  We investigate the metastable repulsive branch of a mobile impurity coupled
to a degenerate Fermi gas via short-range interactions. We show that the
quasiparticle lifetime of this repulsive Fermi polaron can be experimentally
probed by driving Rabi oscillations between weakly and strongly interacting
impurity states. Using a time-dependent variational approach, we find that we
can accurately model the impurity Rabi oscillations that were recently measured
for repulsive Fermi polarons in both two and three dimensions. Crucially, our
theoretical description does not include relaxation processes to the
lower-lying attractive branch. Thus, the theory-experiment agreement
demonstrates that the quasiparticle lifetime is determined by many-body
dephasing within the upper repulsive branch rather than by the metastability of
the upper branch itself. Our findings shed light on recent experimental
observations of persistent repulsive correlations, and have important
consequences for the nature and stability of the strongly repulsive Fermi gas.
