## 2 Simulation Configurations  
The scope of this work is to focus on the witness beam dynamics in a stable wakefield. It's therefore convenient to use the toy model that was first introduced by [PERSON] et al. [7]. It employs a single, non-evolving \"proton\" bunch rather than the self-modulated proton bunch train as the driver travelling in an initially homogeneous plasma and an externally injected electron bunch as the witness beam trailing behind, as shown in Figure 1. This model can significantly reduce the simulation cost since we do not need to simulate the proton self-modulation every time, while the latter can be ultra time-consuming. The nominal density of the uniform plasma is \(n_{0}=7\times 10^{14}\) cm\({}^{-3}\). The non-evolving proton driver has a Lorentz factor of \(\gamma_{p_{0}}=426.29\), an RMS bunch length of \(\sigma_{zp}=40\) um, an RMS transverse size of \(\sigma_{rp}=200\) um and a charge of 2.34 nC. The driver parameters are the same as those in Ref. [7], which allow us to mimic the quasi-linear wakefield driven by the self-modulated SPS proton bunches. Figure 1: The simplified scheme of AWAKE Run 2 acceleration stage. The densities of plasma electrons (\(n_{e}\), green), the proton driver bunch (\(n_{pb}\), pink) and the electron witness bunch (\(n_{eb}\), black) are shown in colourmaps, with values normalized by the unperturbed plasma electron density \(n_{0}=7\times 10^{14}\) cm\({}^{-3}\). Particle beams propagate from left to right. \(\xi=z-ct\) is the longitudinal coordinate in the co-moving frame. The blue solid line represents the loaded longitudinal wakefield \(E_{z}\), and the red dashed line is the transverse wakefield \(W_{x}=E_{x}-cB_{y}\) at one \(\sigma_{re}\) off the longitudinal axis. \(\sigma_{re}\) is the RMS beam radius of the electron witness bunch. The baseline witness electron beam in this study has the following parameters: a charge of \(Q_{e}=120\) pC, an RMS bunch length of \(\sigma_{ze}=60\) um, an initial energy of 150 MeV (\(\gamma_{e_{0}}=295.54\)) and an initial normalized emittance of \(e_{n_{0}}=6.84\) um. The change of baseline witness beam parameters compared with those in previous studies [7,8] is a result of the evolution of the electron beamline design [14,16]. The increase in the initial emittance is due to the Coulomb scattering of the witness electrons when they penetrate through two aluminium foils before the injection point (one for the vacuum window and the other one for laser beam dump) [17]. The witness bunch radius at the injection point is chosen as the matched radius in the pure plasma ion column, which is given by [18]:  
\[\sigma_{r,ic}=\left(\frac{2\epsilon_{n0}^{2}}{\gamma_{e0}k_{p}^{2}}\right)^{1/ 4}, \tag{1}\]  
where \(\epsilon_{n0}\) is the normalized emittance at the injection point, and \(\gamma_{e0}\) is the Lorentz gamma factor of the electron beam. The plasma wave number \(k_{p}\) is given as \(k_{p}=\omega_{p}/c\), where \(c\) is the speed of light. \(\omega_{p}=\sqrt{n_{0}e^{2}/m_{0}\epsilon_{0}}\) is the plasma frequency, \(\epsilon_{0}\) is the vacuum permittivity, \(m_{0}\) and \(e\) are the rest mass and charge of an electron, respectively. For the witness parameters we considered here, Equation (1) gives the same matched beam size \(\sigma_{r,ic}=10.64\) um for both the low-charge (120-pC) and high-charge (400-pC) cases. Then the peak density \(n_{e0}=N_{e}/((2\pi)^{3/2}\sigma_{ze}\sigma_{r,ic}^{2})\) of a Gaussian bunch for the two cases can be calculated as 10 (120 pC) and 20 (400 pC) times the plasma density \(n_{0}\), respectively. Here, \(N_{e}=Q_{e}/e\) is the number of bunch electrons. Alternatively, for the bubble-regime, a more significant parameter for evaluating the bubble formation could be the normalized beam charge \(\widetilde{Q}=N_{e}k_{p}^{3}/n_{0}\)[19]. Then for the 120-pC and 400-pC electron beam, the normalized charges are calculated as 0.13 and 0.44, respectively.