  When light and matter are weakly coupled, they can be described as two
distinctive systems exchanging quanta of energy. By contrast, for very large
coupling strength, the systems hybridize and form compounds that cannot be
described in terms of light or matter only. In this Thesis, we study some
exotic properties which arise in this regime. In particular, we are interested
in the possibility to engineer quantum phase transitions in these systems. One
direction we explore is the study of two-photon coupling, a mechanism in which
matter creates or absorb photons by pairs. This mechanism creates a rich phase
diagram containing both phase transitions and instabilities. A second topic is
the use of these transitions for sensing applications. Indeed, near the
critical point, the system becomes extremely sensitive to external
perturbations. We study a setup in which a single qubit is coupled to a bosonic
field. We show that even this finite-size system displays a phase transition,
which can be used to measure the frequency of the qubit and the field with
improved accuracy. This protocol could be used to develop small-scale sensors.
Finally, we study how the ability of a system to perform certain metrological
tasks could be used to characterize and quantify nonclassicality, by using the
formalism of resource theories. The first three chapters present the main
concepts and key results in the fields of ultrastrong light-matter coupling,
superradiant phase transitions, and quantum metrology. I have strived to write
these chapters pedagogically; they can be used by non-specialists or students
as an introduction to these domains. The three other chapters present my own
research contributions. Although most of those have already been published
elsewhere, this manuscript contains additional results, remarks, and
perspectives, and should be considered as an improved version of the original
papers.
