Hollow Condensates, Topological Ladders and Quasiperiodic Chains.

다국어 초록 (Multilingual Abstract)

This thesis presents three distinct topics pertaining to the intersection of condensed matter and atomic, molecular and optical (AMO) physics. We theoretically address the physics of hollow Bose-Einstein condensates and the behavior of vortices within them then discuss localization-delocalization physics of one-dimensional quasiperiodic models, and end by focusing on the physics of localized edge modes and topological phases in quasi-one-dimensional ladder models. For all three topics we maintain a focus on experimentally accessible, physically realistic systems and explicitly discuss experimental implementations of our work or its implications for future experiments. First, we study shell-shaped Bose-Einstein condensates (BECs). This work is motivated by experiments aboard the International Space Station (ISS) in the Cold Atom Laboratory (CAL) where hollow condensates are being engineered. Additionally, shell-like structures of superfluids form in interiors of neutron stars and with ultracold bosons in three-dimensional optical lattices. Our work serves as a theoretical parallel to CAL studies and a step towards understanding these more complex systems. We model hollow BECs as confined by a trapping potential that allows for transitions between fully-filled and hollow geometries. Our study is the first to consider such a real-space topological transition. We find that collective mode frequencies of spherically symmetric condensates show non-monotonic features at the hollowing-out point. We further determine that for fully hollow spherically symmetric BECs effects of Earth's gravity are very destructive and consequently focus on microgravity environments. Finally, we study quantized vortices on hollow condensate shells and their response to system rotation. Vortex behavior interesting as a building block for studies of more complicated quantum fluid equilibration processes and physics of rotating neutron stars interiors. Condensate shells' closed and hollow geometry constrains possible vortex configurations. We find that those configurations are stable only for high rotation rates. Further, we determine that vortex lines nucleate at lower rotation rates for hollow condensates than those that are fully-filled.Second, we analyze the effects of quasiperiodicity in one-dimensional systems. Distinct from truly disordered systems, these models exhibit delocalization in contrast to well-known facts about Anderson localization. We study the famous Aubry-Andre-Harper (AAH) model, a one-dimensional tight-binding model that localizes only for sufficiently strong quasiperiodic on-site modulation and is equivalent to the Hofstadter problem at its critical point. Generalizations of the AAH modelhave been studied numerically and a generalized self-dual AAH model has been proposed and analytically analyzed by S. Ganeshan, J. Pixley and S. Das Sarma (GPD). For extended and generalized AAH models the appearance of a mobility edge i.e. an energy cut-off dictating which wavefunctions undergo the localization-delocalization transition is expected. For the GPD model this critical energy has been theoretically determined. We employ transfer matrices to study one-dimensional quasiperiodic systems. Transfer matrices characterize localization physics through Lyapunov exponents. The symplectic nature of transfer matrices allows us to represent them as points on a torus. We then obtain information about wavefunctions of the system by studying toroidal curves corresponding to transfer matrix products. Toroidal curves for localized, delocalized and critical wavefunctions are distinct, demonstrating a geometrical characterization of localization physics. Applying the transfer matrix method to AAH-like models, we formulate a geometrical picture that captures the emergence of the mobility edge. Additionally, we connect with experimental findings concerning a realization of the GPD model in an interacting ultracold atomic system.Third, we consider a generalization of the Su-Schrieffer-Heeger (SSH) model. The SSH chain is a one-dimensional tight-binding model that can host localized bound states at its ends. It is celebrated as the simplest model having topological properties captured by invariants calculated from its band-structure. We study two coupled SSH chains i.e. the SSH ladder. The SSH ladder has a complex phase diagram determined by inter-chain and intra-chain couplings. We find three distinct phases: a topological phase hosting localized zero energy modes, a topologically trivial phase having no edge modes and a phase akin to a weak topological insulator where edge modes are not robust. The topological phase of the SSH ladder is analogous to the Kitaev chain, which is known to support localized Majorana fermion end modes. Bound states of the SSH ladder having the same spatial wavefunction profiles as these Majorana end modes are Dirac fermions or bosons. The SSH ladder is consequently more suited for experimental observation than the Kitaev chain. For quasiperiodic variations of the inter-chain coupling, the SSH ladder topological phase diagram reproduces Hofstadter's butterfly pattern. This system is thus a candidate for experimental observation of the famous fractal. We discuss one possible experimental setup for realizing the SSH ladder in its Kitaev chain-like phase in a mechanical meta-material system. This approach could also be used to experimentally study the Hofstadter butterfly in the future.Presented together, these three topics illustrate the richness of the intersection of condensed matter and AMO physics and the many exciting prospects of theoretical work in the realm of the former combining with experimental advances within the latter.