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Creating Extended Landau Levels of Large Degeneracy with Photons Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Kuan-Hao (Waylon) Chen, Physics, M.S. Graduate Program in Physics The Ohio State University 2018 Dissertation Committee: Prof. Tin-Lun Ho, Advisor Prof. Ilya Gruzberg Prof. Stuart Raby Prof. Rolando Valdes-Aguilar c Copyright by Kuan-Hao (Waylon) Chen 2018 Abstract Large degeneracy in Landau levels is a key to many quantum Hall phenomena. Geometric effects on quantum Hall states is another interesting problem that can probe the correlations in the quantum Hall states. A recent experiment has reported a result in creating the energy levels and the wave functions of Landau problem in a cone with photons. Based on their system, we generalize the scheme and discover a way to create extended degenerate levels with considerably larger degeneracy than that of the conventional Landau levels. To fully understand how to achieve this degenerate levels with photons, we also present the relevant topics in optics that are not familiar to condensed matter community to make it self-contained. The reason of this dramatically large degeneracy is that each degenerate level contains the whole spectrum of a Landau problem in a cone. In another words, we compress the spectrum of a two-dimensional system into one single energy. This considerably large degeneracy is expected to cause dramatic phenomena in quantum Hall and many-body physics. We suggest experimental measurements that could show this discovery. ii To the fifteen year-old boy that wants to be a physicist. iii Acknowledgments I am the most grateful to my advisor, Dr. Tin-Lun Ho. Every discussion with him ensouls my research. Countless efforts that he has put in to advise me and to shape me into a physicist, and most importantly, a responsible man are invaluable. I also would like to express my thanks to the committee. I thank Dr. Stuart Raby for the advice he gave in my first year when I studied high energy physics with him. I thank Dr. Ilya Gruzberg for numerous valuable discussions about my research and the excellent classes he offered to build my knowledge. I also thank Dr. Rolando Valdes-Aguilar for his input of ideas from an experimentalist's viewpoints every now and then. I appreciate the support and resources from the Department of Physics. My research would have been impossible without them. I especially would like to thank Dr. Jon Pelz for offering the financial support in my third year and this last semester when the funding of my research was short. I also thank the Ministry of Education Taiwan for the study abroad loan. I also thank the amazing superwoman Mrs. Kris Dunlap for taking care of every administrative affair for all physics graduate students. Many teacher's inspiration, encouragement and support are essential along my path in pursuing a PhD in physics. I thank my physics teacher at my fifteen, Mr. Ching-Peng Kuo, for the enlightenment and inspiration that made me start my jour- ney in physics. I thank Mrs. Chu-Fong Yen for the everlasting care, encouragement, iv and setting an example of dedication for me. I thank Dr. Jin-Tzu Chen for sharing his profound knowledge with the young mind and the great support when I applied for the PhD programs. I enjoyed scientific discussions and friendships with many colleagues. The discus- sions are always meaningful and fun, and sometimes they even turn out to be fruitful. I thank Dr. Joe McEwen, Bowen Shi and Alex Davis for the fun time we had in the geometry club in my first year. I thank Cheng Li, Jiaxin Wu, and James Roland for many valuable discussions about my research and ideas in physics. My first year in graduate school was not a total mess only because of the help and company from these people. I especially thank Dr. Xiaolin Zhu for the countless helps with my first year life in a new city. I thank Rachel Hsiao, Yun-Hao Hsiao, Michelle Lee and Zijie Poh for the friendship since my first year. I thank Dr. Fuyan Lu for selflessly sharing the information when I applied for the internship. To the amazing people I became close to after my fourth year in Columbus during my roughest time, your company is indispensable. I thank all fitness classes and the instructors in the RPAC and every amazing friend I made there that made my life wholesome. I thank Dr. Xianyu Yin, Hsiu-Chen Chang, Pei-Zu Hsieh, Dr. Kuyung- min Lee, and Dr. Kyusung Hwang for many good times together. In particular, I thank Jorge Torres, Derek Everett, Noah Charles and Mary-Frances Miller for every laughter and all the delicious food we have shared and the irreplaceable friendship. I also thank Dr. Chris Ehemann for the valuable internship experience and the amazing friendship. I am the most blessed person to be loved and cared for by all my dear friends, near and far. Love, support and company from them make my life meaningful. To v the dearest friends I knew since I was in Taiwan, I thank Kuo-Wei Chen and Tang Lee for a listening ear and sincere advice that are never absent, Yao-Yu Lin for the honest sharing of our cynical attitude to physics and life, Chi Lin for the most candid comments and sarcasm on things, Shao-Yu Chi for the lifelong friendship, and everyone in CK60th 109 class. I thank Alice Chi for her presence in my life, which has made me a different per- son, and unquestionably a better and happier one. I thank my parents for always trusting me, loving me and giving me the freedom to explore the world in my own way. I thank Kai-Wen Hsiao for starting this journey with me and ever taking the biggest part in my life. Washington D.C., November 22, 2018 Waylon Chen vi Vita November 21, 1989 . Born - Taipei, Taiwan 2012 . .B.S. Physics, Mathematics, The Ohio State University. 2016 . .M.S. Physics, The Ohio State University. 2013-present . .Graduate Associate, The Ohio State University. Fields of Study Major Field: Physics vii Table of Contents Page Abstract . ii Dedication . iii Acknowledgments . iv Vita......................................... vii List of Figures . xi 1. Introduction . .1 1.1 Motivation . .1 1.2 Overview . .3 2. Landau problem in cone and anti-cone surfaces . .6 2.1 Introduction . .6 2.2 Landau problem with electrons in cone surfaces . .7 2.2.1 The cone surface . .7 2.2.2 The Schr¨odinger'sequation and the solutions . .8 2.2.3 Gaussian curvature for cone surfaces . 12 2.3 Landau problem in anti-cone surfaces . 16 2.3.1 The Schr¨odinger'sequation and the solutions . 18 2.3.2 The response of the energy levels to the strength of curvature singularity . 20 3. The Landau levels of a rotating harmonic oscillator: the synthetic gauge 22 3.1 Introduction . 22 3.2 The quantum Hamiltonian for a uniformly rotating potential . 23 viii 3.3 A rotating quantum harmonic oscillator . 26 3.3.1 wave functions in polar coordinate . 26 3.3.2 the ladder operator approach . 31 3.4 the two labeling schemes of the spectrum . 35 4. Realization of rotating quantum harmonic oscillators with paraxial optics 37 4.1 Introduction . 37 4.2 Simulating the Schr¨odingerequation of massive particle using laser beam and generation of harmonic oscillator wave functions . 38 4.2.1 Description of laser beams in paraxial approximation . 38 4.2.2 Laser beams in a two-mirror cavity . 43 4.3 Realization of synthetic magnetic fields with the optical resonator . 48 4.3.1 paraxial ray optics . 49 4.3.2 paraxial wave optics . 59 4.3.3 Summary . 75 5. Generalized synthetic Landau levels with photons . 80 5.1 Introduction . 80 5.2 Simulating Landau levels on a cone with three-fold symmetry . 81 5.2.1 The energy levels . 81 5.2.2 Wave functions on a cone . 85 5.2.3 Detection of the level structure . 86 5.3 Create extended landau levels in a cone with dramatically larger degeneracy . 88 5.3.1 The energy levels and wave functions of Landau problem for general σ ............................ 89 5.3.2 Extend the Landau level degeneracy to simulate the energy level structure of a σ ! 1 anti-cone . 90 5.3.3 Detection of the extended level structure . 95 5.4 Characterizations of the states in the (χ1 = 2π=3; χ2 = 4π=3) de- generate manifold . 97 5.4.1 Finding all degenerate states in an extended level . 98 6. The Poincar´ehit patterns . 103 6.1 Introduction . 103 6.2 The hit patterns from the round-trip matrix of a photonic resonator 105 6.2.1 The formalism . 105 6.2.2 Hit patterns of the experimental resonator . 107 ix 6.3 The hit patterns from the Hamilton formalism: the stroboscopic evolution . 111 6.3.1 The Hamilton equations in matrix form . 111 6.3.2 Hamilton equations for a harmonic oscillator . 112 6.3.3 Hamilton equation for a rotating harmonic oscillator . 113 6.4 Study the rotational symmetry of the hit patterns with the Gouy phases . 118 6.4.1 The three copies of cyclotron motion in stroboscopic dynamics121 6.5 Using hit patterns to find the critical point for extended Landau levels125 Appendices 130 A. Landau problem on cone and anti-cone surfaces embedded in R3 ..... 130 A.1 The Landau problem with magnetic fields on a cone surface in R3 .
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