Heat and spin transport in a cold atomic Fermi gas

Motivated by recent experiments measuring the spin transport in ultracold unitary atomic Fermi gases [Sommer et al., Nature (London) 472, 201 (2011); Sommer et al., New J. Phys. 13, 055009 (2011)], we explore the theory of spin and heat transport in a three-dimensional spin-polarized atomic Fermi gas. We develop estimates of spin and thermal diffusivities and discuss magnetocaloric effects, namely the the spin Seebeck and spin Peltier effects. We estimate these transport coefficients using a Boltzmann kinetic equation in the classical regime and present experimentally accessible signatures of the spin Seebeck effect. We study an exactly solvable model that illustrates the role of momentum-dependent scattering in the magnetocaloric effects.

Figure 1

Illustration of the spin Seebeck effect. (a) Far away from unitarity, the scattering rate is proportional to the relative speed. Thus, “hot” majority ($\uparrow$) atoms (top) collide more often with minority ($\downarrow$) atoms than do “cold” majority atoms (bottom), giving the spin Seebeck current and the heat current opposite directions. (b) At unitarity, the scattering rate is inversely proportional to the relative speed. Therefore, the direction of the spin Seebeck current is reversed relative to (a).

Figure 2

Normalized Seebeck coefficient $S_s^{\prime }$ as a function of $\lambda /\left|a\right|$ in the classical regime. The normalization is by a factor of $\frac{15}{608\sqrt{2}}\left(\frac{\hslash }{m}\frac{1}{T{\lambda }^3}\frac{1+4\pi {(a/\lambda )}^2}{{(a/\lambda )}^2}\right)\frac{(n_{\uparrow }-n_{\downarrow })}{n}$. This choice of normalization, which is inversely proportional to a typical value of scattering cross section with scaled scattering length $a/\lambda$, gives finite values at both limits and gives the Seebeck coefficient 1 at unitarity [Eq. (15)]. We can see that $S_s^{\prime }$ changes sign near $\lambda /\left|a\right|\simeq 3.62$. Since we factored out all explicit temperature and polarization dependence in the normalization choice and scaled the scattering length, this plot remains the same for all temperature and polarization ranges at this order of approximation ($L=1$).

Figure 3

$\delta n_i(t,0)$ normalized by initial deviation $\delta n_{i0}$ as a function of time near unitarity. (a) Majority density deviation relaxes monotonically for any polarization. (b) Minority density deviation shows nonmonotonic relaxation for $p>0$ due to the spin Seebeck effect. Here we assume ${}^6$Li atoms with $T/T_F=4$ and the longitudinal length of the trap $L=200\mu \mathrm{m}$ to set the time scale. $p$ increases from top to bottom.

Figure 4

$\delta n_i(t,0)$ normalized by initial deviation $\delta n_{i0}$ as a function of time far away from unitarity. (a) Majority density deviation relaxes nonmonotonically for $p>0$ due to the spin Seebeck effect. Inset figure magnifies the nonmonotonic part of majority density deviations which are very weak compared to the minority density deviations at unitarity shown in Fig. 3. (b) Minority density deviation relaxes monotonically for any polarization. Here we assume ${}^6$Li atoms with $T/T_F=4$ and $\lambda /\left|a\right|=4$ and the longitudinal length of the trap $L=200\mu \mathrm{m}$ to set the time scale.

Figure 5

Normalized polarization deviation as a function of time for three global polarization values, $p=0.9$ (lower curve), 0.6 (highest curve), and 0.3 (center curve). Near unitarity (solid lines) the polarization deviation is positive while far away from unitarity (dashed lines) it is negative. We found that the polarization deviation is the largest near $p=0.6$ in both limits. Same physical parameters as Figs. 3 and 4 were used to fix the time scale. The polarization deviation $\delta p(t,0)$ at the initially cold end of the cloud is normalized by the initial total density deviation, $\delta n_0/n_0$.

Figure 6

Extremum density deviation normalized by initial deviation, $|\frac{\delta n_i(t_{i,\text{ext}},0)}{\delta n_{i0}}|$ as a function of polarization. Near unitarity (solid line), $i=\downarrow$; far from unitarity (dashed line), $i=\uparrow$. $t_{i,\text{ext}}$ is the time when the density deviation of species $i$ reaches its extremum. Far from unitarity (dashed line), the normalized extremum density deviation vanishes in the high polarization limit since there are no minority atoms to scatter from.