We demonstrate theoretically that photon-photon attraction can be engineered in the continuum of scattering states for pairs of photons propagating in a hollow-core photonic crystal fiber filled with cold atoms. The atoms are regularly spaced in an optical lattice configuration and the photons are resonantly tuned to an internal atomic transition. We show that the hard-core repulsion resulting from saturation of the atomic transitions induces bunching in the photonic component of the collective atom-photon modes (polaritons). Bunching is obtained in a frequency range as large as tens of GHz, and can be controlled by the inter-atomic separation. We provide a fully analytical explanation for this phenomenon by proving that correlations result from a mismatch of the quantization volumes for atomic excitations and photons in the continuum. Even stronger correlations can be observed for in-gap two-polariton bound states. Our theoretical results use parameters relevant for current experiments and suggest a simple and feasible way to induce interactions between photons.
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