Spectroscopy of momentum state lattices

We explore a technique for probing energy spectra in synthetic lattices that is analogous to scanning tunneling microscopy. Using one-dimensional synthetic lattices of coupled atomic momentum states, we explore this spectroscopic technique and observe qualitative agreement between the measured and simulated energy spectra for small two- and three-site lattices as well as a uniform many-site lattice. Finally, through simulations, we show that this technique should allow for the exploration of the topological bands and the fractal energy spectrum of the Hofstadter model as realized in synthetic lattices.