Mean-eld theory of spinor Bose-Einstein Condensates

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Vacuum system and experimental control

In Fig. 2.1, we show an overall view of the setup around the science chamber. Six crossed beams of 589 nm laser combined with the anti-Helmholtz coils form the Magneto-Optical Trap, which will be discussed in section A. Two crossed beams of 1064 nm infrared laser form the Large-Crossed dipole trap, which will be discuss in section B. Pumps (not shown in Fig. 2.1) are used to maintain the vacuum inside the science chamber. A Bose-Einstein condensation experiment requires very high vacuum in the science chamber. The collisions between trapped cold atoms and atoms of the residual gas at room temperature will immediately \kick » the cold atoms out of the trap. Such collisions thus decrease the life time of the trapped cloud and must be avoided. Normally, in our experiment, the background pressure in the science chamber is about 10􀀀11 mbar [52, 43], which is maintained by two sets of pumps, a getter pump and an ion pump.
The science chamber is made of Titanium which is paramagnetic with a low magnetic susceptibility. This is crucial for the spinor condensate experiment because of the sensitivity to magnetic eld. The science chamber is equipped with several viewports allowing wide optical access, antire ection coated for 589 nm (MOT and imaging) and 1064 nm (optical dipole trap). Lateral viewports (CF25) gives access for the six MOT beams and optical dipole trap beams. Two larger viewports (CF63) along the vertical axis allow us to install a large numerical aperture (NA) objective for high resolution imaging [52]. A Bose-Einstein condensation experiment requires also very precise response timing of each optical and electronic elements, for example, the timing of the switch on/o of current in the coil for the magnetic eld. We use the input/output cards made by National Instrument to communicate between a computer, giving the commands, and the instruments. The precise sequence of instructions is managed by a software from MIT (Cicero, Atticus) which also handles the communication with the National Instrument cards [53]. These cards, analog or digital, are all synchronized with a precision better than 1 s which guarantees the response timing.

Evaporative cooling to BEC

The rst step to the Bose-Einstein condensation is the Magneto-Optical Trap (MOT), which is summarized in appendix A. After the MOT, we have about 2 107 atoms with temperature T 200 K. After the MOT, we load the atoms into the large crossed dipole trap and do the compression, which is summarized in appendix B. After the compression, we have about 1:4 105 atoms with temperature T 100 K.
After the compression in the Large-CDT, we start evaporative cooling. Evaporative cooling is proved to be very ecient for many kinds of atoms [54, 55, 22]. During the evaporation, we will reach the regime of Bose-Einstein condensation. The most crucial point in the evaporative cooling is to keep the elastic collision rate high which helps us to evaporate the hot atoms out of the trap [33]. In our experiment, as the evaporation goes on, the Large-CDT can not keep the eciency always high. This is the reason why we introduce the second evaporation in the more conned, deeper composite dipole trap, Small Crossed Dipole Trap (Small-CDT), which is composed by a Small Vertical Dipole Trap (Small-VDT) and a Small Horizontal Dipole Trap (Small-HDT). At the end of the evaporation in the Small-CDT, we realize an almost pure Bose-Einstein Condensate with about 5000 atoms.

Experimental setup of the Small-CDT

In Fig. 2.2, we illustrate the conguration of the Small-CDT together with the Large- CDT by a view from the top. The Small-VDT propagates in the +z direction, and the Small-HDT propagates in the +u direction. Both Small-VDT and Small-HDT are focused and crossed at the waist of each beams at the center of the science chamber, the same as the Large-CDT.

Small Vertical Dipole Trap (Small-VDT)

The dipole trap \Small-VDT » is generated by a 500 mW laser with wavelength D = 1064 nm, the same wavelength as the ber laser for the Large-CDT. The laser is focused by a large numerical aperture (NA) objective into the science chamber. This objective also serves for imaging system, which will be discussed in section 2.5.2. The trap can be switched o rapidly by a Acousto-Optical Modulator (AOM) within several s. In order to stabilize the power of the Small-VDT, we realize a power feedback system, which is similar to that of the Large-CDT, by measuring the power by a photo-diode (before the laser enters the science chamber) as the feedback signal. We measure the waist of the Small-VDT trap by parametric oscillation [57], which gives the size of the waist of Small-VDT wSV 9:05 0:02 m: (2.6)

Small Horizontal Dipole Trap (Small-HDT)

The dipole trap \Small-HDT » is generated by a 20 W ber laser with wavelength SH = 1070 nm. The laser is focused by a f = 200 mm lens as illustrated in Fig. 2.2. As the Small-VDT trap, the Small-HDT is also controlled by a AOM and a power feedback system (feedback signal is measured after the laser passing the science chamber). The waist of Small-HDT trap is measured together with the Small-VDT trap by parametric oscillation [57], which gives the size of the waist of Small-HDT wSH 11:00 0:01 m: (2.7)

Two-step evaporation

As discussed in section 2.3.1, our evaporative cooling is realized by two steps. We illustrate in Fig. 2.3 the global view of the laser powers of the three dipole traps as a function of t (t=0 means the beginning of the evaporation) during the experiment, including the Large- CDT loading, compression, the rst and the second evaporation. The sub-gure (a), (b), (c) represent the power of the Large-CDT, Small-VDT and Small-HDT, respectively.

First evaporation

The rst evaporation lasts 2 seconds. The main purpose of this process is to ll the atoms from the Large-CDT to the Small-CDT. In Fig. 2.4, we plot the composed potential of the Small-VDT+Large-CDT (blue line) and the potential of the Large-CDT only (red line). We suppose the power of the Large-CDT PCDT = 1:5 W and the power of the Small-VDT PD = 250 mW, which corresponds to the powers around the middle of the rst evaporation (t = 1:5 s in Fig. 2.3). We can see clearly from Fig. 2.4 that when we decrease the power of the Large-CDT, the trap depth of the Small-VDT is much larger than that of the Large-CDT. As the temperature decreases during the evaporative cooling, atoms will gradually ll in the more conned, deeper trap, the Small-VDT.

Table of contents :

Contents
Introduction
1 Mean-eld theory of spinor Bose-Einstein Condensates 
1.1 Introduction
1.2 Elements for scalar condensate
1.2.1 The ideal Bose gas
1.2.2 Bose gas with interactions
1.2.3 Calculation for scalar interacting Bose gas
1.3 Spinor BEC : Pure condensate at zero temperature
1.3.1 Hyperne structure of 23Na
1.3.2 Hamiltonian of the interacting spin-1 Bose gas
1.3.3 Mean-eld approach to the spinor Hamiltonian – HSMA
1.4 Spinor BEC : Condensate with thermal cloud at nite temperature
1.4.1 Semi-ideal HF approximation for spinor BEC
1.4.2 Simulation results for mz = 0
1.5 Conclusion
2 Experimental realization and diagnosis of spinor Bose-Einstein Condensates 
2.1 Introduction
2.2 Vacuum system and experimental control
2.3 Evaporative cooling to BEC
2.3.1 Elements of evaporative cooling
2.3.2 Experimental setup of the Small-CDT
2.3.3 Two-step evaporation
2.4 Spinor condensate preparation and diagnosis
2.4.1 Magnetic eld control
2.4.2 Magnetization controlled spinor gas preparation
2.4.3 Spin diagnosis
2.5 Imaging
2.5.1 Absorption imaging
2.5.2 Imaging systems
2.5.3 Kinetics mode
2.6 Image analysis
2.6.1 Fitting
2.6.2 Counting spin populations
2.6.3 Imaging noises
2.6.4 Methods to reduce structural noise
2.7 Conclusion
3 Phase diagram of spin 1 antiferromagnetic Bose-Einstein condensates 
3.1 Introduction
3.2 Experimental conguration
3.3 Experimental results and interpretation
3.4 Conclusion and perspectives
4 Collective uctuations of spin-1 antiferromagnetic Bose-Einstein condensates 
4.1 Introduction
4.2 Quantum analysis of a spin-1 antiferromagnetic BEC
4.2.1 Formulation in the basis of total spin eigenstates jN; S;Mi
4.2.2 Thermal equilibrium for h ^ Szi = 0
4.2.3 Broken-symmetry approach
4.3 Generalization to arbitrary distribution of M
4.4 Hartree-Fock Approach
4.4.1 Semi-ideal Hartree Fock approximation for spinor BEC
4.4.2 Simulation results and analysis
4.5 Analysis of experimental results
4.5.1 Data analysis
4.5.2 Experimental results of temperature during evaporation
4.5.3 Experimental results of temperature during hold time
4.5.4 Fluctuation of magnetization mz at q = 0
4.6 Discussion of the results
4.7 Conclusion
Conclusion and perspectives
A Magneto-Optical Trap (MOT) 
A.1 Elements of Doppler cooling and MOT
A.2 589 nm laser system
A.3 Sodium MOT
B Loading the Large Crossed Dipole Trap (Large-CDT) 
B.1 Elements of optical dipole traps
B.2 Experimental setup
B.2.1 Conguration of the Large-CDT
B.2.2 Feedback system of the Large-CDT
B.3 Loading the Large Crossed Dipole Trap
B.3.1 Optimization of the MOT lasers
B.3.2 Optimization of the Large-CDT power
B.4 Compression in the Large-CDT
C Supplementary Material : Phase diagram of spin 1 antiferromagnetic Bose-Einstein condensates 
C.1 Sample preparation
C.2 Stern-Gerlach expansion
C.3 Spin interaction energy
C.4 Conservation of magnetization
D Calculation details for Broken symmetry picture 
D.1 Spin nematic states jN :i
D.2 Calculation of ^ in section 4.3
D.3 Fluctuation of magnetization mz at q = 0
E Published articles 
Bibliography 

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