Unitaryn-designs via random quenches in atomic Hubbard and spin models: Application to the measurement of Rényi entropies

We present a general framework for the generation of random unitaries based on random quenches in atomic Hubbard and spin models, forming approximate unitary n-designs, and their application to the measurement of R\'enyi entropies. We generalize our protocol presented in Elben et al. [Phys. Rev. Lett. 120, 050406 (2018)] to a broad class of atomic and spin-lattice models. We further present an in-depth numerical and analytical study of experimental imperfections, including the effect of decoherence and statistical errors, and discuss connections of our approach with many-body quantum chaos.