We begin by briefly discussing how to solve Eq. 37 exactly, thus giving an exact solution to the polaritonic quantum dynamics. There are, in principle, many possible strategies for exact quantum dynamics propagation, and we only outline one of the most commonly used strategies based on the Born-Huang expansion and subsequent propagation in the energy basis. We describe the total wavefunction of the electron-photon-nuclear DOFs using the adiabatic-Fock basis as,