To illustrate the advantage of such an approach in comparison to the, perhaps more standard, polaritonic picture, let us consider a simple scenario involving a system with one nucleus, one photonic mode, and two electronic states. In our formulation, we would propagate the photon and nucleus classically using coupled trajectories under the effect of the two electronic states. Alternatively, if we were to adopt a different EF form of the PEN wavefunction, for instance χ N (R,t)Φ N (r, q,t; R), where the nuclear wavefunction is the marginal amplitude and the photon-electronic wavefunction is the conditional amplitude, the two electronic states would be dressed by the photons, yielding the polaritonic states. The mixing of the electronic and photonic degrees of freedom would yield at least four polaritonic states, if one were to consider only the zero-photon and one-photon states. The onedimensional nuclear evolution then takes place under the effect of these four polaritonic states. This alternative representation, while conceptually interesting and widely used in the literature, might prove computationally more demanding for the CTMQC algorithm as well as for the coupled-trajectory Tully surface hopping (CTTSH) algorithm -derived as well from EF. Specifically, CTMQC requires the calculation of nonadiabatic coupling vectors at all times to evaluate nuclear forces, while both CTMQC and CTTSH require calculations of nuclear gradients of the cavity-adiabatic/polaritonic PESs at all times to evaluate the force accumulated along the trajectories. Thus, treating four polaritonic states instead of two electronic states is clearly more expensive from the computational perspective. Furthermore, we stress that we are not pioneering the idea of treating photons in terms of trajectories; other authors have utilized such an idea, for instance in the context of the vibrational strong-coupling regime .