Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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3. Disorder The poles of the averaged Green function (3.44) define the complex dispersion relation �ω = ɛk + Σ B (k, �ω). (3.46) In the Born approximation, the self-energy is of order V 2 0 /µ 2 , and its energy argument �ω can be consistently replaced with its on-shell value ɛk. In the following, we separate the self-energy into the real and the imaginary correction of the dispersion relation �ωk = ɛk + Σ B (k, ɛk) = ɛk 3.4.2. Mean free path � 1 + 2 V0 µ 2 Λ(k) − i γk 2ɛk � . (3.47) According to the reasoning in section 3.2 the Bogoliubov modes are expressed in a basis that is characterized by the momentum k and is not the eigenbasis of the Bogoliubov Hamiltonian. Thus, k is “not a good quantum number” and the Bogoliubov modes suffer scattering, which is reflected in a finite lifetime and the broadening of their dispersion relation. Scattering rate and mean free path A Bogoliubov excitation, which evolves like e −iɛkt/� in the unperturbed case, evolves in the disordered case like e −i(ɛk+ReΣ)t/� e −|ImΣ|t/� . That means, its intensity gets damped by elastic scattering events to other modes. The imaginary part of the self-energy defines the inverse lifetime or scattering rate �γk = −2ImΣ B (k, ɛk). (3.48) In equation (3.39), the only possibility for an imaginary part to occur is the imaginary part of the Green function Im(ɛk − ɛk+q + i0) −1 = −πδ(ɛk − ɛk+q). This restricts the integral to the energy shell. Physically, this is not surprising. Because of energy conservation, a Bogoliubov quasiparticle can only be scattered to modes with the same energy. Other modes can only be virtually excited, but the quasiparticle has to come back. Such virtual scattering events will enter the real part of the self-energy, see subsection 3.4.5 below. On the energy shell, the scattering element simplifies, section 2.4. Thus the scattering rate is given as the d-dimensional angular integral �γk = 74 2 V0 2µ 2(kσ)d � kξ � 2 + (kξ) 2 (1 + (kξ) 2 ) dΩd (2π) d−1Cd � � 2 2kσ sin θ/2 A (kξ, θ) . (3.49)
3.4. Deriving physical quantities from the self-energy With the group velocity vg = ∂kɛk/�, the scattering rate is converted to the scattering mean free path ls = vg/γk 1 = kls V 2 0 4µ 2 kdσd (1 + k2ξ2 ) 2 � dΩd (2π) d−1Cd � � 2 2kσ sin θ/2 A (kξ, θ). (3.50) The fraction in front of the integral assures that the mean free path diverges both for very low momenta kσ ≪ 1 and for high momenta kξ ≫ 1. In- between, the mean free path scales with µ 2 /V 2 0 . Thus, describing Bogoliubov excitations as plane waves also in the inhomogeneous case is well justified. 3.4.3. Boltzmann transport length In equation (3.50), elastic scattering in all directions contributes to the inverse mean free path. The forward scattering events, however, do not randomize the direction of the quasiparticles and do not affect the diffusive transport. The relevant length for diffusion is the Boltzmann transport length lB [22, 23], the length of randomization of direction. It is obtained from equation (3.50) by introducing a factor [1 − cos(θ)], which suppresses forward scattering: 1 klB = V 2 0 4µ 2 kdσd (1 + k2ξ2 ) 2 � dΩd � � 2 [1 − cos(θ)] Cd 2kσ sin θ/2 A (kξ, θ), (2π) d−1 (3.51) In subsection 4.1.2, Boltzmann transport length and mean free path are compared in the regime of a hydrodynamic Bose-Einstein condensate. 3.4.4. Localization length In order to determine the localization lengths of the Bogoliubov states in a disordered Bose-Einstein condensate, the single-particle Green functions of the previous section are not sufficient. Rather, the intensity propagator should be considered [22, 23, 123], in order to derive the weak-localization correction to the diffusion constant. This is beyond the scope of this work, but nevertheless, we can estimate the localization lengths of Bogoliubov excitations, based on general results on localization of particles and phonons. In one-dimensional disordered systems, the backscattering process k ↦→ −k that amounts to the inverse Boltzmann transport length (3.51) is known to induce strong, Anderson localization of the excitation in the disordered potential [17]. The inverse localization length Γloc = 1/(2lB), which describes exponential localization, is directly proportional to the inverse 75
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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
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 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: 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
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
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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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