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The present numerical investigation of a Plasma Wakefield Accelerationscenario in the weakly non linear regime with external injection is motivatedby the upcoming campaigns at the SPARC\_LAB test facility where the final goalis to demonstrate modest gradient acceleration ($\sim$1 GV/m) with no qualityloss. The accelerated bunch can be envisioned to seed a free electron laser.The numerical study has been conducted with the particle-in-cell code ${\ttALaDyn}$, an exhaustive description of the plasma-acceleration version isprovided. The configuration consider a two bunches setup with parameters in thefacility range, the bunches are generated and pre-accelerated up to 100 MeV bya high brightness photo-injector prior plasma injection. To verify the workingpoint robustness we have considered case scenario where the driver bunchreaches the plasma or with a larger dimension or with large emittance. We alsopresent an analytical approach based on the envelope equation that allows toreduce the matching condition in the presence of a ramp. Here, we limit ourinterest to a simplified theoretical case with a linear plasma ramp. As a finalaspect we propose to combine classical integrated bunch diagnostics with thetest by Shapiro-Wilk, a mathematical test to diagnose bunch deviation from aGaussian distribution.

Design of high brightness Plasma Wakefield Acceleration experiment at SPARC\_LAB test facility with particle-in-cell simulations