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

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CHAPTER 5. EXPERIMENTAL RESULTS Ellipticity 1.35 1.30 1.25 1.20 1.15 1.10 1.05 1.00 0.95 80 85 90 95 100 105 110 115 120 Magnetic Field / mT Figure 5.14: The ellipticity in the crossover at higher temperatures: here T = 0,3TF. The step has disappeared. The residual ellipticity is lower than its superfluid counterpart in figure 5.11 resonance. Our data are complementary to the data taken by the Innsbruck group [117]. They measured the quadrupole mode in a cigar-shaped harmonic trap along the long axis. They found its frequency and damping to change around 1/kFa = −0,8. They attributed this to the transition from a hydrodynamic to collisionless behavior. Their gas is colder than ours, below T = 0,1TF. Studying the temperature behavior, they found that heating up the gas to T = 0,2TF, the transition moves above 1/kFa = −0,66, which is in agreement with our measurements. 5.4 Molecular condensate On the molecular side of the Feshbach resonance, we performed additional studies on the molecular Bose-Einstein condensates expected there. As a BEC is superfluid, we expect the same behavior as in the 106
5.4. MOLECULAR CONDENSATE last sections. But more than that, we expect the momentum distribution after time of flight to show a bimodality, a landmark of Bose-Einstein condensation. Indeed, this behavior was observed both in 40 K at JILA [24], as well as in 6 Li at MIT [25]. In our experiments, however, we saw that the ellipticity corresponds to condensation, but the bimodality was not obvious, as it can be seen in the paper in appendix A.1. This is due to the much stronger interactions that we have compared to the usual condensation of bosonic atoms. The group in Innsbruck ramped the magnetic field down to 68 mT, a value where the bimodality becomes visible [72]. We attempt to model the expansion of the cloud with strong interactions. The theory of Bose-Einstein condensation is too wide a field to be treated here in depth, it can be found in textbooks (e.g. reference [40]). The condensate wave function follows the Gross-Pitaevskii equation i¯h ∂ψ ∂t = (−¯h2 ∇ 2 2m + Vext(r) + g|ψ| 2 )ψ (5.15) This equation can be seen as a Schrödinger equation with a kinetic and potential energy and an interaction term in the Hamiltonian. The interaction part is proportional to the condensate density nc = |ψ| 2 . The proportionality constant is the coupling g = 4π¯h 2 am/m. Here, am is the scattering length between molecules. In section 2.4.6 we saw that am = 0,6a, where a is the atom-atom scattering length. We suppose that there is a BEC in the center of the trap, which is expanding after switching off the trap. Around this condensate is a cloud of thermal molecules. We only consider interactions between the molecules in the thermal cloud and the BEC as a whole. Then the BEC acts as a potential for the thermal cloud. This way, we can write the Hamiltonian of the molecules in the thermal cloud: H(r, p) = p2 2m 1 � + m ω 2 i∈{x,y,z} 2 i r 2 i + 2gnc(r) (5.16) The term gnc is exactly twice the interaction energy which appears in the Gross-Pitaevskii equation. The factor two is owed to the exchange interaction of bosonic particles. One might note here that the two is not the BCS result of the molecule-molecule interaction being twice the atom-atom interaction, as our basic particle is already a molecule. In the trap, this gives a Mexican hat potential as illustrated in figure 5.15. We start our simulation by generating a random distribution of up to one million atoms following the Bose-Einstein distribution function 107
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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,
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3.1. THE VACUUM CHAMBER Figure 3.1:
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3.2. THE ZEEMAN SLOWER Figure 3.2:
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3.3 The laser system 3.3. THE LASER
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2 2 P 3/2 10 GH z 2 2 P 1/2 671 nm
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input to Fabry- Pérot input non-po
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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 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: ellipticity 1.4 1.3 1.2 1.1 1.0 0.9
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
Page 145 and 146: RESONANT SCATTERING PROPERTIES CLOS
Page 151 and 152: A.5 Expansion of an ultra-cold lith
Page 153 and 154: 2 L. Tarruell, et al. to image the
Page 155 and 156: 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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