Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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4. Disorder—Results and Limiting Cases 20 15 10 5 0 20 15 10 5 0 v = −0.03 v = −0.15 v = +0.03 v = +0.15 -0.6 -0.4 -0.2 0 0.2 -0.6 -0.4 -0.2 0 0.2 ΛN ΛN Figure 4.9.: Typical histograms of the correction to the speed of sound ΛN = ∆cµ 2 /(cV 2 0 ) at kξ = 0.05, kσ = 1, σ/ξ = 20 over 50 realizations of disorder. The mean value of the respective distribution is marked with a solid line, and the Born prediction (4.15) ΛN = 1 1 ln 2 − 4 2 is shown as a dashed line. For V0/µ � 0.1, the width of the distribution is clearly narrower than the mean value, i.e. the speed of sound shows self-averaging behavior. For V0 = +0.15µ, the distribution becomes very broad, extending even to positive values. The repulsive potential is so strong that the condensate density gets depressed nearly to zero below the highest peaks. Then, it is impossible to imprint a plane-wave density modulation (4.34). In two out of 50 realizations, no result could be obtained. evolution is computed using again the fourth-order Runge-Kutta algorithm. The excitation propagates with a modified speed of sound and is slightly scattered at the same time. In order to extract the relevant information, the deviations of the wave function from the ground state are Bogoliubov transformed. Then the phase velocity is extracted from the complex phase of γk ∝ e −iɛkt/� . This is done for many realizations of disorder and averaged over. By comparison with the phase velocity in the clean system, the change in the speed of sound is obtained. Furthermore, one can monitor the lifetime of the excitations by observing the elastically scattered amplitude γ−k. 4.3.2. Disorder average and range of validity of the Born prediction The numerical results for the speed of sound and other quantities (and experimental results as well) depend on the particular realization of disorder. For an infinite system, the results would be perfectly self-averaging. For practical reasons the size of the system was chosen to be 200 disorder corre- 98
ΛN �0.1 �0.2 �0.3 �0.4 4.3. Numerical study of the speed of sound V0/µ �0.2 �0.1 0.0 0.1 0.2 0.0 Figure 4.10.: Correction to the speed of sound at kξ = 0.05, kσ = 1, σ/ξ = 20 as function of the disorder strength. The perturbative prediction appears as a horizontal line. The values were obtained by averaging over 50 realizations of disorder, the error bars show the estimated error of this mean value, which is given by the width of the distribution divided by the square root of the number of realizations. The histograms corresponding to the points at v = ±0.03, ±0.15 are shown in figure 4.9. At large values of v, the potential is likely to fragment the condensate. At v = 0.15, this happened in two out of 50 realizations; At v = 0.2 it happened already in five out of 30 realizations, making the disorder average questionable. lation lengths. In figure 4.9 the distribution of results is shown for different disorder strengths V0 at kξ = 0.05 and kσ = 1. As long as the disorder is not too strong, the distributions are clearly single-peaked with well-defined averages. For larger values of |V0| � 0.1µ, the highest potential peaks or wells reach the value of the chemical potential µ, which violates the assumptions of perturbation theory of the Born approximation from chapter 3. In the case of an attractive (red-detuned) potential with deep wells, V0 < 0, the effect is not very dramatic. In the opposite case of a repulsive (blue-detuned) speckle potential, however, the rare high peaks of the speckle potential may fragment the condensate. In the corresponding panel of V0 = +0.15µ, the distribution is strongly broadened, including even some points with opposite sign. Also the mean values are shifted as function of the disorder strength V0. This is investigated in figure 4.10, where the mean values of the distributions in figure 4.9 are shown together with their estimated error. The correction is shown in units of v 2 = V 2 0 /µ 2 , such that the Born approximation from subsection 3.4.5 appears as the horizontal line as function of the disorder strength V0. At small values of |V0/µ|, the agreement is very good. Then there is a clear negative linear trend, which is due to the third moment of the speckle distribution function (3.10). At larger values v � 0.15, the 99
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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
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 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: 1 0.5 0 -0.5 0 20 40 60 80 100 x/σ
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 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
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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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