  We investigate the crossover of the entanglement entropy towards its thermal
value in nearly integrable systems. We employ equation of motion techniques to
study the entanglement dynamics in a lattice model of weakly interacting
spinless fermions after a quantum quench. For weak enough interactions we
observe a two-step relaxation of the entanglement entropies of finite
subsystems. Initially the entropies follow a nearly integrable evolution,
approaching the value predicted by the Generalized Gibbs Ensemble (GGE) of the
unperturbed model. Then, they start a slow drift towards the thermal stationary
value described by a standard Gibbs Ensemble (GE). While the initial relaxation
to the GGE is independent of the interaction, the slow drift from GGE to GE
values happens on time scales proportional to the inverse interaction squared.
For asymptotically large times and subsystem sizes the dynamics of the
entropies can be predicted using a modified quasiparticle picture that keeps
track of the evolution of the fermionic occupations caused by the integrability
breaking. This picture gives a quantitative description of the results as long
as the integrability-breaking timescale is much larger than the one associated
with the (quasi) saturation to the GGE. In the opposite limit the quasiparticle
picture still provides the correct late-time behaviour, but it underestimates
the initial slope of the entanglement entropy.
