Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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4. Disorder—Results and Limiting Cases 4.4. Particle regime In the previous sections of this chapter, we have investigated in detail the sound-wave excitations in the low-energy regime kξ ≪ 1. The predictions of the disordered Bogoliubov theory, however, hold more generally and allow the transition to the particle regime kξ ≫ 1. There, the interaction can be neglected with respect to the kinetic energy. The Hamiltonian (2.11) becomes non-interacting, and one could expect that the entire Bogoliubov problem passes over to the problem of free particles in disorder [22, 23]. However, in the non-interacting limit some subtleties occur because of as- sumptions made in the derivation of the Bogoliubov Hamiltonian. In this section, we firstly discuss the predictions from k = ∞ section 3.4 in the particle regime kξ = 10 ≫ 1. In parameter space, the curve kξ = 10 is opposite to that of section 4.3. The elastic scattering properties agree with those of free particles, but the real corrections to the spectrum turn out to differ. The leading order tends to a constant that can be absorbed in a shift of the chemical potential. In subsection 4.4.3, we compute the disorder averaged dispersion relation in the Schrödinger particle limit, where the condition µ ≫ V from the Bogoliubov regime is reversed. Finally, we study numerically the transition between the two regimes. 4.4.1. Mean free path Already in the single scattering problem [subsection 2.4.3], we have learned that in the limit ξ → ∞, the first-order scattering element (2.45c) ap- proaches the scattering amplitude of a free particle in disorder W (1) k+q k ≈ σ = 0 ξ = ∞ kξ = 10 � Vq qξ ≫ 1 , (4.35a) −Vq qξ ≪ 1 . (4.35b) Only the forward scattering element W (1) kk = −V0 [cf. (2.58)] deviates, but can be absorbed in the chemical potential. Consequently, the imaginary part of the self-energy, which consists of the angular integral over the elastic scattering shell, passes over to the results of free particles in disorder. Equation (3.53) reduces to 102 1 kls = � �2 V0 2ɛ 0 k (kσ) d � dΩd (2π) d−1Cd σ = ∞ � � θ 2kσ sin 2 , (4.36)
4.4. Particle regime which coincides with [23, eq. (34)]. All quantities related to the interaction have disappeared. The energy scale µ has been replaced with the kinetic energy ɛ0 k of the particle and the only remaining length scales are combined in the reduced momentum kσ. In contrast to the sound-wave regime [subsection 4.1.2], the inverse mean free path of particles diverges at low energies. Slow particles are scattered strongly, and the perturbation theory breaks down [22, 23]. 4.4.2. Renormalization of the dispersion relation in the Bogoliubov regime Let us calculate the real part of the self-energy (3.39), i.e. the disorder correction of the dispersion relation, in the non-interacting limit µ → 0. Apart from W (1) , we also need the anomalous scattering element Y (1) (2.45d), which does not vanish as one might expect for non-interacting particles. Instead, we find Y (1) k+q k ≈ � Vq � k 2 + (k + q) 2 � /q 2 Vq(kξ) 2 qξ ≫ 1 , (4.37a) qξ ≪ 1 . (4.37b) The real part of the self-energy (3.39), together with the correction fixing the particle number (3.59), is an integral over the momentum transfer q. The correlator Cd(qσ) provides a sharp cutoff at q = 2/σ, and also the smoothing factors contained in the envelope functions suppress the integrand for qξ ≫ 1. Thus, we can approximate the integrand for small q. Using (4.35b), (4.37b) and (2.50), we find M (0) N := ɛk µ ΛN = � d d q (2π) dσd q Cd(qσ) 2ξ2 (2 + q2ξ2 ) 2. (4.38) Only the anomalous coupling y (1) and the second order w (2) have contributed to this leading order, but not the normal scattering w (1) . Equation (4.38) holds for kξ ≫ 1 and kσ not too small. It is a strictly positive function of σ/ξ. In the leading order, there is no dependence on the momentum kξ, thus it does not affect quantities like the group velocity and the effective mass (inverse of the second derivative of the dispersion relation). It is interpreted as a correction of the chemical potential in � ɛk (2.36) = ɛ 0 k + µ ↦→ ɛ 0 k + µ 1 + M (0) N V 2 0 /µ 2 � . (4.39) The integral (4.38) has the same form as the correction (3.59), due to fixing the particle number. However, it has the opposite sign. The correction 103
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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
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2.5. Exact diagonalization of the B
Page 63 and 64: 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 and 112: ΛN �0.1 �0.2 �0.3 �0.4 4.3
Page 113: nnc(V )−nnc(0) nnc(0) 1.2 1.15 1.
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
Page 162 and 163: 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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