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Ultracold atoms in optical lattices offer a unique platform for investigatingdisorder-driven phenomena. While static disordered site potentials have beenexplored in a number of optical lattice experiments, a more general controlover site-energy and off-diagonal tunneling disorder has been lacking. The useof atomic quantum states as "synthetic dimensions" has introduced thespectroscopic, site-resolved control necessary to engineer new, more tailoredrealizations of disorder. Here, by controlling laser-driven dynamics of atomicpopulation in a momentum-space lattice, we extend the range ofsynthetic-dimension-based quantum simulation and present the first explorationsof dynamical disorder and tunneling disorder in an atomic system. By applyingstatic tunneling phase disorder to a one-dimensional lattice, we observeballistic quantum spreading as in the case of uniform tunneling. When theapplied disorder fluctuates on timescales comparable to intersite tunneling, weinstead observe diffusive atomic transport, signaling a crossover from quantumto classical expansion dynamics. We compare these observations to the case ofstatic site-energy disorder, where we directly observe quantum localization inthe momentum-space lattice.

Ballistic, diffusive, and arrested transport in disordered momentum-space lattices