I think it’s typically better to understand what’s happening at the classical level before the quantum level for questions like this.

Classically, when an electromagnetic wave enters a material, the material itself responds to the electromagnetic wave because it’s composed of charged particles. The collective oscillation of this macroscopic number of charged particles itself creates an electromagnetic field. The field that you can physically observe and measure is the total electromagnetic field. Through the superposition principle, the new total field will be moving slower, and you can analyze the properties of this total field.

The reason I point this out is that intuition about quantum physics breaks down. For instance, it doesn’t necessarily makes sense to label a photon a specific ID number; photon number is not conserved. Moreover, you cannot distinguish photons of the same energy from each other: to ask if it’s the “same” photon is thus not a meaningful question to ask. Not to mention that photons don’t have classical trajectories in the usual sense, and so on.

In particular, what we conceive as a “photon” is a freely propagating quantum of light in a vacuum, without undergoing interactions. But in a medium, light is clearly very strongly with the material. Indeed, the light in the medium is physically the result of the original light wave interacting with the material. So whatever quantum particle (which is really an excitation of a quantum field) you want to use to describe what’s happening won’t behave like “ordinary” photons.

Some people will even say that you shouldn’t of think of photons like physical objects you can touch and manipulate, but rather the footprint of a quantum mechanical interaction between the electromagnetic field and whatever it is you’re talking about.

Which is all to say: photons don’t act like classical billiard balls of light, and unlike electrons are purely relativistic, so ordinary non-relativistic quantum mechanics won’t work either.