Collective modes and stability of Bose-Fermi mixtures with a BCS-BEC crossover

We investigate an ultracold Bose-Fermi mixture with a Feshbach resonance between two hyperfine states of fermions. Using a functional integral method, we calculate collective modes associated with fermion pairs and bosons in the superfluid phase. We derive a stability condition of the mixtures that is valid in the entire region of the BCS-BEC crossover. This stability condition, which smoothly connects the well-known results obtained in both the BCS and Bose-Einstein Condensate (BEC) limits, shows that sufficiently strong fermion-boson (FB) interactions destabilize the mixtures. In order to investigate the consequence of this instability, we study the ground-state energy of both the uniformly mixed and the phase-separated states. We find that FB repulsion induces a phase separation, whereas FB attraction should cause a collapse of the mixture. The possibilities for the experimental observation of such instabilities are also discussed.

Figure 1

Sound velocities of the Bose-Fermi mixture in units of $v_F$, the Fermi velocity in the BCS limit, calculated using $m_B∕m_F=7∕6$, $k_F{a}_I=0.2$, $k_F{a}_B=0.4$, and $n_B∕n_{F\sigma }=0.1$ with $k_F$ being the Fermi wave number in the BCS limit.

Figure 2

Stability condition for Bose-Fermi mixtures plotted from Eq. (36) (solid line). The dashed and dotted lines correspond to the BCS limit Eq. (37) and the BEC limit Eq. (38), respectively.

Figure 3

Fermi pressure $P_F^{\left(0\right)}+P_F^{\left(\mathrm{sc}\right)}$ plotted against interaction parameter (solid line). In the BCS limit, it approaches $2∕5{\mu }_F{n}_F$ (gray dashed line), whereas in the BEC limit, it approaches $g_F{n}_C^2∕2$ (gray dotted line).