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The observed quasiparticle structure can be understood in terms of Floquet theory, where a stationary state is expanded into a basis of Fourier components of multiples of the photon frequency $\Omega$: $|\psi(t)\rangle=\sum_{m}\exp(-i(\epsilon+m\Omega)t)|u_m\rangle$, where $\epsilon$ is the equilibrium energy of the state. With this ansatz the time-dependent Schr\"odinger equation becomes an eigenvalue problem $\sum_{n}\mathcal{H}^{mn} |u_n\rangle = \epsilon |u_m\rangle$ of the static Floquet Hamiltonian $\mathcal{H}^{mn} = \frac{\Omega}{2 \pi}\int_{2\pi/\Omega} dt e^{i(m-n)\Omega t} H(t) + \delta_{mn}m\Omega \mathbf{1}$. The eigenstates of this Hamiltonian span a Hilbert space with the dimension of the original electronic Hilbertspace times the multiphoton dimension. The contribution of the latter is in principle infinite, but here can be truncated.   1. Our calculations show that depending on the delay between pump and probe lasers Floquet bands appear in the photo electron spectrum at energies controlled by the parameters of the pump pulse. By simulating a slow switching off process, we show how these Floquet bands collapse and finally combine with the equilibrium bands.   Fig. 1 "movie" (including pump-probe overlap cartoons)  2. The (The  observed photo-dressing of the electronic bands can be systematically explained by Floquet theory, including the subsequent non-stationary undressing phase. Using Floquet TDDFT we can give a complete picture of the effects of the pump parameters, including the hybridization of Floquet bands and off-resonant effects. effects)  Fig. 2 "Laser parameters" a) two examples from ARPES ARPES b) parameter scans with Flquet theory c) stepping of the switch off phase. d) with Floquet theory.  \section{scaling}  3. In the case that the frequency of the pump field is on resonance with the equilibrium quasiparticle gap, the observed shift of energies resulting from hybridization of Floquet-bands are known as the ac-Stark effect. We discuss how this follows from Floqeut theory and explain the limits in which the well known formula for the shift as a function of energy holds. Among others, this gives a systematic explanation of the mechanism behind recently observerd optical Stark effects in TMDs\cite{sie_valley-selective_2014}.  Fig. 3. "beyond ac-Stark" scaling and dependence of shift on pump parameters