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arXiv:cond-mat/0610437 · Physical Review Letters

Attractive Fermi Gases with Unequal Spin Populations in Highly Elongated Traps

Giuliano Orso

Published Feb 14, 2007• 262 citations
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Abstract

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.

DOI
10.1103/physrevlett.98.070402

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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.
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.
Based on the integrable Gaudin model and local density approximation, we discuss the ground state of a one-dimensional trapped Fermi gas with imbalanced spin population, for an arbitrary attractive interaction. A phase separation state, with a polarized superfluid core immersed in an unpolarized superfluid shell, emerges below a critical spin polarization. Above it, coexistence of polarized superfluid matter and a fully polarized normal gas is favored. These two exotic states could be realized experimentally in highly elongated atomic traps, and diagnosed by measuring the lowest density compressional mode. We identify the polarized superfluid as having an Fulde-Ferrell-Larkin-Ovchinnikov structure, and predict the resulting mode frequency as a function of the spin polarization.
We numerically study imbalanced two component Fermi gases with attractive interactions in highly elongated harmonic traps. An accurate parametrization formula for the ground state energy is presented for a spin-polarized attractive Gaudin-Yang model. Our studies are based on an accurate microscopic spin-density-functional theory through the Kohn-Sham scheme which employs the one-dimensional homogeneous Gaudin-Yang model with a Luther-Emery-liquid ground-state correlation as a reference system. A Thomas-Fermi approximation is examined incorporating the exchange-correlation interaction. By studying the charge and spin density profiles of the system based on these methods, we gain a quantitative understanding of the role of attractive interactions and polarization on the formation of the two-shell structure, with the coexisted Fulde-Ferrell-Larkin-Ovchinnikov-type phase in the center of the trap and either the BCS superfluid phase or the normal phase at the edges of the trap. Our results are in good agreement with the recent theoretical consequences.
We calculate the density profiles of a trapped spin-imbalanced Fermi gas with attractive interactions in a one-dimensional optical lattice, using both the local-density approximation (LDA) and density-matrix renormalization-group (DMRG) simulations. Based on the exact equation of state obtained by Bethe ansatz, the LDA predicts that the gas phase separates into shells with a partially polarized core and fully paired wings, the latter occurring below a critical spin polarization. This behavior is also seen in numerically exact DMRG calculations at sufficiently large particle numbers. We show that, unlike in the continuum case, the critical polarization is a nonmonotonic function of the interaction strength and vanishes in the limit of large interactions.
We study the properties of a one-dimensional (1D) gas of fermions trapped in a lattice by means of the density matrix renormalization group method, focusing on the case of unequal spin populations, and strong attractive interaction. In the low-density regime, the system phase separates into a well-defined superconducting core and a fully polarized metallic cloud surrounding it. We argue that the superconducting phase corresponds to a 1D analog of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state, with a quasicondensate of tightly bound bosonic pairs with a finite center-of-mass momentum that scales linearly with the magnetization. In the large density limit, the system allows for four phases: in the core, we either find a Fock state of localized pairs or a metallic shell with free spin-down fermions moving in a fully filled background of spin-up fermions. As the magnetization increases, the Fock state disappears to give room for a metallic phase, with a partially polarized superconducting FFLO shell and a fully polarized metallic cloud surrounding the core.
We propose and analyze a variational wave function for a population-imbalanced one-dimensional Fermi gas that allows for Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type pairing correlations among the two fermion species, while also accounting for the harmonic confining potential. In the strongly interacting regime, we find large spatial oscillations of the order parameter, indicative of an FFLO state. The obtained density profiles versus imbalance are consistent with recent experimental results as well as with theoretical calculations based on combining Bethe ansatz with the local density approximation. Our variational wave function displays no signature of the FFLO state in the densities of the two fermion species. Nonetheless, the oscillations of the order parameter appear in density-density correlations, both in situ and after free expansion. Furthermore, above a critical polarization, the value of which depends on the interaction, we find the unpaired Fermi-gas state to be energetically more favorable.