  Ultracold atoms in optical lattices offer a unique platform for investigating
disorder-driven phenomena. While static disordered site potentials have been
explored in a number of optical lattice experiments, a more general control
over site-energy and off-diagonal tunneling disorder has been lacking. The use
of atomic quantum states as "synthetic dimensions" has introduced the
spectroscopic, site-resolved control necessary to engineer new, more tailored
realizations of disorder. Here, by controlling laser-driven dynamics of atomic
population in a momentum-space lattice, we extend the range of
synthetic-dimension-based quantum simulation and present the first explorations
of dynamical disorder and tunneling disorder in an atomic system. By applying
static tunneling phase disorder to a one-dimensional lattice, we observe
ballistic quantum spreading as in the case of uniform tunneling. When the
applied disorder fluctuates on timescales comparable to intersite tunneling, we
instead observe diffusive atomic transport, signaling a crossover from quantum
to classical expansion dynamics. We compare these observations to the case of
static site-energy disorder, where we directly observe quantum localization in
the momentum-space lattice.
