Engineering Topological Quantum Matter with Patterned Light

Topological phases are intriguing phases of matter which cannot be described with traditional characterization methods, and numerous efforts has been put to achieve these exotic phases of matter in a variety of quantum platforms. In this thesis, we discuss how topological quantum states of matter can be engineered by utilizing spatially patterned light, which has become available thanks to the recent advances in beam shaping techniques. First, we discuss a scheme to construct an optical lattice to confine ultracold atoms on the surface of torus. We investigate the feasibility of this construction with numerical calculations including the estimation of tunneling strengths. We then propose a supercurrent generation experiment to verify the non-trivial topology of the created surface. We propose a scheme to construct fractional quantum Hall states which can demonstrate topological degeneracy. We show how our scheme can be generalized to surfaces with higher genus for exploration of richer topological physics. Next, we extend our effort for creation of topologically non-trivial surfaces for ultracold atoms to the surfaces with open boundaries. This becomes possible by constructing a bilayer optical lattice with multiple pairs of twist defects. We explain how a spin-dependent optical lattice can serve as the bilayer optical lattice for this purpose. We discuss how fractional quantum Hall states can be loaded on this surface, as well as manipulation and measurement techniques via optical protocols. Then we turn our attention to electronic systems irradiated by spatially patterned light. In particular, we investigate a way to imprint the superlattice structure in the two-dimensional electronic systems by shining circularly-polarized light. We demonstrate the wide optical tunability of this system allows one to realize a wide variety of band properties. We show that these tunable band properties lead to exotic physics ranging from the topological transitions to the creation of nearly flat bands, which can allow the realization of strongly correlated phenomena in Floquet systems. Finally, we investigate the Floqut vortex states created by shining light carrying non-zero orbital angular momentum on a 2D semiconductor. We analytically and numerically study the properties of those vortex states, with the methods analogous tothe ones applied to superconducting vortex states. We show that such Floquet vortex states exhibit a wide range of tunability, and illustrate the potential utility of such tunability with an example application in quantum state engineering