Stepwise Bose-Einstein Condensation in a Spinor Gas 

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The crossed dipole trap and the dimple optical traps

The crossed dipole trap is produced by a single laser beam from a ! = 1070 nm high power fibre laser (up to 40W) with a linear polarisation assured by a Glan-Taylor polariser. The laser is focused at the centre of the chamber with a waist !0 ≈ 40μm and then folded and sent again back towards the atoms at an angle ✓ = 45○ with respect to the initial direction, see Figure 3.1(a). To produce a crossed dipole trap, we rotate the light polarisation by ⇡ between the two crossing arms to suppress interferences.
The power of the laser is controlled by a servo loop on the pump diode current in combination with a motorised !￿2 wave-plate and a Glan-Taylor polariser. The motorised wave-plate + Glan-Taylor polariser can change the laser power without any distortion of the beam2 profile and allow us to have powers below the lasing threshold; the drawback of this method is the low bandwidth of 10Hz limited by the rotation speed of the wave-plate. The faster servo loop on the pump diode current is used to stabilise the laser power around a value given by the rotating wave-plate and, thanks to a bandwidth of a few KHz, it can corrects high frequency fluctuations of the laser power. A thorough description of our stabilisation system for the laser power can be found in the PhD thesis [160]. To continue in our description of the sequence and explain the need for a second cross dipole trap, we recall here the potential felt by neutral atoms given by a laser beam propagating along the direction z [57].

Stern-Gerlach time of flight

As described in Chapter 1, we actually reach degeneracy in a spin-1 system. To probe our spinor system we need a spin dependent imaging technique. We implemented a Stern-Gerlach (SG) imaging technique, see[51, 144], which consists in applying a spin dependent force during the ballistic expansion.

Manipulating internal states with spin degrees of freedom

The study of a spinor condensate requires a very good control on the spin degrees of freedom of the trapped atoms. In particular the applied magnetic field, which defines the energy difference between the Zeeman sub-levels of our spin-1 system, must be very well controlled. In this section, we introduce the methods we use to characterise the ambient magnetic fields as well as the bias magnetic field generated by our coils. We then present the methods we use to set the magnetisation of the spinor gas and to change the spin state of the system.

Magnetic field control

The control of external magnetic field is of paramount importance in spinor condensate physics. External magnetic fields determine the ground state of the system and a good control on them is required to prepare the system in any desired configuration. During the experimental sequence, we need to change magnitude and orientation of the external magnetic field to perform different tasks; for example, apply a field along x with a certain magnitude and then rotate it towards z to image. This section is divided into two parts: in the first one we derive an adiabaticity criterium for any magnetic field variation and then we illustrate the spectroscopy method we use to characterise the external and applied magnetic fields.
We consider a single 23Na atom plunged in a magnetic field B aligned along z. We use a semi-classical description of the atom, for which its internal degrees of freedom are described with quantum mechanics and the external degrees of freedom, (r,p), are described with classical mechanics. Therefore, the internal state of the atom will see a magnetic field moving with B(r) → B(r0 + pt m) and we can describe the system of an atom moving in a set magnetic field like a fixed atom plunged in a magnetic field changing in modulus and orientation in time . We suppose the atom to be in the spin eigenstate mF = +1 at time t = 0. If the magnetic field, at t > 0, starts to rotate in space, the atom will try to remain in the same eigenstate while the latter will try to follow the magnetic field movement rotating in spin space. To remain in the same state, the atom must follow the magnetic field adiabatically.
To obtain a criterium for the adiabaticity, we can confront the velocity at which the atomic spin can rotate in a magnetic field B, expressed by the Larmor frequency !L = p ￿h , where p is the linear Zeeman shift (2.62), with the magnetic field rate of change ✓˙. The atom follows adiabatically the magnetic field if !L ￿ ✓˙, or equivalently, if ￿ ˙B ￿ ￿￿B￿￿ ￿ μBB ￿h (3.8).

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Table of contents :

1. Introduction 
2. Elements of Bose-Einstein condensation 
2.1. The scalar Bose-Einstein condensate in a 3d harmonic trap
2.1.1. The ideal Bose gas
2.1.2. The role of interactions
2.1.3. The mean-field approximation at T = 0
2.1.4. Mean field approximation at T > 0
2.2. The scalar Bose gas in a 1D harmonic trap
2.2.1. Bose gases in one dimension
2.2.2. Quasicondensation in 3D anisotropic trap
2.2.3. Phase Fluctuations in TOF
2.3. The spin-1 Bose Gas
2.3.1. Hyperfine structure of Na atoms
2.3.2. Two-body scattering between Na atoms
2.3.3. The Zeeman shift
2.3.4. The Spinor Many-body Hamiltonian
2.3.5. Spinor BEC in the single spatial mode
3. Production and characterization of a spin-1 Bose-Einstein condensate of Sodium atoms 
3.1. Experimental Setup and cooling techniques
3.1.1. UHV chamber and atomic source
3.1.2. Magneto-optical trap
3.1.3. The crossed dipole trap and the dimple optical traps
3.1.4. Stern-Gerlach time of flight
3.1.5. Imaging after TOF
3.2. Manipulating internal states with spin degrees of freedom
3.2.1. Magnetic field control
3.2.2. Rabi Oscillations
3.2.3. Adiabatic rapid passage
3.2.4. Magnetization preparation
3.3. Image characterisation
3.3.1. Magnification characterisation
3.3.2. Calibration of spin-dependent cross sections
3.4. Image analysis
3.4.1. Noise modelisation
3.4.2. Noise reduction
3.5. From the 3d to the 1d geometry
3.5.1. The adiabatic transfer
3.5.2. Characterisation of the trap frequencies
4. Stepwise Bose-Einstein Condensation in a Spinor Gas 
4.1. Article
4.2. Supplementary Materials
4.2.1. Experimental sequence
4.2.2. Evaporation dynamics
4.2.3. Extracting Tc
4.2.4. Theoretical models of spinor gases at finite temperatures .
5. Spin-1 BEC in 1D: Spin domains and phase transition 
5.1. Stable phases of a 1D Spin-1 antiferromagnetic BEC
5.1.1. The uniform case
5.1.2. Adding an harmonic potential in the LDA approximation
5.1.3. The phase transition
5.1.4. GP simulation vs LDA solution
5.2. 1D-3D crossover
5.3. Preparation and study of spin domains
5.3.1. Minimisation of magnetic field gradients
5.3.2. Fitting the Spin Domains
5.3.3. Equation of State and temperature
5.4. 1D Transition
6. Binary mixtures 
6.1. Spin-dipole polarisability
6.1.1. Polarised cloud response: z0
6.1.2. Spin-dipole polarisability vs magnetisation
7. Conclusions and perspectives 
Appendices
A. Adiabatic Transfer of a quasi-condensate in 1D 
A.1. Theory of quasi-condensate
A.2. The adiabatic transfer
B. Numerical solution of the spin 1 Gross-Pitaevskii equations

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