The interacting two-dimensional Bose gas

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Table of contents

Introduction 
1. Phase transitions with ultracold two-dimensional Bose gases 
1.1. The ideal gas
1.1.1. The infinite uniform two-dimensional Bose gas
1.1.2. Bose-Einstein condensation in a finite system
1.2. The interacting two-dimensional Bose gas
1.2.1. Interactions in two dimensions: the quasi 2D regime
1.2.2. Scale invariance in 2D
1.2.3. The mean-field Hartree–Fock approximation
1.2.4. The Berezinskii-Kosterlitz-Thouless (BKT) transition
2. Producing and imaging two–dimensional Bose gases 
2.1. Experimental setup
2.1.1. Experimental sequence
2.1.2. A new setup: the hybrid trap
2.1.3. Preparing two-dimensional Bose gases and reaching degeneracy
2.2. Imaging two-dimensional Bose gases and processing the data
2.2.1. Using absorption images to determine the density
2.2.2. Imaging setup
2.2.3. An algorithm for image processing: the Principal Component Analysis (PCA)
3. The equation of state of the two-dimensional Bose gas 
3.1. Exploring the thermodynamics of a two-dimensional Bose gas
3.1.1. Experimental preparation of 2D samples
3.1.2. Thermodynamic analysis
3.1.3. Measuring the interaction energy
3.2. A fit-free equation of state: compressibility, density and pressure
3.2.1. Choosing the correct dimensionless variables
3.2.2. Characterizing the trapping potential
3.2.3. Measuring the equation of state with the global method
3.2.4. Thermometry on single images
3.2.5. Excited levels in the transverse direction
4. Superfluidity in two dimensions 
4.1. A brief theoretical overview
4.1.1. The three-dimensional case
4.1.2. The two–dimensional case
4.2. Superfluid character of a two-dimensional Bose gas
4.2.1. Experimental scheme
4.2.2. Observation of a critical velocity
4.2.3. Comparison with theory
4.3. Closing remarks
5. Fluctuations of the two-dimensional Bose gas
5.1. Experimental procedure
5.2. Local density fluctuations
5.2.1. Characterizing the density minima
5.2.2. Quantifying the distribution of minima
5.2.3. Qualitative interpretation
5.3. Density correlation function
5.3.1. Correlation in real space
5.3.2. Correlations in reciprocal space
5.4. Concluding remarks
6. The uniform two-dimensional Bose gas
6.1. A brief theoretical analysis
6.1.1. Ideal gas in a stadium potential
6.1.2. Interacting gas in a stadium potential
6.1.3. Experimental perspectives
6.2. Experimental realization: preliminary studies
6.2.1. Creating a box-like potential: a holographic method
6.2.2. Creating a box-like potential: by forming the image of a mask
7. Single atom imaging scheme 
7.1. Working principles
7.1.1. Optical molasses
7.1.2. The pinning lattice
7.1.2.1. Influence of the hyperfine structure of the excited states
7.1.2.2. Influence of the hyperfine structure of the ground state
7.1.2.3. Heating by spontaneous emission of photons
7.1.2.4. “Gray molasses” effect from the lattice
7.1.3. Choosing the parameters
7.2. Characterizing our implementation
7.2.1. Experimental setup
7.2.2. Preliminary results
Concluding remarks 
Summary
Perspectives
On the two dimensional Bose gas
On strongly correlated states
Appendix A. Contribution of the excited states to the EoS
Appendix B. Conversion of phase fluctuations into density fluctuations
B.1. Density distribution, in real and reciprocal space
B.2. Case of a small perturbation
B.3. Interpretation in terms of Talbot effect
Appendix C. Diagonalization of the vectorial light shift in a lattice
Appendix D. Collection efficiency of the imaging system
Bibliography