This theoretical work investigates the strong light-matter coupling between an optical mode inside a planar microcavity and an embedded doped quantum well. In particular, the properties of the resulting quasi particles, the (exciton or trion)-polaritons being a coherent superposition of a photon and exciton (trion), are studied as a function of electron density. Over the last decade, polaritons have attracted a lot of attention because of their strong optical non-linearity and their bosonic character. Initiated by the milestone experiment that realized a Bose-Einstein condensate of polaritons, much progress has been made on polaritons and their superfluid properties. More recently, the possibility for quantum information processing with polaritons is exploited and engineering microcavity polaritons to use them as a single photon source is of major interest. The thesis only focusses on the linear properties of the polaritons, meaning that polariton-polariton interactions are not considered here.

An historical survey through the many possibilities for realizing strong light-matter coupling is presented. We also give an overview of what kind of experiments have been performed on this systems to characterize the polaritons. The basic concepts and relevant quantities are presented to set the stage for more elaborate approaches used later on. In particular, we will introduce the microcavity, the quantum well and its elementary excitations, the exciton and trion. The exciton-polariton as a coupled two level is introduced in the planar microcavity set-up. Then, we put the focus on the addition of extra charge carriers inside the quantum well. Several interesting many-body effects, such as the Fermi edge singularity (FES) and Anderson Orthogonality Catastrophe (AOC) are briefly discussed to motivate this work.

Then, the polariton formation between an optical mode and a highly doped quantum well containing a quantum degenerate two-dimensional electron gas is discussed. Here, highly doped means that the average inter-electron distance is less than the exciton Bohr radius. The most difficult ingredient in describing the linear polariton properties turns out to be the calculation of the optical properties of the two-dimensional electron gas; due to the absorption of a photon, a valence band hole and conduction band electron are created within the quantum well. The valence band hole then acts as an attractive potential for all the electrons. Within some approximations, we compute the 2DEG optical susceptibility, fully taking into account the above mentioned many-body effects of the FES and AOC. In a second stage we couple the 2DEG with the electromagnetic field. The resulting eigenmodes of the coupled system are discussed and their single particle properties, such as the effective mass and the Rabi frequency are elucidated.

An important assumption in the previous part is for the valence band hole to have infinite mass. The latter is generally not true and we use a perturbative calculation to take into account this finite hole mass. The Mahan exciton will be introduced and we calculate a lowest order contribution in the Lindhard polarization of the 2DEG for the lifetime of the Mahan polaritons. In general, i.e. even without taking into account the finite mass, the results of this part will be accurate if the Fermi time exceeds the period of the Rabi oscillations.

When lowering the electron density, there will be a regime in which the average electron distance exceeds the exciton Bohr radius. The exciton can then be described as a single bosonic entity, from which the electrons can scatter. We introduce this effective description of the exciton and the trion quasiparticle as an electron bound to an exciton. Again, the 2DEG optical susceptibility is computed within this particular model and we investigate the trion-polariton, being the coherent superposition of the optical mode inside the cavity and the trion inside the quantum well. It will be shown that this model captures some major experimentally observed features such as the transfer of oscillator strength and asymmetric lineshapes. The coupling with the photon field is then discussed. We also focus on the spatial structure (`the size') of the trionic component in the trion-photon superposition as a function of Fermi energy and Rabi frequency. The latter is an important quantity to set the stage for further research on polariton-polariton interactions, where it was pointed out that the electrons could enhance the polariton interaction strength. By computing the size of the trion, we obtain a first estimate of how polaritons could interact in presence of a 2DEG.