Optical and electromagnetic signals in general are the most common carriers to communicate information over long distances, because their elementary constituents (the photons) do not interact with each other in free space, allowing a linear and undistorted propagation. On the other hand, to process the carried information, some form of interaction between the signals is required. This can be achieved when light propagates inside a material, where a nonlinear optical response can allow to achieve such effective optical interaction leading to important applications such as optical modulation and switching, nonlinear spectroscopy, and frequency conversion, commonly used in modern science and engineering. All the mentioned processes usually occur at high light intensities, due to the extremely weak nonlinear optical response of the most common materials, and over the years, many different approaches have been de-veloped among the years to increase the efficiency of such schemes, the most common in-volving cavities, nanophotonics structures and ensembles of Rydberg atoms. Despite the great efforts that have been invested along this research line, significant improvements are still necessary to make a definitive breakthrough and to fully achieve the realm of Quantum Non-linear Optics (QNLO), where nonlinear effects occur at the level of individual photons. This not only enables the realization of nonlinear classical devices operating at the lowest possible intensity levels but also allows the generation and manipulation of non-classical states of light, a task that has become even more timely in light of the emerging field of Quantum Technologies. Furthermore, the perspective to generate and control photon-photon interac-tions brings entirely new possibilities, such as potentially realizing quantum many-body physics with light. QUANLUX aimed to tackle this challenge by identifying novel promising light-matter inter-faces for QNLO protocols and by investigating the complex emergent behavior of strongly interacting photons. These objectives have been achieved via the development of theoretical frameworks and advanced numerical methods capable to solve the many-body problem of multi-photons propagation.
