  Two-dimensional semiconductors inside optical microcavities have emerged as a
versatile platform to explore new hybrid light-matter quantum states. The
strong light-matter coupling leads to the formation of exciton-polaritons,
which in turn interact with the surrounding electron gas to form quasiparticles
called polaron-polaritons. Here, we develop a general microscopic framework to
calculate the properties of these quasiparticles such as their energy and the
interactions between them. From this, we give microscopic expressions for the
parameters entering a Landau theory for the polaron-polaritons, which offers a
simple yet powerful way to describe such interacting light-matter many-body
systems. As an example of the application of our framework, we then use the
ladder approximation to explore the properties of the polaron-polaritons. We
furthermore show that they can be measured in a non-demolition way via the
light transmission/reflection spectrum of the system. Finally, we demonstrate
that the Landau effective interaction mediated by electron-hole excitation is
attractive leading to red shifts of the polaron-polaritons. Our work provides a
systematic framework to study exciton-polaritons in electronically doped
two-dimensional materials such as novel van der Waals heterostructures.

Mesoscale and Nanoscale Physics;Quantum Gases