Trapped Two-Dimensional Fermi Gases with Population Imbalance

We study population imbalanced Fermi mixtures under quasi-two-dimensional confinement at zero temperature. Using mean-field theory and the local-density approximation, we study the ground state configuration throughout the BEC-BCS crossover. We find the trapped system to be either fully normal or to consist of a superfluid core surrounded by a normal shell, which is itself either fully or partially polarized. Upon changing the trap imbalance, the trap configuration may undergo continuous transitions between the different ground states. Finally, we argue that thermal equilibration throughout the trap will be considerably slowed down at low temperatures when a superfluid phase is present.

💡 Research Summary

This paper investigates a two‑component Fermi gas with unequal spin populations confined in a quasi‑two‑dimensional harmonic trap at zero temperature. Using the standard BCS mean‑field formalism together with the local‑density approximation (LDA) to account for the spatially varying trapping potential, the authors map out the ground‑state configuration of the system across the entire BEC‑BCS crossover.

The theoretical framework starts from a contact s‑wave interaction characterized by the 2D scattering length (a_{2D}). Within mean‑field theory the order parameter (\Delta), the average chemical potential (\mu), and the effective Zeeman field (h=(\mu_\uparrow-\mu_\downarrow)/2) are introduced. The grand‑canonical free energy (\Omega(\Delta,\mu,h)) is minimized to obtain self‑consistent equations for (\Delta) and (\mu). Because the trap is harmonic, the local chemical potential varies as (\mu(r)=\mu_0-V(r)), where (V(r)=\frac{1}{2}m\omega^2 r^2). LDA treats each radial shell as a locally uniform 2D system, allowing the authors to determine the phase (superfluid or normal) at every radius.

Two qualitatively distinct global configurations emerge. In the first, the entire cloud remains normal (N) – no pairing gap appears anywhere. In the second, a superfluid core with a finite pairing gap (\Delta(r)>0) forms at the trap centre, surrounded by a normal shell. The normal shell can be either fully polarized (NP), containing only the majority spin component, or partially polarized (PP), where both spin species are present but with unequal densities. The boundaries between these regions are set by the local values of (\mu(r)) and (h); when the effective Zeeman field exceeds a critical value (h_c) the superfluid core disappears, leading to a continuous second‑order transition to the fully normal state.

The authors systematically explore how the three control parameters – the central chemical potential (\mu_0), the population‑imbalance field (h), and the interaction strength (encoded in (1/k_F a_{2D}) or equivalently (a_{2D})) – determine which of the three possible trapped configurations (N, S+NP, S+PP) is realized. On the BEC side of the crossover (strong attraction, large positive (a_{2D})), the superfluid core expands and the normal shell becomes thin; on the BCS side (weak attraction, negative (a_{2D})), the opposite trend occurs, with a shrinking core and a thick normal envelope. By continuously varying the trap imbalance (i.e., the ratio (h/\mu_0)), one can drive smooth crossovers between these configurations, providing a clear experimental knob for probing phase separation in 2D.

Beyond the static phase diagram, the paper addresses dynamical aspects, focusing on thermal equilibration. In a superfluid region quasiparticle collisions are strongly suppressed, and heat transport proceeds mainly via collective phonon‑like excitations. In contrast, the normal shell supports ordinary fermionic collisions, leading to a much higher thermal conductivity. Consequently, when a superfluid core is present the overall heat‑flow across the cloud is bottlenecked at the interface, dramatically slowing down the approach to thermal equilibrium at low temperatures. This prediction implies that experimental measurements of temperature or entropy redistribution will be markedly slower in the presence of phase separation, a factor that must be considered in any quantitative comparison with theory.

The work is directly relevant to current ultracold‑atom experiments that can realize quasi‑2D geometries using highly anisotropic optical traps or lattice confinement, and that can tune both the interaction strength (via a Feshbach resonance) and the spin imbalance (via radio‑frequency or optical pumping). The authors suggest that spatially resolved imaging, radio‑frequency spectroscopy, or momentum‑space probes could be used to map out the predicted core‑shell structures and to test the predicted slowdown of thermalization.

In summary, the paper provides a comprehensive mean‑field + LDA analysis of population‑imbalanced Fermi gases in a 2D trap, identifies three distinct trapped ground‑states (fully normal, superfluid core + fully polarized normal shell, and superfluid core + partially polarized normal shell), elucidates how these states evolve across the BEC‑BCS crossover, and highlights the crucial role of suppressed heat transport in the superfluid phase. The results offer clear guidance for future experimental investigations of low‑dimensional imbalanced superfluidity and set the stage for more sophisticated treatments that include beyond‑mean‑field fluctuations or finite‑temperature effects.