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.

Detta examensarbete har gjorts direkt mot båda författarnas inriktning inommaskiningenjörsprogrammet. Inriktningen är produktionsutveckling ochexamensarbetet har inneburit utveckling av ett paketeringssystem från grundenbaserat på ett enklare befintligt system. Det nuvarande systemet som finns påKvibilles mejeri har föråldrad utrustning och dålig kapacitet, vilket mejeriet villeförbättra. Projektet har gett författarna möjlighet att testa och utvidga sina kunskaperinom sitt utbildningsområde. Det som utförts i projektet har inneburit mycket informationssökande, teorin bakominnefattar bl.a. produktionssystem, arbetsplatsergonomi, modeller förproduktionsutveckling, direktiv, regler och utvärderingsmetoder samt givetvis dataom de maskiner och den utrustning som varit intressant för projektet. Det harförutom informationssökning inneburit kontakt med en rad olika maskinleverantörerför att få information och priser. Metodiken som används bestämdes under projektets förstudie, metoden är utveckladav Bellgran & Säfsten (2005) och är en ren produktionsutvecklingsmetod. Metodenhar följts till de delar som varit intressanta och vissa delar såsom driftsättning harförbisetts då resultatet av projektet inte realiserats. Metoden innebar bl.a. en studie avnuvarande och liknande produktionssystem samt intervjuer med dess operatörer. Projektets gång ledde till ett antal koncept för en helt ny paketeringslinje som inte harmånga likheter med den nuvarande linjen och dess arbetssätt. Författarna är mycketnöjda med det resulterande slutgiltiga konceptet och de fördelar konceptet innebärsom är av både teknisk och ekonomisk karaktär. Huvudresultaten inkluderar 50%ökad kapacitet och minskade manuella tunga moment som lett till minskat behov avoperatörer. / This thesis is in line with the two author’s orientation within the mechanicalengineering program. The orientation is production development and the project hasbeen to develop a new packaging line based on a simple existing one. The existingsystem is located in a dairy plant and have old equipment and low capacity which theplant want to improve. The project has given the authors opportunity to use andwiden their knowledge within their area of education.The project includes a lot of research, the theory behind the project is amongst otherthings; production systems, evaluation systems, work ergonomics, models forproduction development, directives, rules and data for the machines and equipmentneeded for the project. Contact has been made with several machine suppliers to findinformation and to get prices. The method used was decided during the projects background study, the method isdeveloped by Bellgran & Säften (2005) and is a method designed purely forproduction development. Only applicable parts of the method has been used whichmeans that some parts where left out such as the part for deployment since theproject won’t be realized as of now. The method included a study of the currentsystem and similar production systems as well as interviews with the staff. This project has led to a number of concepts for a new production system that worksvery different compared to the old system. The authors are very pleased with theresulting new final concept and the improvements it lead to which is technical andeconomical. The main results is 50% improved capacity and reduced heavy manualtasks which has resulted in less need of line operators.

The perspective of procurement and supply chain management is changing dramatically; traditionally, it was seen as a support function; however, the procurement function is receiving increased attention and investment as an essential contributor to the strategic success and a business enabler. While an end-to-end digital supply chain is an opportunity as it unleashes the next level of strategic growth and involves minimal investment in infrastructure, it is still a challenge to optimize and transform. Furthermore, the recent pandemics and geopolitical disruptions of Covid-19, the Ukraine-Russian war, Brexit and the US-China trade war; have structurally changed the global economy and revealed a new risk assessment that will result in the re-introduction of buffers, boundaries across industries and a partial return to regionalization with sort of de-globalization in which existing just-in-time getting replaced by just-in-case strategy.