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arXiv:0704.0385 · Physical Review A

Supershell structures and pairing in ultracold trapped Fermi gases

Magnus Ögren, H. Heiselberg

Published Aug 1, 2007• 4 citations
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Abstract

We calculate level densities and pairing gaps for an ultracold dilute gas of fermionic atoms in harmonic traps under the influence of mean field and anharmonic quartic trap potentials. Supershell nodes, which were found in Hartree-Fock calculations, are calculated analytically within periodic orbit theory as well as from WKB calculations. For attractive interactions, the underlying level densities are crucial for pairing and supershell structures in gaps are predicted.

DOI
10.1103/physreva.76.021601

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We calculate level densities and pairing gaps for an ultracold dilute gas of fermionic atoms in harmonic traps under the influence of mean field and anharmonic quartic trap potentials. Supershell nodes, which were found in Hartree-Fock calculations, are calculated analytically within periodic orbit theory as well as from WKB calculations. For attractive interactions, the underlying level densities are crucial for pairing and supershell structures in gaps are predicted.
We show that a dilute harmonically trapped two-component gas of fermionic atoms with a weak repulsive interaction has a pronounced super-shell structure: The shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the ``magic numbers'' occurring between the beat nodes by half a period. The length and amplitude of this beating mode depend on the strength of the interaction. We give a simple interpretation of the beat structure in terms of a semiclassical trace formula for the symmetry breaking $\mathrm{U}(3)\ensuremath{\rightarrow}\mathrm{SO}(3)$.
It was recently shown in self-consistent Hartree-Fock calculations that a harmonically trapped dilute gas of fermionic atoms with a repulsive two-body interaction exhibits a pronounced {\it super-shell} structure: the shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the ``magic numbers'' occurring between the beat nodes by half a period. The length and amplitude of the beating mode depends on the strength of the interaction. We give a qualitative interpretation of the beat structure in terms of a semiclassical trace formula that uniformly describes the symmetry breaking U(3) →\to→ SO(3) in a 3D harmonic oscillator potential perturbed by an anharmonic term ∝r4\propto r^4∝r4 with arbitrary strength. We show that at low Fermi energies (or particle numbers), the beating gross-shell structure of this system is dominated solely by the two-fold degenerate circular and (diametrically) pendulating orbits.
We calculate the elementary excitations and pairing of a trapped atomic Fermi gas in the superfluid phase. The level spectra and pairing gaps undergo several transitions as the strength of the interactions between atoms and the number of atoms are varied. For weak interactions, the Cooper pairs are formed between particles residing in the same harmonic oscillator shell. In this regime, the nature of the paired state is shown to depend critically on the position of the chemical potential relative to the harmonic oscillator shells and on the size of the mean field. For stronger interactions, we find a region where pairing occur between time-reversed harmonic oscillator states in different shells also.
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 shell structures for weakly interacting fermions in harmonic oscillator traps at zero temperature undergo several transitions depending on the number of particles in the trap and their interaction strength. Calculations of the one and two-particle spectra give the pairing gaps, and the many particle state is constructed by the seniority scheme. For sufficiently few particles, N less than or similar to 10(4), the pairing field exceeds the mean-field splitting of single particle levels at the Fermi surface and the greater number of almost degenerate states due to the SU(3) symmetry in the harmonic oscillator shell lead to supergaps. Deformation and rotation are also discussed.