Quasiparticle scattering rate in a strongly polarized Fermi mixture

We analyze the scattering rate of an impurity atom in a Fermi sea as a function of momentum and temperature in the BCS-BEC crossover. The cross section is calculated using a microscopic multichannel theory for the Feshbach resonance scattering, including finite range and medium effects. We show that pair correlations significantly increase the cross section for strong interactions close to the unitarity regime. They give rise to a molecule pole of the cross section at negative energy on the BEC side of the resonance, which smoothly evolves into a resonance at positive scattering energy with a nonzero imaginary part on the BCS side. The resonance is the analog of superfluid pairing for the corresponding population balanced system. Using Fermi liquid theory, we show that the low temperature scattering rate of the impurity atom is significantly increased due to these pair correlations for low momenta. We demonstrate that finite range and mass imbalance effects are significant for the experimentally relevant ${}^6\mathrm{Li}-{}^{40}\mathrm{K}$ mixture, and we finally discuss how the scattering rate can be measured using radio-frequency spectroscopy and Bose-Fermi mixtures.

Figure 1

The $\mathbf{P}=0$ cross section $\sigma$ in units of $4\pi k_F^{-2}$ as a function of the energy $\omega$ and scattering length $a$ for $T=0.01T_F,m_{\downarrow }=m_{\uparrow }$, and $k_F{r}_{\mathrm{eff}}=0$. The black line is the numerically obtained pole of the scattering matrix, while the red line is the molecule pole in a vacuum given by ${\epsilon }_{\mathrm{B}}=-1/2m_r{a}^2$.

Figure 2

The cross section $\sigma$ in units of $4\pi k_F^{-2}$ as a function of total energy $\omega$ and scattering length $a$ for $T=0.01{\epsilon }_F,m_{\downarrow }/m_{\uparrow }=40/6$, and $k_F{r}_{\mathrm{eff}}=-1.8$ relevant for the recent experiment using a ${}^{40}\mathrm{K}-{}^6\mathrm{Li}$ mixture [11]. The black line is the numerically obtained pole of the scattering matrix, while the red line is the molecule pole in a vacuum given by ${\epsilon }_{\mathrm{B}}=-1/2m_r{a}^2$.

Figure 3

The cross section for $k_{F}a=+1$ (top) and $k_{F}a=-1$ (bottom) relative to vacuum cross section for $T=0.01T_F,m_{\downarrow }=m_{\uparrow }$, and $k_F{r}_{eff}=0$.

Figure 4

The cross section for $k_{F}a=+1$ (top) and $k_{F}a=-1$ (bottom) relative to vacuum cross section for $T=0.01T_F,m_{\downarrow }/m_{\uparrow }=6.6$, and $k_F{r}_{eff}=-1.8$.

Figure 5

The collision rate of the polaron as a function of momentum for $T=0.01{\epsilon }_F$ and for $k_F{r}_{eff}=0$ with equal masses $m_{\downarrow }=m_{\uparrow }$ (top), and the case relevant to the ${}^6\mathrm{Li}-{}^{40}\mathrm{K}$ mixture with $k_F{r}_{eff}=-1.8$ and $m_{\downarrow }/m_{\uparrow }=6.6$ (bottom). In both figures, the solid blue line is for $k_{F}a=+1$, the solid red line is for $k_{F}a=-1$, dashed lines are the vacuum rate, and the dash-dotted lines are the $p\gg k_F$ limit given by Eq. (6).

Figure 6

The collision rate of the polaron as a function of temperature for $p=0$. The dashed lines are the vacuum rate; the dash-dotted lines are the $p\gg k_F$ limit. The remaining lines are for $k_F{r}_{eff}=0$ with equal masses $m_{\downarrow }=m_{\uparrow }$ and $k_{F}a=+1$ (magenta) and $k_{F}a=-1$ (green), and the case relevant to the ${}^6\mathrm{Li}-{}^{40}\mathrm{K}$ mixture with $k_F{r}_{eff}=-1.8,m_{\downarrow }/m_{\uparrow }=6.6$ and $k_{F}a=+1$ (blue) and $k_{F}a=-1$ (red).