Periodic Reporting for period 3 - RYD-QNLO (Quantum nonlinear optics through Rydberg interaction) Reporting period: 2020-10-01 to 2022-03-31 Summary of the context and overall objectives of the project Optical photons - for all practical purposes - do not interact with each other. In fact, this fundamental property of light and other electromagnetic waves underlies all our communication technology from radio transmission to fibre-optic communication lines. More generally, our ability to generate, control and detect light has not only revolutionised modern telecommunication, but has had a broad impact on science and society, from remote sensing in meteorology and climatology, over optical tomography in biomedicine to the recent detection of gravitational waves. On the other hand, the possibility to exploit quantum mechanical phenomena such as coherence, quantum superposition and quantum entanglement of photons opens fundamentally new directions for communication and computation using light. Exploiting these novel concepts requires ability to generate and manipulate light at the level of single photons. Exploring novel approaches to generating effective interaction between individual photons thus forms the core goal of this research project.More specifically, the project employs a combination of state-of-the-art experimental techniques from atomic physics and quantum optics, including ultracold atomic gases, strongly interacting Rydberg atoms, and coherent optical control methods such as electromagnetically induced transparency to realize an optical medium inside which individual photons behave like interacting particles. Building on this novel concept, the project explores both fundamental aspects of the quanta of light as well as technological applications in quantum information. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far In the first period of the project, work has mainly focused on two concepts investigated using two distinct experimental setups.One part of the project has investigated the fundamental processes of absorption and stimulated emission at the level of single photons. Towards this end, we have realized so-called Rydberg superatoms, ensembles of many thousand laser-cooled atoms which collectively behave like a single quantum emitter strongly coupled to a single light mode. So far, we have observed in this setup how a single superatom can mediate correlations between two or even three individual photons. As planned, we have extended our experimental capabilities to produce multiple separated superatoms all coupled to the same probe mode. Currently, we are studying the superradiance of two simultaneously excited superatoms.The second part of the project is concentrated on building up a new apparatus employing ultracold Ytterbium atoms for mediating photon-photon interactions. Here, the construction of the setup is proceeding, in September 2019 we observed the first magneto-optical trap loading the different bosonic isotopes of Yb. At the moment, the work is focused on implementing the probe/control setup required for excitation of Ytterbium Rydberg atoms with single photons. In parallel, we have constructed the full laser system including the Rydberg and probe lasers, all of which are now stabilized to linewidths of a few Hz. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The superatom experiment has observed for the first time nontrivial correlations between 3 photons mediated by a single emitter. With the new capabilities of the system, we expect to push this topic of interaction between discrete numbers of emitters and photons further than any other system has so far, coupling next 2 and then a growing numbers of emitters to a single light mode, implementing a small-scale photonic quantum network.The Yb experiment is well on track to be the world-first setup using Ytterbium (or any other 2-valence electron system) for single-photon interaction. While various questions stated in the initial proposal about this new system remain - such as how high the optical depth can be pushed by exploiting the unique features of Yb - so far, the designed apparatus has fulfilled all design criteria and seems well suited to carry out the proposed research in the next periods. In particular, we hope to perform Rydberg polariton experiments with up to 10 strongly interacting polaritons in the next period, which would exceed current experiments and push beyond the limits of current theory.