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arXiv:cond-mat/0502096 · Physical Review A

Supershell structure in trapped dilute Fermi gases

Y. C. Yu, Magnus Ögren, Sven Åberg, S. M. Reimann, M. Brack

Published Nov 11, 2005• 8 citations
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Abstract

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 $\mathrm{U}(3)\ensuremath{\rightarrow}\mathrm{SO}(3)$.

DOI
10.1103/physreva.72.051602

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Shell effects in the coordinate space can be seen with degenerate Fermi vapors in non-uniform trapping potentials. In particular, below the Fermi temperature, the density profile of a Fermi gas in a confining harmonic potential is characterized by several local maxima. This effect is enhanced for "magic numbers" of particles and in quasi-1D (cigar-shaped) configurations. In the case of a multi-component Fermi vapor, the separation of Fermi components in different spatial shells (phase-separation) depends on temperature, number of particles and scattering length. We derive analytical formulas, based on bifurcation theory, for the critical density of Fermions and the critical chemical potential, which give rise to the phase-separation.

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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 U(3)\ensuremath→SO(3)\mathrm{U}(3)\ensuremath{\rightarrow}\mathrm{SO}(3)U(3)\ensuremath→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 examine the spin asymmetry of ground states for two-dimensional, harmonically trapped two-component gases of fermionic atoms at zero temperature with weakly repulsive short range interactions. Our main result is that, in contrast to the three-dimensional case, in two dimensions a non-trivial spin-asymmetric phase can only be caused by shell structure. A simple, qualitative description is given in terms of an approximate single particle model, comparing well to the standard results of Hartree-Fock or direct diagonalization methods.
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.
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.
Shell effects in the coordinate space can be seen with degenerate Fermi vapors in non-uniform trapping potentials. In particular, below the Fermi temperature, the density profile of a Fermi gas in a confining harmonic potential is characterized by several local maxima. This effect is enhanced for "magic numbers" of particles and in quasi-1D (cigar-shaped) configurations. In the case of a multi-component Fermi vapor, the separation of Fermi components in different spatial shells (phase-separation) depends on temperature, number of particles and scattering length. We derive analytical formulas, based on bifurcation theory, for the critical density of Fermions and the critical chemical potential, which give rise to the phase-separation.