臺灣博碩士論文加值系統

本論文研究了非簡諧位能阱中的玻色凝結體（BEC）的非平衡動力學，並利用主
成分分析（PCA）技術來提高圖像品質。在本實驗中，我們在混合型位能阱（hybrid trap）中產生了銣原子 87 Rb的 BEC，此混合位能阱是磁陷阱(magnetic trap)和光學偶極阱 (optical dipole trap)的組合，產生的BEC溫度低於 100nK。以此為基礎，我們為進一步研 究了BEC的動力學。接著，我們關閉光學偶極阱並保持降低了四極磁場位能阱低的強度 ，以便剛好足以抵銷大部分的重力作用，以減慢 BEC 自由落下的速度。
我們引入了主成分分析 (PCA)，以消除，在光束路徑到達 CCD 相機之前，由光學 元件的動度所產生的雜訊，與干涉效應的條紋。並以低通濾波器去除計算計算過程中產生 的高頻雜訊。
我們觀察到 BEC 在水平方向的振盪在兩種條件下是一致的（高原子數目
：1.2×10^5 個，低原子數目：3×10^4 個），同時也在不同原子密度下比較原子間相互作用 對 BEC 動力學的影響。此外，還觀察到 BEC 的不均勻形狀及其不對稱的密度分佈。然 而這種奇特的動態和密度分佈只有在量子簡併性下才能觀察到。觀察到在磁場作用下落下 的熱原子雲是向四面八方膨脹，其分佈呈高斯均勻分佈。
所觀察到的動態軌跡和集體激發模式提供了豐富的量子動態物理，可進一步進行理 論研究和數值模擬的探討。

This thesis studies BEC non-equilibrium dynamics in anharmonic trap and the implement of a principal component analysis (PCA) technique to enhance the images' quality.
In our experiment, we produced a BEC of $^{87}$Rb atoms in a hybrid trap, a combination of magnetic trap and optical dipole trap, with a typical temperature below 100nK. Subsequently, dynamics of BEC was investigated further.
Firstly, we turned off optical dipole trap and kept quadrupole magnetic field on with low magnitude just enough to slow down the speed of BEC under gravity force.
We introduce the principal component analysis (PCA) to get rid of the fringes coming from small intensity fluctuations and interference effects of many optical elements before beam path gets to the CCD camera. Subsequently, a Kernel low-pass filter was applied to remove the high frequency noise generated during computing calculation.
We observed that the oscillation of BEC in the horizontal direction is consistent for two conditions (high number: 1.2x10^5 atoms, low number: 3x10^4 atoms), while the effect of atom-atom interactions on the dynamics of BEC is compared under various atomic densities. Additionally, the inhomogeneous shapes of BEC and its asymmetrical density distributions are observed as well. On the other hand, such peculiar dynamic and density distribution are only observed under quantum degeneracy. The thermal cloud falling under magnetic field was observed to expand in all directions, and the cloud's distributions are Gaussian and homogeneous.
The observed dynamic trajectories and collective excitation modes provide fruitful quantum dynamic physics that acquires further theoretical investigation and numerical simulation.

Contents
Abstract -------------------------------------------------------I
Abstract (Chinese) ---------------------------------------------II
Acknowledgements -----------------------------------------------III
Contents -------------------------------------------------------IV
List of Figures ------------------------------------------------VII
List of Tables -------------------------------------------------IX
1 Introduction -------------------------------------------------1
1.1 Motivation -------------------------------------------------1
1.2 Outline ----------------------------------------------------2
2 Theoretical background ---------------------------------------3
2.1 Atomic Structure -------------------------------------------3
2.2 Laser Cooling ----------------------------------------------5
2.2.1 Laser (Doppler) Cooling ----------------------------------5
2.2.2 Polarization Gradient Cooling ----------------------------7
2.3 Magneto-Optical Trap (MOT) ---------------------------------8
2.3.1 Principle of Operation -----------------------------------8
2.3.2 MOT Compression ------------------------------------------11
2.4 Magnetic Trap ----------------------------------------------11
2.5 Optical Dipole Trap ----------------------------------------14
2.6 Evaporation Cooling ----------------------------------------18
2.6.1 RF Evaporation from Magnetic Trap ------------------------18
2.6.2 Evaporation from ODT -------------------------------------20
2.7 Bose-Einstein Condensation ---------------------------------20
2.7.1 Time-independent Gross-Pitaevskii equation ---------------21
2.7.2 Time-dependent Gross-Pitaevskii equation -----------------23
2.7.3 Mean-field theory of spinor-1 condenstates ---------------24
2.7.4 Miscibility and immiscibility of spinor condensate components-26
3 Experimental Apparatus----------------------------------------28
3.1 Vaccum system ----------------------------------------------28
3.1.1 MOT chamber ----------------------------------------------29
3.1.2 Science cell----------------------------------------------30
3.2 Laser system -----------------------------------------------31
3.2.1 Frequencies ----------------------------------------------32
3.2.2 Lasers ---------------------------------------------------33
3.2.3 Power amplifier and beam shaping -------------------------34
3.3 Optics around the MOT chamber ------------------------------34
3.4 Coils-------------------------------------------------------39
3.4.1 MOT coils ------------------------------------------------40
3.4.2 Balance coils --------------------------------------------40
3.4.3 RF coil---------------------------------------------------40
3.5 Magnetic transfer ------------------------------------------41
3.6 Optical dipole trap ----------------------------------------41
3.7 Detecting system -------------------------------------------43
3.7.1 Fluorescence imaging -------------------------------------44
3.7.2 Absorption imaging ---------------------------------------45
4 Improvement of imaging system --------------------------------48
4.1 The addition of new slave laser ----------------------------48
4.2 Absorption imaging setup with new camera -------------------49
4.3 Principal component analysis Background Subtraction --------52
4.3.1 Training data --------------------------------------------54
4.3.2 Predicting step ------------------------------------------54
4.3.3 Reconstructed image --------------------------------------54
4.3.4 Kernel low-pass filter -----------------------------------55
5 Polarization of BEC ------------------------------------------57
5.1 Unexpected Components --------------------------------------57
5.2 Manipulation of mixed components ---------------------------59
5.3 Single F = 1,mF = −1 component -----------------------------61
5.3.1 Principle of the Stern-Gerlach separation ----------------61
5.3.2 Stern-Gerlach experiment ---------------------------------62
6 BEC non-equilibrium dynamics in anharmonic potential ---------64
6.1 Magnetic trap potential ------------------------------------64
6.2 Breathing Mode and Trap frequency --------------------------68
6.3 Scissor Mode -----------------------------------------------71
6.4 Comparison to Thermal expansion ----------------------------74
7 Conclusion and future work -----------------------------------76
7.1 Conclusion -------------------------------------------------76
7.2 Future work ------------------------------------------------77

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