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Table of contents
I Theory and experimental setup
1 Overview of the two-dimensional Bose gas
1.1 The ideal uniform Bose gas in three and two dimensions
1.1.1 Does condensation occur?
1.1.2 Coherence length
1.2 The weakly interacting 2D Bose gas: fundamental equations
1.2.1 Adding interactions
1.2.2 From the quantum Hamiltonian to classical field
1.2.3 Classical field formalism
1.2.4 Factorization of the “frozen direction”
1.2.5 The quasi-2D regime
1.2.6 Scale invariance
1.3 The weakly interacting 2D Bose gas: some properties
1.3.1 Reduction of density fluctuations and quasi-condensate
1.3.2 Bogoliubov analysis
1.3.3 Algebraic decay of correlations
1.4 Superfluid transition in the 2D Bose gas
1.4.1 Definition of superfluidity and superfluid fraction
1.4.2 The superfluid universal jump
1.4.3 Connexion to the Berezinskii-Kosterlitz-Thouless transition
1.4.4 Critical temperature of the transition
1.5 Classical field Monte Carlo simulations
1.5.1 One way to implement the classical field Monte Carlo approach
1.5.2 Result on correlation functions
1.5.3 Monte Carlo simulations of Prokof’ev and Svitsunov
1.6 Equation of state
1.6.1 Limiting cases of equation of state
1.6.2 Quasi-condensate and superfluid fractions
1.6.3 Summary of transitions crossed by a finite-size 2D uniform Bose gas
2 Experimental Setup
2.1 Reaching the degenerate regime
2.1.1 Experimental sequence to load the final trap
2.1.2 Computer control
2.1.3 Laser system
2.1.4 Magnetic field control
2.2 Absorption imaging of the cloud
2.2.1 Brief description
2.2.2 Imaging axes of the setup
2.2.3 Optical resolution of the vertical axis
2.2.4 Calibration of intensity on the atoms
2.3 Description of the final trap
2.3.1 Loading of the atoms in a single fringe
2.3.2 Compression in the accordion
2.3.3 In-plane confinement: Potentials of arbitrary shape
2.4 Manipulation of hyperfine states
2.4.1 Hyperfine structure and Breit-Rabi formula
2.4.2 Stabilisation of the magnetic field
2.4.3 Radio-frequency transfers
2.4.4 Micro-wave transfers
2.5 Crossing the condensation threshold of the gas
2.5.1 Blasting
2.5.2 Making a thermal 2D Bose gas
2.5.3 Condensate focussing
2.6 Thermometry of the gas
2.6.1 Difficulty for the uniform 2D Bose gas
2.6.2 Improved thermometry
II Light scattering
3 About light scattering by atomic clouds
3.1 Introduction
3.2 Modelling light scattering
3.2.1 Response of a single dipole
3.2.2 Coherent and incoherent parts
3.2.3 Modelling of the coherent part: dielectric approach
3.2.4 Modelling of the incoherent part: random walk approach
3.2.5 “Full” Modelling: coupled dipoles approach
3.2.6 Definition of “cooperative effects”
3.3 Experimental investigations and interpretations
3.3.1 Different experimental investigations
3.3.2 About the shifts of the resonance
3.3.3 Finite-size scaling for coupled dipoles
3.3.4 About “cooperative effects” in MOTs
3.4 Conclusion
4 Study of light transmission
4.1 Experimental methods
4.1.1 Cloud preparation
4.1.2 Transmission measurement
4.1.3 Computation of the optical depth
4.1.4 Atom number calibration
4.1.5 Experimental protocol
4.2 Theoretical description
4.2.1 Perturbative approach
4.2.2 Coupled dipole simulations
4.3 Experimental results
4.4 Conclusion
4.5 Additional remark: reflection coefficient of a 2D gas
5 Study of light diffusion
5.1 Results
5.1.1 Light spreading in a dense atomic cloud
5.1.2 Diffusion model
5.1.3 Variation of the decay length with density at resonance
5.1.4 A graded-index waveguide
5.2 Discussion
5.3 Methods
5.3.1 Preparation of the atomic cloud
5.3.2 Imaging system
5.3.3 Coupled dipole simulations
5.3.4 Semi-analytical model of light guiding
III Out-of-equilibrium and superfluid studies
6 Merging of N independent condensates
6.1 Motivation for this work
6.1.1 Kibble-Zurek mechanism and its study in bulk systems
6.1.2 Disentangling the Kibble-Zurek mechanism: study of coarsening dynamics
6.2 Study of the relaxation dynamics in the merging of N independent condensates
6.2.1 Experimental protocol
6.2.2 Principle of the measurements
6.2.3 Results for the number of segments
6.2.4 Results on time evolution
6.2.5 Conclusion
6.3 Supplementary information
6.3.1 About the experimental sequence
6.3.2 Phase reference of the inner ring
6.3.3 Independence of BECs
6.3.4 Width of the distribution for different merging times
6.3.5 Lifetime of supercurrents
6.3.6 Center values for the relaxation measurements
6.3.7 Experimental sequence and data analysis for the merging of two condensates
7 Further characterization of the setup
7.1 Calibration of the magnifications
7.1.1 Lattice in situ: measure of M1 ×M2
7.1.2 Bragg diffraction through a lattice: measure of M2/M1
7.1.3 Conclusion of the calibration
7.2 Calibration of the effective atomic cross-section with light
7.2.1 Principle of the calibration
7.2.2 Experimental sequence
7.2.3 Taking shot noise and read-out noise into account for one image
7.2.4 Taking shot noise and read-out noise into account for the difference of two images
7.2.5 A first method of calibration
7.3 Calibration by modeling the optical response
7.3.1 A coarse-grain approximation
7.3.2 Extracting the matrix using correlations
7.3.3 Robustness using subsets of images
8 Study of Sound propagation
8.1 Motivation for this project
8.1.1 Hydrodynamics and two-fluid model
8.1.2 Reminder of thermodynamics
8.1.3 Prediction
8.2 Experimental study
8.2.1 Protocol for travelling waves
8.2.2 Analysis of the data of travelling waves
8.2.3 Results
8.2.4 Standing waves
8.2.5 Conclusion
8.3 Supplementary information
8.4 Experimental setup
8.4.1 Protocol for characterizing standing waves
8.4.2 Determination of the cloud’s degeneracy
8.4.3 Density calibration
8.5 Summary of the measurements
8.5.1 Complementary results
8.5.2 Landau damping
8.5.3 Influence of the excitation
9 Conclusion and perspectives
9.1 Conclusion of the thesis
9.2 Remaining experimental challenges of the 2D Bose gas
9.3 Perspectives
9.3.1 First-order correlation function measurements
9.3.2 Demixing experiments
9.3.3 Broader perspectives
IV Appendices
A Implementation of Monte Carlo simulations
A.1 Modelisation
A.2 Space discretization
A.2.1 Implementation of the ultraviolet cutoff
A.2.2 Normalization of the fields
A.3 Field evolution
A.3.1 Algorithm used
A.4 Useful variables of the model
B PHYSICAL REVIEW A 95, 013632 (2017)
C Clebsch-Gordan coefficients
C.1 Electric dipole
C.2 Magnetic dipole
D Appendix of the atom number calibration
D.1 Monte Carlo simulations for checking validity of linearisation of noise
D.2 Modelling the optical response
D.2.1 Correlations between pixels
D.2.2 Computation of spatial correlations of the differences of ODs
E List of publications
Bibliography
