Exploring the interplay of topology, disorder, kinetic frustration, and interactions in synthetic momentum-space lattices

There has been much success over the past few decades in exploring coherent quantum dynamics of cold atoms in pristine, homogeneous real-space optical lattices. The recent development of highly-tunable synthetic lattices, based on parametric coupling between discrete quantum states, promises to open up myriad new systems and phenomena to experimental investigation. We describe our efforts to create synthetic lattices based on discrete momentum states of neutral atoms, which can be parametrically coupled with interfering Bragg laser fields. The unique spectroscopic control over all state-to-state transitions in our synthetic lattice allows us to create almost any single-particle tight-binding Hamiltonian, featuring nearly arbitrary arrangements of tunneling terms, artificial gauge fields, and site-energy landscapes. In addition, a key aspect of our approach based on cold atoms is the natural presence of nonlinear interactions, which can lead to emergent, correlated phenomena. We describe several unique problems that this synthetic lattice-based approach has allowed us to explore, dealing with the interplay of topology and disorder, disorder and kinetic frustration, and disorder and interactions.