  A resonantly excited coherent phonon leads to a periodic oscillation of the
atomic lattice in a crystal structure bringing the material into a
non-equilibrium electronic configuration. Periodically oscillating quantum
systems can be understood in terms of Floquet theory and we show these concepts
can be applied to coherent lattice vibrations reflecting the underlying
coupling mechanism between electrons and bosonic modes. This coupling leads to
dressed quasi-particles imprinting specific signatures in the spectrum of the
electronic structure. Taking graphene as a paradigmatic material we show how
the phonon-dressed states display an intricate sideband structure revealing
electron-phonon coupling and topological ordering. This work establishes that
the recently demonstrated concept of light-induced non-equilibrium Floquet
phases can also be applied when using coherent phonon modes for the dynamical
control of material properties. The present results are generic for bosonic
time-dependent perturbations and similar phenomena can be observed for plasmon,
magnon or exciton driven materials.
