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

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CHAPTER 2. THEORY resonance r0/a0 Bres/mT ∆B/mT abg/a0 µ/µB γ −1/3 /a0 broad 29,9 83,415 30 -1405 2,0 101 narrow 29,9 54,326 -0,1 61,6 2,1 400 Table 2.1: The parameters of our model for the broad and narrow Feshbach resonances, from reference [55]. a0 is the Bohr radius, a0 = 0,53 · 10 −10 m, and µB is the Bohr magneton, µB = 9,3 · 10 −24 J/T. 0 m ) = ( hMHz E or a = a 4000 2000 0 -2000 -4000 0 20 40 60 80 100 120 140 B = mT Figure 2.6: The scattering length a (solid line) and the molecular binding energy Em (dashed line) by the magnetic field. The large resonance is centered at 83 mT. The narrow resonance at 54,325 mT is depicted by the vertical bar, it is too narrow to be shown otherwise. The scattering length crosses zero at 52,8 mT [19]. 28
2.3. FESHBACH RESONANCES to fit the real scattering length with an accuracy better than 1% in the relevant range from 60 mT to 120 mT using the empirical formula � a = ab 1 + ∆ � (1 + α(B − B0)) (2.21) B − B0 with the fitted parameters ab = −1405a0, B0 = 83,4149 mT, ∆ = 30,0 mT and α = 0,0040 mT −1 . On the left side of the resonance, the existence of the bound state in the closed channel close to resonance leads to a molecular state of atom pairs. One should not get confused between the two: the bound state exists only in the closed channel. The molecular state which we are about to discuss, on the other hand, is a state that depends on both channels and is a stable state that can be occupied for macroscopic time scales. Certainly, the molecular state is also a bound state in the sense that its wave function vanishes at large distances, leading to an exponentially decaying wave function |χ〉 = e −√ εmr |o〉 for r > r0 (2.22) where −Em = −(¯h 2 /m)εm is the binding energy of the molecule (Em > 0), and the only change we have to apply to equation (2.14b) is to replace the wave vectors q by ¯q 2 ± = q 2 ± −εm, and the mixing angle φ by ¯φ. Hence, (2.17) becomes − √ εm = ¯q+ cos 2 θ tan ¯q+r0 + ¯q− sin 2 θ tan ¯q−r0 (2.23) Using the same approximations as above, we can calculate the molecular binding energy as − √ εm = 1 r0 − abg − γ εc + εm (2.24) plotted in figure 2.6. If the molecular binding energy is small, we can compare this to equation (2.18). This gives us a simple relation between the scattering length and the binding energy: Em = ¯h2 ma 2 (2.25) This is the same result we get from a single channel calculation from equation 2.9. As long as the effective range r0 is sufficiently short, we can use an effective single channel model in most cases. 29
Page 1 and 2: École Normale Supérieure Laborato
Page 3 and 4: Contents 1 Introduction 7 2 Theory
Page 5 and 6: Acknowledgments This thesis would n
Page 7 and 8: Chapter 1 Introduction “The inter
Page 9 and 10: T c / T F 1 10 -2 10 -4 cold Fe rm
Page 11 and 12: JILA, where they use magnetic field
Page 13 and 14: Chapter 2 Theory Da steh’ ich nun
Page 15 and 16: 2.2. A REMINDER ON SCATTERING THEOR
Page 17 and 18: 2.2. A REMINDER ON SCATTERING THEOR
Page 19 and 20: pote ntial e ne rgy or probability
Page 21 and 22: 2.3. FESHBACH RESONANCES a total an
Page 23 and 24: energy / mRy 2.0 1.5 1.0 0.5 0.0 -0
Page 25 and 26: E / GH z 0,0 -0,5 -1,0 -1,5 -2,0 0
Page 27: 2.3. FESHBACH RESONANCES relating t
Page 31 and 32: 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
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F T = T r o f e u l a v d e t t i f
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Chapter 5 Experimental results Well
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5.1. MOMENTUM DISTRIBUTION Now we h
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2 ¹h = ¹ 2 a m 2 6 5 4 3 2 1 0 -1
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5.1. MOMENTUM DISTRIBUTION get na 3
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n( k) 3 F k 1.0 0.8 0.6 0.4 0.2 5.2
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n( k) 3 F k 1.0 0.8 0.6 0.4 0.2 0.0
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5.3. HYDRODYNAMIC EXPANSION where A
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5.3. HYDRODYNAMIC EXPANSION The sit
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5.3. HYDRODYNAMIC EXPANSION and tak
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° 0.67 0.66 0.65 0.64 0.63 0.62 0.
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ellipticity 1.6 1.4 1.2 1.0 0.8 clo
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Ellipticity 1.55 1.50 1.45 1.40 1.3
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ellipticity 1.4 1.3 1.2 1.1 1.0 0.9
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5.4. MOLECULAR CONDENSATE last sect
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density (a. u.) 70 60 50 40 30 20 1
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E / GH z 0 -1 -2 |1,1〉 ⊗ |1/2,-
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number of atoms / 1000 25 20 15 10
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5.6. OTHER EXPERIMENTS AND OUTLOOK
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Tem perature T c 0 Sarm a Ph ase 5.
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5.6. OTHER EXPERIMENTS AND OUTLOOK
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Chapter 6 Conclusions In this thesi
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Cold fermions in optical lattices c
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Appendix A Articles A.1 Experimenta
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VOLUME 93, NUMBER 5 FIG. 1. Onset o
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VOLUME 93, NUMBER 5 gas. The platea
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P-wave Feshbach resonances of ultra
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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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