Recent progress in various areas of physics has demonstrated our ability to control quantum effects in customized systems and materials, thus paving the way for a promising future for quantum technologies. The emergence of such quantum devices, however, requires one to understand fundamental problems in non-equilibrium statistical physics, which can pave the way towards full control of quantum systems, thus reinforcing new applications and providing innovative perspectives. This project is dedicated to the study and the control of out-of-equilibrium properties of quantum many-body systems which are driven across phase transitions. Among several approaches, it will mainly focus on slow quenches and draw on the understanding delivered by the Kibble-Zurek (KZ) mechanism. This rather simple paradigm connects equilibrium with out-of-equilibrium properties and constitutes a benchmark for scaling hypothesis. It could pave the way towards tackling relevant open questions, which lie at the heart of our understanding of out-of-equilibrium dynamics and are key issues for operating in a robust way any quantum simulator. Starting from this motivation, we will test the limits of validity of the KZ dynamics by analyzing its predictions, thus clarifying its predictive power, and extend this paradigm to quantum critical systems with long-range interactions and to topological phase transitions. We will combine innovative theoretical ideas of condensed-matter physics, quantum optics, statistical physics and quantum information, with advanced experiments with ultracold atomic quantum gases. Quantum gases are a unique platform for providing model systems with the level of flexibility and control necessary for our ambitious goal. Their cleanness and their robustness to decoherence will greatly enhance the efficient interplay between theory and experiments, and provide a platform of studies whose outcomes are expected to have a strong scientific impact over a wide range of disciplines. On the short time scale we will exploit this knowledge to develop viable protocols for quantum simulators. In general, we expect that the results of this project will lay the ground for the development of the next generation of quantum devices and simulators.n of the proposed research, which would lay the ground for future device/simulator development in the mid-term. Our proposed work lies deeply within the “Quantum Technologies” theme. More specifically, by providing a deeper understanding and direct control of out-of-equilibrium phenomena in quantum many-body systems, we will make impactful contributions to the areas of “Quantum simulation” and “Quantum metrology, sensing and imaging”. Firstly, we will make significant advances to the initialization of a quantum system in a well-controlled initial state (ground state, without defects) and optimize the adiabatic control of its time evolution to an “interesting” target state, both of which are crucial features for adiabatic quantum computing. It is expected that the initial state and target state could be separated by a phase transition, which brings to the fore the question of the time evolution of a quantum system near a critical point. The response of a system when driven across critical points is further relevant for developing atomtronic devices and quantum sensors at the limit, that may find applications to the detection of extremely weak signals. This could include applications in diverse fields such as the detection of dark matter. Our consortium is composed of world-leading scientists with pioneering contributions in non-equilibrium dynamics of ultracold atomic systems and possesses a unique combination of the relevant expertise and tools for the successful completion of the proposed research. We expect that the results of this project will lay the ground for the development of the next generation of quantum devices and simulators.