Chemical potential in the degenerate interacting regime

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Table of contents

Introduction
1 the bose gas from three to two dimensions
1.1 Statistics and Bose gases
1.1.1 Non-interacting bosons in the grand-canonical ensemble
1.1.2 Bose-Einstein condensation
1.1.3 Validity of the derivation of Bose-Einstein condensation
1.2 Dimensional crossover from two to three dimensions
1.2.1 Experimental realization of a 2D Bose gas
1.2.2 Transverse condensation
1.2.3 Coherence length at the transverse condensation point
1.3 Behaviour of a 2D plane of atoms
1.3.1 Non-interacting Bose gas
1.3.2 The interacting 2D Bose gas
1.3.3 Superfluid regime: Berezinskii-Kosterlitz-Thouless transition versus Bose-Einstein condensation
1.4 Conclusion
2 the kibble-zurek mechanism
2.1 Phase transitions and critical slowing down
2.1.1 Static critical exponents
2.1.2 Dynamical exponent
2.2 The Kibble-Zurek prediction
2.2.1 Correlation length and thermalization time
2.2.2 Freezing out of the system
2.2.3 Detection of the order parameter variation: topological defects
2.3 Limitations to the observation of the Kibble-Zurek prediction
2.4 Conclusion
3 quenching the bose gas between three and two dimensions
3.1 Experimental set-up
3.1.1 Laser set-up
3.1.2 Production of degenerate gases
3.1.3 Parameter estimation of Bose gases in box potentials
3.2 Experimental evidence for the dimensional crossover
3.2.1 Phase coherence revealed by velocity distribution measurements
3.2.2 Phase coherence revealed by matter-wave interference
3.2.3 Scaling laws for the emergence of coherence
3.3 Creation of topological defects by quench cooling the gas
3.3.1 Vortices in square geometries
3.3.2 Supercurrents in ring geometries
3.3.3 Discussion on possible improvements on the measurements
3.4 Conclusion
4 a new experimental set-up for 2d physics
4.1 Producing degenerate gases of rubidium
4.1.1 Design principle of the experiment
4.1.2 Laser system
4.1.3 Vacuum system
4.1.4 Laser cooling
4.1.5 Quadrupole trap and radio-frequency evaporation
4.1.6 Production of Bose-Einstein condensates in a crossed dipole trap
4.1.7 Imaging the cloud
4.1.8 Obtaining degenerate gases in shaped potentials
4.2 Shaping the cloud
4.2.1 Making box potentials
4.2.2 Confining the gas to two dimensions
4.3 Conclusion
5 collective effects in light-matter interaction
5.1 Position of the problem
5.1.1 Importance of collective effects in atom-light interactions
5.1.2 Observation of collective effects
5.1.3 Relevance of collective effects for our systems
5.2 Modelling multiple and recurrent scattering effects
5.2.1 Choice of the model
5.2.2 Coupled classical dipoles
5.2.3 Programs
5.3 Preparing a sample
5.3.1 Calibration of the imaging set-up
5.3.2 Computing the optical density of the cloud
5.3.3 Preparation and properties of the atomic sample
5.4 Resonances
5.4.1 Resonance curves for dilute clouds
5.4.2 Lorentzian fits
5.4.3 Wing fits
5.5 Local excitation of a cloud of atom
5.6 Conclusion
6 prospective experiment: evaporation in a tilted lattice
6.1 Solving the scattering problem of an atom in a tilted lattice
6.1.1 Position of the problem—Outline of the resolution
6.1.2 Scattering matrix in real and reciprocal space
6.1.3 Definition of the Bloch-Stark states
6.1.4 Expression of the scattering matrix
6.1.5 Results
6.2 Evaporation using particle interactions
6.2.1 Principle
6.2.2 Simulations
6.2.3 Results
6.3 Conclusion
7 prospective experiment: using magnetic texture to produce supercurrents
7.1 A ring of atoms in a quadrupole field
7.1.1 A neutral atom in a real magnetic field interpreted as a charge in an artificial magnetic field
7.1.2 Case of a spin 1 atom
7.1.3 Higher order spins
7.1.4 Case of the quadrupole field
7.1.5 Higher order fields
7.1.6 Artificial magnetic field
7.2 Condensation in presence of an artificial gauge field
7.2.1 Computing the ground state
7.2.2 Higher order spins or multipolar fields
7.2.3 Spinor and choice of gauge
7.2.4 Detecting the angular momentum
7.3 Measuring Berry’s phase
7.4 A vortex pump
7.4.1 Basic idea
7.4.2 Topological interpretation of the vortex pump
7.4.3 Simulation of experimentally relevant parameters
7.5 Conclusion
Conclusion
a fit-free determination of scale invariant equations of state: application to the two-dimensional bose gas
b calculation of the probability distribution of the relative velocity
c description of the coupled dipole programs
c.1 drawPositions
c.1.1 Inputs
c.1.2 Output
c.2 Heff
c.2.1 Inputs
c.2.2 Output
c.3 excVector
c.3.1 Inputs
c.3.2 Output
c.4 transmission
c.4.1 Inputs
c.4.2 Output
c.5 doResonancesVaryOD
c.5.1 Inputs
c.5.2 Output
d estimating confidence intervals using the bootstrapping method
d.1 Position of the problem
d.2 The bootstrap principle
d.3 Precautions in using the bootstrap
d.3.1 The bootstrap does not provide better estimates of parameters
d.3.2 Bootstrap caveats
d.4 Conclusion
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