Induced interactions in a superfluid Bose-Fermi mixture

We analyze a Bose-Einstein condensate (BEC) mixed with a superfluid two-component Fermi gas in the whole BCS-BEC crossover. Using a quasiparticle random-phase approximation combined with Beliaev theory to describe the Fermi superfluid and the BEC, respectively, we show that the single-particle and collective excitations of the Fermi gas give rise to an induced interaction between the bosons, which varies strongly with momentum and frequency. It diverges at the sound mode of the Fermi superfluid, resulting in a sharp avoided crossing feature and a corresponding sign change of the interaction energy shift in the excitation spectrum of the BEC. In addition, the excitation of quasiparticles in the Fermi superfluid leads to damping of the excitations in the BEC. Besides studying induced interactions themselves, we can use these prominent effects to systematically probe the strongly interacting Fermi gas.

Figure 1

(a) The effective interaction $V(q,\omega )$ (double wavy line) between the bosons. The dashed line is the bare Bose-Bose interaction ${\mathcal{T}}_{\mathrm{B}}$, single wavy lines are the Bose-Fermi interaction ${\mathcal{T}}_{\mathrm{BF}}$, and the solid lines are the Fermi Green's function. (b) The Bose self-energies ${\Sigma }_{11}(q,\omega )$ and ${\Sigma }_{12}(q,\omega )$. The dotted lines are excitations in and out of the BEC.

Figure 2

Speed of sound $c_s$ in the two-component Fermi superfluid as calculated from the pole of the density response function $\chi (\mathbf{q},\omega )$. It approaches $c_s=v_{\mathrm{F}}/\sqrt{3}$ in the BCS limit (green dotted line) and the Bogoliubov result $c_s=v_{\mathrm{F}}\sqrt{k_{\mathrm{F}}{a}_{\mathrm{F}}/3\pi }$ in the BEC limit (blue dash-dotted curve).

Figure 3

The blue (dark gray) and red (light gray) regions show $\mathrm{Re}\chi (q,\omega )$ for $k_{\mathrm{F}}{a}_{\mathrm{F}}=-1$. The black solid line is the weak-coupling Anderson-Bogoliubov mode, and the quasiparticle continuum for $\omega >2\Delta$ is indicated by a dashed region. The green dashed line is the Bogoliubov spectrum ${\varepsilon }_k$ of the atomic BEC, and the red solid line is the Beliaev spectrum $E_k$ for the coupled Bose-Fermi mixture. The damping ${\delta }_k$ of the Beliaev excitations is shown as a red dash-dotted line.

Figure 4

Same as Fig. 3, but for a unitary Fermi gas with $k_{\mathrm{F}}{a}_{\mathrm{F}}=\infty$. Here $\Delta \approx 0.69E_{\mathrm{F}}$.

Figure 5

Same as Figs. 3 and 4, but for the Fermi superfluid in the BEC regime with $k_{\mathrm{F}}{a}_{\mathrm{F}}=1$. Here $\Delta \approx 1.35{\epsilon }_{\mathrm{F}}$.

Figure 6

(top) Correction $(E_k-{\varepsilon }_k)/{\epsilon }_{\mathrm{F}}$ to the BEC dispersion due to the induced interaction as a function of Fermi-Fermi scattering length $k_{\mathrm{F}}{a}_{\mathrm{F}}$ and momentum $k$. (bottom) Decay ${\delta }_k/{\epsilon }_{\mathrm{F}}$ of the excitations in the BEC.