Atom-atom interactions around the band edge of a photonic crystal waveguide

Tailoring the interactions between quantum emitters and single photons constitutes one of the cornerstones of quantum optics. Coupling a quantum emitter to the band edge of a photonic crystal waveguide (PCW) provides a unique platform for tuning these interactions. In particular, the crossover from propagating fields $E(x) propto e^{pm ik_x x}$ outside the bandgap to localized fields $E(x) propto e^{-kappa_x |x|}$ within the bandgap should be accompanied by a transition from largely dissipative atom-atom interactions to a regime where dispersive atom-atom interactions are dominant. Here, we experimentally observe this transition for the first time by shifting the band edge frequency of the PCW relative to the $rm D_1$ line of atomic cesium for $bar{N}=3.0pm 0.5$ atoms trapped along the PCW. Our results are the initial demonstration of this new paradigm for coherent atom-atom interactions with low dissipation into the guided mode.