This thesis covers most of my work in the field of ultracold atoms loaded in optical lattices. This thesis can be divided into five different parts. In Chapter 1, after a brief introduction to the field of optical lattices I review the fundamental aspects pertaining to the physics of systems in periodic potentials and a short overview of the experiments on ultracold atoms in an optical lattice. In Chapter 2 we develop an inhomogeneous mean-field theory for the extended Bose-Hubbard model with a quadratic, confining potential. In the absence of this poten¬tial, our mean-field theory yields the phase diagram of the homogeneous extended Bose-Hubbard model. This phase diagram shows a superfluid (SF) phase and lobes of Mott-insulator(MI), density-wave(DW), and supersolid (SS) phases in the plane of the chemical potential and on-site repulsion ; we present phase diagrams for representative values of , the repulsive energy for bosons on nearest-neighbor sites. We demonstrate that, when the confining potential is present, superfluid and density-wave order parameters are nonuniform; in particular, we obtain, for a few representative values of parameters, spherical shells of SF, MI ,DW ,and SSphases. We explore the implications of our study for experiments on cold-atom dipolar con¬densates in optical lattices in a confining potential. In Chapter3 we present an extensive study of Mottinsulator( MI) and superfluid (SF) shells in Bose-Hubbard (BH) models for bosons in optical lattices with har¬monic traps. For this we develop an inhomogeneous mean-field theory. Our results for the BH model with one type of spinless bosons agrees quantitatively with quan¬tum Monte Carlo(QMC) simulations. Our approach is numerically less intensive than such simulations, so we are able to perform calculations on experimentally realistic, large three-dimensional(3D) systems, explore a wide range of parameter values, and make direct contact with a variety of experimental measurements. We also generalize our inhomogeneous mean-field theory to study BH models with har¬monic traps and(a) two species of bosons or(b) spin-1bosons. With two species of bosons we obtain rich phase diagrams with a variety of SF and MI phases and as¬sociated shells, when we include a quadratic confining potential. For the spin-1BH model we show, in a representative case, that the system can display alternating shells of polar SF and MI phases; and we make interesting predictions for experi¬ments in such systems. . In Chapter 4 we carry out an extensive study of the phase diagrams of the ex-tended Bose Hubbard model, with a mean filling of one boson per site, in one dimension by using the density matrix renormalization group and show that it contains Superfluid (SF), Mott-insulator (MI), density-wave (DW) and Haldane ¬insulator(HI) phases. We show that the critical exponents and central charge for the HI-DW,MI-HI and SF-MI transitions are consistent with those for models in the two-dimensional Ising, Gaussian, and Berezinskii-Kosterlitz-Thouless (BKT) uni¬versality classes, respectively; and we suggest that the SF-HI transition may be more exotic than a simple BKT transition. We show explicitly that different bound¬ary conditions lead to different phase diagrams.. In Chapter 5 we obtain the excitation spectra of the following three generalized of Bose-Hubbard(BH) models:(1) a two-species generalization of the spinless BH model, (2) a single-species, spin-1 BH model, and (3) the extended Bose-Hubbard model (EBH) for spinless interacting bosons of one species. In all the phases of these models we show how to obtain excitation spectra by using the random phase approximation (RPA). We compare the results of our work with earlier studies of related models and discuss implications for experiments.

Le travail théorique présenté dans cette thèse concerne la formation de molécules ultra-froides bialcalines et le contrôle de leurs degrés de liberté externes et internes. Cette étude est motivée par les nombreuses expériences en cours visant à l'obtention d'un gaz quantique dégénéré de molécules dans leur état fondamental absolu. Le schéma de formation étudié repose sur le processus de transfert adiabatique stimulé (STIRAP) réalisé en présence d'un potentiel optique de piégeage (réseau optique) des atomes et des molécules.Nous avons déterminé les paramètres du réseau optique (intensité et fréquence du champ laser) qui permettent de piéger efficacement des dimères d'alcalins en évaluant la polarisabilité dynamique acquise par les molécules soumises à un champ externe. Ces calculs reposent en particulier sur la connaissance détaillée de la structure électronique des molécules. Nous avons identifié des plages de longueur d'ondes dites « magiques » où la polarisabilité est la même pour chaque niveau peuplé au cours du transfert adiabatique, permettant ainsi un transfert optimal. Ce formalisme nous a également permis d'obtenir les coefficients Van der Waals de l'interaction à longue portée nécessaires pour étudier les taux de collisions entre molécules.Nous avons réalisé une étude plus détaillée de la molécule RbCs. En étudiant précisément la probabilité de transition de la molécule vers un niveau excité, nous avons proposé un schéma STIRAP pour transférer des molécules de RbCs, initialement dans un niveau vibrationnel excité, vers leur état rovibrationnel fondamental.Ces travaux ont montré l'importance de la connaissance précise de la structure hyperfine de l'état électronique moléculaire excité pour réaliser un gaz dégénéré de molécules dans un état quantique bien défini. Un modèle asymptotique nous a permis d'obtenir une première estimation de la structure hyperfine des courbes d'énergies potentielles des premiers états moléculaires excités des molécules Cs2 et RbCs. / The theoretical work presented in this thesis is focused on the formation of ultracold bialcaline molecules and on the control of their external and internal degrees of freedom. This study is motivated by the increasing number of experiments aiming at obtaining a quantum degenerate gas of molecules in their absolute ground state. The formation scheme we worked on is based on the Stimulated Raman Adiabatic Passage (STIRAP) technique operated while molecules are trapped inside an optical lattice.We have determined the parameters of the optical lattice (intensity and wavelength of the laser) that allow for an efficient trapping of the alkali dimers by evaluating the dynamic polarizability of molecules in the presence of an external field. Such calculations require the accurate knowledge of the electronic structure of the molecules. We have identified the so-called ``magic'' wavelength for which all levels populated during the STIRAP sequence have the same polarizability, thus ensuring an optimal transfer. The same approach has also been used to compute the strength of the long-range interaction between polar bialkali molecules needed to evaluate collision rates.The particular case of the RbCs molecule has been investigated. We have selected a radiative transition allowing for an efficient STIRAP scheme yielding molecules in their rovibrational ground state. These works have raised the need for the precise knowledge of the hyperfine structure of the excited electronic molecular state involved in the STIRAP scheme. We have developed an asymptotic model to obtain a first estimate of the hyperfine structure for the potential curves of the lowest excited states of Cs2 and RbCs.

Depuis plusieurs décennies un intérêt est né pour combiner deux systèmes quantiques pour former unsystème hybride quantique (SHQ) aux qualités qu’il serait impossible d’atteindre avec un seul des deuxsous-constituants. Parmi les systèmes quantiques, les atomes froids se distinguent par leur fort découplagede l’environnement, permettant un contrôle précis de leurs propriétés intrinsèques. En outre, les simulateursquantiques réalisés en piégeant des atomes froids dans des réseaux optiques présentent des propriétéscontrôlables (échelle d’énergie, géométrie,...) qui permettent d’étudier de nouveaux régimes intéressants enphysique de la matière condensée. Dans cette quête de phases quantiques exotiques (e.g., antiferromagnétisme),la réduction de l’entropie thermique est un défi crucial. Le prix à payer pour atteindre de si faiblestempérature et entropie est un long temps de thermalisation qui limite la réalisation expérimentale. La réductionde la période du réseau est une solution prometteuse pour augmenter la dynamique du système.Les SHQs avec des atomes froids offrent de riches perspectives mais requiert d’interfacer des systèmes quantiquesdans des états différents (solide/gaz) à des distances très proches, ce qui reste un défi expérimental.Le projet AUFRONS, dans lequel s’inscrit cette thèse, vise à refroidir un gaz d’atomes froids jusqu’aurégime de dégénérescence quantique puis de transporter et piéger ce nuage en champ proche d’une nanostructure.L’idée est d’obtenir un gaz d’atomes froids piégé dans un réseau bidimensionnel aux dimensionssublongueur d’onde, à quelques dizaines de nm de la structure. Un des objectifs est d’étudier les interactionsau sein du réseau mais également le couplage des atomes avec les modes de surface.Le travail réalisé durant cette thèse se décompose en une partie expérimentale et une partie théorique.Dans la première nous présentons le refroidissement d’atomes de 87Rb jusqu’au régime de dégénérescencequantique. La seconde partie est consacrée aux simulations théoriques d’une nouvelle méthode que nousavons implémentée pour piéger et manipuler des atomes froids à moins de 100 nm d’une nanostructure.Cette méthode, qui tire profit de la résonance plasmonique et des forces du vide (effet Casimir-Polder),permet de créer des potentiels sublongueur d’onde aux paramètres contrôlables. Nous détaillons ainsi lescalculs des forces optiques et des forces du vide que nous appliquons au cas d’un atome de 87Rb en champproche d’une nanostructure 1D. / An interest for hybrid quantum systems (HSQs) has been growing up for the last decades. This object combines two quantum systems in order to take advantage of both systems’ qualities, not available withonly one. Among these quantum systems, ultracold atoms distinguish themselves by their strong decoupling from environment which enables an excellent control of their intrinsic properties. Optical lattice quantum simulators with tunable properties (energy scale, geometry,...) allows one to investigate new regimes incondensed matter physics. In this quest for exotic quantum phases (e.g., antiferromagnetism), the reduction of thermal entropy is a crucial challenge. The price to pay for such low temperature and entropy is a longthermalization time that will ultimately limit the experimental realization. Miniaturization of lattice spacingis a promising solution to speed up the dynamics. Engineering cold atom hybrids offers promising perspectives but requires us to interface quantum systems in different states of matter at very short distances, which still remains an experimental challenge.This thesis is part of the AUFRONS project, which aims at cooling down an atomic gas until the quantum degeneracy regime then transport and trap this cloud in the near field of a nanostructure. The idea is to trapcold atoms in a two-dimensional subwavelength lattice, at a few tenth of nm away from the surface. One goal is to study atom-atom interactions within the lattice but also atom-surface modes coupling.The work realized during this thesis splits into an experimental part and a theoretical part. In the firstone, we present the cooling of 87Rb atoms until the quantum degeneracy regime. The second part is dedicated to theoretical simulations of a new trapping method we have implemented to trap and manipulate cold atoms below 100 nm from structures. This method takes advantage of plasmonic resonance and vacuum forces (Casimir-Polder effect). It allows one to create subwavelength potentials with controllable parameters.We detail the calculations of optical and vacuum forces to apply them to an atom of 87Rb in the vicinity of a 1D nanostructure.

Das Ziel dieser Arbeit ist es die mathematische Theorie der Begriffsalgebren zu entwickeln. Wir betrachten dabei hauptsaechlich das Repraesentationsproblem dieser vor Kurzem eingefuehrten Strukturen. Motiviert durch die Suche nach einer geeigneten Negation sind die Begriffsalgebren entstanden. Sie sind nicht nur fuer die Philosophie oder die Wissensrepraesentation von Interesse, sondern auch fuer andere Felder, wie zum Beispiel Logik oder Linguistik. Das Problem Negationen geeignet einzufuehren, ist sicher eines der aeltesten der wissenschaftlichen oder philosophischen Gemeinschaft und erregt auch zur Zeit die Aufmerksamkeit vieler Wissenschaftler. Verschiedene Typen von Logik (die sich sehr stark durch die eigefuehrte Negation unterscheiden) unterstreichen die Wichtigkeit dieser Untersuchungen. In dieser Arbeit beschaeftigen wir uns hauptsaechlich mit der kontextuellen Logik, eine Herangehensweise der Formalen Begriffsanalyse, basierend auf der Idee, den Begriff als Einheit des Denkens aufzufassen. / The aim of this investigation is to develop a mathematical theory of concept algebras. We mainly consider the representation problem for this recently introduced class of structures. Motivated by the search of a "negation" on formal concepts, "concept algebras" are of considerable interest not only in Philosophy or Knowledge Representation, but also in other fields as Logic or Linguistics. The problem of negation is surely one of the oldest problems of the scientific and philosophic community, and still attracts the attention of many researchers. Various types of Logic (defined according to the behaviour of the corresponding negation) can attest this affirmation. In this thesis we focus on "Contextual Logic", a Formal Concept Analysis approach, based on concepts as units of thought.

Im Rahmen dieser Arbeit wurden die Kondo-Gitter YbT2Si2 (T = Rh, Ir, Co) und CeTPO (T = Ru, Os, Fe, Co) untersucht. In diesen Systemen treten starke ferromagnetische Korrelationen der 4f-Momente zusammen mit ausgeprägter Kondo-Wechselwirkung auf, deren theoretische Beschreibung bislang sehr kontrovers diskutiert wird. Diese Arbeit liefert damit einen essentiellen experimentellen Beitrag zur Physik von ferromagnetischen Kondo-Gittern. So konnten qualitativ hochwertige Einkristalle von YbRh2Si2 hergestellt und erstmalig an einem Schwere-Fermion-System deren kritische Fluktuationen um den magnetischen Phasenübergang analysiert werden. Weiterhin konnte das bis dahin unverstandene Auftreten einer Elektron-Spin-Resonanz (ESR)-Linie in YbT2Si2 auf ferromagnetische Korrelationen zurückgeführt werden. Außerdem wurde mit CeFePO ein neues Schwere-Fermion-System mit starken ferromagnetischen Korrelationen entdeckt sowie mit dem isoelektronischen CeRuPO der seltene Fall eines ferromagnetisch geordneten Kondo-Gitters realisiert. / Within the context of this thesis the Kondo lattices YbT2Si2 (T = Rh, Ir, Co) and CeTPO (T = Ru, Os, Fe, Co) were investigated. In these systems strong ferromagnetic correlations of the 4f-moments together with pronounced Kondo interactions are present, whose theoretical description are pres-ently controversial discussed. Therefore, this work gives an essential experimental contribution to the physics of ferromagnetic Kondo lattices. The main results include the growth of high-quality single crystals of YbRh2Si2 and the first analysis of the critical fluctuations around the magnetic phase transition in a heavy fermion system. Furthermore, the unexpected observation of an electron spin resonance in YbT2Si2 could be ascribed to ferromagnetic correlations. Moreover, a new heavy fermion system CeFePO with strong ferromagnetic correlations was found and with the isoelec-tronic CeRuPO the rare case of a ferromagnetic Kondo-lattice discovered.

In this thesis, we generalize clones (as well as their relational counterparts and the relationship between them) to categories. Based on this framework, we introduce a general duality theory for clones and apply it to obtain new results for clones on finite sets.

This thesis concerns the mathematical modelling of phase transformations with a special emphasis on martensitic phase transformations and their application to the modelling of steels. In Chapter 1, we develop a framework that determines the optimal transformation strain between any two Bravais lattices and use it to give a rigorous proof of a conjecture by E.C. Bain in 1924 on the optimality of the so-called Bain strain. In Chapter 2, we review the Ball-James model and related concepts. We present some simplification of existing results. In Chapter 3, we pose a conjecture for the explicit form of the quasiconvex hull of the three tetragonal wells, known as the three-well problem. We present a new approach to finding inner and outer bounds. In Chapter 4, we focus on highly compatible, so called self-accommodating, martensitic structures and present new results on their fine properties such as estimates on their minimum complexity and bounds on the relative proportion of each martensitic variant in them. In Chapter 5, we investigate the contrary situation when self-accommodating microstructures do not exist. We determine, whether in this situation, it is still energetically favourable to nucleate martensite within austenite. By constructing different types of inclusions, we find that the optimal shape of an inclusion is flat and thin which is in agreement with experimental observation. In Chapter 6, we introduce a mechanism that identifies transformation strains with orientation relationships. This mechanism allows us to develop a simpler, strain-based approach to phase transformation models in steels. One novelty of this approach is the derivation of an explicit dependence of the orientation relationships on the ratio of tetragonality of the product phase. In Chapter 7, we establish a correspondence between common phenomenological models for steels and the Ball-James model. This correspondence is then used to develop a new theory for the (5 5 7) lath transformation in low-carbon steels. Compared to existing theories, this new approach requires a significantly smaller number of input parameters. Furthermore, it predicts a microstructure morphology which differs from what is conventionally believed.

No description available.

Dans une première partie de cette thèse, nous donnons des minorations universelles ne dépendant que de la dimension – explicites, de trois invariants globaux des quotients des espaces hyperboliques quaternioniques : leur rayon maximal, leur volume, ainsi que leur caractéristique d’Euler. Nous donnons également une majoration de leur constante de Margulis, montrant que celle-ci décroit au moins comme une puissance négative de la dimension. Dans une seconde partie, nous étudions un réseau remarquable des isométries du plan hyperbolique quaternionique, le groupe modulaire d’Hurwitz. Nous montrons en particulier qu’il est engendré par quatres éléments, et construisons un domaine fondamental pour le sous-groupe des isométries de ce réseau qui stabilisent un point à l’infini. / In the first part of this thesis, we derive explicit universal – that is, depending only on the dimension – lower bounds on three global invariants of quaternionic hyperbolic sapces : their maximal radius, their volume, and their Euler caracteristic. We also exhibit an upper bound on their Margulis constant, showing that this last quantity decreases at least like a negative power of the dimension. In the second part, we study a specific lattice of isometries of the quaternionic hyperbolic plane : the Hurwitz modular group. In particular, we show that this group is generated by four elements, and we construct a fundamental domain for the subgroup of isometries of this lattice stabilising a point on the boundary of the quaternionic hyperbolic plane.

We prove that there exist three distinct, comprehensive classes of (formal) contexts with polynomially many concepts. Namely: contexts which are nowhere dense, of bounded breadth or highly convex. Already present in G. Birkhoff's classic monograph is the notion of breadth of a lattice; it equals the number of atoms of a largest boolean suborder. Even though it is natural to define the breadth of a context as being that of its concept lattice, this idea had not been exploited before. We do this and establish many equivalences. Amongst them, it is shown that the breadth of a context equals the size of its largest minimal generator, its largest contranominal-scale subcontext, as well as the Vapnik-Chervonenkis dimension of both its system of extents and of intents. The polynomiality of the aforementioned classes is proven via upper bounds (also known as majorants) for the number of maximal bipartite cliques in bipartite graphs. These are results obtained by various authors in the last decades. The fact that they yield statements about formal contexts is a reward for investigating how two established fields interact, specifically Formal Concept Analysis (FCA) and graph theory. We improve considerably the breadth bound. Such improvement is twofold: besides giving a much tighter expression, we prove that it limits the number of minimal generators. This is strictly more general than upper bounding the quantity of concepts. Indeed, it automatically implies a bound on these, as well as on the number of proper premises. A corollary is that this improved result is a bound for the number of implications in the canonical basis too. With respect to the quantity of concepts, this sharper majorant is shown to be best possible. Such fact is established by constructing contexts whose concept lattices exhibit exactly that many elements. These structures are termed, respectively, extremal contexts and extremal lattices. The usual procedure of taking the standard context allows one to work interchangeably with either one of these two extremal structures. Extremal lattices are equivalently defined as finite lattices which have as many elements as possible, under the condition that they obey two upper limits: one for its number of join-irreducibles, other for its breadth. Subsequently, these structures are characterized in two ways. Our first characterization is done using the lattice perspective. Initially, we construct extremal lattices by the iterated operation of finding smaller, extremal subsemilattices and duplicating their elements. Then, it is shown that every extremal lattice must be obtained through a recursive application of this construction principle. A byproduct of this contribution is that extremal lattices are always meet-distributive. Despite the fact that this approach is revealing, the vicinity of its findings contains unanswered combinatorial questions which are relevant. Most notably, the number of meet-irreducibles of extremal lattices escapes from control when this construction is conducted. Aiming to get a grip on the number of meet-irreducibles, we succeed at proving an alternative characterization of these structures. This second approach is based on implication logic, and exposes an interesting link between number of proper premises, pseudo-extents and concepts. A guiding idea in this scenario is to use implications to construct lattices. It turns out that constructing extremal structures with this method is simpler, in the sense that a recursive application of the construction principle is not needed. Moreover, we obtain with ease a general, explicit formula for the Whitney numbers of extremal lattices. This reveals that they are unimodal, too. Like the first, this second construction method is shown to be characteristic. A particular case of the construction is able to force - with precision - a high number of (in the sense of "exponentially many'') meet-irreducibles. Such occasional explosion of meet-irreducibles motivates a generalization of the notion of extremal lattices. This is done by means of considering a more refined partition of the class of all finite lattices. In this finer-grained setting, each extremal class consists of lattices with bounded breadth, number of join irreducibles and meet-irreducibles as well. The generalized problem of finding the maximum number of concepts reveals itself to be challenging. Instead of attempting to classify these structures completely, we pose questions inspired by Turán's seminal result in extremal combinatorics. Most prominently: do extremal lattices (in this more general sense) have the maximum permitted breadth? We show a general statement in this setting: for every choice of limits (breadth, number of join-irreducibles and meet-irreducibles), we produce some extremal lattice with the maximum permitted breadth. The tools which underpin all the intuitions in this scenario are hypergraphs and exact set covers. In a rather unexpected, but interesting turn of events, we obtain for free a simple and interesting theorem about the general existence of "rich'' subcontexts. Precisely: every context contains an object/attribute pair which, after removed, results in a context with at least half the original number of concepts.