Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

Contents
2.5. Exact diagonalization of the Bogoliubov problem . . . . . . 47
2.5.1. Analogy to other bosonic systems . . . . . . . . . . . 48
2.5.2. Bogoliubov-de-Gennes equations . . . . . . . . . . . . 48
2.5.3. Zero-frequency mode . . . . . . . . . . . . . . . . . . 50
2.5.4. Eigenstates of non self-adjoint operators . . . . . . . 50
2.5.5. Bogoliubov eigenstates with non-zero frequency . . . 51
2.5.6. Non-condensed atom density . . . . . . . . . . . . . . 54
2.6. Conclusions on Gross-Pitaevskii and Bogoliubov . . . . . . . 56
3. Disorder 57
3.1. Optical speckle potential . . . . . . . . . . . . . . . . . . . . 58
3.1.1. Speckle amplitude . . . . . . . . . . . . . . . . . . . . 59
3.1.2. Generalization to 3D . . . . . . . . . . . . . . . . . . 60
3.1.3. Intensity and potential correlations . . . . . . . . . . 61
3.2. A suitable basis for the disordered problem . . . . . . . . . . 63
3.2.1. Bogoliubov basis in terms of free particle states . . . 64
3.2.2. Bogoliubov basis in terms of density and phase . . . 64
3.3. Effective medium and diagrammatic perturbation theory . . 66
3.3.1. Green functions . . . . . . . . . . . . . . . . . . . . . 66
3.3.2. The self-energy . . . . . . . . . . . . . . . . . . . . . 68
3.3.3. Computing the self-energy in the Born approximation 71
3.4. Deriving physical quantities from the self-energy . . . . . . . 73
3.4.1. The physical meaning of the self-energy . . . . . . . . 73
3.4.2. Mean free path . . . . . . . . . . . . . . . . . . . . . 74
3.4.3. Boltzmann transport length . . . . . . . . . . . . . . 75
3.4.4. Localization length . . . . . . . . . . . . . . . . . . . 75
3.4.5. Renormalization of the dispersion relation . . . . . . 76
3.4.6. Density of states . . . . . . . . . . . . . . . . . . . . 79
4. Disorder—Results and Limiting Cases 81
4.1. Hydrodynamic limit I: ξ = 0 . . . . . . . . . . . . . . . . . . 83
4.1.1. Direct derivation from hydrodynamic equations of
motion . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.1.2. Transport length scales . . . . . . . . . . . . . . . . . 85
4.1.3. Speed of sound . . . . . . . . . . . . . . . . . . . . . 87
4.1.4. Density of states . . . . . . . . . . . . . . . . . . . . 90
4.2. Hydrodynamic limit II: towards δ-disorder . . . . . . . . . . 92
4.2.1. Mean free path . . . . . . . . . . . . . . . . . . . . . 92
4.2.2. Speed of sound . . . . . . . . . . . . . . . . . . . . . 93
4.3. Numerical study of the speed of sound . . . . . . . . . . . . 96
4.3.1. The numerical scheme . . . . . . . . . . . . . . . . . 96
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