We analyze the physics of Bose-Einstein condensates confined in two dimensional (2D) quasiperiodic optical lattices, which offer an intermediate situation between ordered and disordered systems. First, we analyze the time-of-flight interference pattern that reveals quasiperiodic long-range order. Second, we demonstrate localization effects associated with quasidisorder as well as quasiperiodic Bloch oscillations associated with the extended nature of the wave function of a Bose-Einstein condensate in an optical quasicrystal. In addition, we discuss in detail the crossover between diffusive and localized regimes when the quasiperiodic potential is switched on, as well as the effects of interactions.
We analyze the physics of Bose-Einstein condensates confined in two dimensional (2D) quasiperiodic optical lattices, which offer an intermediate situation between ordered and disordered systems. First, we analyze the time-of-flight interference pattern that reveals quasiperiodic long-range order. Second, we demonstrate localization effects associated with quasidisorder as well as quasiperiodic Bloch oscillations associated with the extended nature of the wave function of a Bose-Einstein condensate in an optical quasicrystal. In addition, we discuss in detail the crossover between diffusive and localized regimes when the quasiperiodic potential is switched on, as well as the effects of interactions.
Quasicrystals are long-range ordered and yet non-periodic. This interplay results in a wealth of intriguing physical phenomena, such as self-similarity, the inheritance of topological properties from higher dimensions, and the presence of non-trivial structure on all lengthscales. However, quasicrystalline materials are notoriously hard to synthesise, as defects and impurities may greatly alter their final microscopic composition. The field of quantum simulation with ultracold atoms offers a solution to this problem, namely the use of optical lattices – standing waves of light. Optical lattices are free of impurities and therefore ideally suited to study quasicrystals, enabling unprecedented access to observables that are unattainable in condensed matter systems. This study presents the first experimental realisation of a two-dimensional quasicrystalline potential for ultracold atoms, based on an eightfold symmetric optical lattice. Features pertaining to both ordered and disordered phases are observed, from sharp diffraction peaks in the matter-wave interference pattern, to a disorder-induced localised phase emerging at a critical lattice depth V_loc ~ 1.78 E_rec. The localised phase seems to be resilient against moderate interactions, which would make this the first experimental realisation of a 2D Bose glass.
The higher-order interactions of Bose-Einstein condensate in multi-dimensional optical lattices are discussed both analytically and numerically. It is demonstrated that the effects of the higher-order atomic interactions on the sound speed and the stabilities of Bloch waves strongly depend on the lattice strength. In the presence of higher-order effects, tighter and high-dimensional lattices are confirmed to be two positive factors for maintaining the system's energetic stability, and the dynamical instability of Bloch waves can take place simultaneously with the energetic instability. In addition, we find that the higher-order interactions exhibit a long-range behavior and the long-lived coherent Bloch oscillations in a tilted optical lattice exist. Our results provide an effective way to probe the higher-order interactions in optical lattices.
A Bose-Einstein condensate loaded in a one-dimensional bichromatic optical lattice with constituent sublattices having incommensurate periods is considered. Using the rational approximations for the incommensurate periods, we show that below the mobility edge the localized states are distributed nearly homogeneously in the space and explore the versatility of such potentials. We show that superposition of symmetric and antisymmetric localized states can be used to simulate various physical dynamical regimes, known to occur in double-well and multiwell traps. As examples, we obtain an alternative realization of a bosonic Josephson junction, whose coherent oscillations display beatings or switching in the weakly nonlinear regime, and describe self-trapping and four-mode dynamics, mimicking coherent oscillations and self-trapping in four-well potentials. These phenomena can be observed for different pairs of modes, which are localized due to the interference rather than due to a confining trap. The results obtained using few-mode approximations are compared with the direct numerical simulations of the one-dimensional Gross-Pitaevskii equation. The localized states and the related dynamics are found to persist for long times even in the repulsive condensates. We also described bifurcations of the families of nonlinear modes, the symmetry breaking, and stable minigap solitons.
By direct numerical simulation of the time-dependent Gross-Pitaevskii equation, we study different aspects of the localization of a noninteracting ideal Bose-Einstein condensate (BEC) in a one-dimensional bichromatic quasiperiodic optical-lattice potential. Such a quasiperiodic potential, used in a recent experiment on the localization of a BEC [Roati et al., Nature (London) 453, 895 (2008)], can be formed by the superposition of two standing-wave polarized laser beams with different wavelengths. We investigate the effect of the variation of optical amplitudes and wavelengths on the localization of a noninteracting BEC. We also simulate the nonlinear dynamics when a harmonically trapped BEC is suddenly released into a quasiperiodic potential, as done experimentally in a laser speckle potential [Billy et al., Nature (London) 453, 891 (2008)]. We finally study the destruction of the localization in an interacting BEC due to the repulsion generated by a positive scattering length between the bosonic atoms.
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 long-range ordered but not periodic, representing an interesting middle ground between order and disorder. We experimentally and numerically study the localization transition in the ground state of noninteracting and weakly interacting bosons in an eightfold symmetric quasicrystalline optical lattice. In contrast to typically used real space in situ techniques, we probe the system in momentum space by recording matter wave diffraction patterns. Shallow lattices lead to extended states whereas we observe a localization transition at a critical lattice depth of V_{0}≈1.78(2)E_{rec} for the noninteracting system. Our measurements and Gross-Pitaevskii simulations demonstrate that in interacting systems the transition is shifted to deeper lattices, as expected from superfluid order counteracting localization. Quasiperiodic potentials, lacking conventional rare regions, provide the ideal testing ground to realize many-body localization in 2D.