在本篇論文之中，我們將討論如何使{_\ ^{87}}Rb原子團更加穩定製備成量子簡併玻色子，即玻色—愛因斯坦凝聚態 (Bose-Einstein condensate, BEC）。

在我們的實驗設備中，影響BEC的穩定與品質有幾個重要因素：原子團載入磁光陷阱（MOT）中的初始溫度與數量，將原子團載入至光學偶極陷阱（ODT）的效率，並對蒸發冷卻進行優化。

首先，我們對3對反向的光束組成的光學蜜糖裝置進行微調，並保證每一對反向光束重合無間。隨後，光學蜜糖被對齊並與磁四極陷阱中心重合，因此形成穩固的磁光陷阱（MOT）。接著調整磁場強度與修正冷卻雷射光的失諧頻率，以此來優化連續冷卻過程中的壓縮MOT（CMOT）與極化梯度冷卻（PGC）階段。

其次，我們對射頻蒸發冷卻（RFC）進行優化，目標是將溫度冷卻至 20µK 以下，並保持原子數量約為107。對此我們優化了RF訊號的時間常數與振幅，期望在盡可能低的溫度下維持原子數量約107。

第三，我們希望提高原子載入ODT的效率，並且優化ODT的蒸發冷卻。整個蒸發冷卻由數個步驟共同組成。我們透過逐步降低光位能阱深度，來讓較熱原子逐漸離開位能阱，來達成近一步冷卻原子，此步驟稱為光學式蒸發式冷卻（Optical Evaporative Cooling）。在最終的蒸發冷卻步驟時，我們會有原子數量約105凝結成BEC。

上述實驗流程中，我針對每個站點的參數做最佳化實驗，包含各站點所需的持續時間，及在每個站點中各個雷射光所需的強度與頻率的控制，藉此可利用上述優化每個站點的參數，來達到穩定製造BEC的目的。

In this thesis, I aim to investigate stabilizing methods for the Bose-Einstein condensate (BEC) system of 87Rb.
The stability and quality of the BEC primarily depend on several key factors in our experimental setup: the initial loading of atoms into the magneto-optical trap (MOT), initial temperatures of the atoms into MOT, the efficiency of loading atoms into the optical dipole trap (ODT), optimized evaporative cooling.
Firstly, the optical molasses setup is fine-tuned by utilizing six counterpropagating beams and ensuring their corresponding retroreflections overlap seamlessly. Subsequently, the optical molasses is aligned such that it shares the same center as the magnetic quadrupole trap resulting in a stable MOT. Tuning the magnetic field and adjusting the detuning frequency of the cooling beam enables the optimization of the compresses MOT and polarization gradient cooling (PGC) stages in the successive cooling process.
Secondly, we perform an optimization process for RF-evaporative cooling (RFC) to reduce the temperature below 20 µK while maintaining an atom number of approximately 107. The RF frequency is exponentially decreased from 30 MHz to 3 MHz over a fixed number of steps. We optimize both the time constant and amplitude of the RF signal to achieve an atom number around 107 at the lowest possible temperature.
Thirdly, we improve the loading efficiency to the ODT and optimize evaporative cooling in ODT. The evaporative cooling process is consisting of several stages. Each evaporative cooling stage is tuned by controlling the atom loss while reducing the temperature of the atoms. In the final evaporative cooling stage, we condense ~105 atoms forming the BEC.
Throughout the optimization process described, we were able to reach a benchmark of 18 milliseconds for PGC by adjusting the intensity and duration of frequency detuning in the molasses. The evaporation cooling phase was executed in 5 seconds, divided into 5 equal steps. This collective optimization effort led to a consistently stable preparation of the BEC.

[1] H.k. kowalski, V. cao long, k. dinh xuan, M. głódź, B. nguyen huy, & J. szonert. (2010). Magneto-Optical Trap: Fundamentals and Realization. COMPUTATIONAL METHODS IN SCIENCE AND TECHNOLOGY Special Issue (2) 115-129 (2010), SI (2), 115–129.
[2] Chu, S. (1998). Nobel Lecture: The Manipulation of Neutral Particles. Reviews of Modern Physics, 70(3), 685.
[3] Metcalf, H. J., & van der Straten, P. (1999). Laser cooling and trapping. Springer Science & Business Media.
[4] Wineland, D. J., & Itano, W. M. (1979). Laser cooling of atoms. Physical Review A, 20(4), 1521.
[5] H. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer Verlag, 1999).
[6] Petrich, W., Anderson, M. H., Ensher, J. R., & Cornell, E. A. (1994). Behavior of atoms in a compressed magneto-optical trap. JOSA B, 11(8), 1332-1335.
[7] Metcalf, H. J., & van der Straten, P. (1999). Laser Cooling and Trapping. Springer Science & Business Media.
[8] Griffin, A., Snoke, D. W., and Stringari, S. (eds). 1995. Bose-Einstein Condensation. Cambridge University Press.
[9] Second International Conference on Bose-Einstein Condensation, Mt. Ste. Odile, France, 1995; organized by J. T. M. Walraven, J. Treiner, and M. W. Reynolds.
[10] “Universal Themes of Bose-Einstein Condensation,” Lorentz Center, Leiden, The Netherlands; organized by K.Burnett, P. B. Littlewood, N. P. Proukakis, D. W. Snoke, and H. T. C. Stoof.
[11] Blatt, J. M. 1964. Theory of Superconductivity: Pure and Applied Physics. Academic Press.
[12] Pethick, C. J., & Smith, H. (2012). Bose-Einstein Condensation in Dilute Gases (2nd ed.). Cambridge University Press.
[13] Johnson, R. C. (1950). Antenna Engineering Handbook (THIRD.). New York: McGraw-Hill.
[14] IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-25, NO. 6, NOVEMBER 1977
[15] Analog Devices公司官方網站：https://www.analog.com/en/products/adf4351.html
[16] Henri de Bellescize, "La réception synchrone," L'Onde Électrique (later: Revue de l'Electricité et de l'Electronique), vol. 11, pages 230–240 (June 1932).
[17] Leonov, G. A.; Kuznetsov, N. V.; Yuldashev, M. V.; Yuldashev, R. V. Hold-in, pull-in, and lock-in ranges of PLL circuits: rigorous mathematical definitions and limitations of classical theory.. IEEE Transactions on Circuits and Systems I: Regular Papers (IEEE). 2015, 62 (10): 2454–2464.
[18] J. D. Kraus, "The Helical Antenna," in Proceedings of the IRE, vol. 37, no. 3, pp. 263-272, March 1949, doi: 10.1109/JRPROC.1949.231279.,
[19] J. D. Kraus "Helical beam antenna" Electronics vol. 20 pp. 109-111 April 1947
[20] D. Kraus, Antennas 2nd ed. , McGraw-Hill, 1988.
[21] Johnson, R. C. (1950). Antenna Engineering Handbook (THIRD.). New York: McGraw-Hill.
[22] IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-25, NO. 6, NOVEMBER 1977
[23] https://www.microwaves101.com/encyclopedias/helix-antennas
[24] J. D. Kraus "Helical beam antenna" Electronics vol. 20 pp. 109-111 April 1947
[25] Ferran Martin Ciurana, QUANTUM CONTROL OF COLD ATOMS USING MICROWAVES, ICFO Master's Thesis, ICFO Photonics Institute, (8th September 2011)
[26] Chris leahy, J. todd hastings, & P. m. wilt. (1997). Temperature Dependence of Doppler-Broadening in Rubidium: An Undergraduate Experiment. Am. J. Phys., 65(5), 367–371.
[27] Lin, Y. J., Perry, A. R., Compton, R. L., Spielman, I. B., & Porto, J. V. (2009). Rapid production of R 87 b Bose-Einstein condensates in a combined magnetic and optical potential. Physical Review A, 79(6), 063631.