In Fig. , we present in the top panel the evolution of the expectation value of the photon number N and in the bottom panel the time trace of the electronic population in the excited state S 1 . In the figure, the reference QD results are in black, CTMQC is in blue, MTE in red and TSH in orange. We remind that we used here the parameters ω = 0.17 E h and g = 0.01 (ea 0 ) -1 ; the photon-nuclear density is centered initially at R = 2.0 a 0 and q = 0.0 a 0 . Since the cavity is in resonance with the S 0 -to-S 1 transition at the Franck-Condon point, where the dynamics starts, electronic population is immediately transferred from the excited state to the ground state by emitting photons, as the initial increase of the average photon number confirms. All trajectory methods underestimate these effects, i.e., the transfer of population and the creation of photons. However, CTMQC and MTE agree better with the reference than TSH. Overall, the fast oscillations of the average photon number, related to the oscillation of the velocity field along q discussed above in Fig. , are well-captured by CTMQC and MTE, while they appear to be unstable in TSH.