## 3 Results and Discussion
Figure 2: TARPES snapshots of six BSTS samples: (**a**) Sb\({}_{2}\)Te\({}_{3}\), (**b**) Bi\({}_{0.5}\)Sb\({}_{1.5}\)TeSe\({}_{2}\), (**c**) Bi\({}_{0.8}\)Sb\({}_{1.2}\)Te\({}_{3}\), (**d**) Bi\({}_{2}\)Te\({}_{3}\), (**e**) Bi\({}_{2}\)Te\({}_{2.65}\)Se\({}_{0.35}\), and (**f**) Bi\({}_{2}\)Te\({}_{0.5}\)Se\({}_{2.5}\). For each sample three different pump-probe delays are shown: negative (“–” label, left) i.e., the equilibrium condition, zero (“0” label, center), +1 ps (“+” label, right). Fermi level is shown by the gray horizontal lines. Dashed black lines are guides to the eye, showing the topological surface states (TSS). Bulk bands (BB1 and BB2) are clearly visible at zero delay. Passing from panel (**a**) to panel (**f**) the samples are progressively n-doped. Color intensity maps are on a logarithmic scale. Figure 4a shows the dynamics of BB1 (the high-energy bulk band). Within experimental noise all curves are identical and can be fit by a simple phenomenological function: an exponential decay convoluted with the IRF of the instrument (red dashed line). This fit is reproduced in all other panels (b-f) for comparison. Previous investigations reported similar fast dynamics of BB1 [31]. Here we notice that, upon pumping, electrons are optically transferred to BB1 states and subsequently relax to lower energy states with a characteristic time of 220 fs, regardless of the doping level. Therefore, bulk relaxation seems to be unaffected by sample doping and in particular by the filling degree of the TSS. Moreover, in all analyzed BSTS samples, the BB2 state (the low-energy bulk band) is largely unoccupied before pumping and it is only mildly photo-populated. Thus, it can easily accommodate most of the electrons decaying from higher energy levels. According to this observation, it seems that BB1-to-BB2 is the dominant interband scattering process leading to the electronic relaxation of BB1 states. In other words, the TSS does not significantly affect the BB1 dynamics. On the other hand, there seems to be an interplay between BB2 and TSS. Their electronic dynamics are shown in panels (b-c) of Figure 4. Several relaxation mechanisms have been proposed in the past, including electron-phonon interactions [22,31], spin-mediated channels [32], bulk diffusion [18], the photovoltage effect [23,33] and surface resonant states [34]. By comparing Figure 4b, the BB2, and Figure 4c, the TSS, on the same time scale we can clearly see that: (i) both relaxation dynamics are sensibly slower than BB1; (ii) there is a clear intensity build-up for p-doped samples (i.e., darker lines) (Figure 4b); and (iii) the slowest relaxation, especially for the TSS, takes place when the DN is at the Fermi level (i.e., the Bi\({}_{0.8}\)Sb\({}_{1.2}\)Te\({}_{3}\) sample). These three considerations point out the similarities, and thus suggest a possible entanglement, between the dynamics of BB2 and TSS. In fact, some authors have already pointed out that BB2 might have a strong surface character, sharing the spin texture of the TSS [34,35]. This aspect is beyond the scope of the present study, but each of the three previous points deserve some comments. Figure 3: Photo-induced electronic population (red) and depletion (green) of the BSTS samples at 0 delay (top row) and 1 ps past the pump (bottom row). Same sample order of Figure 2 from (**a**) to (**f**). The black dashed lines highlight the TSS. The thick gray arrows connect the Dirac nodes (DN) of the BSTS compounds. The small boxes labeled 1 to 4 show the integration windows used to estimate the temporal dynamics on the different bands. The box color becomes lighter as the sample passes from p-doped (panel **a**) to n-doped (panel **f**). Point (i) is probably the most trivial. The fact that relaxation times in BB2 and TSS are longer than in BB1 can be simply understood in terms of [PERSON]'s golden rule: the higher the energy of the excited particle, the larger the number of available (empty) states it can decay into, and consequently a larger decay rate (i.e., faster decay time).