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

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18 CHAPTER 2. FORMALISM AT WEAK COUPLING
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
Page 1: Hubbard Model for AsymmetricUltraco
Page 4 and 5: coupling regime. The trapping poten
Page 6 and 7: 4.5 Critical temperatures . . . . .
Page 8 and 9: 2 CHAPTER 1. INTRODUCTIONlattice si
Page 10 and 11: 4 CHAPTER 1. INTRODUCTION1.2 Unbala
Page 12 and 13: 6 CHAPTER 1. INTRODUCTION
Page 14: 8 CHAPTER 2. FORMALISM AT WEAK COUP
Page 17 and 18: 2.3. SELF-CONSISTENCY EQUATIONS 11w
Page 19 and 20: 2.3. SELF-CONSISTENCY EQUATIONS 13w
Page 21 and 22: 2.4. PROPERTIES OF THE SELF-CONSIST
Page 23: 2.4. PROPERTIES OF THE SELF-CONSIST
Page 33 and 34: 3.3. NUMERICAL METHOD 27- In Figure
Page 35 and 36: 3.3. NUMERICAL METHOD 29Input of
Page 37 and 38: 3.4. NUMERICAL RESULTS 31x-position
Page 39 and 40: 3.4. NUMERICAL RESULTS 33x-position
Page 41 and 42: Chapter 4Model with spin-dependent
Page 43 and 44: 4.2. SUPERFLUIDITY AWAY FROM HALF F
Page 45 and 46: 4.3. NUMERICAL METHOD FOR THE PROBL
Page 47 and 48: 4.4. NUMERICAL RESULTS FOR THE GROU
Page 49 and 50: 4.5. CRITICAL TEMPERATURES 430.4∆
Page 51 and 52: 4.5. CRITICAL TEMPERATURES 454.5.2
Page 53 and 54: Chapter 5Strong coupling limitIn th
Page 55 and 56: 5.2. GENERAL PROCEDURE 49with an an
Page 57 and 58: = ∑ ijlσ= − ∑ ijlσ= ∑ ijl
Page 59 and 60: 5.4. LOW TEMPERATURE LIMIT 53At fir
Page 61 and 62: 5.4. LOW TEMPERATURE LIMIT 555.4.2
Page 63 and 64: Summary and OutlookIn this thesis w
Page 65 and 66: Bibliography[1] C. Wieman and E. Co
Page 67 and 68: DanksagungAn erster Stelle möchte

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