We propose a means to realize two-dimensional quasiperiodic structures by trapping atoms in an optical potential. The structures have eight-fold symmetry and are closely related to the well-known quasiperiodic octagonal (Ammann-Beenker) tiling. We describe the geometrical properties of the structures obtained by tuning parameters of the system. We discuss some features of the corresponding tight-binding models, and experiments to probe quantum properties of this optical quasicrystal.
We propose a means to realize two-dimensional quasiperiodic structures by trapping atoms in an optical potential. The structures have eight-fold symmetry and are closely related to the well-known quasiperiodic octagonal (Ammann-Beenker) tiling. We describe the geometrical properties of the structures obtained by tuning parameters of the system. We discuss some features of the corresponding tight-binding models, and experiments to probe quantum properties of this optical quasicrystal.
We describe a way to obtain a two-dimensional quasiperiodic tiling with eight-fold symmetry using cold atoms. One can obtain a series of such optical tilings, related by scale transformations, for a series of specific values of the chemical potential of the atoms. A theoretical model for the optical system is described and compared with that of the well-known cut-and-project method for the Ammann–Beenker tiling. The relation between the two tilings is discussed. This type of cold atom structure should allow the simulation of several important lattice models for interacting quantum particles and spins in quasicrystals.
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.
Quasicrystals are nonperiodic structures having no translational symmetry but nonetheless possessing long-range order. The material properties of quasicrystals, particularly their low-temperature behavior, defy easy description. We present a compact optical setup for creating quasicrystal optical potentials with five-fold symmetry using interference of nearly co-propagating beams for use in ultracold atom quantum simulation experiments. We verify the optical design through numerical simulations and demonstrate a prototype system. We also discuss generating phason excitations and quantized transport in the quasicrystal through phase modulation of the beams.
We describe a quasiperiodic optical lattice, created by a physical realization of the abstract cut-and-project construction underlying all quasicrystals. The resulting potential is a generalization of the Fibonacci tiling. Calculation of the energies and wave functions of ultracold atoms loaded into such a lattice demonstrate a multifractal energy spectrum, a singular continuous momentum-space structure, and the existence of controllable edge states. These results open the door to cold atom quantum simulation experiments in tunable or dynamic quasicrystalline potentials, including topological pumping of edge states and phasonic spectroscopy.
Quasicrystals are long-range ordered and yet nonperiodic. This interplay results in a wealth of intriguing physical phenomena, such as the inheritance of topological properties from higher dimensions, and the presence of nontrivial structure on all scales. Here, we report on the first experimental demonstration of an eightfold rotationally symmetric optical lattice, realizing a two-dimensional quasicrystalline potential for ultracold atoms. Using matter-wave diffraction we observe the self-similarity of this quasicrystalline structure, in close analogy to the very first discovery of quasicrystals using electron diffraction. The diffraction dynamics on short timescales constitutes a continuous-time quantum walk on a homogeneous four-dimensional tight-binding lattice. These measurements pave the way for quantum simulations in fractal structures and higher dimensions.
We investigate the necessary features of the pair interaction for the stabilization of self-assembled quantum quasicrystals in two-dimensional bosonic systems. Unlike the classical scenario, our results show that two-dimensional octagonal, decagonal, and dodecagonal aperiodic phases require a distinct number of properly tuned characteristic length scales for their stabilization. By using a mean-field spectral variational approach and Gross-Pitaevskii numerical calculations, we determine that the dodecagonal quasicrystal structure requires at least two characteristic length scales for its stabilization, while the decagonal and octagonal patterns need at least three and four length scales, respectively. The family of pair interaction potentials considered, albeit simple, is well justified in terms of a novel experimental platform based on laser-painted interactions in a cavity QED setup. Finally, we perform a structural characterization of the quasicrystal patterns obtained and show that these phases coexist with a finite superfluid fraction, forming what can be called a super quasicrystal phase.