## 3 Simulation Results
The electron injection process is shown by the shaded area, which occurred once the general injection threshold of a\({}_{0}\)\(\sim\)2.4, marked by the green shaded area, was reached. (**b,c**) The corresponding pulse duration, peak energy, charge, and RES of \(e\) beams for different cases. (**d–f**) Snapshots of the laser-driven plasma electron distribution in the r-z plane for the three cases at the same time. \(e\) beam with a shorter duration of less than 1 fs can be produced with a brightness of \(\sim 3\times 10^{15}\) A/m\({}^{2}\)/0.1%. Compared with our previous work [27; 37] based on the cascaded acceleration, in which \(e\) beams were injected from the second bubble into the first bubble to realize further acceleration, these \(e\) beams were injected and accelerated in the first bubble in this scheme. Using this specific plasma-density profile, the density plateau and up-ramp before the peak of the density profile offer an appropriate evolution of the self-focusing of the laser to increase its intensity and the amplitude of the driven wake. In close proximity to the injection threshold, the transient decrease in the wake phase occurs, and the injection only occurs near the end of the density down-ramp, resulting in a very localized injection and, thus, formation of an \(e\) beam with an ultrashort length. For a certain point, this may be used to tune the injection and the duration length of the injected \(e\) beam, which may be approximated as the one in [36], indicating that the injection of ultrashort bunches with attosecond durations is possible here. The overall evolution of the laser pulse is plotted in Figure 4a for the case of zb = 450 \(\upmu\)m, where the laser's evolution in the presented plasma density has played an important role in keeping the energy spread at the per-mille level. As shown in Figure 4b, the defocusing of the laser induced the contraction of the bubble and the transformation of the accelerating field from a highly nonlinear type to a sawtooth-like type. With a proper injection phase, the accelerated \(e\) beam could gradually slip back into the rear of the bubble from the positive-slope accelerating field to the negative one. The chirp of the \(e\) beam, which would be positive due to the highly nonlinear accelerating field, would be mitigated during this process, as shown in Figure 4c. The whole evolution of the relative and absolute energy spread is included in Figure 4d, which indicates that the dechirping process benefits the low energy spread in this scheme (the underlying physics will be explained in more detail in a separate publication in the future). As observed, the bright \(e\) beam with a peak energy of 821 MeV, a peak current of 8 kA, and an rms energy spread of 0.5% can be achieved, which is in good agreement with the measured one. Figure 4: (**a**) The evolution of the transverse beam intensity along the laser propagation for \(z_{b}=450\)\(\upmu\)m. The self-focusing process of the laser can be identified. (**b**) Demonstration of a snapshot of electron density distribution in the r-z plane for z = 3.8 mm and the on-axis plasma field lineouts (blue line) in the PIC simulation. (**c**) The schematic of a dechirping process. The chirps of \(e\) beams at different propagation distances are marked with dashed lines. Shaped areas show the spectra of corresponding \(e\) beams at different z. (**d**) The evolution of the peak energy, the RES, and the absolute energy spread (AES) with respect to the propagation distance.