Counterflow in a doubly superfluid mixture of bosons and fermions

In this article, we calculate the friction between two counterflowing bosonic and fermionic superfluids. In the limit where the boson-boson and boson-fermion interactions can be treated within the mean-field approximation, we show that the force can be related to the dynamical structure factor of the fermionic component. Finally, we provide asymptotic expressions for weakly and strongly attractive fermions and show that the damping rate obeys simple scaling laws close to the critical velocity.

Figure 1

Sketch of the structure factor of an interacting Fermi gas: $S_{\mathrm{st}}$ vanishes outside the gray area and $\mathrm{\Delta }$ is the excitation gap. Dotted line: low momentum phonon branch $E=\hslash c_{f}q$. Dashed line: Landau's critical velocity $E=\hslash v_{L}q$ below which the structure factor vanishes. Note that this sketch corresponds to a weakly interacting system where Landau's velocity is fixed by pair breaking, and $v_L<c_f$. In the molecular regime of the crossover, $v_L=c_f$ [2, 12].

Figure 2

Construction of the dynamic structure factor in momentum space. The shaded area represents the Fermi sphere. Energy-momentum conservation fixes the initial $k_z$ to a value $K_z$ (see text). Since the initial state must lie inside the Fermi sea, we must have $\kappa \le {\kappa }_1=\sqrt{k_F^2-K_z^2}$. Moreover, the final momentum must be above the Fermi surface, hence $\kappa \ge \sqrt{k_F^2+{(K_z+q)}^2}$. The thick solid line represents the momenta contributing to the dynamical structure factor. Upper panel, case (1): $K_z+q$ larger than $k_f$. The final momentum is always above the Fermi surface. Middle panel, case (2): $K_z+q$ is lower than $k_F$, with ${\kappa }_2\le {\kappa }_1$. The permitted values of $\mathit{k}$ are located in the ring ${\kappa }_2\le \kappa \le {\kappa }_1$. Lower panel, case (3): For ${\kappa }_2\ge {\kappa }_1$, the contribution to the structure factor vanishes.

Figure 3

Dynamical structure factor of the ideal gas. The shaded area corresponds to the region where the structure factor takes finite values. The solid lines corresponds to condition $-k_F\le K_z\le k_F$. The dashed line marks the limit between cases (1) and (2) and is associated with the condition $K_z+q=k_F$.