  Understanding the behavior of an impurity strongly interacting with a Fermi
sea is a long-standing challenge in many-body physics. When the interactions
are short-ranged, two vastly different ground states exist: a polaron
quasiparticle and a molecule dressed by the majority atoms. In the
single-impurity limit, it is predicted that at a critical interaction strength,
a first-order transition occurs between these two states. Experiments, however,
are always conducted in the finite temperature and impurity density regime. The
fate of the polaron-to-molecule transition under these conditions, where the
statistics of quantum impurities and thermal effects become relevant, is still
unknown. Here, we address this question experimentally and theoretically. Our
experiments are performed with a spin-imbalanced ultracold Fermi gas with
tunable interactions. Utilizing a novel Raman spectroscopy combined with a
high-sensitivity fluorescence detection technique, we isolate the quasiparticle
contribution and extract the polaron energy, spectral weight, and the contact
parameter. As the interaction strength is increased, we observe a continuous
variation of all observables, in particular a smooth reduction of the
quasiparticle weight as it goes to zero beyond the transition point. Our
observation is in good agreement with a theoretical model where polaron and
molecule quasiparticle states are thermally occupied according to their quantum
statistics. At the experimental conditions, polaron states are hence populated
even at interactions where the molecule is the ground state and vice versa. The
emerging physical picture is thus that of a smooth transition between polarons
and molecules and a coexistence of both in the region around the expected
transition.
