Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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2. The Inhomogeneous Bogoliubov Hamiltonian function is important for the choice of a suitable Bogoliubov basis for the disorder average in section 3.2. The explicit knowledge of the Bogoliubov eigenstates reveals more about the condensate than the Gross-Pitaevskii ground state: The condensate depletion, i.e. the number of particles that are not in the Gross-Pitaevskii state, can be computed. 2.5.1. Analogy to other bosonic systems The commutator relation of density and phase fluctuations � δˆn(r), δ ˆϕ(r ′ ) � = iδ(r − r ′ ) [see section 2.3] is analogous to the commutator of coordinate q and momentum p of a particles [xi, pj] = i�δij. The Hamiltonian (2.38) can schematically be written as F = 1 � T T p , q 2 � � M C CT K � � p q � , (2.63) which is the general quadratic bosonic Hamiltonian [107]. It describes a particle or a set of particles in a harmonic potential. The mass matrix and the matrix of spring constants are given as M = ɛ 0 +S and K = 2µ+ɛ 0 +R, respectively. In the general Hamiltonian, off-diagonal blocks C may be present. Their absence in the Bogoliubov problem (2.38) corresponds to time-reversal symmetry, which is compatible with the absence of magnetic fields or rotations. The quadratic “potential” q T Kq is harmonic, which is suitable for describing phonons in random elastic media [31, 107]. By defining creation and annihilation operators of the harmonic oscillators, the Hamiltonian (2.63) takes a structure analogous to the Bogoliubov Hamiltonian (2.42) F = 1 � � † ˆγ , ˆγ 2 � Γ ∆ ∆ † ΓT � � ˆγ ˆγ † � . (2.64) Note the different sign convention for the second component of the pseudospinors, such that Γ k ′ k = W k ′ k, but ∆ k ′ k = Y k ′ (−k). From the structure of (2.64), rather general properties follow, like the chiral symmetry (i.e. the occurrence of frequencies in pairs ±ω) [107]. In the following, we will derive this and some more properties at the concrete Bogoliubov problem. 2.5.2. Bogoliubov-de-Gennes equations How can the general Bogoliubov Hamiltonian (2.63) or (2.64) be diagonalized? For this task, the representation in terms of the field operator δ ˆ Ψ is 48
2.5. Exact diagonalization of the Bogoliubov problem used, instead of density δˆn and phase δ ˆϕ. This makes the structure better comparable with the Schrödinger equation and helps to point out the differences. The expansion of the grand canonical Hamiltonian (2.11) reads ˆF = 1 � d 2 d r � δ ˆ Ψ † (r), δ ˆ Ψ(r) � � L G G∗ � � δΨ(r) ˆ L δ ˆ Ψ † � + cst., (2.65) (r) with the Hermitian operator L = − �2 2m∇2 + V (r) − µ + 2g|Φ(r)| 2 and G = gΦ(r) 2 . The goal is now to find a transformation to quasiparticles ˆ βν, such that the Bogoliubov Hamiltonian ˆ F takes the diagonal form ˆF = � �ων ˆ β † ν ˆ βν. (2.66) ν The Bogoliubov Hamiltonian was obtained by the expansion around the saddlepoint. It turns out that the saddlepoint is indeed an energy minimum, not only in the homogeneous problem (2.3.1), but also in inhomogeneous systems. Therefore, the spectrum �ων is positive. The transformation mixes δ ˆ Ψ(r) and δ ˆ Ψ † (r), similar to the homogeneous Bogoliubov transformation in subsection 2.3.1. Furthermore, the inhomogeneous potential and its density imprint will mix the plane waves, such that the eigenstates will have some unknown shape. The general transformation to a set of eigenmodes reads δ ˆ Ψ(r) = �� uν(r) ˆ βν − v ∗ ν(r) ˆ β † � ν , (2.67) ν with a set of initially unknown functions uν(r) and vν(r) [106]. Note that there are different conventions concerning the sign in front of v ∗ ν. Here and e.g. in [54, 105, 106], the minus sign is used, with vk(r) = vke ik·r in the homogeneous case. In other literature, vν(r) is defined with opposite sign [55, 79, 85, 86]. The quasiparticle annihilators ˆ βν appearing in equation (2.67) and their creators obey bosonic commutator relations, as will be verified below. The equation of motion for δ ˆ Ψ(r) is set up using both forms of the Hamiltonian (2.65) and (2.66). By equating the coefficients, we find the following equation on the uν(r) and vν(r) � L −G η −G∗ � � uν(r) L vν(r) � �� =:Q � = �ων � uν(r) vν(r) � , (2.68) where η = diag(1, −1). This equation is known as the Bogoliubov-de-Gennes equation for the Bogoliubov eigenmodes [86]. Although both operators Q 49
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Bogoliubov Excitations of Inhomogen
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Abstract In this thesis, different
Page 8 and 9:
Contents 2.5. Exact diagonalization
Page 10 and 11: Contents Bibliography 141 List of F
Page 13 and 14: 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: T 1 B 2 2.5. Exact diagonalization
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
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ΛN �0.1 �0.2 �0.3 �0.4 4.3
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nnc(V )−nnc(0) nnc(0) 1.2 1.15 1.
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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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