Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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4. Disorder—Results and Limiting Cases condensate gets already fragmented, due to rare high potential peaks from the exponential tail in (3.10), see also figure 4.8. This is reflected in the positive deviation of the mean values and also in the increased width of the distribution. To sum things up: Beyond-Born effects are clearly visible, but the Born result remains useful in a rather wide range. The correction stays negative, even for nearly fragmented condensates. 4.3.3. Non-condensed fraction Even in the ground state, there are particles that are not in the Gross- Pitaevskii condensate function Φ, but in excited states (subsection 2.5.6). In order to verify the preconditions for treating the one-dimensional disordered problem on grounds of the Gross-Pitaevskii equation, we compute the fraction of non-condensed particles. We start with the homogeneous (quasi)condensate and evaluate the sum (2.87). In the thermodynamic limit, this sum diverges in one dimension. Nonetheless, we can compute the sum for systems of finite size. For the system size L = 200ξ, used in the previous subsections, we find nnc|L=200ξ = 0.3488/ξ. Note that the non-condensed fraction diverges logarithmically with L: for L = 105ξ, for example, we find nnc|L=10 5 ξ = 1.046/ξ. We need to relate the non-condensed density to the total density n1D of the Bose gas. This non-condensed fraction scales like (n1Dξ) −1 , i.e. the average particle spacing in relation to the healing length [55, chapter 17]. The numerical procedure for computing the non-condensed density in presence of disorder is as follows. In the same manner as in the Gross- Pitaevskii integration (subsection 4.3.1), the system is discretized into l points, the disorder potential is generated, and the Gross-Pitaevskii ground state is computed. Then, the 2l × 2l real-space matrix (2.68) is set up and diagonalized numerically. As the matrix is not symmetric, the eigenvectors (uν(r), vν(r)) are not pairwisely orthogonal. Instead, they satisfy the biorthogonality relation (2.74). The non-condensed density (2.85) is then evaluated from the relevant modes with positive frequency (section 2.5). In figure 4.11, results for the parameters used in the previous subsections are shown. The initial increase is quadratic in the reduced disorder strength V0/µ. Strong deviations occur at V0 � 0.1µ. At the disorder strength V0/µ = 0.1, the fraction of non-condensed atoms is increased by about 25%, i.e. the non-condensed fraction is still of the same order as in the homogeneous case. In conclusion, the 1D validity condition for Gross-Pitaevskii theory in the homogeneous case is nnc/n1D ∝ (ξn1D) −1 ≪ 1 [55, chapter 17] (with a 100
nnc(V )−nnc(0) nnc(0) 1.2 1.15 1.1 1.05 0.002 0.004 0.006 0.008 0.01 4.3. Numerical study of the speed of sound V 2 µ 2 Figure 4.11: Relative increase of the non-condensed density nnc due to disorder. With the parameters of the present section σ = ξ = L/200. Note that the increase and the reference nnc(V = 0) both depend on the system size L/ξ. proportionality factor that depends logarithmically on L). As function of disorder strength, the non-condensed fraction increases, but it stays within the same order of magnitude, at least in the range of parameters of the present section. 4.3.4. Speed of sound as function of the correlation length Having checked the prediction at one particular value of the disorder correlation length, we now investigate different regimes of length scales. Given the condition kξ = 0.05 ≪ 1, the correlation length σ can be the shortest of all length scales, larger than the healing length but shorter than the wave length, or the largest of all length scales. In the intermediate range, the full integral (3.60) interpolates between the hydrodynamic limit (4.15) and the limit of low-energy excitations (4.31). In the first case, the prediction (4.31) from section 4.2 holds (green dashed line in figure 4.2). In the last case, the hydrodynamic limit (4.14) from section 4.1 is applicable (blue dotted line in figure 4.2). In-between, the full formula from subsection 3.4.5 has to be evaluated (solid line in figure 4.2). In figure 4.2 numerical data averaged over 50 realizations of disorder is shown together with the perturbative prediction from subsection 3.4.5. We find good agreement, both for V0 > 0 and V0 < 0. 101
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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
Page 61 and 62: 2.5. Exact diagonalization of the B
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: ΛN �0.1 �0.2 �0.3 �0.4 4.3
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 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
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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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