Atom-dimer scattering and stability of Bose and Fermi mixtures

Motivated by a recent experiment by the Ecole Normale Supérieure de Lyon (ENS) group on the mixture of Bose and Fermi superfluids [I. Ferrier-Barbut et al., Science 345, 1035 (2014)], we investigate the effective scattering between a bosonic atom and a molecule (dimer) of fermion atoms. It is found that the mean-field prediction of the atom-dimer scattering length $\left(a_{\mathrm{ad}}\right)$, as simply given by the boson-fermion scattering length $\left(a_{\mathrm{bf}}\right)$, generically fails. Instead, $a_{\mathrm{ad}}$ crucially depends on the ratio between $a_{\mathrm{bf}}$ and $a_{\mathrm{ff}}$ (the fermion-fermion scattering length), and in addition it log-periodically depends on the three-body parameter. We identify the universal parameters in characterizing $a_{\mathrm{ad}}$ for a wide range of $a_{\mathrm{ff}}$ in the molecular side of the fermion-fermion Feshbach resonance, and further demonstrate that the atom-dimer many-body system can become unstable against either phase separation or collapse as tuning $a_{\mathrm{ff}}$. Our results have some implications for the ENS experiment.

Figure 1

Verification of universal formula for $a_{\mathrm{ad}}$ [in units of $a_{\mathrm{bf}}$, see Eq. (1)] at $x=a_{\mathrm{bf}}/a_{\mathrm{ff}}=2$. The square points show numerical results by solving Eqs. (6) and (7) for different momentum cutoffs ${\kappa }^{*}$. The green curve is the fit to Eq. (1) with $s_0=1.00624,C_1=0.29,C_2=0.41,\Phi =-0.34\pi$. The fit gets more accurate for larger ${\kappa }^{*}{a}_{\mathrm{bf}}$.

Figure 2

Universal parameters $C_1,C_2$ (upper panel), and $\Phi$ (lower panel) as functions of $x$. In (a1) and (b1), $x<0$; in (a2) and (b2), $x>1$.

Figure 3

$a_{\mathrm{ad}}$ in the unitary limit of $a_{\mathrm{bf}}$. The length unit here is $a_{\mathrm{ff}}$. (a) $a_{\mathrm{ad}}$ as changing $a_{\mathrm{bf}}$ to $+\infty$ (black square) or $-\infty$ (red circle) for fixed ${\kappa }^{*}{a}_{\mathrm{ff}}=100$. The dashed horizontal line denotes the universal value $a_{\mathrm{ad}}/a_{\mathrm{ff}}=-0.513$ at unitarity $\left(|a_{\mathrm{bf}}|=\infty \right)$. (b) $a_{\mathrm{ad}}$ as a function of ${\kappa }^{*}$ at $|a_{\mathrm{bf}}|=\infty$. The green curve is the fit to Eq. (8).

Figure 4

Resonance of $a_{\mathrm{ad}}$ by tuning $x$. (a) $a_{\mathrm{ad}}/a_{\mathrm{bf}}$ as functions of $x$ for fixed ${\kappa }^{*}{a}_{\mathrm{bf}}=100$. The resonance position is $x=2.7$. (b) Resonance width $W$ for positive (black line) and negative (red line) resonance position $x$.

Figure 5

Many-body phase diagram of atom-dimer system for negative(a) and positive(b) $a_{\mathrm{bf}}$ and with a fixed ${\kappa }^{*}\left|a_{\mathrm{bf}}\right|=100$. Three phases are shown: homogenous mixture (MIX), phase separation (PS), and collapse phase (CL). Black solid lines denote the phase boundaries between MIX and PS/CL. Gray vertical lines mark the atom-dimer resonances at $x=-1.21\left(a\right),2.7\left(b\right)$, which also serve as PS-CL phase boundaries. Red dashed lines shows phase boundaries based on the mean-field prediction of $a_{\mathrm{ad}}$; above the boundary is MIX phase and below is CL (a) or PS (b) phase.