In conventional nonlinear optics, linear quantum optics 1,2 , and cavity quantum electrodynamics 3,4 to create e ective photonphoton interactions photons must have, at one time, interacted with matter inside a common medium. In contrast, in Rydberg quantum optics [5][6][7][8][9][10] , optical photons are coherently and reversibly mapped onto collective atomic Rydberg excitations 11 , giving rise to dipole-mediated e ective photon-photon interactions that are long range 12,13 . Consequently, a spatial overlap between the light modes is no longer required. We demonstrate such a contactless coupling between photons stored as collective Rydberg excitations in spatially separate optical media. The potential induced by each photon modifies the retrieval mode of its neighbour 7,9,14,15 , leading to correlations between them. We measure these correlations as a function of interaction strength, distance and storage time, demonstrating an e ective interaction between photons separated by 15 times their wavelength. Contactless e ective photon-photon interactions 16 are relevant for scalable multichannel photonic devices 15,17 and the study of strongly correlated many-body dynamics using light 18 . In vacuum, photon-photon interactions are so weak that they are discernible only at cosmological scales 19 . This is a remarkable asset for astronomy, imaging and communications, but a serious obstacle for all-optical processing. In linear quantum optics, one can engineer an effective interaction via measurement 1 which enables entanglement swapping between remote photons 2 ; however, this relies on probabilistic post-selective measurements. Inside a medium, interactions between light fields are possible if the lightmatter coupling is nonlinear. For sufficiently strong nonlinearities, such as in cavity quantum electrodynamics (cQED) 3 or Rydberg quantum optics 5 , one enters a quantum nonlinear regime 20 , allowing deterministic effective interactions at the single-photon level 4,[7][8][9][10] . This has paved the way towards prototypical single-photon transistors 21,22 , phase-shifters 23 , and photon gates 24 . A unique feature of Rydberg-mediated quantum nonlinear optics 5 is that the interaction is long range 12,13 , and can be mediated through free space by virtual microwave photons. This adds new geometric degrees of freedom to the nonlinear optics tool box which could facilitate scalable photonic networks, quantum simulation with photons, and circumvent 'no-go' theorems that limit the scope for all-optical quantum information processing (QIP) [15][16][17]25 . To directly demonstrate the long-range character of Rydberg-mediated nonlinear optics, we realize two strongly interacting photonic channels in independent optical media separated by an adjustable distance d of more than 10 µm. While stored as collective Rydberg excitations, van der Waals (vdW) interactions imprint non-uniform phase shifts 14,15 that lead to reduced retrieval in the photons' initial modes (Fig. 1). By counting photons in the unperturbed modes, the red...

The combination of electromagnetically-induced transparency (EIT) and Rydberg excitations in atomic media is a compelling and versatile platform for both applications in quantum information, and fundamental studies of nonlinear quantum optics and non-local quantum dynamics. In this paper, we study the dynamics of a Rydberg-EIT system in a medium that allows for more than one Rydberg excitation in the propagation direction. We study the cross-over between coherent collective emission ('flash') of two-level atoms to a Rydberg dressed regime. The complex dynamics are studied using both intensity and time correlation measurements. We show that while steady-state EIT gives a second order correlation g (2) = 0.79 ± 0.04, the Rydberg-dressed flash exhibits anti-bunching down to 0.2 ± 0.04.

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