Rising time of entanglement between scattering spins

We investigate the time evolution of entanglement in a process where a mobile particle is scattered by static spins. We show that entanglement increases monotonically during a transient and then saturates to a steady-state value. For a quasimonochromatic mobile particle, the transient time depends only on the group velocity and width of the incoming wave packet and is insensitive to the interaction strength and spin number of the scattering particles. These features do not depend on the interaction model and can be seen in various physical settings.