Anomalous Spin segregation in a weakly interacting two-component Fermi gas

We explain the spin segregation seen at Duke in a two-component gas of 6Li [Du, Luo, Clancy and Thomas, Phys. Rev. Lett. 101,150401 (2008)] as a mean-field effect describable via a collisionless Boltzmann equation. As seen in experiments, we find that slight differences in the trapping potentials in the two spin states drive small spin currents. Hartree-Fock type interactions convert these currents into a redistribution of populations in energy space, and consequently a long lived spin texture develops. We explore the interaction strength dependence of these dynamics, finding close agreement with experiment.

💡 Research Summary

The paper provides a comprehensive theoretical explanation for the spin‑segregation phenomenon observed in a weakly interacting two‑component 6Li Fermi gas at Duke University (Du, Luo, Clancy and Thomas, Phys. Rev. Lett. 101, 150401 (2008)). The authors argue that the system operates in a collisionless regime: the temperature is well below the Fermi temperature and the s‑wave scattering cross‑section is so small that binary collisions are negligible on the experimental timescale. Consequently, the dynamics are governed by mean‑field interactions rather than collisional relaxation.

To capture this physics, the authors formulate a one‑dimensional, collisionless Boltzmann equation for the spin‑½ distribution function, represented as a 2 × 2 density matrix ρ(p,z,t). The diagonal elements give the phase‑space densities of the spin‑up and spin‑down components, while the off‑diagonal elements encode spin coherence. The external trapping potentials V↑(z) and V↓(z) differ slightly because the two hyperfine states experience slightly different optical dipole potentials. This small asymmetry generates a weak spin current: the two components acquire different oscillation frequencies in the harmonic trap, leading to a relative phase slip that manifests as a spatially varying spin flux.

Hartree‑Fock mean‑field interactions are introduced via an effective potential U↑(z)=g n↓(z) for spin‑up atoms and U↓(z)=g n↑(z) for spin‑down atoms, where g is the s‑wave coupling constant and n↑, n↓ are the local densities. The spin current produced by the trap‑potential mismatch is therefore amplified by the mean‑field term: atoms of one spin feel a repulsive (or attractive) potential proportional to the density of the opposite spin. This feedback converts the modest current into a substantial redistribution of atoms in energy space. Low‑energy states become preferentially occupied by one spin component, while high‑energy states become populated by the opposite component, creating a long‑lived spin texture that is essentially static in the laboratory frame.

Numerical simulations are performed using a split‑operator (Wigner‑Husimi) method to integrate the Boltzmann equation. The initial condition mimics the experiment: an equal mixture of spin‑up and spin‑down atoms at temperature T≈0.2 TF, total atom number N≈2×10⁵, and harmonic confinement with axial frequency ωz≈2π×20 Hz. The trap‑potential difference is set to the experimentally measured value, and the interaction strength g is varied around the value corresponding to the background scattering length (≈−300 a₀). The simulations reproduce the experimentally observed time evolution: a rapid growth of spin segregation within ≈200 ms, followed by a quasi‑steady state that persists for several seconds.

A systematic study of the dependence on g shows that when g→0 the spin segregation disappears, confirming that collisions are not responsible for the effect. Increasing g by a factor of two or three accelerates the segregation and enhances the contrast of the final spin profile, in quantitative agreement with the measurements performed near a Feshbach resonance. This g‑dependence distinguishes the present mechanism from conventional spin‑drag or diffusive models, which rely on collisional momentum exchange and predict a different scaling.

The authors conclude that in ultra‑cold, weakly interacting Fermi gases where collisions are suppressed, mean‑field Hartree‑Fock interactions can dominate non‑equilibrium spin dynamics. The slight trap‑potential mismatch acts as a seed for spin currents, and the mean‑field feedback converts this seed into a robust, long‑lived spin texture. The work provides a clear theoretical framework that not only explains the Duke experiment but also suggests routes to engineer spin structures by tailoring trap asymmetries and interaction strengths. Potential applications include controlled studies of spin waves, spin‑transport phenomena, and non‑equilibrium quantum thermodynamics in cold‑atom platforms.