Quantum Science of Correlated Light and Matter

The collective behaviour of matter as well as its interplay with light is one of the most important topics of modern science. Understanding it is crucial in basic research, as it holds the key to a variety of correlated quantum many-body phenomena like spin liquids or superconductvity. At the same time, this understanding forms the basis for many (quantum) technological applications which define the modern era. We discuss light-matter systems, where strong matter-matter and strong light-matter interactions are present simultaneously. From a condensed matter perspective, one might expect to tune the properties of quantum materials by quantum light and from a quantum optics perspective one might engineer interesting novel facets of quantum light originating from such entangled light-matter systems.  Our main focus is to investigate the quantum cooperativity of these correlated light-matter systems and the physical consequences of the induced long-range interactions. Specifically, I will discuss the cooperative properties of (i) the paradigmatic Dicke-Ising model being the sum of a matter-matter Ising interaction and a quantum Rabi (Dicke) Hamiltonian, (ii) quantum spin models with algebraically decaying long-range interactions, and (iii) subwavelength emitter arrays .