  Quantum simulation of many-body quantum systems using Rydberg-atom platforms
has become of extreme interest in the last years. The possibility to realize
spin Hamiltonians and the accurate control at the single atom level paved the
way for the study of quantum phases of matter and dynamics. Here, we propose a
quantum optimal control protocol to engineer current states: quantum states
characterized by Rydberg excitations propagating in a given spatially closed
tweezer networks. Indeed, current states with different winding numbers can be
generated on demand. Besides those ones with single winding number,
superposition of quantum current states characterized by more winding numbers
can be obtained. The single current states are eigenstates of the current
operator that therefore can define an observable that remains persistent at any
time. In particular, the features of the excitations dynamics reflects the
nature of current states, a fact that in principle can be used to characterize
the nature of the flow experimentally without the need of accessing high order
correlators.
