Abstract: Hamiltonian engineering can provide insights into the control and manipulation of quantum systems, which may pave the way for progress in quantum sensing and quantum many-body physics. Here, I will focus on a specific type of Hamiltonian engineering technique based on the spectroscopic coupling of atomic states. I will describe how we use two-photon Bragg resonances to couple discrete momentum states of our BEC, forming a synthetic lattice of momentum states in one dimension. Taking advantage of the site-resolved control afforded by our synthetic lattice technique, I will show some experimental results regarding transport in a one-dimensional quasiperiodic mosaic lattice. I will then discuss two-dimensional momentum state lattices. Finally, I will offer prospective directions in leveraging atomic interactions to probe many-body topological physics and squeezed states in our experimental platform.