What mode of atom excitation is responsible for light slowing in transparent media? Refraction index (or speed of light in a transparent medium) is often explaned as light interacting with atoms in a way such that the atoms create a secondary wave that once added up to the primary ends up delaying it.
This supposes that light is able to excite atoms one way or another. Considering that transparency is not quantized in the sense that a large bandwidth of colors  can pass through a window, how is this atom excitation described ?
 A: Two things. Energy levels are not precise. They are precise only in the idealized world where there is nothing in the universe except electrons and protons. But in fact there are other things in the universe. Atoms couple to these other things and what was an discrete energy level spreads out a little. Or a lottle, depending on circumstances. There is a classical analog that's mathematically just about identical: two coupled pendula. Uncoupled, both have the same frequency. When coupled, the system has two frequencies, one higher the other lower than the uncoupled frequency. In a solid, there are a very large number of atom-oscillators, and what was a single natural frequency for a single atom, becomes a very large number of oscillators that spread to become a band.
Even in a single atom far from other atoms in a vacuum the energy levels are not perfectly narrow.  They have a small amount of spread.  This is also due to coupling, in this case coupling between the atom and the electromagnetic field.
(Aside #1: recall two coupled pendula.  If I start one oscillating while the other is at rest, the other pendulum will start to oscillate.  Energy is transferred from one to the other under those conditions.  Similarly, an excited atom can transfer energy to the objects that it is coupled to, that is, the EM field.  This is a semi-classical [atom quantum, field classical] way of describing spontaneous emission)
(Aside #2:  in a solid the electronic states also couple to states of motion of the nuclei, so-called phonons.  Another mechanism for broadening.)
The second point is somewhat semantic.  What do we mean by excite.  Usually we mean drive an atom into a higher energy level.  This is clearly not happening in a transparent solid.  Actually, there is a tiny absorption/excitation on account of the coupling and spreading. The lack of (this meaning of) excitation does not mean that there is no interaction.  If you admit a different, non-standard, definition of excite to mean interact you will have "excitation".  The oscillating EM field can drive the electron cloud in the atom to slosh back and forth while the much heavier, lattice-locked nucleus remains relatively quiet.  An oscillating dipole.  And from there we go to the cited Q&A above.
A: As you are asking about atoms, a transparent medium is a lattice of atoms, and quantum mechanically the photons interact with the whole lattice. If the medium is truly transparent and keeps the frequencies , i.e. images are transmitted and color is not changed  it means that the "photon -lattice" scattering is elastic , in the center of mass of the system "photon-lattice ". Because of the mass of the lattice, the laboratory system practically coincides  with the center of mass system, very very small energy is needed to go from the center of mass to the laboratory system. Thus colors (energy) and phases are practically unchanged.

transparency is not quantized in the sense that a large bandwidth of colors can pass through a window,

As the interaction is with the lattice and the process practically elastic a large band width, depending on the material, is allowed. The limits would come from the wavelength of the light, i.e. the energy of the photon. High energy photons (xray) would interact with individual atoms and very low energy ones would be absorbed in heat, vibrational levels of the lattice.
In opaque solids, the photons interact with the surface atoms, there is no organized lattice to have an elastic scattering with.
