Proton beam self-modulation seeded by electron bunch in plasma with density ramp

Seeded self-modulation in a plasma can transform a long proton beam into a train of micro-bunches that can excite a strong wakefield over long distances, but this needs the plasma to have a certain density profile with a short-scale ramp up. For the parameters of the AWAKE experiment at CERN we numerically study which density profiles are optimal if the self-modulation is seeded by a short electron bunch. With the optimal profiles, it is possible to 'freeze' the wakefield at approximately half the wavebreaking level. High-energy electron bunches (160 MeV) are less efficient seeds than low-energy ones (18 MeV), because the wakefield of the former lasts longer than necessary for efficient seeding.