  We study excitonic states of an atomic impurity in a Fermi gas, i.e., bound
states consisting of the impurity and a hole. Previous studies considered bound
states of the impurity with particles from the Fermi sea where the holes only
formed part of the particle-hole dressing. Within a two-channel model, we find
that, for a wide range of parameters, excitonic states are not ground but
metastable states. We further calculate the decay rates of the excitonic states
to polaronic and dimeronic states and find they are long lived, scaling as
$\Gamma^{\rm{Exc}}_ {\rm{Pol}} \propto ( \Delta\omega)^{5.5}$ and
$\Gamma^{\rm{Exc}}_ {\rm{Dim}} \propto (\Delta\omega)^{4}$. We also find that a
new continuum of exciton-particle states should be considered alongside the
previously known dimeron-hole continuum in spectroscopic measurements. Excitons
must therefore be considered as a new ingredient in the study of metastable
physics currently being explored experimentally.
