Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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4. Disorder—Results and Limiting Cases Concordance with Goldstone theorem It is important to notice that the relative correction of the Bogoliubov dispersion relation (4.14) has been found to be finite for any value of kσ and in any dimension. That means, it is indeed possible to cast ReΣ into a correction of the speed of sound, ck + ReΣ = c ′ k. The spectrum remains gapless. This is a necessity, because disorder does not affect the U(1) symmetry of the (quasi)condensate condensate and the Bogoliubov excitations must remain gapless Goldstone modes [72]. 4.1.4. Density of states In the present sound-wave regime, phase velocity and group velocity coincide. Thus the average density of states (3.67) reduces to [115] � 2 V0 ρ(ω) = ρ0(ω) 1 − µ 2 � d + k ∂ � � Λ(k) , (4.22) ∂k with ρ0(ω) = Sd(2πc) −d ω d−1 . Similarly to [127], we consider the scaling function gd(ωσ/c) = ρ(ω)/ρ0(ω) − 1, which is computed from the correction (4.14) of the speed of sound shown in figure 4.4(b). The limiting values from (4.19) translate to gd(κ) = � 2 V × 2µ 2 1, κ ≪ 1, d 4 (2 + d), κ ≫ 1. (4.23) Gurarie and Altland [127] suggested that one should be able to deduce from the asymptotic values and the curvatures of this scaling function whether the average density of state exhibits a “boson peak” at intermediate frequency ω ≈ c/σ. The asymptotics of the scaling function in our case allow for a smooth, monotonic transition between the limiting values in any dimension d. Thus one has no reason to expect any extrema in-between. In figure 4.5(a), numerical results of the full scaling function (4.22), derived from the speed-of-sound correction Λ due to speckle disorder, are shown. In three dimensions, the scaling function is indeed smooth and monotonic. In one dimension, however, the scaling function shows a strikingly nonmonotonic behavior. Using (4.15) and (4.22), we can write down this function analytically 90 g1(κ) = v2 2 � 1 + κ 2 ln � � � � κ − 1� 3κ2 � �κ + 1� − 4 ln � � � 1 − κ � 2 κ2 �� , κ = ωσ/c. (4.24)
gd(κ) v 2 1.75 1.5 1.25 1 0.75 0.5 0.25 0.5 1 1.5 2 2.5 3 κ (a) Speckle disorder d = 3 d = 2 d = 1 gd(κ) v 2 4.1. Hydrodynamic limit I: ξ = 0 1.75 1.5 1.25 1 0.75 0.5 0.25 0.5 1 1.5 2 2.5 3 κ (b) Gaussian disorder d = 3 d = 2 d = 1 Figure 4.5.: Correction of the density of states gd(κ) = (ρ − ρ0)/ρ0 divided by the squared disorder strength v = V/µ as function of reduced momentum κ = ωσ/c in dimension d = 1, 2, 3. (a) Speckle disorder. At κ = 1, the momentum beyond which elastic backscattering becomes impossible in the Born approximation, there is a logarithmic divergence in d = 1, a kink in d = 2, and a curvature discontinuity in d = 3. (b) Gaussian model of disorder with spatial correlation V (r)V (0) = V 2 0 exp [−r 2 /(2σ 2 )] and C G 1 (kσ) = √ 2π exp(−k 2 σ 2 /2) as used in [122]. The sharp features of the speckle disorder are washed out, but the structure of a local minimum followed by a local maximum in d = 1 still exists. It shows a pronounced dip around κ ≈ 0.7 and a sharp logarithmic divergence at κ = 1. This particular structure is a consequence of the Born approximation, more specifically the non-analyticity of the speckle pair correlation function (3.12a) at the boundary of its support. The existence of this “boson moat” could not be inferred from the asymptotics of g1(κ) alone [127]. Indeed, expanding the asymptotic behavior as gd(κ) = v 2 � β × < d (1 + α< d κ2 + . . . ), κ ≪ 1, β > d (1 + α> d κ−2 (4.25) + . . . ), κ ≫ 1, we find α < 3 1 = −1 − 2 | ln κ| < 0 and α> 1 1 = 18 > 0 of opposite sign. Together with the fact that β < 1 is larger than β> 1 , these asymptotics would be compatible with a monotonic behavior and thus are not sufficient to infer the existence of intermediate extrema. In two dimensions, a kink separates a monotonic range for κ < 1 from a totally flat plateau for kσ > 1 figure 4.5(a). This can be traced back to the exact formula for the correction of the speed of sound (4.21), where the k−2 term cancels when (4.22) is applied. The coefficient α > 2 = 0 vanishes, which seems to happen also in other two-dimensional cases [127]. In three dimensions, the logarithmic singularity has moved to the second derivative of g3(κ), which is hardly resolvable in the figure. The asymptotics 91
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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 +
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 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: ǫk 0.25 0.5 0.75 1 1.25 1.5 1.75 k
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
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
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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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