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Ground State Entanglement Entropy for Discrete-Time Coupled Harmonic Oscillators

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Ground State Entanglement Entropy for Discrete-Time Coupled Harmonic Oscillators GROUND STATE ENTANGLEMENT ENTROPY FOR DISCRETE-TIME COUPLED HARMONIC OSCILLATORS MR. WATCHARANON KANTAYASAKUN A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE (PHYSICS) FACULTY OF SCIENCE KING MONGKUT'S UNIVERSITY OF TECHNOLOGY THONBURI 2017 Ground State Entanglement Entropy for Discrete-Time Coupled Harmonic Oscillators Mr. Watcharanon Kantayasakun B.Sc. (Applied Physics) A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science (Physics) Faculty of Science King Mongkut's University of Technology Thonburi 2017 Thesis Committee ................................................................... Chairman of Thesis Committee (Lect. Watchara Liewrian, Ph.D.) ................................................................... Member and Thesis Advisor (Lect. Monsit Tanasittikosol, Ph.D.) ................................................................... Member and Thesis Co-Advisor (Asst. Prof. Sikarin Yoo-Kong, Ph.D.) ................................................................... Member (Lect. Thana Sutthibutpong, Ph.D.) ................................................................... Member (Asst. Prof. Pichet Vanichchapongjaroen, Ph.D.) Copyright reserved ii หัวข้อวิทยานิพนธ์ เอนโทรปีความพวั พนั ของคู่ตวั แกวง่ กวดั ฮาร์มอนิคสา หรับเวลาที่ไม่ ตอ่ เนื่อง หน่วยกิต 12 ผู้เขียน นายวชั รนนท ์ กนั ตยาสกุล อาจารย์ที่ปรึกษา ดร.มนตส์ ิทธ์ิ ธนสิทธิโกศล ผศ. ดร.สิขรินทร์ อยคู่ ง หลักสูตร วิทยาศาสตรมหาบัณฑิต สาขาวิชา ฟิสิกส์ ภาควิชา ฟิ สิกส์ คณะ วิทยาศาสตร์ ปีการศึกษา 2560 บทคดั ยอ่ ความพัวพันของสถานะพ้ืนของระบบท้งั ในกรณีเวลาที่ต่อเนื่องและเวลาไม่ต่อเนื่องถูกวดั โดยใชเ้ อน โทรปีเชิงเส้น ผลการศึกษาแสดงใหเ้ ห็นวา่ ความพวั พนั ของระบบจะเพ่มิ ข้ึนถา้ อตั รกิริยาระหวา่ งระบบ ย่อยน้ันเพ่ิมข้ึนท้ังสองกรณี นอกจากน้ียงั พบว่าความแข็งแรงของอัตรกิริยาของระบบย่อยกับ ส่ิงแวดล้อมน้ันส่งผลกับอัตราการเข้าสู่ความพัวพนั ของระบบแต่ส่ิงที่บ่งช้ีความแตกต่างของ พฤติกรรมความพัวพันของระบบที่เวลาไม่ต่อเนื่องกบั เวลาที่ต่อเนื่องคือ การมีอยขู่ องเงื่อนไขที่ทา ให้ ไมเ่ กิดการพวั พนั สา หรับระบบที่เวลาไม่ต่อเนื่อง ซ่ึงส่ิงน้ีเป็นตวั บ่งบอกวา่ พฤติกรรมความพวั พนั ของ ระบบไมส่ ามารถข้ึนไปถึงคา่ สูงสุดได ้ คาส าคัญ : ความพัวพัน/ เวลาที่ตอ่ เนื่อง / เวลาไมต่ ่อเนื่อง iii Thesis Title Ground State Entanglement Entropy for Discrete-Time Coupled Harmonic Oscillators Thesis Credits 12 Candidate Mr. Watcharanon Kantayasakun Thesis Advisors Dr. Monsit Tanasittikosol Asst. Prof. Dr. Sikarin Yoo-Kong Program Master of Science Field of Study Physics Department Physics Faculty Science Academic Year 2017 Abstract The ground state entanglement of the system, both in discrete-time and continuous- time cases, is quantified through the linear entropy. The result shows that the entanglement increases as the interaction between the particles increases in both time scales. It is also found that the strength of the harmonic potential affects the entanglement behaviour of the system. The different feature of the entanglement between continuous-time and discrete-time scales is that, for discrete-time entangle- ment, there is a cut-off condition. This condition implies that the system can never be in a maximally entangled state. Keywords : Continuous-Time/ Discrete-Time/ Entanglement iv ACKNOWLEDGEMENTS This thesis is supported by my thesis advisor, Dr. Monsit Tanasittikosol for editing and supporting my thesis writing. Next, I would like to thank another advisor, Asst. Prof. Dr. Sikarin Yoo-Kong for invaluable teaching and giving the motiva- tion in my thesis. Without their advices, this thesis would not have been completed. In addition, I am grateful to my thesis committee, Dr. Watchara Liewrian, Dr. Tanapat Deesuwan and Dr.Thana Sutthibutpong for the discussion and their help on numerical computation and Asst. Prof. Dr. Pichet Vanichchapongjaroen for his suggestion and help to improve the thesis. Moreover, I would like to thank Mr.Kittipong Aimsamer and Mr. Kittikun Surawut- tinack for his help to template fixing. Finally, I also thank every member in the lab \Theoretical and Computational Physics (TCP)" in KMUTT. v CONTENTS PAGE ABSTRACT IN THAI ii ABSTRACT IN ENGLISH iii ACKNOWLEDGMENTS iv CONTENTS v LIST OF FIGURES vii CHAPTER 1. INTRODUCTION 1 1.1 Motivation . .1 1.2 Structure of thesis . .2 2. THEORETICAL BACKGROUND 3 2.1 Lagrangian Mechanics . .3 2.1.1 Hamilton's principle . .3 2.1.2 Method to discretisation connect to discrete-time Euler-lagrange equation . .6 2.2 Systems of harmonic oscillator . .9 2.3 Quantum continuous-time harmonic oscillator . 10 2.4 Quantum discrete-time harmonic oscillator . 13 2.5 Chapter summary . 15 vi PAGE 3. Entanglement and Entanglement Measure 16 3.1 Entanglement . 16 3.2 Density matrix . 19 3.3 Bipartite systems . 22 3.4 Reduced Density Matrix . 22 3.5 Entanglement entropy . 24 3.6 Chapter summary . 26 4. Ground state entanglement for coupled harmonic oscillators 27 4.1 Continuous-time coupled harmonic oscillators . 27 4.2 Euler-Lagrange equation and equation of motion of continuous time 29 4.3 Lagrangian equation and Equation of motion of Discrete time . 31 4.4 Continuous-time quantum coupled harmonic oscillators . 32 4.5 Discrete-time quantum coupled harmonic oscillators . 34 4.6 Linear-entropy for continuous-time coupled harmonic oscillators . 39 4.7 Linear-entropy for discrete-time coupled harmonic oscillators . 40 4.8 Result and discussion . 41 4.9 Chapter summary . 43 5. CONCLUDING REMARKS 44 BIOGRAPHY 46 vii LIST OF FIGURES FIGURE PAGE 2.1 Example of infinitely many paths connecting between the initial point (qi; ti) and final point (qf ; tf ).......................4 2.2 The deformation of the path. .5 2.3 The geometrical implication of the derivative of a function. .6 2.4 The forward difference scheme of the function with respect to time. .7 2.5 Variation of discrete-time action. .8 2.6 The discrete-time flow. 13 3.1 The light cone structure at point A on 1+1 space time diagram. 17 3.2 Entanglement pair of two particles is separated. The positron is given to Bob and the electron is given to Alice. 18 4.1 The local coupled harmonics oscillators. 27 4.2 The contour plots of the potential function (a) σ = 0; k 6= 0 , (b) σ = 0:3; k = 0:1............................... 28 4.3 Two normal modes of motion. (a) Mode 1 is the center of mass motion. (b) Mode 2 is the relative motion. 30 4.4 Contours of the probability density for the ground state and first excited states. 34 4.5 Contours of the discrete-time probability density for the ground state and first excited states. 38 4.6 The relation between the linear entropy and the interaction (σ) with different amount of the discrete-time scale and the external interac- tion (k)................................... 42 CHAPTER 1 INTRODUCTION 1.1 Motivation The dynamics of a physical system is the study of the system that evolves in time. Originally introduced by Sir Issac Newton in Philosophisae Naturalis Prin- cipia Mathematica, the flow of time is stated as follow: \absolute time flows equably (continuous) without regard to anything external" [? ]. Therefore, it is natural to consider the continuous time flow in any physical systems one is interested, even, in quantum mechanics. In this thesis, the case of continuous-time flow is called the usual case. To physicists, time is the mysterious quantity and this leads to the main question of this thesis which is \can the time based on the idea of continuous-time flow be discrete?". Many physicists have proposed the idea that the continuous time flow constitutes from discrete-time steps [???? ]. In 1985, Carl M. Bender et al. constructed the discrete time lattice using the finite element method [? ]. Further- more, in 2007, Annick Lesne suggested that the discrete and continuous behaviours of time coexist in any natural phenomena depending on the scale of the observation [? ]. Recently, the latest study about the new phase of matter considered space and time being discrete, which was known as time crystal. This time crystal was suggested by Frank Wilczek [?? ]. There was an experiment to detect time crystal using ion trap [? ], confirming the theoritical idea proposed by Wilczeks[? ]. Then, Mir Faizal et al. showed that the deformation of Heisenberg uncertainty principle led to the discrete spectrum of time [? ]. Therefore, being motivated by all of these ideas, the question that this thesis would like to answer is \does the behaviours, such as the entanglement, of both classical and quantum systems remain the same for both discrete and continuous time flow, if not what is the extra features arising from having time being discrete?". To investigate this, the system of coupled harmonic oscillators is used as the model in this thesis. 2 1.2 Structure of thesis There are five chapters in this thesis. In chapter 2, we give the theoretical back- ground of Lagrangian mechanics of continuous time and discrete time. Then, we explain Schr¨odingermechanics in continuous-time and discrete-time systems. We explain the idea of quantum entanglement in chapter 3. In chapter 4, we try to answer the question arising from the motivation of this thesis by using the coupled harmonic oscillators and then its discrete-time Schr¨odingerequation and its solution are computed. Then, the discrete-time wave function is used to calculate the linear entropy for measuring the entanglement behaviour of the system. The last chapter summarises and discusses the results obtained in this thesis. CHAPTER 2 THEORETICAL BACKGROUND In this chapter, we explain about the theoretical background of this thesis. We start with
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