Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

More documents

Recommendations

Info

1. Interacting Bose Gases Strategy of this work and a short peek at the main results We are interested in the disordered problem, where the particular potential is unknown. This makes it impossible and also undesirable to compute the spectrum and the eigenstates explicitly. Instead, the spectrum is computed in the disorder average, by means of a diagrammatic approach. The structure of part I “The Disordered Bogoliubov Problem” is as follows. In chapter 2, the general framework is set up for the treatment of a Bose- Einstein condensate and its Bogoliubov excitations in presence of a weak external potential. Starting from the very concepts of Bose statistics and Bose- Einstein condensation, we derive the Gross-Pitaevskii mean-field framework (subsection 2.2.2). Subsequently, the ground state is treated in a mean-field manner, but the excited particles are described fully quantized. The expansion of the many-particle Hamiltonian around the mean-field ground state leads to the inhomogeneous Hamiltonian for Bogoliubov excitations (section 2.3). Via the Gross-Pitaevskii equation, this Hamiltonian depends nonlinearly on the external potential. As a first application, the scattering of Bogoliubov quasiparticles at a single impurity is discussed in detail (section 2.4). Finally, the general structure of the Bogoliubov Hamiltonian is discussed, in particular the orthogonality relations of its eigenstates. Chapter 3 is dedicated to the disordered Bogoliubov problem. The experimentally relevant speckle disorder potential and its statistical properties are discussed in section 3.1. Then, in section 3.2, a suitable basis for the disordered Bogoliubov problem is found. All findings then enter in the diagrammatic perturbation theory of section 3.3, which leads to the concept of the effective medium with the disorder-averaged dispersion relation ɛk, determined by the self energy Σ. In physical terms, this yields corrections to quantities like the density of states, the speed of sound and the mean free path. The theory derived is indeed valid in a large parameter space: the excitations considered can be particle-like or sound-like, the disorder potential can be correlated or uncorrelated on the length scale of the wave length, and the condensate can be in the Thomas-Fermi regime or in the smoothing regime, depending on the ratio of condensate healing length and disorder correlation length. 12
1.6. Standing of this work The discussion of the results in the different regimes of the parameter space deserves a separate chapter, which is given in chapter 4. Concerning the speed of sound, the main results are: • The correction due to uncorrelated disorder depends on the dimension. The leading correction is positive in three dimensions and negative in one dimension. In two dimensions, the speed of sound remains unaffected. • For correlated disorder, the speed of sound is reduced, independent of the dimension. In the case of uncorrelated disorder in three dimensions, the result of this work reproduces previous results [75, 78, 80]. To my knowledge, the results in lower dimensions and in correlated disorder are new. The corrections to the speed of sound depend non-monotonically on the disorder correlation length. In the density of states, this results in an interesting signature. In one dimension, a sharp peak in the density of states is found at kσ = 1. The “condensate depletion” computed by Huang and Meng and Giorgini et al. [77, 78] is interpreted as a mere deformation of the Gross-Pitaevskii condensate. We compute numerically the “Beyond-Huang-Meng” noncondensed fraction (subsection 2.5.6 and 4.3.3). Part II: Bloch oscillations with time dependent interactions In part II, lattice dynamics are discussed, more precisely Bloch oscillations of Bose-Einstein condensates in tilted lattice potentials with time-dependent interactions. This topic might appear rather exotic and far-fetched at this point, but as pointed out above, all experimental requirements are available and interesting new physics waits to be discovered. The main finding of this part is, that by modulating the interaction in a suitable way it is not only possible to maintain the Bloch oscillation, but also to make it more robust against certain perturbations. A more detailed introduction to the subject is given in section 6.1. 13
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: 1.6. Standing of this work are rest
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 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 81 and 82:
Page 83 and 84:
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
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 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
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
Page 148 and 149:
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,
Page 164 and 165:
Bibliography [131] F. Bloch, Über
Page 167 and 168:
Acknowledgments I thank everybody w
show all

Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?