behaviors concurrently. Numerical simulations provide transmittance of individual beams in Ag-TiO2 metamaterials under different configurations. In regimes of the effective-medium theory predicting elliptic dispersion, negative refraction may be stronger than the expected positive refraction.