Photonic Molecules Photons are known to be able to form 'molecules' through Rydberg Blockade, gaining some emergent mass, slowing down to subluminal speed.
But what is the force binding them together? Each photon doesn't have a charge so I assume they are not bound by electromagnetism. (Or are they?)
Or is it quantum entanglement that keeps them bound to one another? If so, how does it work exactly?
 A: When light is shined through a gas of Rydberg atoms the light experiences Rydberg atom mediated photon-photon interactions. That is, the photons are not interacting directly with each other. The cross section for any proposed direct photon-photon interactions (see the Schwinger limit) are orders of magnitude below what is experimentally accessible at the moment.
Instead, similar to how photons mediate charge-charge interactions in the Standard Model, the interactions are mediated by Rydberg atoms.
The story is something like this. One photon enters the Rydberg matter medium and alters the state of the Rydberg matter. The presence of the Rydberg matter also affected the propagation of this first photon, perhaps slowing it down. Now, if a second photon comes in a short time later (such that the Rydberg matter has not relaxed from the effect of the first photon) this second photon will see a different matter configuration and thus have different propagation than it would have without the presence of the first photon. For example, perhaps it will propagate closer to its vacuum speed of light.
In the scenario I described above the first photon was slowed down by the Rydberg medium, but "cleared a path" for the second photon to propagate almost at full speed. This means that on the other side of the Rydberg medium the second photon will be 'closer' to the first photon than it would have been without the Rydberg medium. If you put a black box around the Rydberg medium and just look at photons going in and photons going out it would look to you like the photons have undergone a photon-photon attractive interaction inside the black box.
Another way to describe this is using "Rydberg polaritons". Here, when the photon enters the medium you note that, because of the photon-Rydberg interaction (mediated by the interaction of the photon electric field with the bound electron in the Rydberg atom) the eigenstates of the system are now superpositions of ground state and Rydberg state atoms with the photon either unabsorbed or absorbed. That is, the excitation within the system has some Rydberg character and some photon character. The state is a superposition of Rydberg atom and photon. This is a Rydberg polariton quasiparticle. If a second photon comes in, it forms a second Rydberg polariton. Now, these two Rydberg polaritons both have a little bit of Rydberg character to them and we know that two Rydberg atoms can have a very strong dipole-dipole interaction. This means the two Rydberg polaritons undergo an interaction with each other. Because the Rydberg polaritons have some photon character this means the photons are affected by this interaction as well.
See Repulsive photons in a quantum nonlinear medium and references therein for more information.
