Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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Contents<br />
Bibliography 141<br />
List <strong>of</strong> Figures<br />
x<br />
1.1. Time-<strong>of</strong>-flight images <strong>of</strong> <strong>Bose</strong>-<strong>Einstein</strong> condensation . . . . . 7<br />
1.2. Observation <strong>of</strong> sound propagation . . . . . . . . . . . . . . . 8<br />
2.1. Bunching <strong>of</strong> bosons in a minimal system . . . . . . . . . . . 18<br />
2.2. Condensate density pr<strong>of</strong>ile in presence <strong>of</strong> an impurity . . . . 29<br />
2.3. <strong>Bogoliubov</strong> dispersion relation . . . . . . . . . . . . . . . . . 32<br />
2.4. <strong>Bogoliubov</strong> scattering vertex . . . . . . . . . . . . . . . . . . 35<br />
2.5. 2D single-scattering setup . . . . . . . . . . . . . . . . . . . 39<br />
2.6. First-order scattering envelope functions . . . . . . . . . . . 43<br />
2.7. Fourier analysis <strong>of</strong> the stationary scattering state . . . . . . 44<br />
2.8. Elastic scattering amplitude . . . . . . . . . . . . . . . . . . 46<br />
2.9. Transmission <strong>of</strong> <strong>Bogoliubov</strong> excitations across a narrow impurity<br />
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47<br />
3.1. Schematic representation <strong>of</strong> the disordered <strong>Bogoliubov</strong> setting 57<br />
3.2. Principle <strong>of</strong> the speckle phenomenon . . . . . . . . . . . . . 58<br />
3.3. Speckle correlation functions in d = 1, 2, 3 . . . . . . . . . . 62<br />
3.4. Geometry <strong>of</strong> the scattering process . . . . . . . . . . . . . . 78<br />
3.5. <strong>Bogoliubov</strong> density <strong>of</strong> states . . . . . . . . . . . . . . . . . . 79<br />
4.1. Parameter space <strong>of</strong> the full <strong>Bogoliubov</strong> problem . . . . . . . 81<br />
4.2. Relative correction <strong>of</strong> the speed <strong>of</strong> sound (at kξ = 0.05) . . . 82<br />
4.3. Mean free path and Boltzmann transport length . . . . . . . 86<br />
4.4. Disorder-averaged dispersion relation (at ξ = 0) . . . . . . . 89<br />
4.5. Correction <strong>of</strong> the density <strong>of</strong> states (at ξ = 0) . . . . . . . . . 91<br />
4.6. Typical virtual scattering event in the regime kσ ≪ 1, kξ ≪ 1 93<br />
4.7. Relative correction <strong>of</strong> the dispersion relation (at k = 0) . . . 95<br />
4.8. Speckle potential and ground-state density pr<strong>of</strong>ile . . . . . . 97<br />
4.9. Histograms <strong>of</strong> the correction to the speed <strong>of</strong> sound . . . . . . 98<br />
4.10. Correction to the speed <strong>of</strong> sound as function <strong>of</strong> the disorder<br />
strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99<br />
4.11. Increase <strong>of</strong> the non-condensed density nnc due to disorder . . 101<br />
4.12. Relative disorder correction MN = (ɛk − ɛk)µ/V 2<br />
0 (at kξ = 10) 104<br />
4.13. Real part <strong>of</strong> the self-energy for individual atoms . . . . . . . 106<br />
4.14. Transition from the <strong>Bogoliubov</strong> regime to free-particle plane<br />
waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107