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arXiv:physics/0403125

Scaling theory of excitations in trapped fermionic condensates

G. F. Bertsch, Aurel Bulgac, Ricardo A. Broglia

Published Mar 26, 2004• 0 citations
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Abstract

Formulas are derived for the coupled quadrupolar and monopolar oscillations of a fermion condensate trapped in a axially symmetric harmonic potential. We consider two-component condensates with a large particle-particle scattering length and in the limits of frequencies either small (adiabatic) or large (diabatic) compared to the pairing gap. Using the method of coordinate scaling, we reproduce Stringari's results for adiabatic excitations and present new results for the diabatic limit. Unlike the adiabatic oscillations, the diabatic ones show sensitivity to the interaction between particles.

DOI
10.48550/arxiv.physics/0403125

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Formulas are derived for the coupled quadrupolar and monopolar oscillations of a fermion condensate trapped in a axially symmetric harmonic potential. We consider two-component condensates with a large particle-particle scattering length and in the limits of frequencies either small (adiabatic) or large (diabatic) compared to the pairing gap. Using the method of coordinate scaling, we reproduce Stringari's results for adiabatic excitations and present new results for the diabatic limit. Unlike the adiabatic oscillations, the diabatic ones show sensitivity to the interaction between particles.
We use a variational ansatz to derive a closed dynamics involving only the monopole and quadrupole shape degrees of freedom of the condensate, which is a generalization of the work of Perez-Garcia and collaborators. Taking the limit of small oscillations around the equilibrium configuration, we calculate the excitation energies of the low-lying even-parity states of the condensate. In this limit, the diagonal and non-diagonal quadrupole degrees of freedom are decoupled. For the diagonal elements, our results agree exactly with those of Perez-Garcia and collaborators. For the non-diagonal elements we obtain analytical expressions for the excitation energies which are valid for any trap geometry and any particle number and that agree, in the Thomas-Fermi limit and for axially symmetric traps, with Stringari's results. For negative scattering lengths we show that the collapsing state is the breathing mode, independent of the trap geometry.
We calculate the ground-state energy and the collective excitation frequency of trapped bosons at large scattering length interacting via the realistic two-body van der Waals potential. Our many-body method keeps two-body correlations produced by all interacting pairs. When the scattering length is small compared to the trap size and the number of bosons in the trap is of the order of a few thousands, the mean-field results are in good agreement with the many-body results. However for large particle numbers, even when the condensate is sufficiently dilute, the interatomic correlation comes into the picture. When the scattering length is quite large near the Feshbach resonance, the Bose gas becomes highly correlated. The many-body results are close to the Gross-Pitaevskii results for a small number of bosons, however, large deviations are noted in the large particle limit. We also calculate the lowest collective excitation and the interaction energy for large scattering lengths. The monopole excitation frequency exhibits a pronounced dependence on the scattering length. We also observe a universal behavior for the interaction energy at the limit of large scattering length.
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.
The oscillation frequencies of collective excitations of a trapped Bose-Einstein condensate, when calculated in the mean-field approximation and in the Thomas-Fermi limit, are independent of the scattering length aaa. We calculate the leading corrections to the frequencies from quantum fluctuations around the mean field. The semiclassical correction is proportional to N1/5a6/5{N}^{1/5}{a}^{6/5}N1/5a6/5, where NNN is the number of atoms in the condensate. The correction is positive semidefinite and is zero for surface modes whose eigenfunctions have a vanishing Laplacian. The shift in the frequency of the lowest quadrupole mode for an axially symmetric trap is large enough that it should be measurable in future experiments.
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: $\ensuremath{\Delta}\ensuremath{\simeq}1.1\mathrm{MeV}.$