  We study the time evolution of the entanglement entropy in bosonic systems
with time-independent, or time-periodic, Hamiltonians. In the first part, we
focus on quadratic Hamiltonians and Gaussian initial states. We show that all
quadratic Hamiltonians can be decomposed into three parts: (a) unstable, (b)
stable, and (c) metastable. If present, each part contributes in a
characteristic way to the time-dependence of the entanglement entropy: (a)
linear production, (b) bounded oscillations, and (c) logarithmic production. In
the second part, we use numerical calculations to go beyond Gaussian states and
quadratic Hamiltonians. We provide numerical evidence for the conjecture that
entanglement production through quadratic Hamiltonians has the same asymptotic
behavior for non-Gaussian initial states as for Gaussian ones. Moreover, even
for non-quadratic Hamiltonians, we find a similar behavior at intermediate
times. Our results are of relevance to understanding entanglement production
for quantum fields in dynamical backgrounds and ultracold atoms in optical
lattices.
