  We present a scheme for measuring R\'enyi entropies in generic atomic Hubbard
and spin models using single copies of a quantum state and for partitions in
arbitrary spatial dimension. Our approach is based on the generation of random
unitaries from random quenches, implemented using engineered time-dependent
disorder potentials, and standard projective measurements, as realized by
quantum gas microscopes. By analyzing the properties of the generated unitaries
and the role of statistical errors, with respect to the size of the partition,
we show that the protocol can be realized in exisiting AMO quantum simulators,
and used to measure for instance area law scaling of entanglement in
two-dimensional spin models or the entanglement growth in many-body localized
systems.
