Fermi-Fermi mixtures in the strong-attraction limit

The phase diagrams of low density Fermi-Fermi mixtures with equal or unequal masses and equal or unequal populations are described at zero and finite temperatures in the strong attraction limit. In this limit, the Fermi-Fermi mixture can be described by a weakly interacting Bose-Fermi mixture, where the bosons correspond to Feshbach molecules and the fermions correspond to excess atoms. First, we discuss the three and four fermion scattering processes, and use the exact boson-fermion and boson-boson scattering lengths to generate the phase diagrams in terms of the underlying fermion-fermion scattering length. In three dimensions, in addition to the normal and uniform superfluid phases, we find two stable nonuniform states corresponding to (i) phase separation between pure unpaired (excess) and pure paired fermions (molecular bosons); and (ii) phase separation between pure excess fermions and a mixture of excess fermions and molecular bosons. Lastly, we also discuss the effects of the trapping potential in the density profiles of condensed and noncondensed molecular bosons, and excess fermions at zero and finite temperatures, and discuss possible implications of our findings to experiments involving mixtures of ultracold fermions.

Figure 1

Diagrammatic representation of the integral equation for the boson-fermion scattering $T$ matrix $T_{\mathbf{k}}^{BF}(\mathbf{p},p_0)$. Here, we use the notation $(-\uparrow )\equiv \downarrow$ and vice versa.

Figure 2

The boson-fermion scattering length $a_0^{BF}\left(0\right)∕a_F$ versus mass anisotropy $m_{\uparrow }∕m_{\downarrow }$ is shown in (a) when lighter ↑-type (hollow circles) or heavier (solid circles) ↓-type fermions are in excess. The boson-boson scattering length $a^{BB}\left(0\right)∕a_F$ versus mass anisotropy $m_{\uparrow }∕m_{\downarrow }$ is shown in (b). In addition, the scattering lengths in the Born approximation are shown as solid lines.

Figure 3

Phase diagram of population imbalance $P=(n_{\uparrow }-n_{\downarrow })∕(n_{\uparrow }+n_{\downarrow })$ versus scattering parameter $1∕\left(K_F{a}_F\right)$ for (a) equal masses when $m_{\uparrow }=m_{\downarrow }$, and (b) unequal masses when $m_{\uparrow }=0.15m_{\downarrow }$ in the BEC side. We show normal $\left(N\right)$, uniform superfluid $\left(U\right)$, and phase-separated nonuniform superfluid phases PS(1) and PS(2).

Figure 4

Phase diagram of population imbalance $P=(n_{\uparrow }-n_{\downarrow })∕(n_{\uparrow }+n_{\downarrow })$ versus scattering parameter $1∕\left(K_F{a}_F\right)$ in the Born approximation for (a) equal masses when $m_{\uparrow }=m_{\downarrow }$, and (b) unequal masses when $m_{\uparrow }=0.15m_{\downarrow }$ in the BEC side. We show normal $\left(N\right)$, uniform superfluid $\left(U\right)$, and phase-separated nonuniform superfluid phases PS(1) and PS(2).

Figure 5

Density (in units of $K_F^3$) of condensed $\left(n_C\right)$ and noncondensed $\left(n_{\mathit{NC}}\right)$ molecular bosons, and excess $\left(n_E\right)$ fermions versus trap radius $r∕R_{\mathit{TF}}$ is shown for (a) $T=0$, (b) $T=0.2E_F$, (c) $T=0.35E_F$, and (d) $T=0.41E_F$. Here, $m_{\uparrow }=m_{\downarrow }$, the population imbalance parameter is $P=0.5$ and the scattering length parameter is $1∕\left(K_F{a}_F\right)=2$.

Figure 6

Density (in units of $K_F^3$) of ↑- $\left(n_{\uparrow }\right)$ and ↓-type $\left(n_{\downarrow }\right)$ fermions versus trap radius $r∕R_{\mathit{TF}}$ is shown for (a) $T=0$, (b) $T=0.2E_F$, (c) $T=0.35E_F$, and (d) $T=0.41E_F$. Here, $m_{\uparrow }=m_{\downarrow }$, population imbalance parameter is $P=0.5$ and scattering length parameter is $1∕\left(K_F{a}_F\right)=2$.