Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

7 Bogoliubov excitations of Bloch state condensates 83
7.1 Bogoliubov equations . .............................. 84
7.2 Bogoliubov bands and Bogoliubov Bloch amplitudes . .............. 85
7.3 Tight binding regime of the lowest Bogoliubov band . .............. 92
7.4Velocityofsound ................................. 97
8 Linear response - Probing the Bogoliubov band structure 101
8.1Dynamicstructurefactor .............................102
8.2Staticstructurefactorandsumrules.......................109 9 Macroscopic Dynamics 115
9.1Macroscopicdensityandmacroscopicsuperfluidvelocity ............116
9.2Hydrodynamicequationsforsmallcurrents....................118 9.3Hydrodynamicequationsforlargecurrents....................120 9.4SoundWaves ...................................121
9.5 Small amplitude collective oscillations in the presence of harmonic trapping . . 122
9.6Center-of-massmotion:Linearandnonlineardynamics .............129
10 Array of Josephson junctions 133
10.1Current-phasedynamicsinthetightbindingregime...............134
10.2 Lowest Bogoliubov band . . . ..........................136
10.3JosephsonHamiltonian ..............................138
11 Sound propagation in presence of a one-dimensional optical lattice 143
11.1GenerationofSoundSignals ...........................144
11.2Current-Phasedynamics..............................145
11.3Nonlinearpropagationofsoundsignals......................146
11.4 Experimental observability . . . ..........................157
12 Condensate fraction 159
12.1 Quantum depletion within Bogoliubov theory ..................160
12.2Uniformcase....................................161
12.3Shallowlattice...................................165
12.4Tightbindingregime ...............................166
Acknowledgements 187