Mean-field study of an antiferromagnetic spinor condensate 

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Transfer and evaporation in a small volume dipole trap

In a dipolar potential created by a gaussian beam, the trapping frequencies evolves as the square root of the light intensity. Thus, with a given power, by focusing a laser beam on a smaller waist we can obtain a stronger confinement. The idea to reach the degenerate regime is to transfer the atoms from the CDT to a second dipolar trap of smaller volume called the dimple trap where we then continue the evaporative cooling [47]. This second dipolar trap consists of two laser beams created by two different infrared power lasers at 1064 nm. They are transmitted through acousto-optic modulators and optical fibers to the chamber. The first beam propagates through the chamber vertically and is focused on the atomic cloud by an objective of large numerical aperture placed in the lower entring flange. Its waist is 9 μm. We denote this beam as vertical dimple. The second beam propagates in the horizontal plane perpendicularly to the second beam of the CDT (see figure (2.2) and (2.3)) and is focused at the same position by a 200mm lens on a waist of 11 μm. We denote it as horizontal dimple. Here again the precise crossing of the two beams and the superposition of the two waists is crucial. The vertical dimple is kept fixed at all time, and we adjust the pointing of the horizontal dimple with a motorized mirror and a lens mounted on a translation stage. The dimple trap is actually the spatially most stable of all our traps, and its position is used as the reference position on which the alignment of the CDT and of the MOT is optimized.
At its maximum level the dimple trap has a depth of 54 μK. It is then negligible compared to the confinement induced by the MOT and does not perturb neither the potential created by CDT before its evaporation. The dimple trap is switched on from the beginning of the experimental sequence but is of little effect until the CDT is evaporated. Then, during this step as the depth of the CDT decreases the presence of the dimple is progressively revealed and forms a narrow peaked potential at the center of the softer CDT potential, as shown in figure (2.3). In the same time it is filled by the least energetic atoms that fall in it. At the end of the evaporation of the CDT the atoms are mainly held by the dimple potential. Once the CDT is switched off the evaporative cooling can now start in the dimple trap.

Diagnostic of the spinor gas

To study the magnetic properties of a spinor condensate we need to be able to probe its spin state. Two main techniques are used in spinor experiments.
A first technique uses dispersive imaging and takes advantage of the dependence of the dielectric tensor of the atomic sample in its local spin state. By sending a far off-resonant linearly polarized laser beam through the atomic gas, and by measuring the rotation of its polarization we can retrieve the local magnetization of the cloud. This technique allows one to acquire several images from the same experimental run. It was demonstrated in [73] where the authors relied on its principle to characterize the three components of the local spin of the gas.
A second technique consists in performing a Stern-Gerlach experiment followed by absorption imaging. The application of a gradient of magnetic field associated with an homogeneous bias field realises a spin-dependent force on the atoms that allows to spatially separate the three Zeeman states during a time-of-flight experiment. Absorption imaging then gives access to the populations of the three spin states and to their spatial distribution. This technique is destructive.
These two techniques give different informations on the spin state of the atomic sample. The Stern-Gerlach method being easier to implement, we choose to use this technique, before additionally implementing the dispersive imaging in the future.

Application of magnetic fields

The Stern-Gerlach experiment requires the application of external magnetic fields. We here briefly describe our coil system which we use to create these fields. The calibration of the field created by these coils is presented in section 2.3.1.
We are able to apply an homogeneous static magnetic field at the center of the chamber using three pairs of coils in Helmoltz configuration placed around the chamber along the three axis x, y, z and that create the three components of a controllable field (Bx,By,Bz). We are also able to create a quadrupolar magnetic field using the pair of coils already used for the MOT. The strong axis of this quadrupolar field is along the y direction.

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Calibration of the scattering cross-sections

We describe in this section how the atomic density is extracted from the two absorption images. We note I1(x, y) the intensity measured on the first image (with atoms) and I2(x, y) the intensity on the second image (without atoms). The atomic density is noted n(x, y, z). We suppose that the probe beam propagates along the z axis.

Nematic order in spinor condensates

The magnetic phase transition between the antiferromagnetic and the broken-axisymmetry phases is revealed by the presence or the absence of the mF = 0 component. We understood these two phases as the consequence of the predominance of the spin interactions (which favors the coexistence of |mF = +1i and |mF = −1i) in the first one and of the predominance of the quadratic Zeeman effect (which favors |mF = 0i) in the second one (see chapter 1). The phase diagram actually hides a more profound order. The study of spin correlations allows to describe the transition in terms of an orientational order of the total spin of the system. The two competing effects indeed favor different kinds of spin anisotropy: the spin interactions favor a nematic ordering of the spin while the quadratic Zeeman effect favors transverse ferromagnetism. In this section we first define the nematic order parameter for a mean-field spin state and then calculate it in the case of an antiferromagnetic spin-1 condensate.

Table of contents :

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 con- densate 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
4 Contents
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 
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 


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