Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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4. Disorder—Results and Limiting Cases α > 3 < 0 leaves an all but structureless average density of state, as expected [127]. As a rule, specific correlation-related features, like the non-analyticities at κ = 1, tend to be washed out by integration in higher dimensional k-space. Thus we expect arguments on general grounds [127] to hold more reliably in higher dimensions. Conversely, the low-dimensional behavior may escape a bird’s-view approach and require detailed calculations. We have presented such a calculation for spatially correlated speckle disorder, so that our results should be of immediate use for cold-atom experiments. The non-analyticities at κ = 1 are particular features of speckle disorder. The limiting values (4.23) and the asymptotics, however, are generic and hold also for other models of disorder, like the Gaussian model [122]. In figure 4.5(b), the correction of the density of states due to such a Gaussian disorder is shown. The sharp features of the speckle disorder are washed out, but limiting values, asymptotics and even the structure of an intermediate minimum and maximum in one dimension are the same. 4.2. Hydrodynamic limit II: towards δ-disorder The low-energy regime is defined by excitation energies ɛk much smaller than the chemical potential µ. In terms of length scales, this is phrased as ξ/λ ∝ kξ ≪ 1. In the previous section, this has been achieved by setting the healing length ξ to zero. For the third length scale of the system, σ σ = 0 ξ k = 0 = ∞ ξ the disorder correlation length σ, this implied ξ ≪ σ. In order to cover also the low-energy excitations in truly uncorrelated disorder σ < ξ, we change the point of view in this section and realize kξ ≪ 1 by setting k to zero. This allows describing the low-energy excitations of a disordered BEC with arbitrary ratio σ/ξ of correlation length and healing length. The price to pay is that kσ is constrained to be small. Compared with the Bogoliubov wave length, the bare disorder potential is uncorrelated. The effective potential, i.e. the density profile, is smoothed to different degrees, depending on the ratio of ξ and σ. 4.2.1. Mean free path In the present limit k ≪ σ −1 , ξ −1 , the inverse mean free path (4.11) evaluates exactly the same way as in the limit (4.12), where ξ ≪ σ, k −1 and kσ ≪ 1 implied kξ ≪ 1 and kσ ≪ 1. According to (4.12), the inverse mean free path (kls) −1 scales like (V0/µ) 2 (kσ) d . Note that the scaling l −1 s ∝ k d−1 is 92
4.2. Hydrodynamic limit II: towards δ-disorder proportional to the surface of the energy shell, equivalently to the density of states counting the states available for elastic scattering, see equation (3.64). In the limit k → 0, the elastic energy shell shrinks and the scattering mean free path diverges, even when measured in units of k −1 . 4.2.2. Speed of sound In the momentum integration of the self-energy (3.39) [equivalently in (3.60)], an effective cutoff is provided either by the disorder correlator Cd(qσ) at q = σ −1 or by the smoothing functions included in w (1) , y (1) and w (2) at q = ξ −1 . Both cutoffs are much larger than k, such that the accessible kspace volume of virtual states is much larger than the volume enclosed by the elastic scattering shell (figure 3.4). A typical virtual scattering event is sketched in figure 4.6. Let us now consider the relative correction of the dispersion relation (3.60) in the limit k = 0. The first part in the bracket reduces to q 2 ξ 2 and the function h(kξ, qξ) simplifies to h(0, qξ) = − � 2 cos 2 β + (qξ)4 2 + (qξ) 2 � , (4.26) with β = ∡(k, q). Altogether, equation (3.60) becomes ΛN = 2σ d � d d qσ (2π) d Cd(q) (2 + q2ξ2 ) 2 � 2 2 q ξ 2 + q2ξ2 − cos2 � β . (4.27) The elastic-scattering pole, which was originally present in (3.39), has disappeared from the formula, because practically all relevant virtual scattering states k ′ = k+q are outside the elastic scattering sphere, see figure 4.6. Also, the angle β = ∡(k, q) becomes equivalent to the angle θ = ∡(k, k ′ ). So, cos 2 β represents the p-wave scattering of sound waves, again. In contrast to the hydrodynamic case in the previous section, where the correction to the speed of sound is found to be always negative, there are now two competing k β θ k ′ q Figure 4.6: Typical virtual scattering event in the regime kσ ≪ 1, kξ ≪ 1. The cutoffs σ −1 and ξ −1 given by the correlator and by the smoothing factor, respectively, are much larger than k. So, most of the virtual scattering states k ′ = k + q are far outside the elastic scattering shell |k ′ | = k. 93
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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
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 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: gd(κ) v 2 1.75 1.5 1.25 1 0.75 0.5
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
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
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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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