Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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Contents 2.5. Exact diagonalization of the Bogoliubov problem . . . . . . 47 2.5.1. Analogy to other bosonic systems . . . . . . . . . . . 48 2.5.2. Bogoliubov-de-Gennes equations . . . . . . . . . . . . 48 2.5.3. Zero-frequency mode . . . . . . . . . . . . . . . . . . 50 2.5.4. Eigenstates of non self-adjoint operators . . . . . . . 50 2.5.5. Bogoliubov eigenstates with non-zero frequency . . . 51 2.5.6. Non-condensed atom density . . . . . . . . . . . . . . 54 2.6. Conclusions on Gross-Pitaevskii and Bogoliubov . . . . . . . 56 3. Disorder 57 3.1. Optical speckle potential . . . . . . . . . . . . . . . . . . . . 58 3.1.1. Speckle amplitude . . . . . . . . . . . . . . . . . . . . 59 3.1.2. Generalization to 3D . . . . . . . . . . . . . . . . . . 60 3.1.3. Intensity and potential correlations . . . . . . . . . . 61 3.2. A suitable basis for the disordered problem . . . . . . . . . . 63 3.2.1. Bogoliubov basis in terms of free particle states . . . 64 3.2.2. Bogoliubov basis in terms of density and phase . . . 64 3.3. Effective medium and diagrammatic perturbation theory . . 66 3.3.1. Green functions . . . . . . . . . . . . . . . . . . . . . 66 3.3.2. The self-energy . . . . . . . . . . . . . . . . . . . . . 68 3.3.3. Computing the self-energy in the Born approximation 71 3.4. Deriving physical quantities from the self-energy . . . . . . . 73 3.4.1. The physical meaning of the self-energy . . . . . . . . 73 3.4.2. Mean free path . . . . . . . . . . . . . . . . . . . . . 74 3.4.3. Boltzmann transport length . . . . . . . . . . . . . . 75 3.4.4. Localization length . . . . . . . . . . . . . . . . . . . 75 3.4.5. Renormalization of the dispersion relation . . . . . . 76 3.4.6. Density of states . . . . . . . . . . . . . . . . . . . . 79 4. Disorder—Results and Limiting Cases 81 4.1. Hydrodynamic limit I: ξ = 0 . . . . . . . . . . . . . . . . . . 83 4.1.1. Direct derivation from hydrodynamic equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.1.2. Transport length scales . . . . . . . . . . . . . . . . . 85 4.1.3. Speed of sound . . . . . . . . . . . . . . . . . . . . . 87 4.1.4. Density of states . . . . . . . . . . . . . . . . . . . . 90 4.2. Hydrodynamic limit II: towards δ-disorder . . . . . . . . . . 92 4.2.1. Mean free path . . . . . . . . . . . . . . . . . . . . . 92 4.2.2. Speed of sound . . . . . . . . . . . . . . . . . . . . . 93 4.3. Numerical study of the speed of sound . . . . . . . . . . . . 96 4.3.1. The numerical scheme . . . . . . . . . . . . . . . . . 96 viii
Contents 4.3.2. Disorder average and range of validity of the Born prediction . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3.3. Non-condensed fraction . . . . . . . . . . . . . . . . . 100 4.3.4. Speed of sound as function of the correlation length . 101 4.4. Particle regime . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.4.1. Mean free path . . . . . . . . . . . . . . . . . . . . . 102 4.4.2. Renormalization of the dispersion relation in the Bogoliubov regime . . . . . . . . . . . . . . . . . . . . . 103 4.4.3. Transition to really free particles . . . . . . . . . . . 105 4.4.4. Closing the gap with a Gross-Pitaevskii integration . 106 4.4.5. Conclusions on the particle limit . . . . . . . . . . . 108 5. Conclusions and Outlook (Part I) 109 5.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2. Experimental proposals . . . . . . . . . . . . . . . . . . . . . 110 5.3. Theoretical outlook . . . . . . . . . . . . . . . . . . . . . . . 111 II. Bloch Oscillations 113 6. Bloch Oscillations and Time-Dependent Interactions 115 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.1.1. Bloch oscillation of a single particle . . . . . . . . . . 116 6.1.2. Experimental realization . . . . . . . . . . . . . . . . 117 6.1.3. Time dependent interaction g(t) . . . . . . . . . . . . 118 6.2. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.2.1. Tight binding approximation . . . . . . . . . . . . . . 119 6.2.2. Smooth-envelope approximation . . . . . . . . . . . . 120 6.3. Periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . 122 6.4. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 123 6.5. Collective coordinates . . . . . . . . . . . . . . . . . . . . . . 124 6.5.1. Breathing dynamics in the stable cases . . . . . . . . 127 6.5.2. Unstable cases—decay mechanisms and other dynamics127 6.6. Dynamical instabilities . . . . . . . . . . . . . . . . . . . . . 130 6.6.1. Linear stability analysis of the infinite wave packet . 130 6.6.2. Bloch periodic perturbations . . . . . . . . . . . . . . 131 6.6.3. Unstable sine . . . . . . . . . . . . . . . . . . . . . . 132 6.6.4. Robustness with respect to small perturbations . . . 133 6.7. Conclusions (Part II) . . . . . . . . . . . . . . . . . . . . . . 136 A. List of Symbols and Abbreviations 137 ix
Page 1: Bogoliubov Excitations of Inhomogen
Page 5: Abstract In this thesis, different
Page 10 and 11: Contents Bibliography 141 List of F
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
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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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