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

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CHAPTER 5. EXPERIMENTAL RESULTS Γ/γ 5 4 3 2 1 0 0,0 0,5 1,0 1,5 2,0 Te m pe rature T/T F Figure 5.7: The correction factor between the typical scattering rate γ resulting from the calculations in the text and the real scattering rate Γ as a function of the temperature. The scattering rate decreases for high temperatures as does the scattering cross section. At low temperatures the Pauli exclusion principle reduces the scattering rate. This can decrease scattering down to values found in the noninteracting regime, but as the gas condenses at these low temperatures, it will become hydrodynamic. Figure taken from reference [106] point where a = 0, so that we can expect to enter into the collisionless regime at some point. In the experiment, one has to be careful about judging whether an expansion is anisotropic or not: the ellipticity of the expanded cloud is not a clear signature, as the cloud will always be elliptic initially if the trap is not isotropic. An anisotropically expanding cloud will eventually invert its ellipticity. As this will never happen for an isotropically expanding cloud, this ellipticity inversion is a clear signature of an anisotropic expansion. Our final goal is to distinguish the superfluid state from a normal state. Originally, it was proposed to take the same path as for bosons 96
5.3. HYDRODYNAMIC EXPANSION and take the ellipticity inversion as a signature for superfluidity [108]. For the collisional regime, however, it was pointed out by Jackson et al. [109] 2 that at very strong interactions the cloud will also show ellipticity inversion, but this effect is much smaller than the ellipticity resulting from a superfluid cloud. Therefore we calculate the ellipticity that we expect from a hydrodynamic theory and compare to the experiment, and take the gas to be superfluid once the ellipticity inversion is close to the predicted value. 5.3.2 The scaling ansatz We now want to calculate the ellipticity of an expanding superfluid. In a superfluid, all particles have condensed into one macroscopic wavefunction ψ. This wavefunction follows the continuity equation of quantum mechanics, ∂|ψ| 2 + ∇ · j = 0 (5.9) ∂t where the probability current j is defined by 2mj = ψ∇ψ∗ − ψ∗∇ψ. We write the wave function as ψ = |ψ| exp iφ. Then the density is n = |ψ| 2 and the probability current becomes j = (¯h/m)n∇φ. After having defined a velocity v = j/n, we are left with the continuity equation of classical fluid mechanics: ∂n ∂t + ∇ · (nv) = 0 (5.10) A superfluid is a liquid with vanishing viscosity. Such a liquid is described by the Euler equation, which reads, after having used the Gibbs-Duhem relation [40], as m ∂v ∂t � + ∇ µ(n) + Vext(r) + 1 2 mv2 � = 0 (5.11) Now we are supposed to plug in the results from BCS theory, see section 2.4, for the equation of state µ(n). Unfortunately, this leads to coupled differential equations which are only numerically solvable. A good approximation is to use a polytropic equation of state, µ(n) ∝ n γ , where γ is the polytropic exponent [108]. Recently, a large amount of literature was published regarding this exponent since it is crucial in describing the Innsbruck [29] and Duke 2 This paper uses the term “hydrodynamic” only for the superfluid state as we do, opposed to Gehm et al., see last footnote 97
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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:
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 and 94: 5.3. HYDRODYNAMIC EXPANSION where A
Page 95: 5.3. HYDRODYNAMIC EXPANSION The sit
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
Page 145 and 146: RESONANT SCATTERING PROPERTIES CLOS
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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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