Martin Teichmann Atomes de lithium-6 ultra froids dans la ... - TEL

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CHAPTER 5. EXPERIMENTAL RESULTS cloud, µ decreases. On the BCS side, x0 increases, reaching infinity at the border of the cloud. The BEC side is more interesting: x0 decreases and even crosses zero, so that µ is negative at the cloud’s border. As a negative µ leads to a bosonic behavior, we can thus have the situation that the center of our cloud behaves femionic, while the border shows bosonic behavior. From equation (5.4) we can calculate the dependences x0(r) and ∆(r), and substitute into (5.1). This leads to � n(k) = � εk/∆(r) − x0(r) 1 − � (εk/∆(r) − x0(r)) 2 � d + 1 3r (2π) 3 = a6 ho 2a3 � � ∞ a 1 − x0∆(x0)=µ0 2k2G2 (x0) − x0 � (a2k2G 2 (x0) − x0) 2 � + 1 � 4µ0a · 2m ¯h 2 − x0 G2 G(x0) − 2x0G (x0) ′ (x0) G3 dx0 (x0) (2π) 3 (5.5) This function is plotted in figure 5.2 for different values of kFa. In the same graph, we show the distribution in the unitarity limit (equation (5.3)) and in the limiting cases of a noninteracting gas, using the same equations as in chapter 4. Furthermore, we show the momentum distribution of a molecular condensate that we will discuss in the next section. One sees that the transition between the different parameter ranges are smooth, although they are derived from different formulae. The steepest curve corresponds to the noninteracting gas. It actually reaches zero at the Fermi momentum. The central density decreases while traversing the crossover, and the distribution broadens. 5.1.3 Molecular condensate For weak repulsive interactions, meaning that 0 < kFa ≪ 1, the atoms form deeply bound molecules, as seen at the end of section 2.3. In this case the binding energy of the molecules dominates over the interaction energy, and the momentum distribution is determined by the molecular wavefunction. Here we want to show that this also follows from the presented theory. In section 2.4.5 we saw that in the molecular regime, the chemical potential becomes negative and the gap becomes large, as shown in figure 2.7. In combination with figure 5.1 we see that in this case x0 = µ/∆ � −∞. We can develop the equations 2.38 for x−1 0 � −0 and 84
2 ¹h = ¹ 2 a m 2 6 5 4 3 2 1 0 -1 5.1. MOMENTUM DISTRIBUTION -2 0 2 ¹ = ¢ 4 6 Figure 5.1: An illustration of the gap equation. We present the normalized chemical potential as a function of the ratio x0 of the chemical potential and the gap. The left branch corresponds to the BEC side of the resonance, while the right one corresponds to BCS. 85
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École Normale Supérieure Laborato
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Contents 1 Introduction 7 2 Theory
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Acknowledgments This thesis would n
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Chapter 1 Introduction “The inter
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T c / T F 1 10 -2 10 -4 cold Fe rm
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JILA, where they use magnetic field
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Chapter 2 Theory Da steh’ ich nun
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2.2. A REMINDER ON SCATTERING THEOR
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2.2. A REMINDER ON SCATTERING THEOR
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pote ntial e ne rgy or probability
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2.3. FESHBACH RESONANCES a total an
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energy / mRy 2.0 1.5 1.0 0.5 0.0 -0
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E / GH z 0,0 -0,5 -1,0 -1,5 -2,0 0
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2.3. FESHBACH RESONANCES relating t
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2.3. FESHBACH RESONANCES to fit the
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second quantization, this is writte
Page 33 and 34: equation (2.30) can easily be calcu
Page 35 and 36: 2.4. BCS THEORY Using the substitut
Page 37 and 38: occupation probability 1.0 0.8 0.6
Page 39 and 40: 2.4. BCS THEORY loose the last smal
Page 41 and 42: Chapter 3 Experimental setup LASER,
Page 43 and 44: 3.1. THE VACUUM CHAMBER Figure 3.1:
Page 45 and 46: 3.2. THE ZEEMAN SLOWER Figure 3.2:
Page 47 and 48: 3.3 The laser system 3.3. THE LASER
Page 49 and 50: 2 2 P 3/2 10 GH z 2 2 P 1/2 671 nm
Page 51 and 52: input to Fabry- Pérot input non-po
Page 53 and 54: 7 P Am p to m agnetic trap im aging
Page 55 and 56: atom ic be am pinch coils M O T coi
Page 57 and 58: 3.5. THE OPTICAL DIPOLE TRAP The he
Page 59 and 60: heating time / s 10 4 10 3 10 2 10
Page 61 and 62: Physics laser for the vertical. 3.6
Page 63 and 64: frequency / MHz 1000 950 900 850 3.
Page 65 and 66: 3.6. THE COOLING STRATEGY atoms in
Page 67 and 68: 3.7 MOT imaging 3.7. MOT IMAGING Wh
Page 69 and 70: 3.8. COMPUTER CONTROL where the cod
Page 71 and 72: from scipy.special import erf from
Page 73 and 74: Chapter 4 Data analysis A correct d
Page 75 and 76: 4.1. DETERMINATION OF THE TEMPERATU
Page 77 and 78: optical density 14 12 10 4.1. DETER
Page 79 and 80: F T = T r o f e u l a v d e t t i f
Page 81 and 82: Chapter 5 Experimental results Well
Page 83: 5.1. MOMENTUM DISTRIBUTION Now we h
Page 87 and 88: 5.1. MOMENTUM DISTRIBUTION get na 3
Page 89 and 90: n( k) 3 F k 1.0 0.8 0.6 0.4 0.2 5.2
Page 91 and 92: n( k) 3 F k 1.0 0.8 0.6 0.4 0.2 0.0
Page 93 and 94: 5.3. HYDRODYNAMIC EXPANSION where A
Page 95 and 96: 5.3. HYDRODYNAMIC EXPANSION The sit
Page 97 and 98: 5.3. HYDRODYNAMIC EXPANSION and tak
Page 99 and 100: ° 0.67 0.66 0.65 0.64 0.63 0.62 0.
Page 101 and 102: ellipticity 1.6 1.4 1.2 1.0 0.8 clo
Page 103 and 104: Ellipticity 1.55 1.50 1.45 1.40 1.3
Page 105 and 106: ellipticity 1.4 1.3 1.2 1.1 1.0 0.9
Page 107 and 108: 5.4. MOLECULAR CONDENSATE last sect
Page 109 and 110: density (a. u.) 70 60 50 40 30 20 1
Page 111 and 112: E / GH z 0 -1 -2 |1,1〉 ⊗ |1/2,-
Page 113 and 114: number of atoms / 1000 25 20 15 10
Page 115 and 116: 5.6. OTHER EXPERIMENTS AND OUTLOOK
Page 117 and 118: Tem perature T c 0 Sarm a Ph ase 5.
Page 119 and 120: 5.6. OTHER EXPERIMENTS AND OUTLOOK
Page 121 and 122: Chapter 6 Conclusions In this thesi
Page 123 and 124: Cold fermions in optical lattices c
Page 125 and 126: Appendix A Articles A.1 Experimenta
Page 127 and 128: VOLUME 93, NUMBER 5 FIG. 1. Onset o
Page 129 and 130: VOLUME 93, NUMBER 5 gas. The platea
Page 131 and 132: P-wave Feshbach resonances of ultra
Page 133 and 134: P-WAVE FESHBACH RESONANCES OF ULTRA
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A.3 Expansion of a lithium gas in t
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Ry/Rx = 2.0(1) is consistent with t
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× ��
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Bth (G) Bexp (G) 218 not seen 230 2
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Resonant scattering properties clos
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RESONANT SCATTERING PROPERTIES CLOS
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A.5 Expansion of an ultra-cold lith
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2 L. Tarruell, et al. to image the
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4 L. Tarruell, et al. of mean field
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6 L. Tarruell, et al. The starting
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8 L. Tarruell, et al. for 0.5 ms in
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10 L. Tarruell, et al. [11] M. H. A
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Appendix B Bibliography [1] H. VOGE
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[29] M. BARTENSTEIN, A. ALTMEYER, S
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[62] P. NOZIÈRES and S. SCHMITT-RI
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[98] M. MARINI, F. PISTOLESI, and G
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[131] S. STRINGARI, Collective Exci
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