Dipole Oscillation in a Bose–Fermi Superfluid Mixture of ^41K and ^6Li

We study the relation between the bosonic frequency shift of a dipole oscillation and the fermionic compressibility in a Bose–Fermi superfluid mixture. Our results show that the \it \Lambda transition occurs in such a system, which interprets the perplexing phenomenon observed in the recent experiment arXiv:1705.04496. In that experiment, the frequency shift of the bosonic gas was measured with different fermion-fermion s-wave scattering lengths a_\rm f. The most striking feature of their measurement is that the frequency shift shows a puzzling non-monotonic behavior around 1/k_\rm Fa_\rm f\simeq-0.2 with k_\rm F being the Fermi momentum. In the present work, the relation between the bosonic frequency shift and the compressibility of the Fermi gas is revealed by means of the equation of state of the superfluid mixture. Making use of the relation, we find that the compressibility of Fermi gas shows a \it \Lambda-like shape around 1/k_\rm Fa_\rm f\simeq-0.2. We would like to stress that no free parameter is used in our calculation and our results can be further testified in a future experiment.