Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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4. Disorder—Results and Limiting Cases ΛN � 1 �� 8 � 1 �� 4 � 3 �� 8 � 1 �� 2 kσ 0 0 0.5 1 1.5 2 kξ = 0.05 10 20 30 40 σ/ξ Figure 4.2.: Relative correction of the speed of sound ΛN = ∆c µ 2 /(c V 2 0 ) for 1D speckle disorder. Full formula (3.60) for kξ = 0.05 (black line). Limiting formulae ΛN(kσ)|ξ=0 [section 4.1, (4.15)] (dotted blue) and ΛN(σ/ξ)|k=0 [section 4.2, (4.31)] (dashed green). Numerical results of a Gross-Pitaevskii integration, as discussed in section 4.3, are shown for V0/µ = +0.03 (blue straight marks) and V0/µ = −0.03 (red triangular marks). fixes the value of the lightness, leaving the two-dimensional space of hue and saturation, as shown in Fig. 4.1. Limiting cases, where one of the length scales is zero or infinity, i.e. much shorter or much longer than the other length scales, are found on the edges. We focus mainly on the low-energy excitations kξ ≪ 1. Only in the last section, section 4.4, we consider the transition to particle-like excitations kξ ≫ 1. Low-energy excitations We are interested in the low-energy features, i.e. the sound-wave regime kξ ≪ 1. For practical purposes, we choose a small but finite value kξ = 0.05. In the parameter space, the curve defined by kξ = 0.05 appears as a smooth curve close to the ξ = 0 edge and the k = 0 edge, k = 0 as shown in the illustration on the right. Let us consider the speed of sound, which is a significant physical quantity. In figure 4.2, the correction (3.60) of the speed of sound due to a one-dimensional speckle disorder potential is shown. The curve is rather complicated with three different regimes: a linear increase at very short correlation lengths, a nonmonotonic intermediate range and saturation at long correlation lengths. In 82 σ = 0 kξ = 0.05 ξ = 0 σ = ∞
4.1. Hydrodynamic limit I: ξ = 0 the following sections, we will illuminate the different regimes and elaborate limiting results. In addition to excitation wave length and healing length, the disorder correlation length provides a third length scale σ with no a priori constraints. Thus, the sound limit kξ → 0 can be understood in two different ways: 1. Interaction dominates, and the healing length is the shortest of all length scales and drops out, ξ → 0. This so-called hydrodynamic limit leaves kσ as free variable. It is found at the lower right edge in the parameter space figure 4.1 and is discussed in detail in section 4.1. 2. Conversely, the excitation wave length can be taken as the longest of all length scales, i.e. k → 0, leaving σ/ξ as free variable. This limit is found at the lowermost edge of the parameter space and discussed in section 4.2. Analytical results from both limits reproduce the exact correction (3.60) for large σ/ξ and small kσ, respectively, cf. figure 4.2. Finally, in section 4.3, we verify the perturbative predictions in all regimes with a numerical integration of the time-dependent Gross-Pitaevskii equation in a one-dimensional disordered system. 4.1. Hydrodynamic limit I: ξ = 0 Here, the so-called hydrodynamic regime is discussed, where the interaction gn = µ dominates over the kσ = ∞ quantum pressure, i.e. kinetic energy. For the ground state this means that the Thomas-Fermi approximation (2.20) holds, and the excitations are in the sound-wave kσ = 0 regime kξ ≪ 1. The fact that the interaction µ is the largest energy scale implies that the healing length ξ = �/ √ 2mµ is the shortest length scale. Thus, the physical results from section 3.4 can be taken in the limit ξ → 0. All results will then depend only on the ratio σ/λ of the remaining length scales, respectively on kσ = 2πσ/λ. Physically, it is more instructive to perform the hydrodynamic limit in the very beginning and to derive the results from the much simpler hydrodynamic equations of motion [115]. ξ = 0 83
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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
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
Page 59 and 60: 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: 4. Disorder—Results and Limiting
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 and 124: 5.3. Theoretical outlook potential.
Page 125: Part II. Bloch Oscillations 113
Page 128 and 129: 6. Bloch Oscillations and Time-Depe
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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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6. Bloch Oscillations and Time-Depe
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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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