Control of material properties under non-equilibrium condition has emerged as a promising avenue for the development of new functionalities and for the discovery of materials hidden phases. To this end we will look into the emerging (vacuum) dressed states and their connection to Floquet states in driven systems. Light dressing offers the possibility to engineer symmetry breaking which can lead to novel properties of materials (e.g.topology, light-matter hybrid condensation, etc). In this project, we seek to bring together, within the novel theoretical framework of quantum electrodynamics density-functional formalism (QEDFT), recent advances in electronic structure, quantum optics and the nonlinear spectroscopy of many-body systems to study the complex dynamics in quantum systems beyond classical limits and explore the fundamentals of materials that are embedded in quantum cavities or driving by external time dependent electromagnetic fields. Such hybrid light-matter states offer new possibilities of materials engineering by designing specific modes of the electromagnetic field in the optical cavities. Quantum cavities and Floquet materials engineering offers many opportunities for tailoring the properties of light, that is, by shaping the emission spectrum of materials or its coherence and condensation properties. We might as well ask the question if optical cavities can serve as platforms for further solid-state realization of concepts from high-energy physics, such as broken PT symmetry. Also quantum cavities can be further useful in investigating the BEC-BCS crossover in manybody quasiparticle condensation.