Fig. 1. The 1D plot shows the maximum energy gained by ions (in GeV) with time in all three cases labelled. Here, a region in time is identified as , which starts at the onset of transparency and extends until the enhanced ion acceleration slows down (after which the slope of maximum ion energy begins to change to a smaller value).

Fig. 2. These subplots show 2D spatial distributions of electrons (top row, a(i)–d(i)) and ions (bottom row, a(ii)–d(ii)) in the region (only no-QED case shown).

Fig. 3. Energy-angular distribution of electrons (in a.u.) in the BOA phase without radiation reaction (a), with radiation reaction (b) and with pair production as well (c) (excluding the produced Breit–Wheeler electron density) at 80 fs.

Fig. 4. The electron phase space in the no-QED case (a), the RR modelled by the corrected Landau–Lifschitz (LL) method (b) and the RR modelled by the Monte Carlo method (c) at the onset of the BOA phase.

Fig. 5. Energy-angle distribution of photons (a), BW electrons (b) and BW positrons (c) at 80 fs.

Fig. 6. Energy-angle distribution of carbon ions in the BOA phase without the radiation reaction (a), with the radiation reaction (b) and with pair production as well (c) at 80 fs.

Fig. 7. Spectral power as a function of wave number (normalized by Debye’s length with initial temperature) and frequency (normalized by plasma frequency), , in log scale for fs and μm for all three cases (no-QED (a), RR (b) and RR+PP (c)) obtained from the simulations. The real and imaginary roots of Equation (1) (solid and dotted lines, respectively) are over-plotted to facilitate comparison.

Fig. 8. Energy-angle distribution of carbon ions in the BOA phase without the radiation reaction (a), with the radiation reaction (b) and with pair production as well (c) at 130 fs. (d) The angle-averaged ion energy distribution at the same time.

Fig. 9. Photons and pairs saturate after which the direct impact of QED effects can be assumed to be less significant. This justifies the dropping of the RR term in the Lorentz force from the Vlasov equation. QED effects are still captured in form of changes in plasma distribution.