Entanglement entropy production at early times

We study the early-time evolution of interacting subsystems initially prepared in a non-entangled state. We show that the entanglement entropy initially grows as $-\alpha t^2 \log(\alpha t^2)$ and determine the scale $\alpha$ in terms of the interaction strength for the following cases: (i) random-matrix Hamiltonians, (ii) random quadratic Hamiltonians, and (iii) Hamiltonians with a finite number of interactions. The result shows that, in these systems, the entanglement entropy reaches its equilibrium value on a time-scale much shorter than the decoherence time.