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arXiv:1404.4081 · Physical Review A

Pairing correlations in a trapped one-dimensional Fermi gas

Stephen Kudla, Dominique Gautreau, Daniel E. Sheehy

Published Apr 9, 2015• 5 citations
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Abstract

We use a BCS-type variational wave function to study attractively interacting quasi-one-dimensional fermionic atomic gases, motivated by cold-atom experiments that access the one-dimensional regime using an anisotropic harmonic trapping potential (with trapping frequencies ${\ensuremath{\omega}}_{x}={\ensuremath{\omega}}_{y}\ensuremath{\gg}{\ensuremath{\omega}}_{z}$) that confines the gas to a cigar-shaped geometry. To handle the presence of the trap along the $z$ direction, we construct our variational wave function from the harmonic oscillator Hermite functions, which are the eigenstates of the single-particle problem. Using an analytic determination of the effective interaction among harmonic oscillator states along with a numerical solution of the resulting variational equations, we make specific experimental predictions for how pairing correlations would be revealed in experimental probes, such as the local density and the momentum correlation function.

DOI
10.1103/physreva.91.043612

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We use a BCS-type variational wave function to study attractively interacting quasi-one-dimensional fermionic atomic gases, motivated by cold-atom experiments that access the one-dimensional regime using an anisotropic harmonic trapping potential (with trapping frequencies \ensuremathωx=\ensuremathωy\ensuremath≫\ensuremathωz{\ensuremath{\omega}}_{x}={\ensuremath{\omega}}_{y}\ensuremath{\gg}{\ensuremath{\omega}}_{z}\ensuremathωx​=\ensuremathωy​\ensuremath≫\ensuremathωz​) that confines the gas to a cigar-shaped geometry. To handle the presence of the trap along the zzz direction, we construct our variational wave function from the harmonic oscillator Hermite functions, which are the eigenstates of the single-particle problem. Using an analytic determination of the effective interaction among harmonic oscillator states along with a numerical solution of the resulting variational equations, we make specific experimental predictions for how pairing correlations would be revealed in experimental probes, such as the local density and the momentum correlation function.
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.
We theoretically investigate equal-mass spin-balanced two-component Fermi gases in which pairs of atoms with opposite spins interact via a short-range isotropic model potential. We probe the distinction between two-dimensional and quasi-two-dimensional harmonic confinement by tuning the effective range parameter within two-dimensional scattering theory. Our approach, which yields numerically exact energetic and structural properties, combines a correlated Gaussian basis-set expansion with the stochastic variational method. For systems containing up to six particles, we: 1) Present the ground- and excited-state energy spectra; 2) Study non-local correlations by analysing the one- and two-body density matrices, extracting from these the occupation numbers of natural orbitals, the momentum distributions of atoms and pairs, and the molecular `condensate fraction’; 3) Study local correlations by computing the radial and pair distribution functions. This paper extends current theoretical knowledge on the properties of trapped few-fermion systems as realised in state-of-the-art cold-atom experiments.
We investigate low-density, quantum-degenerate gases in the presence of a localized attractive potential in the centre of a one-dimensional harmonic trap. The attractive potential is modelled using a parameterized δ-function, allowing us to determine all single-particle eigenfunctions analytically. From these we calculate the ground-state many-body properties for a system of spin-polarized fermions and, using the Bose–Fermi mapping theorem, extend the results to strongly interacting bosonic systems. We discuss the single-particle densities, the pair-correlation functions, the reduced single-particle density matrices and the momentum distributions as a function of the particle number and strength of the attractive point potential. As an important experimental observable, we place special emphasis on spatial coherence properties of such samples.
Two-component Fermi gases with tunable repulsive or attractive interactions inside quasi-one-dimensional (Q1D) harmonic wells may soon become the cleanest laboratory realizations of strongly correlated Luttiger and Luther-Emery liquids under confinement. We present a microscopic Kohn-Sham density-functional theory of these systems, with specific attention to a gas on the approach to a confinement-induced Feshbach resonance. The theory employs the one-dimensional Gaudin-Yang model as the reference system and transfers the appropriate Q1D ground-state correlations to the confined inhomogeneous gas via a suitable local-density approximation to the exchange and correlation energy functional. Quantitative understanding of the role of the interactions in the bulk shell structure of the axial density profile is thereby achieved. While repulsive intercomponent interactions depress the amplitude of the shell structure of the noninteracting gas, attractive interactions stabilize atomic-density waves through spin pairing. These should be clearly observable in atomic clouds containing of the order of up to 100 atoms.
We investigate low-density, quantum-degenerate gases in the presence of a localised attractive potential in the centre of a one-dimensional harmonic trap.The attractive potential is modelled using a parameterised delta-function, allowing us to determine all single particle eigenfunctions analytically. From these we calculate the ground state many-body properties for a system of spin-polarised fermions and, using the Bose-Fermi mapping theorem, extend the results to strongly interacting bosonic systems. We discuss the single particle densities, the pair correlation functions, the reduced single particle density matrices and the momentum distributions as a function of particle number and strength of the attractive point potential. As an important experimental observable, we place special emphasis on spatial coherence properties of such samples.
The canonical thermodynamic properties of a one-dimensional system of interacting spin-1/2 fermions with an attractive zero-range pseudopotential are investigated within an exact approach. The density operator is evaluated as the statistical average of dyadics formed from a stochastic mean-field propagation of independent Slater determinants. For a harmonically trapped Fermi gas and for fermions confined in a 1D-like torus, we observe the transition to a quasi-BCS state with Cooper-like momentum correlations and an algebraic long-range order. For a few trapped fermions in a rotating torus, a dominant superfluid component with quantized circulation can be isolated.