How to produce a uniform 2d bose gas

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Table of contents

1 introduction 
i producing and manipulating 2d bose gases
2 experimental set-up 
2.1 Overview of a cold atom experiment
2.1.1 General features
2.1.2 Taking pictures of atoms
2.2 How to produce a uniform 2D Bose gas
2.2.1 Cooling atoms down to quantum degeneracy
2.2.2 Confining the gas in two dimensions
2.2.3 Creating a cloud with a uniform atomic density
2.3 How to control the initial state of the cloud
2.3.1 The internal state of the atoms
2.3.2 The phase-space density of the cloud and its temperature
2.4 Conclusion
3 implementation of spatially-resolved spin transfers
3.1 How to induce Raman processes on cold atoms
3.1.1 Elements of theory about two-photon transitions
3.1.2 Our experimental set-up
3.2 Raman transitions without momentum transfer
3.2.1 Measuring Rabi oscillations
3.2.2 Focus and size of the DMD
3.2.3 Local spin transfers
3.3 Raman transitions with momentum transfer
3.3.1 Calibrating the momentum transfer
3.3.2 Local spin transfers with a momentum kick
3.4 Conclusion
ii measuring the first correlation function of the 2d bose gas
4 theoretical considerations on the first correlation function 
4.1 The first-order correlation function of infinite 2D systems
4.1.1 The XY-model and the BKT transition
4.1.2 An ideal gas of bosons in 2D
4.1.3 Interacting bosons in 2D
4.2 Developments for realistic experimental measurements
4.2.1 Exciton-polaritons and out-of-equilibrium effects
4.2.2 Cold atoms and trapping effects
4.2.3 Finite-size effects
4.2.4 Conclusion
5 probing phase coherence by measuring a momentum distribution 
5.1 Measuring the momentum distribution of our atomic clouds
5.1.1 Creating an harmonic potential with a magnetic field
5.1.2 Evolution of atoms in the harmonic potential
5.2 Investigating the width of the momentum distribution
5.2.1 Influence of the initial size of the cloud
5.2.2 Influence of the temperature of the cloud
5.2.3 Determining the first-order correlation function?
5.3 Conclusion
6 measuring g1 via atomic interferometry 
6.1 Interference between two separated wave packets
6.1.1 Free expansion of two wave packets in one dimension
6.1.2 Free expansion of two wave packets in two dimensions
6.2 Setting up and characterising the experimental scheme
6.2.1 The experimental sequence
6.2.2 Measuring the expansion of one line
6.2.3 Measuring the expansion of two lines
6.3 Measuring the phase ordering across the BKT transition
6.3.1 Extracting the contrast of the averaged interference pattern
6.3.2 Results of the measurements across the critical temperature
6.3.3 Discussion and effects that may affect the measurements
6.4 Conclusion
iii dynamical symmetry of the 2d bose gas
7 elements of theory on dynamical symmetries 
7.1 Symmetries of a physical system
7.1.1 The symmetry group as a Lie group
7.1.2 Linking different solutions of a differential equation
7.1.3 Linking solutions of two differential equations
7.2 Dynamical symmetry of weakly interacting bosons in 2D
7.2.1 Symmetry group of the free Gross-Pitaevskii equation
7.2.2 Symmetry group with a harmonic trap
7.2.3 Link between different trap frequencies
7.3 More symmetries in the hydrodynamic regime
7.4 Conclusion
8 an experimental approach of dynamical symmetries
8.1 Experimental sequence
8.1.1 The course of events
8.1.2 The measured observables
8.1.3 Some calibrations
8.2 Verification of the SO(2,1) symmetry
8.2.1 Evolution of the potential energy
8.2.2 Evolution in traps of different frequency
8.3 Universal dynamics in the hydrodynamic regime
8.3.1 Evolution with different interaction parameters
8.3.2 Evolution with different sizes and atom numbers
8.4 Conclusion
9 breathers of the 2d gross-pitaevski i equation 
9.1 Experimental hints
9.1.1 Initial triangular shape
9.1.2 Initial disk shape
9.2 Numerical simulations
9.2.1 Initially triangular-shaped cloud
9.2.2 Initially disk-shaped cloud
9.2.3 Other initial shapes
9.3 Towards an analytical proof?
9.4 Conclusion
10 conclusion 
Appendices
a coupling two hyperfine states with raman beams
b correlation function of an ideal 2d bose gas
c details on the interferometric measurements of g1
d details on the scaling laws of the 2d bose gas
d.1 Free Gross-Pitaevskii equation
d.2 Gross-Pitaevskii equation with a harmonic trap
d.2.1 General case: a variable trap frequency
d.2.2 Particular case: a constant trap frequency
d.2.3 Invariant transformations
d.3 Hydrodynamic equations
e publications
f résumé en français
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