Bose-Einstein condensates (BECs) in light-induced synthetic gauge fields and spaces can provide a highly-tunable platform for quantum simulations. Chapter 1 presents a short introduction to the concepts of BECs and our BEC machine. Chapter 2 introduces some basic ideas of how to use light-matter interactions to create synthetic gauge fields and spaces for neutral atoms. Three main research topics of the thesis are summarized below.

Chapter 3: Recently, using bosonic quasiparticles (including their condensates) as spin carriers in spintronics has become promising for coherent spin transport over macroscopic distances. However, understanding the effects of spin-orbit (SO) coupling and many-body interactions on such a spin transport is barely explored. We study the effects of synthetic SO coupling (which can be turned on and off, not allowed in usual materials) and atomic interactions on the spin transport in an atomic BEC.

Chapter 4: Interplay between matter and fields in physical spaces with nontrivial geometries can lead to phenomena unattainable in planar spaces. However, realizing such spaces is often impeded by experimental challenges. We synthesize real and curved synthetic dimensions into a Hall cylinder for a BEC, which develops symmetry-protected topological states absent in the planar counterpart. Our work opens the door to engineering synthetic gauge fields in spaces with a wide range of geometries and observing novel phenomena inherent to such spaces.

Chapter 5: Rotational properties of a BEC are important to study its superfluidity. Recent studies have found that SO coupling can change a BEC's rotational and superfluid properties, but this topic is barely explored experimentally. We study rotational dynamics of a SO-coupled BEC in an effective rotating frame induced by a synthetic magnetic field. Our work may allow for studying how SO coupling modify a BEC's rotational and superfluid properties.

Chapter 6 presents some possible future directions.