Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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4. Disorder—Results and Limiting Cases 4.4.5. Conclusions on the particle limit Bogoliubov quasiparticles at high energies and free particles without any condensate background have the same clean dispersion relation ɛ0 k and the same elastic scattering amplitude W (1) . Concerning the disorder correction to the dispersion relation, however, their respective behaviors appear contradictory. The correction derived in the Schrödinger regime (4.43) is not recovered by the correction in the Bogoliubov regime (4.38). The leading-order of the latter can be absorbed in an overall shift of the chemical potential (4.39), which leaves physical quantities like the effective mass, i.e. the curvature of the dispersion relation untouched. This is similar to section 2.4, page 44, where we also shifted the chemical potential in order to recover the equations of motion for free particles. It is misleading to expect that the disordered Bogoliubov problem as treated in chapter 3 should pass over to the problem of single atoms in disordered potential [22, 23]. Though the underlying Hamiltonians are the same in the limit g = 0 (or µ = 0, or ξ = ∞), differences occur because of the approximations made. In the expansion of the Bogoliubov Hamiltonian, the smoothed potential (2.21) was used, with the validity condition µ ≫ V . The interactions of Bogoliubov excitations with the condensate background scale like gn0(r) ∼ µ and will never be negligible compared to the disorder potential. After these problems with the real renormalization of the dispersion relation, it is astonishing that there are no differences in the elastic scattering processes. This can be understood in the following simplified reasoning: Schrödinger particles are subject to the external potential only, whereas Bogoliubov excitations in the particle regime see both, the external potential and the condensate background. At relevant wave numbers kξ ≫ 1, the condensate is so smooth that it cannot contribute to the scattering, thus, the elastic properties of Schrödinger particles and Bogoliubov excitations are the same. However, the homogeneous background does shift the mean energy, therefrom the shift of the chemical potential found in equation (4.39). Beyond the leading correction (4.38), there should be k-dependent corrections, but these are at least one order (kξ) −1 smaller. 108
5. Conclusions and Outlook (Part I) 5.1. Summary A general formalism for the description of the Bogoliubov excitations of inhomogeneous Bose-Einstein condensates has been set up. The difficulty of the inhomogeneous many-particle problem has been solved by means of an inhomogeneous Bogoliubov approximation. Taking advantage of the macroscopically populated condensate state, we have separated the problem into the mean-field condensate function Φ(r) and the quantized inhomogeneous Bogoliubov problem. Via the Gross-Pitaevskii equation (2.16), the external potential enters the condensate function Φ(r) nonlinearly. The condensate function then enters the Bogoliubov Hamiltonian and makes it inhomogeneous. The Bogoliubov Hamiltonian (2.42) depends in a simple way on Φ(r) and ¯ Φ(r) = Φ 2 0/Φ(r), i.e., if one could solve the Gross-Pitaevskii equation exactly, one could write down the Bogoliubov Hamiltonian exactly, too. For practical purposes, however, the Bogoliubov Hamiltonian is expanded in powers of the external potential in a systematic way. Already in the first order, interesting physics is observed, namely the transition from p-wave scattering characteristics in the sound-wave regime to s-wave scattering in the particle regime (section 2.4). For the disordered problem, including the renormalization of the dispersion relation, some more efforts have been necessary. Firstly, the self-energy is of second order in the disorder potential, so also the second order of the Bogoliubov Hamiltonian had to be included in the correction. Secondly, the basis in presence of disorder had to be chosen correctly (section 3.2). With the reasoning of section 2.5 and section 3.2, this question has been answered once and for all: Bogoliubov modes (2.33) defined in terms of plane waves in density and phase are the only reasonable choice, unless one wants to exactly diagonalize the disordered Bogoliubov Hamiltonian (2.29). With the disordered Bogoliubov formalism, physical quantities like the mean free path and the renormalized speed of sound can be calculated in the whole parameter space spanned by the excitation wave length, the condensate healing length and the disorder correlation length. 109
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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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Page 39 and 40:
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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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Page 69 and 70: 3. Disorder In the previous chapter
Page 71 and 72: 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: 0.02 0.015 0.01 0.005 MSg 0.01 0.1
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 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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