  We discuss a scheme to measure the many-body entanglement growth during
quench dynamics with bosonic atoms in optical lattices. By making use of a 1D
or 2D setup in which two copies of the same state are prepared, we show how
arbitrary order Renyi entropies can be extracted using tunnel-coupling between
the copies and measurement of the parity of on-site occupation numbers, as has
been performed in recent experiments. We illustrate these ideas for a
Superfluid-Mott insulator quench in the Bose-Hubbard model, and also for
hard-core bosons, and show that the scheme is robust against imperfections in
the measurements.

Quantum Gases;Quantum Physics