Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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2. The Inhomogeneous Bogoliubov Hamiltonian Figure 2.3: Bogoliubov dispersion relation (2.35). Insets: schematic representation of the amplitudes δΨ⊥ and δΨ� (2.37). ǫk µ 10 5 1 δ ˆ Ψ⊥ Ψ0 0 2δ ˆ Ψ� δ ˆ Ψ⊥ 1 kξ 2 transformed to ɛk − �k· v. The static obstacle cannot transfer energy, so excitations can only be created if v > ɛk/(�k). Thus, there is a critical velocity vc = mink ɛk/�k = c. Superfluid flow is one of the key features of Bose-Einstein condensates and has been subject of many experiments. For example stirring experiments, where dissipation was observed when the condensate was stirred with velocities above the critical velocity [56], or persistent flow in a toroidal trap [100]. In the ideal Bose gas (section 2.1), the excitations are still bare particles with dispersion relation ɛ 0 k Ψ0 0 δ ˆ Ψ� and zero critical velocity. The phase transition to the Bose-Einstein condensate phase, together with the interactions, results in superfluidity. Signature of a Bogoliubov excitation Let us split the fluctuation operator δ ˆ Ψ(r) = ˆ Ψ(r) − Φ(r) into its components parallel and perpendicular to the order parameter Φ(r) δ ˆ Ψ� = 1� δΨ ˆ + δ ˆ † Ψ 2 � , δ ˆ Ψ⊥ = 1 � δΨˆ − δ ˆ † Ψ 2i � . (2.37) With equation (2.27) and the Bogoliubov transformation (2.33), the components take the form δ ˆ Ψ� = 1 L d � ak � ik·r e ˆγk + e 2 2 k −ik·r ˆγ † � k , � 1 � ik·r e ˆγk − e −ik·r ˆγ † � k , δ ˆ Ψ⊥ = 1 L d 2 k 2iak again with the coefficient ak = � ɛ0 k /ɛk. The signature of a Bogoliubov , respectively. excitation is a plane wave with amplitudes ak and a −1 k 32
2.3. Bogoliubov Excitations At high energies, the coefficient ak tends to one and the components of the wave parallel and perpendicular to the order parameter are equal. Then, the Bogoliubov wave resembles the plane wave of a free particle δΨ ∝ exp � i(k0 · r − ɛk0 t/�)� (inset in figure 2.3). At low energies, the spectrum (2.35) changes from quadratic to linear, but it is still gapless: at low momenta, excitations with arbitrarily low energy exist. The coefficient ak tends to zero, which means, that the Bogoliubov excitation γk has hardly any signature in the density, but all the more in the phase [see inset in figure 2.3]. In the limit k → 0, the Bogoliubov excitations pass over to a homogeneous shift of the condensate phase. This connects the Bogoliubov excitations to the broken U(1) symmetry of Bose- Einstein condensate. The Bogoliubov excitations are the Goldstone modes [72] associated to this broken symmetry. 2.3.2. The inhomogeneous Bogoliubov problem Let us come back to the Hamiltonian (2.29) and include the external potential V (r) and its imprint of the external potential in the condensate density n0(r). This inhomogeneity breaks translation invariance, which allowed the diagonalization by the Bogoliubov transformation (2.33) in the free problem, with two consequences: Firstly, there will be scattering among the Bogoli- ubov modes, and secondly, the anomalous terms ˆγ † kˆγ† −k ′ and their Hermitian conjugates do not vanish, which leads to anomalous coupling. As the Hamiltonian (2.29) is quadratic in the fluctuations without any term mixing δˆn and δ ˆϕ, the general structure of the disorder part F (V ) = F − F (0) is ˆF (V ) [δˆn, δ ˆϕ] = 1 2L d 2 � � n∞δ ˆϕ † k ′S ′ k kδ ˆϕk + δˆn† k,k ′ k ′R k ′ kδˆnk 4n∞ � . (2.38) The density and phase coupling matrices S and R vanish in absence of disorder. Comparing with (2.29), we find that both the phase couplingelement and the density coupling element are related to the imprint of the external potential in the condensate wave-function � n0(r) = Φ(r). Non-perturbative couplings The phase coupling is proportional to the condensate density n0(r) = Φ(r) 2 and can be written as � �2 m k′ ·k Φ ′ k −qΦq−k. (2.39) S ′ k k = 1 L d 2 q 33
Page 1: Bogoliubov Excitations of Inhomogen
Page 5: 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: 2.3. Bogoliubov Excitations and pha
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 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
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4.1. Hydrodynamic limit I: ξ = 0 T
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4.1. Hydrodynamic limit I: ξ = 0 I
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ǫk 0.25 0.5 0.75 1 1.25 1.5 1.75 k
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gd(κ) v 2 1.75 1.5 1.25 1 0.75 0.5
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4.2. Hydrodynamic limit II: towards
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∆ǫk ǫk �0.1 �0.2 �0.3 �
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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
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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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Page 136 and 137:
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Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
λTB t/TB 0.15 0.05 70 60 50 40 30
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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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