The dynamics of a mobile quantum impurity in a degenerate Fermi system is a fundamental problem in many-body physics. The interest in this field has been renewed due to recent ground-breaking experiments with ultra-cold Fermi gases. Optical creation of an exciton or a polariton in a two-dimensional electron system embedded in a microcavity constitutes a new frontier for this field due to an interplay between cavity-coupling favoring ultra-low mass polariton formation and exciton-electron interactions leading to polaron or trion formation. Here, we present cavity spectroscopy of gate-tunable monolayer MoSe$_2$ exhibiting strongly bound trion and polaron resonances, as well as non-perturbative coupling to a single microcavity mode. As the electron density is increased, the oscillator strength determined from the polariton splitting is gradually transferred from the higher-energy repulsive-exciton-polaron resonance to the lower-energy attractive-polaron manifold. Simultaneous observation of polariton formation in both attractive and repulsive branches indicate a new regime of polaron physics where the polariton impurity mass is much smaller than that of the electrons. Our findings shed new light on optical response of semiconductors in the presence of free carriers by identifying the Fermi polaron nature of excitonic resonances and constitute a first step in investigation of a new class of degenerate Bose-Fermi mixtures.
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