Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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5. Conclusions and Outlook (Part I) Figure 5.1: Density of states in three dimensions (compare to figure 3.5(c)). For sufficiently strong disorder, the Bogoliubov excitations in the crossover region at kξ ≈ 1 are expected to localize (gray). ρ 0.25 0.5 0.75 1 1.25 1.5 1.75 2 ¯hω/µ are now combined to the Bogoliubov dispersion relation (2.35), figure 2.3. Depending on the disorder strength, two regimes are possible: 1. Weak disorder µ < ω ∗ , E ∗ < µ. Phonon and particle regime are combined without any localized states. 2. Strong disorder ω ∗ < µ < E ∗ . There is a range of localized states around the crossing from phonon to particle excitations. The characterization of these localized states between phonons and particles is an interesting topic for further research. A possible approach is a diagrammatic perturbation theory analogous to Vollhardt’s and Wölfle’s approach [123], which allows computing the weak-localization to the diffusion constant. This approach has already been applied to non-interacting cold atoms in speckle potentials [22, 128]. With the block-matrix notation employed in subsection 3.3.1, the Bogoliubov propagator follows equivalent equations of motion. It should be possible to compute the weak-localization correction to the diffusion constant analogously. Finite temperature With the Bogoliubov formalism set up in this work, it is in principle possible, to consider finite-temperature problems. Similar to the zero-temperature Green functions in the block-matrix formalism of subsection 3.3.2, also Matsubara Green functions could be set up and computed. However, the Bogoliubov formalism relies on the macroscopic occupation of the condensate mode and neglects the interactions between the Bogoliubov excitations. At higher temperatures, these interaction effects become relevant and one should take beyond-Bogoliubov terms into account. These interactions appear as additional perturbations and might be accessible with methods similar to those used by Hugenholtz and Pines [76]. However, two perturbations at once, disorder and the interaction among quasiparticles should make the problem extremely complicated. 112
Part II. Bloch Oscillations 113
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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
Page 73 and 74: 3.1. Optical speckle potential In c
Page 75 and 76: 3.2. A suitable basis for the disor
Page 77 and 78: 3.2. A suitable basis for the disor
Page 79 and 80: 3.3. Effective medium and diagramma
Page 85 and 86: 3.4. Deriving physical quantities f
Page 87 and 88: 3.4. Deriving physical quantities f
Page 89 and 90: Fixed particle number 3.4. Deriving
Page 91 and 92: ρµξ d 0.2 0.15 0.1 0.05 1 2 3 4
Page 93 and 94: 4. Disorder—Results and Limiting
Page 95 and 96: 4.1. Hydrodynamic limit I: ξ = 0 t
Page 97 and 98: 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: 5.3. Theoretical outlook potential.
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 150 and 151: A. List of Symbols and Abbreviation
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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