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