Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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Contents<br />
4.3.2. Disorder average and range <strong>of</strong> validity <strong>of</strong> the Born<br />
prediction . . . . . . . . . . . . . . . . . . . . . . . . 98<br />
4.3.3. Non-condensed fraction . . . . . . . . . . . . . . . . . 100<br />
4.3.4. Speed <strong>of</strong> sound as function <strong>of</strong> the correlation length . 101<br />
4.4. Particle regime . . . . . . . . . . . . . . . . . . . . . . . . . 102<br />
4.4.1. Mean free path . . . . . . . . . . . . . . . . . . . . . 102<br />
4.4.2. Renormalization <strong>of</strong> the dispersion relation in the <strong>Bogoliubov</strong><br />
regime . . . . . . . . . . . . . . . . . . . . . 103<br />
4.4.3. Transition to really free particles . . . . . . . . . . . 105<br />
4.4.4. Closing the gap with a Gross-Pitaevskii integration . 106<br />
4.4.5. Conclusions on the particle limit . . . . . . . . . . . 108<br />
5. Conclusions and Outlook (Part I) 109<br />
5.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109<br />
5.2. Experimental proposals . . . . . . . . . . . . . . . . . . . . . 110<br />
5.3. Theoretical outlook . . . . . . . . . . . . . . . . . . . . . . . 111<br />
II. Bloch Oscillations 113<br />
6. Bloch Oscillations and Time-Dependent Interactions 115<br />
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 115<br />
6.1.1. Bloch oscillation <strong>of</strong> a single particle . . . . . . . . . . 116<br />
6.1.2. Experimental realization . . . . . . . . . . . . . . . . 117<br />
6.1.3. Time dependent interaction g(t) . . . . . . . . . . . . 118<br />
6.2. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118<br />
6.2.1. Tight binding approximation . . . . . . . . . . . . . . 119<br />
6.2.2. Smooth-envelope approximation . . . . . . . . . . . . 120<br />
6.3. Periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . 122<br />
6.4. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 123<br />
6.5. Collective coordinates . . . . . . . . . . . . . . . . . . . . . . 124<br />
6.5.1. Breathing dynamics in the stable cases . . . . . . . . 127<br />
6.5.2. Unstable cases—decay mechanisms and other dynamics127<br />
6.6. Dynamical instabilities . . . . . . . . . . . . . . . . . . . . . 130<br />
6.6.1. Linear stability analysis <strong>of</strong> the infinite wave packet . 130<br />
6.6.2. Bloch periodic perturbations . . . . . . . . . . . . . . 131<br />
6.6.3. Unstable sine . . . . . . . . . . . . . . . . . . . . . . 132<br />
6.6.4. Robustness with respect to small perturbations . . . 133<br />
6.7. Conclusions (Part II) . . . . . . . . . . . . . . . . . . . . . . 136<br />
A. List <strong>of</strong> Symbols and Abbreviations 137<br />
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