We use a microwave field to control the quantum state of optical photons stored in a cold atomic cloud. The photons are stored in highly excited collective states (Rydberg polaritons) enabling both fast qubit rotations and control of photon-photon interactions. Through the collective read-out of these pseudospin rotations it is shown that the microwave field modifies the long-range interactions between polaritons. This technique provides a powerful interface between the microwave and optical domains, with applications in quantum simulations of spin liquids, quantum metrology and quantum networks.

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...

Recent developments in the study of ultracold Rydberg gases demand an advanced level of experimental sophistication, in which high atomic and optical densities must be combined with excellent control of external fields and sensitive Rydberg atom detection. We describe a tailored experimental system used to produce and study Rydberg-interacting atoms excited from dense ultracold atomic gases. The experiment has been optimized for fast duty cycles using a high flux cold atom source and a three beam optical dipole trap. The latter enables tuning of the atomic density and temperature over several orders of magnitude, all the way to the Bose-Einstein condensation transition. An electrode structure surrounding the atoms allows for precise control over electric fields and single-particle sensitive field ionization detection of Rydberg atoms. We review two experiments which highlight the influence of strong Rydberg-Rydberg interactions on different many-body systems. First, the Rydberg blockade effect is used to pre-structure an atomic gas prior to its spontaneous evolution into an ultracold plasma. Second, hybrid states of photons and atoms called dark-state polaritons are studied. By looking at the statistical distribution of Rydberg excited atoms we reveal correlations between dark-state polaritons. These experiments will ultimately provide a deeper understanding of many-body phenomena in strongly-interacting regimes, including the study of strongly-coupled plasmas and interfaces between atoms and light at the quantum level.

Far-from-equilibrium dynamics that lead to self-organization are highly relevant to complex dynamical systems not only in physics but also in life, earth, and social sciences. However, it is challenging to find systems with sufficiently controllable parameters that allow quantitatively modeling of emergent properties. Here, we study a nonequilibrium phase transition and observe signatures of self-organized criticality in a dilute thermal vapor of atoms optically excited to strongly interacting Rydberg states. Electromagnetically induced transparency provides excellent control over the population dynamics and enables highresolution probing of the driven-dissipative dynamics, which also exhibits phase bistability. Increased sensitivity compared to previous work allows us to reconstruct the complete phase diagram, including in the vicinity of the critical point. We observe that interaction-induced energy shifts and enhanced decay only occur in one of the phases above a critical Rydberg population. This case limits the application of generic mean-field models; however, a modified, threshold-dependent approach is in qualitative agreement with experimental data. Near threshold, we observe self-organized dynamics in the form of population jumps that return the density to a critical value.