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\section{Floquet theory}  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 Hilbert space  times the multiphoton dimension. The contribution of the latter is in principle infinite, but here can be truncated. The spectrum of this Hamiltonian gives the bandstructure of the dressed quasiparticles. (HH: here more on Flqouet-TDDFT). In Fig.~1d are shown the Floquet bands corresponding to the pump paramters used in the ARPES simulation and the dressed bands observed in the APRES calculation are perfectly reproduced. Thus the dressed states can be interpreted as Floquet-sidebands and the process of measurement in ARPES as an effective averaging of the periodic system. (HH: elaborate this point a bit).  (The observed photo-dressing of the electronic bands can be systematically explained by The striking agreement between  Floquet theory, including the subsequent non-stationary undressing phase. Using Floquet TDDFT we can give a complete picture of theory and ARPES calculation allows to further explore  the effects parameter space  of the pump parameters, including the hybridization of Floquet bands and off-resonant effects) probe pulse. ...