  Interactions in an ultracold boson-fermion mixture are often manifested by
elastic collisions. In a mixture of a condensed Bose gas (BEC) and spin
polarized degenerate Fermi gas (DFG), fermions can mediate spin-spin
interactions between bosons, leading to an effective long-range magnetic
interaction analogous to Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction in
solids. We used Ramsey spectroscopy of the hyperfine clock transition in a
$^{87}$Rb BEC to measure the interaction mediated by a $^{40}$K DFG. By
controlling the bosons density we isolated the effect of mediated interactions
from mean-field frequency shifts due to direct collision with fermions. We
measured an increase of boson spin-spin interaction by a factor of
$\eta=1.45\pm 0.05^{stat} \pm 0.13^{sys}$ in the presence of the DFG, providing
a clear evidence of spin-spin fermion mediated interaction. Decoherence in our
system was dominated by inhomogeneous boson density shift, which increased
significantly in the presence of the DFG, again indicating mediated
interactions. We also measured a frequency shift due to boson-fermion
interactions in accordance with a scattering length difference of
$a_{bf_2}-a_{bf_1}=-5.36 \pm 0.44^{stat} \pm 1.43^{sys}~a_0$ between the
clock-transition states, a first measurement beyond the low-energy elastic
approximation in this mixture. This interaction can be tuned with a future use
of a boson-fermion Feshbach resonance. Fermion-mediated interactions can
potentially give rise to interesting new magnetic phases and extend the
Bose-Hubbard model when the atoms are placed in an optical lattice.
