Calibration of the interaction energy between Bose and Fermi superfluids

In this paper we study the interaction energy in a mixture of Bose and Fermi superfluids realized in recent cold atom experiments. On the Bose-Einstein-condensate (BEC) side of a Feshbach resonance between fermionic atoms, this interaction energy can be directly related to the scattering length between a bosonic atom and a dimer composed of fermions. We calculate the atom-dimer scattering length from a three-body analysis with both a zero-range model and a separable model, including the van der Waals length scale, and we find significant deviation from the result given by a mean-field approach. We also find that the multiple scattering between atom and dimer can account for such a deviation. Our results provide a calibration to the mean-field interaction energy, which can be verified by measuring the shift of collective oscillation frequency.

Figure 1

The atom-dimer scattering length $a_{\mathrm{ad}}/a_{\mathrm{f}}$ as a function of $a_{\text{bf}}/a_{\mathrm{f}}$ given by three-body calculation with mean-field approximation (purple dash-dotted line), zero-range pseudopotential (red solid line), single-pole approximation (black dashed line), and three-body calculation with a separable potential suggested in Ref. [33] to incorporate the van der Waals effect (blue circle with solid line). Here we illustrate the results for ${}^6\mathrm{Li}\text{−}{}^7\mathrm{Li}$ mixture with $a_{\mathrm{bf}}=40.8a_0,\Lambda =4/a_{\mathrm{bf}},\eta =0,R_{\mathrm{vdW}}=31.26a_0$ for ${}^6\mathrm{Li}\text{−}{}^6\mathrm{Li}$ interaction, and $R_{\mathrm{vdW}}=32.49a_0$ for ${}^6\mathrm{Li}\text{−}{}^7\mathrm{Li}$ interaction [35, 36, 37].

Figure 2

The relative deviation $(a_{\mathrm{ad}}^0-a_{\mathrm{ad}})/a_{\mathrm{ad}}^0$ for different mass ratios $\left(M\equiv m_{\mathrm{b}}/m_{\mathrm{f}}\right)$. Cases shown are, from top to bottom, ${}^{133}\mathrm{Cs}\text{−}{}^6\mathrm{Li} (133/6$, blue line with triangle), ${}^{87}\mathrm{Rb}\text{−}{}^6\mathrm{Li}$ ($87/6$, black line with inverted triangle), ${}^7\mathrm{Li}\text{−}{}^6\mathrm{Li}$ ($7/6$, red line with square), ${}^{23}\mathrm{Na}\text{−}{}^{40}\mathrm{K}$ ($23/40$, purple line with diamond), and ${}^7\mathrm{Li}\text{−}{}^{40}\mathrm{K}$ ($7/40$, cyan line with circle).

Figure 3

Frequency shift of dipole oscillation $\delta {\omega }_{\mathrm{b}}$ for bosonic ${}^7\mathrm{Li}$ atoms as a function of $1/\left(k_{\text{F}}{a}_{\text{f}}\right)$. Dots with error bar are experimental data [6]. Blue dashed line is predication from mean-field theory. Red solid line is predication based on $a_{\text{ad}}$ calculated with zero-range model, and black dash-dotted line is predication based on $a_{\text{ad}}$ from separable potential including the van der Waals effect. Here we have chosen $k_F=4.6\times {10}^6{m}^{-1}$. Other parameters take the same values as in Fig. 1.