  When mobile impurities are introduced and coupled to a Fermi sea, new
quasiparticles known as Fermi polarons are formed. There are two interesting,
yet drastically different regimes of the Fermi polaron problem: (I) the
attractive polaron (AP) branch, connected to pairing phenomena spanning the
crossover from BCS superfluidity to the Bose-Einstein condensation of
molecules; and (II) the repulsive branch (RP), which underlies the physics
responsible for Stoner's itinerant ferromagnetism. Here, we study Fermi
polarons in two dimensional systems, where many questions and debates regarding
their nature persist. The model system we investigate is a doped MoSe$_2$
monolayer. We find the observed AP-RP energy splitting and the quantum dynamics
of attractive polarons agree with the predictions of polaron theory. As the
doping density increases, the quantum dephasing of the attractive polarons
remains constant, indicative of stable quasiparticles, while the repulsive
polaron dephasing rate increases nearly quadratically. The dynamics of Fermi
polarons are of critical importance for understanding the pairing and magnetic
instabilities that lead to the formation of rich quantum phases found in a wide
range of physical systems including nuclei, cold atomic gases, and solids.
