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arXiv:cond-mat/0203192 · Physical Review A

Structure and stability of trapped atomic boson-fermion mixtures

Robert Roth

Published Jul 29, 2002• 118 citations
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Abstract

The structure of trapped binary mixtures of bosonic and fermionic atoms at zero temperature is studied using a modified Gross-Pitaevskii equation for the bosons which self-consistently includes the mean-field interaction generated by the fermionic cloud. The density of the fermionic component is described within the Thomas-Fermi approximation. The influence of the boson-boson and the boson-fermion s-wave interaction on the density profiles and the stability of the mixture is investigated systematically. Critical particle numbers for the mean-field collapse caused either by attractive boson-boson or by attractive boson-fermion interactions and for the onset of spatial component separation are discussed. It is shown that the interplay between the boson-boson and the boson-fermion interaction generates a rich and complex phase diagram. Finally, the specific properties and prospects of the boson-fermion mixtures available in present experiments are addressed.

DOI
10.1103/physreva.66.013614

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The influence of boson-boson and boson-fermion interactions on the stability of a binary mixture of bosonic and fermionic atoms is investigated. The density profiles of the trapped mixture are obtained from direct numerical solution of a modified Gross-Pitaevskii equation that is self-consistently coupled to the mean field generated by the interaction with the fermionic species, the fermions which in turn feel the mean field created by the bosons are treated in the Thomas-Fermi approximation. We study the effects of different combinations of signs of the boson-boson and the boson-fermion scattering lengths and determine explicit expressions for critical particle numbers as a function of these scattering lengths.
Static properties of a bose-fermi mixture of trapped potassium atoms are studied in terms of coupled Gross-Pitaevskii and Thomas-Fermi equations for both repulsive and attractive bose-fermi interatomic potentials. Qualitative estimates are given for solutions of the coupled equations, and the parameter regions are obtained analytically for the boson-density profile change and for the boson/fermion phase separation. Especially, the parameter ratio RintR_{int}Rint​ is found that discriminates the region of the large boson-profile change. These estimates are applied for numerical results for the potassium atoms and checked their consistency. It is suggested that a small fraction of fermions could be trapped without an external potential for the system with an attractive boson-fermion interaction.
We investigate the dynamics of magnetically trapped boson-fermion mixtures at very low temperature. Using the mean field approach, we derive the Gross-Pitaevskii equation for the Bose-Einstein condensate wavefunction. We apply the Thomas-Fermi approximation and the hydrodynamic approach to the degenerate Fermi gas and obtain an equation of motion for its number density perturbation. It is remarkable that coupled time-evolution equations for the mixture can be regarded as an extension of the Zakharov equations, a model for the Langmuir waves in plasma.
The collapse of a trapped boson-fermion mixture with attraction between bosons and fermions is investigated in the framework of the effective Hamiltonian for the Bose system. The properties of an 87Rb^{87}\mathrm{Rb}87Rb and 40K^{40}\mathrm{K}40K mixture are analyzed quantitatively at T=0T=0T=0. We find numerically solutions of the modified Gross-Pitaevskii equation which continuously go from the stable to the unstable branch. We discuss how the conditions for the onset of collapse are related to the macroscopic properties of the system. A comparison with the case of a Bose condensate of the atomic 7Li^{7}\mathrm{Li}7Li system is given.
We study a confined mixture of bosons and fermions in the quantal degeneracy regime with attractive boson-fermion interaction. We discuss the effect that the presence of vortical states and the displacement of the trapping potentials may have on mixtures near collapse, and investigate the phase stability diagram of the K-Rb mixture in the mean-field approximation supposing in one case that the trapping potentials felt by bosons and fermions are shifted from each other, as it happens in the presence of a gravitational sag, and in another case, assuming that the Bose condensate sustains a vortex state. In both cases, we have obtained an analytical expression for the fermion effective potential when the Bose condensate is in the Thomas-Fermi regime, that can be used to determine the maxima of the Fermionic density. We have numerically checked that the values one obtains for the location of these maxima using the analytical formulas remain valid up to the critical boson and fermion numbers, above which the mixture collapses.