  We study the dynamics of up to two Rydberg excitations and the correlation
growth in a chain of atoms coupled to a photonic crystal waveguide. In this
setup, an excitation can hop from one atom to another via exponentially
decaying exchange interactions mediated by the waveguide. An initially
localized excitation undergoes a continuous-time quantum walk for short-range
hopping, and for long-range, it experiences quasi-localization. Besides that,
the inverse participation ratio reveals a super-ballistic diffusion of the
excitation in short times, whereas, at a long time, it becomes ballistic. For
two initially localized excitations, intriguing, and complex dynamical
scenarios emerge for different initial separations due to the competition
between the Rydberg-Rydberg and exchange interactions. In particular, the
two-point correlation reveals a light-cone behavior even for sufficiently
long-range exchange interactions. Additionally, we characterize the growth of
bipartite entanglement entropy, which exhibits a global bound if only one
excitation is present in the dynamics. Finally, we analyze the effect of
imperfections due to spontaneous emission from the Rydberg state into photons
outside the waveguide and show that all physical phenomena we predict are well
within experimental reach.
