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

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CHAPTER 5. EXPERIMENTAL RESULTS number of atoms 35000 30000 25000 20000 15000 10000 5000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 TOF in field / ms Figure 5.6: The number of detected atoms as function of the time of flight τ in magnetic field at the unitarity limit. The error bars are one standard error, the points without error bars are single points, thus the error is not defined. The line is a fit with the Landau-Zener model described in the text. 94
5.3. HYDRODYNAMIC EXPANSION The situation with fermions is more complicated. In order to achieve any kind of condensation we need interactions, as the non-interacting Fermi gas shows no phase transition. Therefore we have to treat three different cases: At very low interaction strength, the gas will behave collisionless and show ballistic expansion, this is the noninteracting regime. For stronger interactions we enter into the collisional regime where the gas behaves like a normal fluid. In both situations, the gas can condense into a superfluid below a certain temperature, which is the third case to consider. We now want to estimate which regime the atoms are in, in our experiment. We calculate the expected collision rate in the gas, and compare it to the relevant time scale in our system, the trapping frequency. The interactions reach their maximum at the unitarity limit, therefore we chose this point as reference. The scattering rate Γ can be calculated as Γ = nσv, where n is the density of scatterers, σ is the scattering cross section and v is the average velocity of the atoms. At the unitarity limit, the scattering cross section is σ = 4π/k 2 . The average velocity can be calculated from the temperature of the gas to be v = √ 3kBT/m. Now we can compare this scattering rate with the trap frequency and define the collisional parameter φ = Γ/ω. As we have two different trapping frequencies, we actually have two parameters φ⊥ = Γ/ω⊥ and φz = Γ/ωz = φ⊥/λ, with the definitions from section 2.1. To get a typical value for this parameter, we calculate it at the Fermi temperature, and use kF to calculate the scattering cross section, which leads to φ⊥ = (3λN)1/3 6π (5.8) This calculation can only be approximate, since we have ignored completely the changes in the scattering length k. A detailed calculation by Gehm et al. [106] 1 showed that we need to correct this value by a temperature-dependent factor shown in figure 5.7. Comparing to their experiments in reference [107], they find that a gas is to be treated as collisional above a collisional parameter of φ = 0,4. Our results usually have an atom number around 2 · 10 4 atoms, and the temperature is typically T = 0,2TF. This means that at unitarity we have φ⊥ ≈ 3, so we are clearly in the collisional regime. This changes once we leave unitarity: the scattering cross section will decrease or even vanish at the 1 This paper uses the term “collisional hydrodynamics” for what we call the collisional regime. Given that there is no well-established theory for this regime, using the term hydrodynamic is a bit adventurous. 95
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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
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equation (2.30) can easily be calcu
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2.4. BCS THEORY Using the substitut
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occupation probability 1.0 0.8 0.6
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2.4. BCS THEORY loose the last smal
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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 and 84: 5.1. MOMENTUM DISTRIBUTION Now we h
Page 85 and 86: 2 ¹h = ¹ 2 a m 2 6 5 4 3 2 1 0 -1
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: 5.3. HYDRODYNAMIC EXPANSION where A
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
Page 135 and 136: A.3 Expansion of a lithium gas in t
Page 137 and 138: Ry/Rx = 2.0(1) is consistent with t
Page 139 and 140: × ��
Page 141 and 142: Bth (G) Bexp (G) 218 not seen 230 2
Page 143 and 144: 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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