Quasicrystals are long-range ordered and yet non-periodic. This interplay results in a wealth of intriguing physical phenomena, such as the inheritance of topological properties from higher dimensions, self-similarity, and the presence of non-trivial structure on all scales. The concept of aperiodic order has been extensively studied in mathematics and geometry, exemplified by the celebrated Penrose tiling. However, the understanding of physical quasicrystals (the vast majority of them are intermetallic compounds) is still incomplete owing to their complexity, regarding both growth processes and stability. Ultracold atoms in optical lattices offer an ideal, yet untested environment for investigating quasicrystals. Optical lattices, i.e. standing waves of light, allow the defect-free formation of a large variety of potential landscapes, including quasiperiodic geometries. In recent years, optical lattices have become one of the most successful tools in the large-scale quantum simulation of condensed-matter problems. This study presents the first experimental realisation of a two-dimensional quasicrystalline potential for ultracold atoms, based on an eightfold symmetric optical lattice. It is aimed at bringing together the fields of ultracold atoms and quasicrystals - and the more general concept of aperiodic order. The first part of this thesis introduces the theoretical aspects of aperiodic order and quasicrystalline structure. The second part comprises a detailed account of the newly designed apparatus that has been used to produce quantum-degenerate gases in quasicrystalline lattices. The third and final part summarises the matter-wave diffraction experiments that have been performed in various lattice geometries. These include one- and two-dimensional simple cubic lattices, one-dimensional quasiperiodic lattices, as well as two-dimensional quasicrystalline lattices. The striking self-similarity of this quasicrystalline structure has been directly observed, in close analogy to Shechtman's very first discovery of quasicrystals using electron diffraction. In addition, an in-depth study of the diffraction dynamics reveals the fundamental differences between periodic and quasicrystalline lattices, in excellent agreement with ab initio theory. The diffraction dynamics on short timescales constitutes a continuous-time quantum walk on a homogeneous four-dimensional tight-binding lattice. On the one hand, these measurements establish a novel experimental platform for investigating quasicrystals proper. On the other hand, ultracold atoms in quasicrystalline optical lattices are worth studying in their own right: Possible avenues include the observation many-body localisation and Bose glasses, as well as the creation of topologically non-trivial systems in higher dimensions.

La théorie des pavages apériodiques a connu des développements rapides depuis les années 1980, avec la découvertes d'alliages métalliques cristallisant dans une structure quasi-périodique.Dans cette thèse, on étudie particulièrement deux méthodes de construction de pavages : par coupe et projection, et par substitution. Deux angles d'approche sont développés : l'étude de la fonction de complexité, et l'étude métrique de l'espace de pavages.Dans une première partie, on calcule l'asymptotique de la fonction de complexité pour des pavages coupe et projection, généralisant ainsi des résultats connus en dynamiques symbolique pour la dimension 1. On montre que pour un pavage coupe et projection canonique N sur d sans période, la complexité croît (à des constantes près) comme n à la puissance a, où a est un entier compris entre d et N-d.Ensuite, on se base sur une construction de Pearson et Bellissard qui construisent un triplet spectral sur les ensembles de Cantor ultramétriques. On suit leur construction dans le cas d'ensembles de Cantor auto-similaires. Elle s'applique en particulier aux transversales d'espaces de pavages de substitution.Enfin, on fait le lien entre la distance usuelle sur l'enveloppe d'un pavage et la complexité de ce pavage. Les liens entre complexité et métrique permettent de donner une preuve directe du fait suivant : la complexité des pavages de substitution apériodiques de dimension d croît comme n à la puissance d.La question de liens entre la complexité et la topologie (et pas seulement avec la distance) reste ouverte. Nous apportons cependant des réponses partielles dans cette direction. / Since the 1980s, the theory of aperiodic tilings developed quickly, motivated by the discovery of metallic alloys which crystallize in an aperiodic structure. This highlighted the need for new models of crystals.Two models of aperiodic tilings are specifically studied in this dissertation. First, the cut-and-project method, then the inflation and substitution method. Two point of view are developed for the study of these objects: the study of the complexity function associated to a tiling, and the metric study of the associated tiling space.In a first part, the asymptotic behaviour of the complexity function for cut-and-project tilings is studied. The results stated here generalize formerly known results in the specific case of dimension 1. It is proved that for an (N,d) canonical projection tiling without periods, the complexity grows like n to the a, with a an integer greater or equal to d but lesser or equal to N-d.A second part is based on a construction by Pearson and Bellissard of a spectral triple for ultrametric Cantor sets. Their construction is applied to self-similar Cantor sets. It applies in particular to the transversal of substitution tiling spaces.In a last part, the links between the complexity function of a tiling and the usual distance on its associated tiling space are made explicit. These links can provide a direct and complete proof of the following fact: the complexity of an aperiodic d-dimensional substitution tiling grows asymptotically as n to the d, up to constants. These links between complexity and distance raises the question of links between complexity and topology. Partial answers are given in this direction.