Hubbard Model for Asymmetric Ultracold Fermionic ... - KOMET 337

Chapter 3Unbalanced Fermi-mixturesIn this chapter we will discuss a Fermi-mixture with unequal occupation numbers for thedifferent spin species. We will assume that the system is in the ground state T = 0 andthat the hopping amplitudes are the same for both spin-species. The parabolic potentialcaused by the magneto-optical trap will be introduced and treated within the local densityapproximation (LDA). We will discuss the phases which can occur in such a system and shownumerical results for given trap parameters.3.1 Broken translational invariance and the LDAIn this section we will introduce the concept of the LDA for treating the trapping potential.Ultracold quantum gases are usually trapped in a magneto-optical trap, which can be describedby a quadratic potential for the atoms. The trapping potential and the superimposedoptical lattice form the effective potential in space which the atoms feel [5]. We illustrate thepotential here in one dimension (arbitrary units):2121-2 -1 1 2-1-2 -1 1 2-1-2-2Figure 3.1: Potentials felt by the atomsRed curve: Lattice potentialGreen curve: Trapping potentialFigure 3.2: Sum of the lattice and trapping potentialsIn our one-band tight-binding approximation we assumed that the atoms can be describedas Wannier-states, which are localized around specific lattice sites. Here the position of alattice site is defined, e.g., by a minimum of the translationally invariant lattice potential.We will now assume that the trapping potential varies slowly enough in space, so that thedescription via the Wannier-states is still acceptable and the hopping of atoms from one site19