We present the zero-temperature phase diagram of bosonic atoms in an optical lattice, using two different mean-field approaches. The phase diagram consists of various insulating phases and a superfluid phase. We explore the nature of the insulating phase by calculating both the quasiparticle and quasihole dispersion relation. We also determine the parameters of our single band Bose-Hubbard model in terms of the microscopic parameters of the atoms in the optical lattice.

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We study the prospects for observing superfluidity in a spin-polarized atomic gas of 6 Li atoms, using state-of-the-art interatomic potentials. We determine the spinodal line and show that a BCS transition to the superfluid state can indeed occur in the (meta)stable region of the phase diagram if the densities are sufficiently low. We also discuss the stability of the gas due to exchange and dipolar relaxation and conclude that the prospects for observing superfluidity in a magnetically trapped atomic 6 Li gas are particularly promising for magnetic bias fields larger than 10 T. PACS numbers: 03.75.Fi, 32.80.Pj, 42.50.Vk Ultracold atomic gases have received much attention in recent years, because of their novel properties. For instance, these gases are well suited for high-precision measurements of single-atom properties and for the observation of collisional and optical phenomena that reflect the (Bose or Fermi) statistics of the constituent particles. Moreover, a large variety of experimental techniques are available to manipulate the atomic gas samples by means of electromagnetic fields, which offers the exciting possibility to achieve the required conditions for quantum degeneracy and to study macroscopic quantum effects in their purest form.At present, most experimental attempts towards quantum degeneracy have been performed with bosonic gases and have been aimed at the achievement of Bose-Einstein condensation. In particular, most of the earlier experiments used atomic hydrogen [1,2]. These experiments provided crucial ingredients for the recent attempts with alkali vapors, for which the experimental advances towards the degeneracy regime were so rapid that Bose-Einstein condensation has actually been reported now for the isotopes 87 Rb [3] and 7 Li [4].In view of these exciting developments it seems timely to investigate theoretically also the properties of spinpolarized atomic 6 Li, since 6 Li is a stable fermionic isotope of lithium that can be trapped and cooled in much the same way as its bosonic counterpart. Therefore, magnetically trapped 6 Li promises to be an ideal system to study degeneracy effects in a weakly interacting Fermi gas, thus providing valuable complementary information on the workings of quantum mechanics at the macroscopic level. Moreover, using a combination of theoretical analysis and experimental results [5][6][7], accurate knowledge of the interparticle (singlet and triplet) potential curves of lithium have recently been obtained, which lead to the prediction of a large and negative s-wave scattering length a of 24.6 3 10 3 a 0 (a 0 is the Bohr radius) for a spin-polarized 6 Li gas. This is important for two reasons: First, the fact that the scattering length is negative implies that at the low temperatures of interest [L ¿ r V , where L ͑2ph 2 ͞mk B T͒ 1͞2 is the thermal de Broglie wavelength of the atoms and r V is the range of the interaction] the effective interaction between the lithium atoms is attractive, and we expect a BCS-like phase transition to a superfluid state ...

The real-space densities of a polarized strongly-interacting two-component Fermi gas of 6 Li atoms reveal two low temperature regimes, both with a fully-paired core. At the lowest temperatures, the unpolarized core deforms with increasing polarization. Sharp boundaries between the core and the excess unpaired atoms are consistent with a phase separation driven by a first-order phase transition. In contrast, at higher temperatures the core does not deform but remains unpolarized up to a critical polarization. The boundaries are not sharp in this case, indicating a partially-polarized shell between the core and the unpaired atoms. The temperature dependence is consistent with a tricritical point in the phase diagram.PACS numbers: 03.75. Ss, 05.70.Fh, 74.25.Dw The formation of pairs consisting of one spin-up and one spin-down electron underlies the phenomenon of superconductivity. While the populations of the two spin components are generally equal in superconductors, an imbalance is readily produced in experiments with gases of trapped, ultracold fermionic atoms, as was recently demonstrated [1,2]. Exotic new states of matter are predicted for the unbalanced system that, if realized, may have important implications for our understanding of nuclei, compact stars, and quantum chromodynamics. Calculations show that phase separation between pairs and the excess unpaired atoms is one possible outcome for a strongly interacting two-component gas [3,4,5,6,7]. We previously reported evidence for a phase separation in a trapped atomic Fermi gas to a state containing a paired central core, with the excess unpaired atoms residing outside this core [2]. Such a phase separation can be detected in the real-space distributions using in-situ imaging, where a uniformly paired region produces a minimum in the difference distribution obtained by subtracting the majority and minority spin densities [2].In our previous work, we noted that the excess unpaired atoms reside primarily at the axial poles of the highly-elongated trap, while relatively few occupy the equatorial shell. As a result, the central minimum in the difference images was accompanied by a corresponding central dip in the axial density profile obtained by integrating the two-dimensional column density along the radial coordinate [2]. Several authors have calculated spatial distributions for phase separation, assuming both a harmonic trapping potential and the local density approximation (LDA) [8,9,10,11,12]. It was pointed out that, under these assumptions, a uniformly paired core would produce a constant axial density difference, rather than a central dip [13,14]. Shin et al. have recently adopted in-situ imaging, and present their resulting images as evidence for phase separation, though in their experiment, no such deformation is observed [15].In this paper, we characterize the properties of the phase-separated state using detailed quantitative measurements of the deformation and reconstructions of the three-dimensional (3D) density distributions. Furthermore, we...

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