Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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Contents Bibliography 141 List of Figures x 1.1. Time-of-flight images of Bose-Einstein condensation . . . . . 7 1.2. Observation of sound propagation . . . . . . . . . . . . . . . 8 2.1. Bunching of bosons in a minimal system . . . . . . . . . . . 18 2.2. Condensate density profile in presence of an impurity . . . . 29 2.3. Bogoliubov dispersion relation . . . . . . . . . . . . . . . . . 32 2.4. Bogoliubov scattering vertex . . . . . . . . . . . . . . . . . . 35 2.5. 2D single-scattering setup . . . . . . . . . . . . . . . . . . . 39 2.6. First-order scattering envelope functions . . . . . . . . . . . 43 2.7. Fourier analysis of the stationary scattering state . . . . . . 44 2.8. Elastic scattering amplitude . . . . . . . . . . . . . . . . . . 46 2.9. Transmission of Bogoliubov excitations across a narrow impurity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1. Schematic representation of the disordered Bogoliubov setting 57 3.2. Principle of the speckle phenomenon . . . . . . . . . . . . . 58 3.3. Speckle correlation functions in d = 1, 2, 3 . . . . . . . . . . 62 3.4. Geometry of the scattering process . . . . . . . . . . . . . . 78 3.5. Bogoliubov density of states . . . . . . . . . . . . . . . . . . 79 4.1. Parameter space of the full Bogoliubov problem . . . . . . . 81 4.2. Relative correction of the speed of sound (at kξ = 0.05) . . . 82 4.3. Mean free path and Boltzmann transport length . . . . . . . 86 4.4. Disorder-averaged dispersion relation (at ξ = 0) . . . . . . . 89 4.5. Correction of the density of states (at ξ = 0) . . . . . . . . . 91 4.6. Typical virtual scattering event in the regime kσ ≪ 1, kξ ≪ 1 93 4.7. Relative correction of the dispersion relation (at k = 0) . . . 95 4.8. Speckle potential and ground-state density profile . . . . . . 97 4.9. Histograms of the correction to the speed of sound . . . . . . 98 4.10. Correction to the speed of sound as function of the disorder strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.11. Increase of the non-condensed density nnc due to disorder . . 101 4.12. Relative disorder correction MN = (ɛk − ɛk)µ/V 2 0 (at kξ = 10) 104 4.13. Real part of the self-energy for individual atoms . . . . . . . 106 4.14. Transition from the Bogoliubov regime to free-particle plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
List of Tables 5.1. Localized Bogoliubov quasiparticles in three dimensions . . . 112 6.1. Schematic representation of the setting for Bloch oscillations 115 6.2. Sketch of the dispersion relation in a lattice . . . . . . . . . 116 6.3. Sketch of the intensity of two counter-propagating laser beams117 6.4. Typical initial state for Bloch oscillations . . . . . . . . . . . 119 6.5. Time evolution of stable Bloch oscillations . . . . . . . . . . 123 6.6. Real-space portraits of Bloch oscillations for several modulations g(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.7. Key quantities of the Bloch oscillation with double Bloch period127 6.8. Contraction of the Bloch oscillating wave packet . . . . . . . 128 6.9. Drift and damping of the centroid motion . . . . . . . . . . 129 6.10. Growth of the most unstable mode . . . . . . . . . . . . . . 133 6.11. Rigid and breathing soliton under harmonic perturbation g1 sin(F t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.12. Stability map . . . . . . . . . . . . . . . . . . . . . . . . . . 135 List of Tables 1.1. Typical temperatures and particle densities in BEC experiments 7 6.1. Collective-coordinates parameters for Gaussian and solitonshaped wave packets . . . . . . . . . . . . . . . . . . . . . . 126 List of Boxes 2.1. Feynman diagrams of the condensate function Φ . . . . . . . 27 3.1. Feynman rules: drawing and computing irreducible diagrams 70 xi
Page 1: Bogoliubov Excitations of Inhomogen
Page 5: Abstract In this thesis, different
Page 8 and 9: Contents 2.5. Exact diagonalization
Page 13 and 14: 1. Interacting Bose Gases—Complex
Page 15 and 16: 1.2. Disorder 1.2. Disorder Idealiz
Page 17 and 18: 1.4. Cold-atoms—Universal model s
Page 19 and 20: 1.5. Cold atoms—History and key e
Page 21 and 22: 1.5. Cold atoms—History and key e
Page 23 and 24: 1.6. Standing of this work are rest
Page 25: 1.6. Standing of this work The disc
Page 29 and 30: 2. The Inhomogeneous Bogoliubov Ham
Page 31 and 32: 2.1. Bose-Einstein condensation of
Page 33 and 34: 2.1. Bose-Einstein condensation of
Page 35 and 36: 2.2. Interacting BEC and Gross-Pita
Page 41 and 42: n(x) n∞ 1 0.8 0.6 0.4 0.2 0 -3 -2
Page 43 and 44: 2.3. Bogoliubov Excitations and pha
Page 45 and 46: 2.3. Bogoliubov Excitations At high
Page 47 and 48: „ Φq ′ ¯Φ q ′ « V „ Φq
Page 49 and 50: (2) k ′ q k = ξ2 � k ′′2 +
Page 51 and 52: k k ′ 2.4.1. Single scattering se
Page 53 and 54: 2.4. Single scattering event back a
Page 55 and 56: k + − w (1) k ′ k (a) 2.4. Sing
Page 57 and 58: 2.4. Single scattering event theory
Page 59 and 60: T 1 B 2 2.5. Exact diagonalization
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2.5. Exact diagonalization of the B
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3. Disorder In the previous chapter
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3.1.1. Speckle amplitude 3.1. Optic
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3.1. Optical speckle potential In c
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3.2. A suitable basis for the disor
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3.2. A suitable basis for the disor
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3.3. Effective medium and diagramma
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3.4. Deriving physical quantities f
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3.4. Deriving physical quantities f
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Fixed particle number 3.4. Deriving
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ρµξ d 0.2 0.15 0.1 0.05 1 2 3 4
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4. Disorder—Results and Limiting
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4.1. Hydrodynamic limit I: ξ = 0 t
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4.1. Hydrodynamic limit I: ξ = 0 T
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4.1. Hydrodynamic limit I: ξ = 0 I
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ǫk 0.25 0.5 0.75 1 1.25 1.5 1.75 k
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gd(κ) v 2 1.75 1.5 1.25 1 0.75 0.5
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4.2. Hydrodynamic limit II: towards
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∆ǫk ǫk �0.1 �0.2 �0.3 �
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1 0.5 0 -0.5 0 20 40 60 80 100 x/σ
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ΛN �0.1 �0.2 �0.3 �0.4 4.3
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nnc(V )−nnc(0) nnc(0) 1.2 1.15 1.
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4.4. Particle regime which coincide
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4.4. Particle regime regime, becaus
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0.02 0.015 0.01 0.005 MSg 0.01 0.1
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5. Conclusions and Outlook (Part I)
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5.3. Theoretical outlook potential.
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Part II. Bloch Oscillations 113
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6. Bloch Oscillations and Time-Depe
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λTB t/TB 0.15 0.05 70 60 50 40 30
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A. List of Symbols and Abbreviation
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140 A. List of Symbols and Abbrevia
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Bibliography [11] P. A. Lee and T.
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Bibliography [35] M. Lewenstein, A.
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Bibliography [58] J. Stenger, S. In
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Bibliography [81] V. I. Yukalov and
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Bibliography [108] A. Mostafazadeh,
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Bibliography [131] F. Bloch, Über
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Acknowledgments I thank everybody w
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