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arXiv:1702.02933 · Physical Review A

Trapped imbalanced fermionic superfluids in one dimension: A variational approach

Kelly R. Patton, Dominique Gautreau, Stephen Kudla, Daniel E. Sheehy

Published Jun 28, 2017• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.95.063623

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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.
Ground-state Hartree-Fock-Bogoliubov (HFB) theory is applied to imbalanced spin-1/21/21/2 one-dimensional (1D) Fermi systems that are spatially confined by either a harmonic or a hard-wall trapping potential. It has been hoped that such systems, which can be realized using ultracold atomic gases, would exhibit the long-sought-after Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid phase. The HFB formalism generalizes the standard Bogoliubov quasiparticle transformation by allowing for Cooper pairing to exist between all possible single-particle states and accounts for the effects of the inhomogeneous trapping potential as well as the mean-field Hartree potential. This provides an unbiased framework to describe inhomogeneous densities and pairing correlations in the FFLO state of a confined 1D gas. In a harmonic trap, numerical minimization of the HFB ground-state energy yields a spatially oscillating order parameter reminiscent of the FFLO state. However, we find that this state has almost no imprint in the local fermion densities (consistent with experiments that found no evidence of the FFLO phase). In contrast, for a hard-wall geometry, we find a strong signature of the spatial oscillations of the FFLO pairing amplitude reflected in the local in situ densities. In the hard-wall case, the excess spins are strongly localized near regions where there is a node in the pairing amplitude, creating an unmistakable crystalline modulation of the density.
The ground-state phase diagram of attractively interacting Fermi gases in two dimensions with a population imbalance is investigated. We find the regime of stability for the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, in which pairing occurs at a finite wave vector, and determine the magnitude of the pairing amplitude $\mathrm{\ensuremath{\Delta}}$ and FFLO wave vector qqq in the ordered phase, finding that $\mathrm{\ensuremath{\Delta}}$ can be of the order of the two-body binding energy. Our results rely on a careful analysis of the zero-temperature gap equation for the FFLO state, which possesses nonanalyticities as a function of $\mathrm{\ensuremath{\Delta}}$ and qqq, invalidating a Ginzburg-Landau expansion in small $\mathrm{\ensuremath{\Delta}}$.
The attractive Fermi-Hubbard Hamiltonian is solved via the Bogoliubov-de Gennes formalism to analyze the ground state phases of population imbalanced two-component fermion mixtures in harmonically trapped optical lattices. In the low density limit, the superfluid order parameter has Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type oscillations in the radial direction towards the trap edges to accommodate the unpaired fermions that are pushed away from the trap center with a single peak in their density. However, in the high density limit, while the order parameter has oscillations in the radial direction towards the trap center for low imbalances, it also has oscillations towards the trap edges with increasing imbalance until the superfluid to normal phase transition occurs beyond a critical imbalance. This leads to a single peak in the density of unpaired fermions for low and high imbalances, but leads to double peaks for intermediate imbalances.
Ultracold atomic gases have emerged as an ideal laboratory system to emulate many-body physics in an unprecedentedly controllable manner. Numerous many-body quantum states and phases have been experimentally explored and characterized using the ultracold atomic gases, offering new insights into many exciting physics ranging from condensed matters to cosmology. In this thesis, we will present a systematic numerical study of a novel experimental system, population imbalanced two-component ultracold Fermi gases. We explore the phase diagram of this system in both 3D and 1D especially focusing on the exotic Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase, which is characterized by a spatially oscillating order parameter. In 3D, we solve for the stationary states of trapped imbalanced Fermi gases in a wide range of parameter space with a home-made parallel eigen-solver for Bogoliubov-de Gennes (BdG) equations. Our results show that there exists a metastable state with a FFLO type oscillating order parameter. In 1D, we simulate the dynamical expansion of the population imbalanced Fermi gases from the trap. A numerically quasi-exact scheme, time-evolving block decimation (TEBD), is introduced for the comparative studies with the solution of the time-dependent BdG equation. Our results predict that the existence of FFLO states will leave conspicuous signatures in the density profiles during the expansion. For further understanding of the interplay between the population imbalance and two-body pairing interaction between two spin components, we also study the spin transport properties through trapped ultracold Fermi gases. The preliminary results will be discussed.
The question whether an imbalanced Fermi gas can accommodate polarized superfluidity has been the subject of intense study. One prominent candidate for a polarized superfluid state is the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state. In a one-dimensional (1D) Fermi gas, indirect evidence for this state has recently been found, but in the three-dimensional (3D) case, this exotic state has hitherto remained elusive. In this brief review, we study the influence of a 1D periodic potential on the FFLO state in an imbalanced 3D Fermi gas. We present a path-integral derivation of the free energy of the system, where the possibility of the FFLO state is included through the saddle-point approximation. Using the free energy, the phase diagram of the system, both at fixed chemical potentials and at fixed densities, is constructed, showing that the stability region of the FFLO state is significantly enlarged by the 1D potential. Subsequently, the influence of the potential on the FFLO pairing mechanism is investigated, leading to a qualitative understanding of the shape of the FFLO region in the different phase diagrams. We conclude by investigating the possibility of an FFLO state with a wave vector that lies skewed with respect to the direction along which the periodic potential varies. Our results provide an additional way to facilitate the experimental observation of the FFLO state in a 3D Fermi gas.
We present an exact quantum Monte Carlo study of the attractive one-dimensional Hubbard model with imbalanced fermion population. The pair-pair correlation function, which decays monotonically in the absence of polarization P, develops oscillations when P is nonzero, characteristic of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. The pair momentum distribution peaks at a momentum equal to the difference in the Fermi momenta. At strong coupling, the minority and majority momentum distributions are shown to be deformed, reflecting the presence of the other species and its Fermi surface. The FFLO oscillations survive the presence of a confining potential, and the local polarization at the trap center exhibits a marked dip, similar to that observed experimentally.