Normal-Superfluid Interface Scattering for Polarized Fermion Gases

We argue that, for the recent experiments with imbalanced fermion gases, a temperature difference may occur between the normal (N) and the gapped superfluid (SF) phase. Using the mean-field formalism, we study particle scattering off the N-SF interface from the deep BCS to the unitary regime. We show that the thermal conductivity across the interface drops exponentially fast with increasing $h/k_{B}T$, where $h$ is the chemical potential imbalance. This implies a blocking of thermal equilibration between the N and the SF phase. We also provide a possible mechanism for the creation of gap oscillations [Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like states] as seen in recent studies on these systems.

Figure 1

The N-SF interface (thick vertical line) with the $a$, $b$ spectra on the N side and the gapped $\alpha$, $\beta$ spectra in the SF. The long dashed line cuts the spectra at particlelike (filled circles) and holelike (open circles) quasiparticle states, all having the same energy. An incoming $a$ particle (curly arrow) with momentum $k_p$ and energy $E>\Delta -h$ has up to four scattering channels: the Andreev reflected $k_h$ hole, the specularly reflected $-k_p$ particle, and the transmitted holelike $-k^h$ and particlelike $k^p$ states.

Figure 2

The various scattering regions for an $a$ particle incident on the interface from the N side, as a function of its energy $E$ and ${\xi }_p=k_p^2/2m$ ($k_p$ is its momentum along the $x$ axis). The heavily shaded region is energetically forbidden, while complete reflection occurs in the lightly shaded regions. Particles in region VI may scatter to all states indicated in Fig. 1 by curly arrows. Above line 4 (regions IV and V), Andreev reflections do not occur, and above curve 3 (region IV), holelike excitations are also impossible. The numbered curves are (1) ${\xi }_p={\xi }_{-}$, (2) ${\xi }_p={\mu }_a+E$, (3) ${\xi }_p={\xi }_{+}$, and (4) ${\xi }_p=2(E+h)$.

Figure 3

The thermal conductivity across the N-SF interface $\kappa$ divided by the normal-phase conductivity ${\kappa }_N$ against $T/\Delta$ for the unitary, BCS, and deep BCS cases. For $T\lesssim 0.1\Delta$, $\kappa /{\kappa }_N$ drops dramatically (notice the logarithmic scale). The curve represents the analytical result obtained by the use of the Andreev approximation, Eq. (8). At unitarity, $\Delta =0.69T_F$, in the BCS case $\Delta =0.25T_F$, and in the deep BCS case $\Delta =0.002T_F$.