Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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2. The Inhomogeneous Bogoliubov Hamiltonian • All finite-frequency modes fulfill the orthogonality (2.76) with respect to the ground state. This excludes a real overlap of δΨ with the ground state Φ. • All eigenstates pairwisely fulfill the bi-orthogonality relation (2.74). Bogoliubov ground state The Bogoliubov quasiparticles satisfy the bosonic commutator relations [ ˆ βµ, ˆ β † ν] = δµ,ν, [ ˆ β † µ, ˆ β † ν] = [ ˆ βµ, ˆ βν] = 0, (2.82) which are easily verified using equations (2.75), (2.79) and (2.80). The operators ˆ β † ν and ˆ βν create and annihilate Bogoliubov excitations. In particular, the Bogoliubov ground state is the Bogoliubov vacuum |vac〉, defined by the absence of Bogoliubov excitations 2.5.6. Non-condensed atom density ˆβν |vac〉 = 0. (2.83) Bogoliubov quasiparticles can be excited thermally or by external perturbations. But even in the ground state, not all particles are in the (Gross- Pitaevskii) condensate state. We are interested in the number of non- condensed atoms nnc = 1 Ld � d d r 〈vac| δ ˆ Ψ † (r)δ ˆ Ψ(r) |vac〉 . (2.84) It is important to distinguish this density nnc of atoms that are not in the condensate Φ(r) from the number of particles that are not in the state k = 0. The latter is sometimes referred to as “condensate depletion” [77, 78], but it is more a condensate deformation. The “condensate depletion due to disorder” nR in [77, Eq. (9)] is only due to the first-order smoothing correction (2.22) of the Gross-Pitaevskii wave function. The shift of the non-condensed density due to disorder, which we are going to compute now, is beyond Huang’s and Meng’s work [77]. In order to evaluate the non-condensed density, the field operator δ ˆ Ψ(r) is expressed in terms of the Bogoliubov creators and annihilators, using equation (2.67) 54 nnc = � ν 1 L d � d d r|vν(r)| 2 . (2.85)
2.5. Exact diagonalization of the Bogoliubov problem In this form, the non-condensed density requires the Bogoliubov eigenstates to be computed explicitly. This will be done in the numerical study of the disordered Bogoliubov problem in subsection 4.3.3. Non-condensed density in the homogeneous BEC In the homogeneous Bogoliubov problem (subsection 2.3.1), we use equation (2.27) and (2.33) and write δ ˆ Ψ(r) = 1 L d 2 � � k uke ik·r ˆγk − vke −ik·r ˆγ † k � , vk = ɛk − ɛ0 k 2 � ɛkɛ0 . (2.86) k Inserting this into equation (2.84), we find the density of non-condensed atoms as nnc = 1 Ld �′ |vk| 2 . (2.87) The prime indicates that the homogeneous condensate mode k = 0 is excluded from the summation. Because of the asymptotics |vk| 2 ∼ (kξ) −4 , there is no UV divergence in the dimensions d = 1, 2, 3. In three dimensions, the sum is approximated by an integral, which leads to nnc = 1 6 √ 2π 2 1 , ξ3 k nnc n 8 = 3 √ � na3 π s. (2.88) The fraction of non-condensed particles scales with the gas parameter � na 3 s [77], where as = mg/(4π� 2 ) is the s-wave scattering length. The non-condensed density nnc (2.88) is a function of the healing length ξ = �/ √ 2mgn. It depends on the product of total density n and interaction parameter g. In the ratio nnc/n this is converted to the gas parameter � na 3 s. This is the small parameter that ensures the applicability of Gross- Pitaevskii theory (subsection 2.2.2). The above reasoning is true not only in the present homogeneous case, but also for inhomogeneous condensates. For low momenta, the summand in (2.87) diverges like 1/(kξ). In in one dimension, this IR divergence forbids evaluating the sum (2.87) in the continuum approximation. This is consistent with the fact that there is no true Bose-Einstein condensate in one dimension (subsection 2.1.2). The long-range order is destroyed by the long wave-length fluctuations. However, in finite systems, where the quasi-condensate concept holds, the sum can always be evaluated. 55
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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
Page 15 and 16: 1.2. Disorder 1.2. Disorder Idealiz
Page 17 and 18: 1.4. Cold-atoms—Universal model s
Page 19 and 20: 1.5. Cold atoms—History and key e
Page 21 and 22: 1.5. Cold atoms—History and key e
Page 23 and 24: 1.6. Standing of this work are rest
Page 25: 1.6. Standing of this work The disc
Page 29 and 30: 2. The Inhomogeneous Bogoliubov Ham
Page 31 and 32: 2.1. Bose-Einstein condensation of
Page 33 and 34: 2.1. Bose-Einstein condensation of
Page 35 and 36: 2.2. Interacting BEC and Gross-Pita
Page 41 and 42: 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 63 and 64: 2.5. Exact diagonalization of the B
Page 65: 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 and 114: nnc(V )−nnc(0) nnc(0) 1.2 1.15 1.
Page 115 and 116: 4.4. Particle regime which coincide
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4.4. Particle regime regime, becaus
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0.02 0.015 0.01 0.005 MSg 0.01 0.1
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5. Conclusions and Outlook (Part I)
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5.3. Theoretical outlook potential.
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Part II. Bloch Oscillations 113
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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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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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