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Physical Review Letters

Exact Solution for a Trapped Fermi Gas with Population Imbalance and BCS Pairing

Zu‐Jian Ying, Mario Cuoco, Canio Noce, Huan-Qiang Zhou

Published Apr 10, 2008• 17 citations
Publisher

Abstract

The problem of a two-component Fermi gas in a harmonic trap, with an imbalanced population and a pairing interaction of zero total momentum, is mapped onto the exactly solvable reduced BCS model. For a one-dimensional trap, the complete ground state diagram is determined with various topological features in ground state energy spectra. In addition to the conventional two-shell density profile of a paired core and polarized outer wings, a three-shell structure as well as a double-peak superfluid distribution are unveiled.

DOI
10.1103/physrevlett.100.140406

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The problem of a two-component Fermi gas in a harmonic trap, with an imbalanced population and a pairing interaction of zero total momentum, is mapped onto the exactly solvable reduced BCS model. For a one-dimensional trap, the complete ground state diagram is determined with various topological features in ground state energy spectra. In addition to the conventional two-shell density profile of a paired core and polarized outer wings, a three-shell structure as well as a double-peak superfluid distribution are unveiled.
We study the two-component repulsive Fermi gas with imbalanced populations in one dimension. Starting from the Bethe ansatz solution we calculate analytically the phase diagram for the homogeneous system. We show that three phases appear: the balanced phase, the fully polarized phase, and the partially polarized phase. By means of the local density approximation and the equation of state for the homogeneous system we calculate the density profile for the harmonically confined case. We show that a two-shell structure appears: at the center of the cloud we find the partially polarized phase and at the edges the fully polarized one. The radii of the inner and outer shells are calculated for different values of the polarization and the coupling strength. We calculate the dependence of the magnetization on the polarization for different values of the coupling strength and we show that the susceptibility is always finite.
We investigate two-component attractive Fermi gases with imbalanced spin populations in trapped one-dimensional configurations. The ground state properties are determined with the local density approximation, starting from the exact Bethe-ansatz equations for the homogeneous case. We predict that the atoms are distributed according to a two-shell structure: a partially polarized phase in the center of the trap and either a fully paired or a fully polarized phase in the wings. The partially polarized core is expected to be a superfluid of the Fulde-Ferrell-Larkin-Ovchinnikov type. The size of the cloud as well as the critical spin polarization needed to suppress the fully paired shell are calculated as a function of the coupling strength.
The superfluid phase diagrams of a two-dimensional cold polarized Fermi gas in the BCS-BEC crossover are systematically and analytically investigated. In the BCS-Leggett mean-field theory, the transition from unpolarized superfluid phase to normal phase is always of first order. For a homogeneous system, the two critical Zeeman fields and the critical population imbalance are analytically determined in the whole coupling parameter region, and the superfluid-normal mixed phase is shown to be the ground state between the two critical fields. The density profile in the presence of a harmonic trap calculated in the local-density approximation exhibits a shell structure, a superfluid core at the center, and a normal shell outside. For weak interaction, the normal shell contains a partially polarized cloud with constant density difference surrounded by a fully polarized state. For strong interaction, the normal shell is in a fully polarized state with a density profile depending only on the global population imbalance. The difermion bound states can survive in the whole highly imbalanced normal phase.
We consider the problem of spin-triplet ppp-wave superfluid pairing with total spin projection ms=0{m}_{s}=0ms​=0 in atomic Fermi gas across the Feshbach resonance. We allow for imbalanced populations and take into account the effects due to the presence of a harmonic trapping potential. Within the mean-field approximation for the one- and two-channel pairing models we show that the ppp-wave superfluid state has the lowest energy up to some distance from the trap center, the distance depending on population imbalance and the side of the Feshbach resonance (BCS or BEC) the system is in. Beyond this, the system is in the normal state. Since the system, even in the presence of the harmonic trap, is invariant under O(3) space rotation, the superfluid cores which emerge in the trap are comprised of states that are degenerate with respect to the ``orbital ferromagnet'' state, ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$, as well as the ones in which all angular momentum components of the order parameter are nonzero, provided the modulus of the order parameter maintains a fixed magnitude.
On the basis of the phase diagram of the uniform system, we calculate the density profiles of a trapped polarized Fermi gas at zero temperature using the local density approximation. By varying the overall polarization and the interaction strength, we analyze the appearance of a discontinuity in the profile, signaling a first-order phase transition from a superfluid inner core to a normal outer shell. The local population imbalance between the two components and the size of the various regions of the cloud corresponding to different phases are also discussed. The calculated profiles are quantitatively compared with the ones recently measured by Shin et al. [Phys. Rev. Lett. 101, 070404 (2008)].
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.