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Interacting Ultracold Bosonic Atoms in Geometrically Frustrated Lattices

Tsz Him Leung

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

Geometric frustration is a phenomenon where local energy constraints cannot be simultaneously satisfied because of geometric reasons. This leads to a manifold of degenerate ground states in which ordering can be highly nontrivial and exotic phases of matter could emerge. In a kagome lattice, both interaction energies of spins and kinetic energies of particles can be frustrated. In the latter case, a single-particle band structure with a flat (dispersionless) energy band results.In this dissertation, we report our experimental studies of interacting bosonic systems in kagome lattices. Such studies are performed with a quantum simulator based on ultracold atoms in optical lattices. We describe upgrades and development of the apparatus that constructs optical kagome lattices in our experiment. We briefly discuss results of an experiment studying the Bose-Hubbard model in a trimerized kagome lattice. We then report the details of an experiment investigating the interplay of kinetic frustration and interactions by probing the band structure of an optical kagome lattice with interacting ultracold bosonic atoms. Finally, we propose experimental ideas for exploring the properties of the singular band touching point in the kagome flat band.

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Geometric frustration is a phenomenon where local energy constraints cannot be simultaneously satisfied because of geometric reasons. This leads to a manifold of degenerate ground states in which ordering can be highly nontrivial and exotic phases of matter could emerge. In a kagome lattice, both interaction energies of spins and kinetic energies of particles can be frustrated. In the latter case, a single-particle band structure with a flat (dispersionless) energy band results.In this dissertation, we report our experimental studies of interacting bosonic systems in kagome lattices. Such studies are performed with a quantum simulator based on ultracold atoms in optical lattices. We describe upgrades and development of the apparatus that constructs optical kagome lattices in our experiment. We briefly discuss results of an experiment studying the Bose-Hubbard model in a trimerized kagome lattice. We then report the details of an experiment investigating the interplay of kinetic frustration and interactions by probing the band structure of an optical kagome lattice with interacting ultracold bosonic atoms. Finally, we propose experimental ideas for exploring the properties of the singular band touching point in the kagome flat band.
Quantum simulation is the study of one quantum mechanical system via analog with another. In this thesis we explore a simple model of interacting bosons on a lattice, known as the Bose-Hubbard model, by experimental investigation of ultracold rubidium atoms in an optical lattice made from laser light. We describe the construction and stabilization of an optical superlattice with threefold symmetry, and its use in studying the Bose-Hubbard model on triangular and trimerized kagome lattices. We study the short range phase coherence of a Mott insulator on the triangular lattice, and develop a scheme to mitigate out-of-equilibrium effects arising from the state preparation. We show the first experimental realization of an optical trimerized kagome lattice for cold atoms, and discuss experiments characterizing this lattice. Finally, we provide evidence for a Mott insulating state with fractional average particle number per site with measurements of the nearest-neighbor phase coherence of strongly interacting atoms in the trimerized kagome lattice.
We derive effective Hamiltonians for lattice bosons with strong geometrical frustration of the kinetic energy by projecting the interactions on the flat lowest Bloch band. Specifically, we consider the Bose Hubbard model on the one-dimensional sawtooth lattice and the two-dimensional kagome lattice. Starting from a strictly local interaction the projection gives rise to effective long-range terms stabilizing a supersolid phase at densities above ${\ensuremath{\nu}}_{c}=1/9$ of the kagome lattice. In the sawtooth lattice on the other hand we show that the solid order, which exists at the magic filling ${\ensuremath{\nu}}_{c}=1/4$, is unstable to further doping. The universal low-energy properties at filling $1/4+\ensuremath{\delta}\ensuremath{\nu}$ are described by the well-known commensurate-incommensurate transition. We support the analytic results by detailed numerical calculations using the density-matrix renormalization group and exact diagonalization. Finally, we discuss possible realizations of the models using ultracold atoms as well as frustrated quantum magnets in high magnetic fields. We compute the momentum distribution and the noise correlations, that can be extracted from time of flight experiments or neutron scattering, and point to signatures of the unique supersolid phase of the kagome lattice.
Geometric frustration of particle motion in a kagome lattice causes the single-particle band structure to have a flat s-orbital band. We probe this band structure by placing a Bose-Einstein condensate into excited Bloch states of an optical kagome lattice, and then measuring the group velocity through the atomic momentum distribution. We find that interactions renormalize the band structure, greatly increasing the dispersion of the third band, which is nearly non-dispersing the single-particle treatment. Calculations based on the lattice Gross-Pitaevskii equation indicate that band structure renormalization is caused by the distortion of the overall lattice potential away from the kagome geometry by interactions.
Quantum simulation using ultracold fermions in optical lattices provides valuable insights for strongly correlated many-body phenomena in regimes challenging for classical simulations. In this dissertation, we report the design and construction of an apparatus capable of producing quantum degenerate Bose and Fermi gases. The atoms are then loaded into hexagonal optical superlattices created using two commensurate wavelengths. The apparatus lays the foundation for studying interacting fermions in geometrically frustrated lattices.We also report our experimental results on the characterization of the quantum geometry of Bloch band structures. Using rubidium Bose-Einstein condensates in the honeycomb lattice, we were able to measure the quantum distance around band touching points using parallel transport. Additionally, by simulating optical excitations using noninteracting fermions in a periodically modulated lattice, we observe quasimomentum-dependent optical selection rules, which stem from interband quantum geometry of the honeycomb lattice.
We propose a general method to realize frustrated tunneling of ultracold atoms in a state-dependent optical lattice. Two typical lattice configurations are considered, the square lattice with competing interaction and the kagom\'e lattice with geometrical frustration. The ideal can be extended to implement frustrated tunneling of ultracold atoms in various geometries, which enable us to investigate the physics of frustration in both bosonic and spin systems. We study the mean-field phase diagrams of the considered models and the experimental situations are also discussed.
Ultracold atomic gases trapped at the interference of coherent beams of light constitute an artificial material. This optical lattice material may be used for controlled quantum simulations of condensed matter theories. The bulk of this dissertation concerns the construction and calibration of an optical superlattice that can form the triangular, honeycomb and kagome crystal structures. The properties and geometry of this artificial material may be dynamically changed, allowing for the experiments discussed in this thesis that would be impossible with real materials. The use of ultracold atoms in optical lattices for quantitative tests is challenging because of the novelty of many of the techniques in the field, and because of the myriad experimental differences between these artificial materials and true materials. This thesis reports the development of a method to characterize optical lattice potentials using matter-wave diffraction. We observe an enhancement of inversion asymmetry in matter-wave diffraction from a honeycomb lattice, which we explain using a time-independent perturbative treatment of the single-particle band structure of the lattice. Our experiment also provides new insight into a commonly used detection technique. This thesis culminates in the development and experimental realization of a quantitative test of a condensed-matter theory. The test is insensitive to the experimental differences between artificial materials and real materials. We focus on a prediction from a mean-field treatment of the Bose-Hubbard model that concerns the difference in behavior of itinerant particles on lattices that are identical but for their geometry. Using the tunable geometry of our quantum simulator, we measure the properties of ultracold atomic gases trapped in the triangular and kagome lattices under otherwise identical conditions and find that they are consistent with the mean-field scaling prediction.