Fermi-Bose mixture across a Feshbach resonance

Abstract

We study a dilute mixture of degenerate bosons and fermions across a Feshbach resonance of the Fermi-Fermi scattering length ${a}_{F}$. This scattering length is renormalized by the boson-induced interaction between fermions and its value is crucial to determine the phase diagram of the system. For the mixture in a box and a positive Bose-Fermi scattering length, we show that there are three possibilities: A single uniform mixed phase, a purely fermionic phase coexisting with a mixed phase, and a purely fermionic phase coexisting with a purely bosonic one. As $1∕{a}_{F}$ is increased from a negative value to the Feshbach resonance $(1∕{a}_{F}=0)$, the region of pure separation increases and the other two regions are strongly reduced. Above the Feshbach resonance $(1∕{a}_{F}>0)$, pairs of Fermi atoms become Bose-condensed molecules. We find that these molecules are fully spatially separated from the bosonic atoms when $1∕{a}_{F}$ exceeds a critical value. For a negative Bose-Fermi scattering length we deduce the condition for collapse, which coincides with the onset of dynamical instability of the fully mixed phase. We consider also the mixture in a harmonic trap and determine the conditions for partial demixing, full demixing, and collapse. The experimental implications of our results are investigated by analyzing mixtures of $^{6}\mathrm{Li}\text{\ensuremath{-}}^{23}\mathrm{Na}$ and $^{40}\mathrm{K}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ atoms.