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arXiv:physics/0304005 · Physical Review A

Pairing of fermions in atomic traps and nuclei

H. Heiselberg

Published Nov 24, 2003• 36 citations
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Abstract

Pairing gaps for fermionic atoms in harmonic-oscillator traps are calculated for a wide range of interaction strengths and particle numbers, and compared to pairing in nuclei. Especially systems where the pairing gap exceeds the single-level spacing but is smaller than the shell splitting $\ensuremath{\Elzxh}\ensuremath{\omega}$ are studied, which applies to most trapped Fermi atomic systems as well as to finite nuclei. When solving the gap equation for a large trap with such multilevel pairing, one finds that the matrix elements between nearby harmonic-oscillator levels and the quasi particle energies lead to a double logarithm of the gap, and a pronounced shell structure at magic numbers. It is argued that neutron and proton pairing in nuclei belongs to the class of multilevel pairing, that their shell structure follows naturally, and that the gaps scale as $\ensuremath{\sim}{A}^{\ensuremath{-}1/3}$---all in qualitative agreement with odd-even staggering of nuclear binding energies. Pairing in large systems is related to that in the bulk limit. For large nuclei the neutron and proton superfluid gaps approach the asymptotic value in infinite nuclear matter: $\ensuremath{\Delta}\ensuremath{\simeq}1.1\mathrm{MeV}.$

DOI
10.1103/physreva.68.053616

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Pairing gaps for fermionic atoms in harmonic-oscillator traps are calculated for a wide range of interaction strengths and particle numbers, and compared to pairing in nuclei. Especially systems where the pairing gap exceeds the single-level spacing but is smaller than the shell splitting \ensuremath\Elzxh\ensuremathω\ensuremath{\Elzxh}\ensuremath{\omega}\ensuremath\Elzxh\ensuremathω are studied, which applies to most trapped Fermi atomic systems as well as to finite nuclei. When solving the gap equation for a large trap with such multilevel pairing, one finds that the matrix elements between nearby harmonic-oscillator levels and the quasi particle energies lead to a double logarithm of the gap, and a pronounced shell structure at magic numbers. It is argued that neutron and proton pairing in nuclei belongs to the class of multilevel pairing, that their shell structure follows naturally, and that the gaps scale as \ensuremath∼A\ensuremath−1/3\ensuremath{\sim}{A}^{\ensuremath{-}1/3}\ensuremath∼A\ensuremath−1/3---all in qualitative agreement with odd-even staggering of nuclear binding energies. Pairing in large systems is related to that in the bulk limit. For large nuclei the neutron and proton superfluid gaps approach the asymptotic value in infinite nuclear matter: \ensuremathΔ\ensuremath≃1.1MeV.\ensuremath{\Delta}\ensuremath{\simeq}1.1\mathrm{MeV}.\ensuremathΔ\ensuremath≃1.1MeV.
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.
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.
In cold atoms and in the crust of neutron stars, the pairing gap can reach values comparable with the Fermi energy. While in nuclei the neutron gap is smaller, it is still of the order of a few percent of the Fermi energy. The pairing mechanism in these systems is due to short range attractive interactions between fermions and the size of the Cooper pair is either comparable to the inter‐particle separation or it can be as big as a nucleus, which is still relatively small in size. Such a strong pairing gap is the result of the superposition of a very large number of particle–particle configurations, which contribute to the formation of the Cooper pairs. These systems have been shown to be the host of a large number of remarkable phenomena, in which the large magnitude of the pairing gap plays an essential role: quantum shock waves, quantum turbulence, Anderson–Bogoliubov–Higgs mode, vortex rings, domain walls, soliton vortices, vortex pinning in neutron star crust, unexpected dynamics of fragmented condensates and role of pairing correlations in collisions on heavy‐ions, Larkin–Ovchinnikov phase as an example of a Fermi supersolid, role of pairing correlations in controlling the dynamics of fissioning nuclei.
The properties of a balanced two-component Fermi gas in a one-dimensional harmonic trap are studied by means of the coupled-cluster method. For few fermions we recover the results of exact diagonalization, yet with this method we are able to study much larger systems. We compute the energy, the chemical potential, the pairing gap, and the density profile of the trapped clouds, smoothly mapping the crossover between the few-body and many-body limits. The energy is found to converge surprisingly rapidly to the many-body result for every value of the interaction strength. Many more particles are instead needed to give rise to the nonanalytic behavior of the pairing gap, and to smoothen the pronounced even-odd oscillations of the chemical potential induced by the shell structure of the trap.
We present an ab initio calculation of small numbers of trapped, strongly interacting fermions using the Green's function Monte Carlo method. The ground-state energy, density profile, and pairing gap are calculated for particle numbers N=2−−22N=2--22N=2−−22 using the parameter-free ``unitary'' interaction. Trial wave functions are taken in the form of correlated pairs in a harmonic oscillator basis. We find that the lowest energies are obtained with a minimum explicit pair correlation beyond that needed to exploit the degeneracy of oscillator states. We find that the energies can be well fitted by the expression ${a}_{\mathit{TF}}{E}_{\mathit{TF}}+\ensuremath{\Delta}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}(N,2)$ where ETF{E}_{\mathit{TF}}ETF​ is the Thomas-Fermi energy of a noninteracting gas in the trap and $\ensuremath{\Delta}$ is the pairing gap. There is no evidence of a shell correction energy in the systematics, but the density distributions show pronounced shell effects. We find the value $\ensuremath{\Delta}=0.7\ifmmode\pm\else\textpm\fi{}0.2\ensuremath{\omega}$ for the pairing gap. This is smaller than the value found for the uniform gas at a density corresponding to the central density of the trapped gas.
Near Feshbach resonances, n | a | 3 ≫ 1, systems of Bose and Fermi particles become strongly interacting/dense. In this unitary limit both bosons and fermions have very different properties than in a dilute gas, e.g., the energy per particle approaches a value ℏ 2 n 2/3 / m times a universal many-body constant. Calculations based upon an approximate Jastrow wavefunction can quantitatively describe recent measurements of trapped Bose and Fermi atoms near Feshbach resonances. The pairing gap between attractive fermions also scales as Δ ∼ ℏ 2 n 2/3 / m near Feshbach resonances and is a large fraction of the Fermi energy—promising for observing BCS superfluidity in traps. Pairing undergoes several transitions depending on interaction strength and the number of particles in the trap and can also be compared with pairing in nuclei.