離子阱是目前在量子計算及量子感測上極具前景的平台之一。在可被囚禁的所有原子中，鈣是一個受歡迎的元素。而在鈣元素的所有同位素中，$^{43}$Ca$^+$是唯一具有超精密結構的。使其能夠成為理想的超精密量子位元（hyperfine qubit）。
然而，$^{43}$Ca在天然的鈣金屬中含量非常的低（0.135\%），並因爲能階的超精細結構使得捕獲$^{43}$Ca$^+$較為困難。本論的重點主要在探討我們是如何捕捉$^{43}$Ca$^+$。首先我會從離子阱系統和我們實驗架設的背景開始介紹，並著重於同位素選擇捕捉上做的改變。
在此論文中，我會講述系統中個雷射的穩頻方式，其中，我們採用一種叫做transfer cavity lock的穩頻方式，將腔體鎖定在銫原子的躍遷上。我也會介紹我們如何選擇性捕捉各種鈣同位素離子，並瞭解不同同位素之間的頻率差。除此之外，此論文也會探討捕獲$^{43}$Ca$^+$時遇到的困難以及我們如何解決。我也將介紹如何提升鈣43離子的螢光強度，包含雷射的邊帶強度和偏振方向。經過這些努力，我們成功克服$^{43}$Ca的低含量，成功捕獲$^{43}$Ca離子。利用此論文所發展出的方法，我們可以成功製造出完全由$^{43}$Ca離子所形成的一維離子列。

Trapped-ion systems offer potential for various technologies and experiments, including quantum computing. Calcium ions, in particular, are widely studied in this context. Among the different calcium isotopes, $^{43}$Ca$^+$ stands out due to its hyperfine structure, which makes it suitable for hyperfine qubits.
However, capturing $^{43}$Ca ions presents challenges due to their low natural abundance of 0.135\% and the complexity of their hyperfine energy levels. This thesis documents our successful trapping of the first $^{43}$Ca ion, detailing the experimental setup and modifications made to selectively trap different calcium isotopes.
We utilize a frequency stabilization scheme called transfer cavity locks, where the cavities are locked to a neutral cesium transition. The scheme is introduced, covering its theory, setup, and experimental results. With the scheme, we are able to selectively trap different even calcium isotopes and investigate frequency shifts. The thesis addresses the difficulties encountered and the strategies employed to overcome them in trapping $^{43}$Ca ions.
Additionally, we investigate the fluorescence of $^{43}$Ca ions. These investigations involve methods such as increasing the power of the electro-optic modulator (EOM) sidebands and optimizing the polarization of the 397 nm cooling beam. Through these efforts, we successfully achieved the capture of pure $^{43}$Ca ions, overcoming the challenge posed by their low natural abundance.

Abstract(Chinese)---I
Acknowledgements(Chinese)---II
Abstract---III
Contents---IV
List of Figures---VII
List of Tables---XII
1 Introduction...................... 1
1.1 Introduction to Trapped-Ion...................... 1
1.2 Reasons for Choosing 43Ca+ ...................... 2
1.3 Thesis Focus............................... 2
2 Ion Trapping...................... 4
2.1 Linear Trap ............................... 4
2.2 Laser Doppler Cooling ......................... 8
2.3 Sympathetic Cooling .......................... 9
3 Properties of Calcium .......................... 10
3.1 Photoionization of Calcium Atoms .................. 11
3.2 Energy Levels of Ca+ .......................... 13
3.3 Isotopes Frequency Shift ........................ 14
3.3.1 Energy Levels of 43Ca+ ..................... 14
4 Experiment Setup ..................... 16
4.1 The Ion Trap .............................. 16
4.2 Laser System .............................. 19
4.2.1 The Light source for photoionization . . . . . . . . . . . . . 19
4.2.2 The 397-nm Light Source for Laser Cooling . . . . . . . . . 20
4.2.3 The Light Source for Repumping ............... 20
4.3 Magnetic Field System......................... 22
5 Locking lasers to a stabilized cavity......................... 23
5.1 Stabilization Techniques and Signal Processing . . . . . . . . . . . . 24
5.1.1 Modulation Technique ..................... 24
5.1.2 Pound Drever Hall Technique ................. 25
5.1.3 Signal Processing ........................ 30
5.2 Stabilization System Setup....................... 32
5.2.1 852-nm Reference Laser Setup................. 32
5.2.2 Stabilizing Cavity........................ 37
5.2.3 Stabilizing 866-nm Laser.................... 40
5.2.4 Stabilizing 397-nm Laser.................... 41
5.3 Results.................................. 44
6 Isotope-selective trapping for Calcium ..................................46
6.1 Optimizing the Frequency of the photoionization Laser . . . . . . . 46
6.2 Ion Trapping of 40Ca+, 42Ca+ and 44Ca+ ............... 49
6.2.1 40Ca+ .............................. 49
6.2.2 44Ca+ .............................. 50
6.2.3 42Ca+ .............................. 51
6.3 Trapped 43Ca+ ............................. 53
6.3.1 Trapping Through Sympathetic Cooling . . . . . . . . . . . 54
6.3.2 Enhance 43Ca+ Fluorescence.................. 57
6.3.3 Direct Trapping......................... 59
7 Summary and Outlook ......................... 61
Bibliography......................... 62

[1] F. Arbes, M. Benzing, Th. Gudjons, F. Kurth, and G. Werth. Precise deter- mination of the ground state hyperfine structure splitting of43ca ii. Z Phys D - Atoms, Molecules and Clusters, 31:27–30, 1994.
[2] D. P. L. Aude Craik, N. M. Linke, M. A. Sepiol, T. P. Harty, J. F. Goodwin, C. J. Ballance, D. N. Stacey, A. M. Steane, D. M. Lucas, and D. T. C. Allcock. High-fidelity spatial and polarization addressing of 43Ca+ qubits using near- field microwave control. Phys. Rev. A, 95:022337, Feb 2017.
[3] Jan Benhelm. Precision spectroscopy and quantum information processing with trapped calcium ions. Ph.D. Thesis, University of Innsbruck, 2008.
[4] S. M. Brewer, J.-S. Chen, A. M. Hankin, E. R. Clements, C. W. Chou, D. J. Wineland, D. B. Hume, and D. R. Leibrandt. 27al+ quantum-logic clock with a systematic uncertainty below 10−18. Phys. Rev. Lett., 123:033201, Jul 2019.
[5] Colin D. Bruzewicz, John Chiaverini, Robert McConnell, and Jeremy M. Sage. Trapped-ion quantum computing: Progress and challenges. Applied Physics Reviews., 6, 6 2019.
[6] Y. Castin, H. Wallis, and J. Dalibard. Limit of doppler coolin. J. Opt. Soc. Am. B, 6, 1989.
[7] J. I. Cirac and P. Zoller. Quantum computations with cold trapped ions. Phys. Rev. Lett., 74:4091–4094, May 1995.
[8] Tyler B. Coplen, John Karl B ̈ohlke, P. De Bi`evre, T. Ding, N. E. Holden, J. A. Hopple, H. R. Krouse, A. Lamberty, H. S. Peiser, K. Revesz, S. E. Rieder, K. J. R. Rosman, E. Roth, P. D. P. Taylor, R. D. Vocke, and Y. K. Xiao. Isotope-abundance variations of selected elements (iupac technical report). Pure and Applied Chemistry, 74, May 2002.
[9] R.W.P. Drever, J.L. Hall, F.V. Kowalski, J. Hough, G.M. Ford, A.J. Mun- ley, and H. Ward. Laser phase and frequency stabilization using an optical resonator. Appl. Phys. B, 31, 1983.
[10] M. Drewsen and A. Brøner. Harmonic linear paul trap: Stability diagram and effective potentials. Phys. Rev. A, 62:045401, Sep 2000.
[11] S. Earnshaw. On the nature of the molecular forces which regulate the consti- tution of the luminferous ether. Transactions of the Cambridge Philosophical Society, 7:97, 1848.
[12] Christopher J. Foot. Atomic physics. Department of Physics of University of Oxford.
[13] S Gulde, D Rotter, P Barton, F Schmidt-Kaler, R Blatt, and W Hogervorst. Simple and efficient photo-ionization loading of ions for precision ion-trapping experiments. Applied Physics B, 73(8):861–863, 2001.
[14] T. P. Harty, M. A. Sepiol, D. T. C. Allcock, C. J. Ballance, J. E. Tarlton, and D. M. Lucas. High-fidelity trapped-ion quantum logic using near-field microwaves. Phys. Rev. Lett., 117:140501, Sep 2016.
[15] B.C. Keitch, N. R. Thomas, and D. M. Lucas. Injection locking of violet laser diodes with a 3.2 ghz offset frequency for driving raman transitions in 43Ca+. Opt. Lett., 38:830–832, 2013.
[16] Benjamin Keitch. A quantum memory qubit in calcium-43. Doctor Thesis, University of Oxford, 2007.
[17] Gerhard Kirchmair. Frequency stabilization of a titanium-sapphire laser for precision spectroscopy on calcium ions. Master Thesis, University of Inns- bruck, 2006.
[18] N. Kjaergaard, L. Hornekaer, and A. et al. Thommesen. Isotope selective loading of an ion trap using resonance-enhanced two-photon ionization. Appl Phys B, 71:207–210, 2000.
[19] D. M. Lucas, A. Ramos, J. P. Home, M. J. McDonnell, S. Nakayama, J.-P. Stacey, S. C. Webster, D. N. Stacey, and A. M. Steane. Isotope-selective photoionization for calcium ion trapping. Phys. Rev. A, 69:012711, Jan 2004.
[20] Ann-Marie M ̊artensson-Pendrill, Anders Ynnerman, H ̊akan Warston, Ludo Vermeeren, Roger E. Silverans, Alexander Klein, Rainer Neugart, Christoph Schulz, Peter Lievens, and The ISOLDE Collaboration. Isotope shifts and nuclear-charge radii in singly ionized 40−48Ca. Phys. Rev. A, 45:4675–4681, Apr 1992.
[21] C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland. Demonstration of a fundamental quantum logic gate. Phys. Rev. Lett., 75:4714–4717, Dec 1995.
[22] T.L. Nicholson, S.L. Campbell, R.B. Hutson, G.E. Marti, B.J. Bloom, R.L. McNally, W. Zhang, M.D. Barrett, M.S. Safronova, G.F. Strouse, W.L. Tew, and J. Ye. Systematic evaluation of an atomic clock at 2 ∗ 10−18 total uncer- tainty. Nature communications, 7(6896), 2015.
[23] NobelPrize.org. The nobel prize in physics 1989. Nobel Prize Outreach AB, Oct 1989.
[24] W. No ̈rtersha ̈user, K. Blaum, K. Icker, P. Mu ̈ller, A. Schmitt, K. Wendt, and B. Wiche. Isotope shifts and hyperfine structure in the transitions in calcium ii. Eur. Phys. J. D, 2:33–39, 1998.
[25] W. No ̈rtersha ̈user, N. Trautmann, K. Wendt, and B.A. Bushaw. Isotope shifts and hyperfine structure in the 4s2 1s0 → 4s4p1p1 → 4s4d 1d2 transitions of stable calcium isotopes and calcium-41,. Spectrochimica Acta Part B: Atomic Spectroscopy, 53:709–721, 1998.
[26] Wolfgang Paul. Electromagnetic traps for charged and neutral particles. Rev. Mod. Phys., 62:531–540, Jul 1990.
[27] Wolfgang Paul. Electromagnetic traps for charged and neutral particles. Rev. Mod. Phys., 62:531–540, Jul 1990.
[28] J. M. Pino, J. M. Dreiling, C. Figgatt, J. P. Gaebler, S. A. Moses, M. S. Allman, C. H. Baldwin, M. Foss-Feig, D. Hayes, K. Mayer, C. Ryan-Anderson, and B. Neyenhuis. Demonstration of the trapped-ion quantum ccd computer architecture. Nature, 592:209–213, Apr 2021.

[29] Daryl W. Preston. Doppler-free saturated absorption: Laser spectroscopy. American Journal of Physics, 64, 1996.
[30] Till Rosenband, DB Hume, PO Schmidt, Chin-Wen Chou, Anders Brusch, Luca Lorini, WH Oskay, Robert E Drullinger, Tara M Fortier, Jason E Stal- naker, et al. Frequency ratio of al+ and hg+ single-ion optical clocks; metrol- ogy at the 17th decimal place. Science, 319(5871):1808–1812, 2008.
[31] Stephan Schiller and Claus L ̈ammerzahl. Molecular dynamics simulation of sympathetic crystallization of molecular ions. Phys. Rev. A, 68:053406, Nov 2003.
[32] M Sepiol. A high-fidelity microwave driven two-qubit quantum logic gate in 43ca+. Doctoral Thesis, University of Oxford, 2016.
[33] Anders Sørensen and Klaus Mølmer. Entanglement and quantum computa- tion with ions in thermal motion. Phys. Rev. A, 62:022311, Jul 2000.
[34] Daniel A. Steck. Cesium d line data. available online at http://steck.us/alkalidata, 2019.
[35] Q. A. Turchette, Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland. Heating of trapped ions from the quantum ground state. Phys. Rev. A, 61:063418, May 2000.
[36] T.B.M. van Leet. Narrow-linewidth external cavity diode lasers for atomic physics. Thesis, 2017.