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Table of contents
Introduction
1 Spin-1 Bose-Einstein condensates
1.1 Bose-Einstein condensates with an internal degree of freedom
1.2 Bose-Einstein condensation in scalar gases
1.2.1 Bose-Einstein transition in an ideal gas
1.2.2 Effect of the interactions: ground-state
1.2.3 Effect of the interactions: excited states
1.3 Spin-1 Bose-Einstein condensates: spin Hamiltonian
1.3.1 Single spin-1 particle
1.3.2 Two-body scattering of two spin-1 particles
1.3.3 Many-body Hamiltonian
1.3.4 Effect of applied magnetic fields
1.4 Mean-field theory of spin-1 condensates
1.4.1 Single-mode approximation
1.4.2 Mean-field approximation
1.4.3 Ground-state in the Single-mode approximation
1.4.4 Validity of the single-mode approximation
1.4.5 Excitations in a spinor condensate
1.5 Conclusion
2 Production, manipulation and detection of a spin-1 Bose-Einstein condensate of Sodium
2.1 Experimental methods
2.1.1 The experimental chamber and the atomic source
2.1.2 Magneto-Optical Trap
2.1.3 Resonant laser
2.1.4 Loading in a Crossed Dipole Trap and two-step evaporation
2.1.5 Condensation in the dimple trap
2.2 Diagnostic of the spinor gas
2.2.1 Application of magnetic fields
2.2.2 Stern Gerlach separation
2.2.3 Imaging set-up
2.2.4 Calibration of the scattering cross-sections
2.2.5 Imaging noise
2.3 Preparation of a controlled magnetization
2.3.1 Magnetic fields control
2.3.2 Spin-mixing
2.3.3 Spin distillation
2.4 Conclusion
3 Mean-field study of an antiferromagnetic spinor condensate
3.1 Nematic order in spinor condensates
3.1.1 Definition of the nematic order parameter
3.1.2 Application to mean-field states
3.1.3 Nematic order of a mean-field ground-state
3.2 Experimental study of the phase diagram
3.2.1 Experimental sequence
3.2.2 Results
3.3 Detection of spin-nematic order
3.3.1 Rotation of the spinor wavefunction
3.3.2 Experimental implementation of three-level Rabi oscillations
3.3.3 Evidence for phase-locking
3.4 Conclusion
4 Spin fragmentation in a spin-1 Bose gas
4.1 Fragmentation of a spinor condensate at zero field
4.1.1 Fragmented Bose-Einstein condensates
4.1.2 Spin fragmentation in an antiferromagnetic spinor BEC at T = 0
4.1.3 Spin fragmentation at finite temperatures
4.2 The broken-symmetry picture
4.2.1 Broken-symmetry picture at T = 0
4.2.2 Broken-symmetry approach at finite temperatures
4.2.3 SU(3) coherent states
4.2.4 Broken symmetry description of a spin-1 gas with constrained magnetization
4.3 Connection to spontaneous symmetry breaking
4.3.1 Spontaneous symmetry breaking in the thermodynamic limit
4.4 Conclusion
5 Observation of spin fragmentation and spin thermometry
5.1 Observation of spin fluctuations
5.1.1 Experimental sequence
5.1.2 Data acquisition
5.1.3 Measured moments of n0
5.2 Statistical analysis of the distributions of n0 and mz
5.2.1 Model and method
5.3 Spin temperature and condensed fraction during the evaporation
5.3.1 Temperatures at fixed trap depth
5.4 Two spinor fluids isolated from each other
5.4.1 Comparison of spin and kinetic temperatures
5.4.2 Large q: the condensate and the thermal gas are coupled
5.4.3 Low q: condensate at equilibrium but decoupled from the thermal gas
5.5 Conclusion
Conclusion and perspectives
A Numerical methods for the spinor Gross-Pitaevskii equations 155
A.1 Gross-Pitaevskii equations in imaginary time
A.1.1 The imaginary time propagation method
A.1.2 Dimensionless coupled Gross-Pitaevskii equations
A.2 Propagation of the finite differences scheme
A.3 Numerical implementation
B Geometrical representation of a spin-1 state
B.1 Bloch-Rabi representation
B.2 Application to the mean-field ground state
C Three-level Rabi oscillation
D Generalized coherent states
D.1 Construction of generalized coherent states
D.2 Spin coherent states
D.3 SU(3) coherent states
D.4 Diagonal representation of few-body operators in the SU(3) coherent states basis
E Spin fragmentation of Bose-Einstein condensates with antiferromagnetic interactions
Bibliography
