8.3.4 the semi—classical approximation 
problems 
references 

9 rotating condensates 
9.1 potential flow and quantized circulation 
9.2 structure of a single vortex 
9.2.1 a vortex in a uniform medium 
9.2.2 vortices with multiple quanta of circulation 
9.2.3 a vortex in a trapped cloud 
9.2.4 an off—axis vortex 
9.3 equilibrium of rotating condensates 
9.3.1 traps with an axis of symmetry 
9.3.2 rotating traps 
9.3.3 vortex arrays 
9.4 experiments on vortices 
9.5 rapidly rotating condensates 
9.6 collective modes in a vortex lattice 
problems 
references 

10 superfluidity 
10.1 the landau criterion 
10.2 the two—component picture 
10.2.1 momentum carried by excitations 
10.2.2 normal fluid density 
10.3 dynamical processes 
10.4 first and second sound 
10.5 interactions between excitations 
10.5.1 landau damping 
problems 
references 

11 trapped clouds at non—zero temperature 
11.1 equilibrium properties 
11.1.1 energy scales 
11.1.2 transition temperature 
11.1.3 thermodynamic properties 
11.2 collective modes 
11.2.1 hydrodynamic modes above tc 
11.3 collisional relaxation above tc 
11.3.1 relaxation of temperature anisotropies 
11.3.2 damping of oscillations 
problems 
references 

12 mixtures and spinor condensates 
12.1 mixtures 
12.1.1 equilibrium properties 
12.1.2 collective modes 
12.2 spinor condensates 
12.2.1 mean—field description 
12.2.2 beyond the mean—field approximation 
problems 
references 

13 interference and correlations 
13.1 tunnelling between two wells 
13.1.1 quantum fluctuations 
13.1.2 squeezed states 
13.2 interference of two condensates 
13.2.1 phase—locked sources 
13.2.2 clouds with definite particle number 
13.3 density correlations in bose gases 
13.3.1 collisional shifts of spectral lines 
13.4 coherent matter wave optics 
13.5 criteria for bose—einstein condensation 
13.5.1 the density matrix 
13.5.2 fragmented condensates 
problems 
references 

14 optical lattices 
14.1 generation of optical lattices 
14.1.1 one—dimensional lattices 
14.1.2 higher—dimensional lattices 
14.1.3 energy scales 
14.2 energy bands 
14.2.1 band structure for a single particle 
14.2.2 band structure for interacting particles 
14.2.3 tight—binding model 
14.3 stability 
14.3.1 hydrodynamic analysis 
14.4 intrinsic non—linear effects 
14.4.1 loops 
14.4.2 spatial period doubling 
14.5 from superfluid to insulator 
14.5.1 mean—field approximation 
14.5.2 effect of trapping potential 
14.5.3 experimental detection of coherence 
problems 
references 

15 lower dimensions 
15.1 non—interacting gases 
15.2 phase fluctuations 
15.2.1 vortices and the berezinskii—kosterlitz—thouless transition 
15.3 microscopic theory of phase fluctuations 
15.3.1 uniform systems 
15.3.2 anisotropic traps 
15.4 the one—dimensional bose gas 
15.4.1 the strong—coupling limit 
15.4.2 arbitrary coupling 
15.4.3 correlation functions 
problems 
references 

16 fermions 
16.1 equilibrium properties 
16.2 effects of interactions 
16.3 superfluidity 
16.3.1 transition temperature 
16.3.2 induced interactions 
16.3.3 the condensed phase 
16.4 pairing with unequal populations 
16.5 boson—fermion mixtures 
16.5.1 induced interactions in mixtures 
problems 
references 

17 from atoms to molecules 
17.1 bose—einstein condensation of molecules 
17.2 diatomic molecules 
17.2.1 binding energy and the atom—atom scattering length 
17.2.2 a simple two—channel model 
17.2.3 atom—atom scattering 
17.3 crossover： from bcs to bec 
17.3.1 wide and narrow feshbach resonances 
17.3.2 the bcs wave function 
17.3.3 crossover at zero temperature 
17.3.4 condensate fraction and pair wave function 
17.4 crossover at non—zero temperature 
17.4.1 thermal molecules 

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