Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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A. List of Symbols and Abbreviations ˆH Hamiltonian η Diagonal matrix diag(1, −1), (section 2.5, subsection 3.3.2) η(r) Random field on rough surface (only section 3.1) i Imaginary unit i 2 = −1 k Wave vector L Distance in speckle geometry (only section 3.1) Λ Relative correction of the dispersion relation [in units of V 2 /µ 2 ] λ Wave length µ Chemical potential n Condensate density P Cauchy principal value Φ(r) Gross-Pitaevskii ground-state ˆϕ Phase operator ˆΨ(r) Field operator R Diffusor radius in speckle geometry (only section 3.1) ρ(ω) Density of states Sd Surface of the d-dimensional unit sphere S(k, ω) Spectral function subsection 3.4.1 Σ Self-energy σ Disorder correlation length T Transmission coefficient (subsection 2.4.5) TF Thomas-Fermi (page 26) 138 t Time uk Coefficient for Bogoliubov transformation uk = (ɛk + ɛk)/(2 √ ɛkɛk)
uν(r) Bogoliubov eigenstate, normal component V External potential ˜V (n) n-th order of the smoothed potential (2.24) �V Scattering amplitude (4.5) in the hydrodynamic regime vν(r) Bogoliubov eigenstate, anomalous component vk Coefficient for Bogoliubov transformation uk = (ɛk − ɛk)/(2 √ ɛkɛk) W Normal coupling element ξ Healing length Y Anomalous coupling element Z Partition function Bloch part Ã(z, t) Envelope function (6.7) b Collective coordinates (section 6.5): conjugate momentum of the width degree of freedom w d Lattice spacing dk Phase modulation with wavenumber k (section 6.6) F Force η Bounded time variable ∆k Quantity related to the Lyapunov exponent, equation (6.27) J Tunneling amplitude (6.4) λ Lyapunov exponent M Monodromy matrix p Momentum of the wave packet, equation (6.7) sk Density modulation with wavenumber k (section 6.6) 139
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Bogoliubov Excitations of Inhomogen
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Abstract In this thesis, different
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Contents 2.5. Exact diagonalization
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Contents Bibliography 141 List of F
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1. Interacting Bose Gases—Complex
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1.2. Disorder 1.2. Disorder Idealiz
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1.4. Cold-atoms—Universal model s
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1.5. Cold atoms—History and key e
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1.5. Cold atoms—History and key e
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1.6. Standing of this work are rest
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1.6. Standing of this work The disc
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2. The Inhomogeneous Bogoliubov Ham
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2.1. Bose-Einstein condensation of
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2.1. Bose-Einstein condensation of
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2.2. Interacting BEC and Gross-Pita
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n(x) n∞ 1 0.8 0.6 0.4 0.2 0 -3 -2
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2.3. Bogoliubov Excitations and pha
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2.3. Bogoliubov Excitations At high
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„ Φq ′ ¯Φ q ′ « V „ Φq
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(2) k ′ q k = ξ2 � k ′′2 +
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k k ′ 2.4.1. Single scattering se
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2.4. Single scattering event back a
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k + − w (1) k ′ k (a) 2.4. Sing
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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
Page 99 and 100: 4.1. Hydrodynamic limit I: ξ = 0 I
Page 101 and 102: ǫk 0.25 0.5 0.75 1 1.25 1.5 1.75 k
Page 103 and 104: gd(κ) v 2 1.75 1.5 1.25 1 0.75 0.5
Page 105 and 106: 4.2. Hydrodynamic limit II: towards
Page 107 and 108: ∆ǫk ǫk �0.1 �0.2 �0.3 �
Page 109 and 110: 1 0.5 0 -0.5 0 20 40 60 80 100 x/σ
Page 111 and 112: ΛN �0.1 �0.2 �0.3 �0.4 4.3
Page 113 and 114: nnc(V )−nnc(0) nnc(0) 1.2 1.15 1.
Page 115 and 116: 4.4. Particle regime which coincide
Page 117 and 118: 4.4. Particle regime regime, becaus
Page 119 and 120: 0.02 0.015 0.01 0.005 MSg 0.01 0.1
Page 121 and 122: 5. Conclusions and Outlook (Part I)
Page 123 and 124: 5.3. Theoretical outlook potential.
Page 125: Part II. Bloch Oscillations 113
Page 128 and 129: 6. Bloch Oscillations and Time-Depe
Page 146 and 147: λTB t/TB 0.15 0.05 70 60 50 40 30
Page 152 and 153: 140 A. List of Symbols and Abbrevia
Page 154 and 155: Bibliography [11] P. A. Lee and T.
Page 156 and 157: Bibliography [35] M. Lewenstein, A.
Page 158 and 159: Bibliography [58] J. Stenger, S. In
Page 160 and 161: Bibliography [81] V. I. Yukalov and
Page 162 and 163: Bibliography [108] A. Mostafazadeh,
Page 164 and 165: Bibliography [131] F. Bloch, Über
Page 167 and 168: Acknowledgments I thank everybody w
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