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.)

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 entanglement, 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 motivation 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 interaction (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 background of Lagrangian mechanics of continuous time and discrete time. Then, we explain Schrödingermechanics 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 Schrödingerequation 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 the Lagrangian mechanics and derive the Euler-Lagrange equation. Then, we present the scheme of quantization of the system through the Schr¨odingerwave equation. We also present the time discretisation on both Lagrangian mechanics and Schr¨odingermechanics.

2.1 Lagrangian Mechanics

In 1687, Sir Isaac Newton introduced the law of motion and they successfully pre- dicted the state of motion of the object. However, Newton’s laws of motion involves vector quantities, then in some situations, it is diﬃcult to deal with. Later, alter- native method Lagrangian Mechanics had been developed by Euler, Lagrange and Hamilton. This method, in some situations, provides a much easier than Newton’s method to obtain the equation of motion since the Lagrangian function contains only the scalar functions, namely, kinetic energy and potential energy. Moreover, one can show that the Euler-Lagrange equation can be derived directly from the Newton’s second law through the D’Alembert’s principle.

2.1.1 Hamilton’s principle

In this section, we start with the derivation of the Euler-Lagrange equation from the variational principle. We consider two point on the q − t diagram, where q and t are generalised coordinates and time. Let (qi, ti) and qf , tf be the initial point and the final points, we have an infinitely many paths connecting between the initial point and the final point as shown in Fig. 2.1. Therefore, the question is the following: which path does the particle actually take when subjected to the external force. The answer to the question comes from the Hamilton’s principle of the least action [? ]. 4

The Hamilton’s principle of least action states that the classical path taken by the particle between (qi, ti) and (qf , tf ) is the one for which the action is extremum.

Figure 2.1 Example of inﬁnitely many paths connecting between the initial point

(qi, ti) and ﬁnal point (qf , tf ).

Therefore, the action functional is deﬁned as

Z tf S[q] = L(q, q˙ , t)dt , (2.1) ti where q = {q1, q2, ..., qj, ..., qn} and q˙ = {q˙1, q˙2, ..., q˙j, ..., q˙n} are the sets of generalised coordinates and velocities. With Eq. (2.1), the action is extremum if

δS = 0, (2.2) where δS = S[q + δq] − S[q] represents an inﬁnitesimal change in the action with respect to the change in the function q(t) 7→ q(t) + δq(t), as shown in Fig. 2.2 5

Figure 2.2 The deformation of the path.

We ﬁnd that the action S[q + δq] can be written as follow

Z tf S[q + δq] = L(q + δq, q˙ + δq˙ , t)dt ti n n ! Z tf X ∂L X ∂L = L + δq + δq˙ dt, (2.3) j ∂q j ∂q˙ ti j=1 j j=1 j where the Lagrangian L is expanded upto ﬁrst order using Taylor’s expansion. Ap- plying the condition given by Eq. (2.3), where Taylor’s expansion is used to convert the ﬁrst line into the second line. Thereafter we apply the least action principle, i.e., Eq. (2.2), one gets

n n ! Z tf X ∂L X ∂L 0 = δq + δq˙ dt j ∂q j ∂q˙ ti j=1 j j=1 j " n n n # Z tf X ∂L X d ∂L X d ∂L 0 = δq + δq − δq dt j ∂q dt j ∂q˙ j dt ∂q˙ ti j=1 j j=1 j j=1 j n n Z tf X ∂L d ∂L X ∂L tf 0 = δqj − dt + δqj . ∂q dt ∂q˙ ∂q˙ t ti j=1 j j j=1 j i

We apply the condition δqj(ti) = δqj(tf ) = 0 which comes from the fact that the end points are ﬁxed and cannot be changed. Therefore, we get the Euler-Lagrange 6 equation ∂L d ∂L − = 0 , j = 1, 2, 3, ..., n . (2.4) ∂qj dt ∂q˙j With Eq. (2.4), to obtain the equation of motion one needs to identify the La- grangian function which may be not so trivial. In the next section, we move onto the case of discrete-time variational principle and obtain the discrete-time Euler-Lagrange equation.

2.1.2 Method to discretisation connect to discrete-time Euler-lagrange equation

In this thesis, difference the method of discretisation is called finite difference method. We start this section with the idea of the derivative of functions. A derivative represents the rate of change of a function f(t) with respect to time t as shown in Fig. 2.3

Figure 2.3 The geometrical implication of the derivative of a function.

According to Fig. 2.3, we calculate the function at three points ti, ti+1 and ti−1.

The ﬁrst points ti represents the time at the present (ti = t). The second point ti+1 represents the subsequent time in 1 unit of step (ti+1 = t + ). The third point ti−1 represents the earlier time in 1 unit of step (ti−1 = t − ). In calculus, ∆t is inﬁnitesimally close to zero and the derivative of a function with respect to time is given by df ∆f = lim . (2.5) dt ∆t→0 ∆t 7

On the other hand, if we discard the limit ∆t → 0 in Eq. (2.5), the derivative becomes the diﬀerence equation, which is

df ∆f ≈ . (2.6) dt ∆t

Figure 2.4 The forward diﬀerence scheme of the function with respect to time.

Fig. 2.4 shows the rate of change of a function with respect to time in the case of the forward diﬀerence scheme. From Fig. 2.4 the rate of change of the function with respect to the variable t between ti and ti+1 is given by

∆f f − f f − f = i+1 i = i+1 i . (2.7) ∆t ti+1 − ti

The second derivative can be discretised as

d2f d df f − 2f − f = = i+2 i+1 i (2.8) d2t dt dt 2

In this thesis, we discretise the equations with the forward diﬀerence method. Then we deﬁne the new notations for convenienct purpose, namely, f(t) = f(ti) = f is the function at initial time. ˜ f(t + ) = fi+1 = f is the function shifted by 1 step in time. ˜ f(t + 2) = fi+2 = f is the function shifted by 2 step in time. With this convention, Eq. (2.7) and Eq. (2.8) can be written as

df f˜− f ≡ , (2.9) dt 8 and d2f f˜− 2f˜− f ≡ . (2.10) d2t 2 The next step is to construct the discrete-time Euler-Lagrange equation based on the forward diﬀerence method. This can be done by discretising the Lagrangian and then perform the variational Calculus [?? ]. In order to construct the discrete- time Lagrangian, we need three points in conﬁguration space for variational method. Then we consider three points q, q˜ and q˜ shown in Fig. 2.5, whereq ˜ ≡ q(t + ) and q˜ ≡ q(t + 2) are the generalised coordinates shifted in the discrete-time direction by one discrete-time step and two discrete-time steps 2, respectively.

Figure 2.5 Variation of discrete-time action.

As shown in Fig. 2.5, the solid line represents the action of the classical path which connects three points and the dashed line represents the variation of the action from the classical path. In the discrete-time case, one cannot continuously vary the path as the system is discrete. What can be done is to vary the point between the two ends, in this particular case it isq ˜ ≡ q(t + ). With Fig. 2.5 the discrete-time action is deﬁned as [?? ] S = L(q, q˜) + L0(˜q, q˜), (2.11) and S0 = L(q, q˜ + δq˜) + L0(˜q + δq,˜ q˜). (2.12)

Using The Taylor’s expansion of functional, Eq. (2.12) becomes ∂L (δq˜)2 ∂2L ∂L0 (δq˜)2 ∂2L0 S0 = L(q, q˜) + δq˜ + + ... + L0(˜q, q˜) + δq˜ + + ... ∂q˜ 2! ∂2q˜ ∂q˜ 2! ∂2q˜ 9

With the condition that the action is extremum (δS = 0), we get ∂L ∂L0 δq˜ (q, q˜) + (˜q, q˜) = 0 (2.13) ∂q ∂q˜

From Eq. (2.13), the Discrete-time Euler-Lagrange equation [?? ] is given by ∂L ∂L0 (q, q˜) + (˜q, q˜) = 0 (2.14) ∂q ∂q˜ The Discrete-time Euler-Lagrange equation is used to determine the discrete-time equation of motion. In the next section, we focus on the classical discrete-time of a harmonic oscillator system.

2.2 Systems of harmonic oscillator

In this section, we consider a discrete-time harmonic oscillator system. The Hamil- tonian of the discrete-time harmonic oscillator is chosen such that [? ] p˜2 1 H(x, p˜) = T (˜p) + V (x) = + ω2x2, (2.15) 2 2 where ω is the angular frequency of the oscillator andp ˜ is used here, rather than p, this is because we would like to get the discrete map which will be shown later in Eq. (2.18). As a consequence, the discrete-time Hamilton’s equations can be written in this form p˜ − p ∂H(x, p˜) = − = −ω2x, (2.16) ∂x and x˜ − x ∂H(x, p˜) = =p. ˜ (2.17) ∂p˜ Eq. (2.16) and Eq. (2.17) can be rearranged into

x˜ = (1 − ω22)x + p , andp ˜ = −ω2x + p . (2.18)

We call Eq. (2.18) as the discrete maps. With this discrete map, the right hand side of the equations represents the initial values of position and momentum, leading to the new values of shifted in the direction of time. This is the reason that in this thesis we choose to work with the discrete Hamiltonian given by Eq. (2.15). Eliminating the momentum p from Eq.(2.18), we obtain 2ω2 x˜ + x = 2 1 − x.˜ (2.19) 2 10

Under the continuum limit ( → 0), we get the standard equation of motion of the harmonic oscillator.

2.3 Quantum continuous-time harmonic oscillator

To derive the quantum dynamics of a particle in the harmonic trap, we solve for the wave function Ψ from the Schr¨odingerequation given by

∂Ψ(x, t) 2 ∂2Ψ(x, t) i = − ~ + V (x)Ψ(x, t), (2.20) ~ ∂t 2m ∂2x where V (x) is the potential energy of the system. The Schr¨odingerequation can be solved by using the technique of the separation of variables. Then the wave function Ψ(x, t) can be written as, Ψ(x, t) = ψ(x)T (t). (2.21)

We substitute Eq. (2.21) into Eq. (2.20) then we get

1 dT (t) 2 1 d2ψ(x) i = − ~ + V (x). (2.22) ~T (t) dt 2m ψ(x) d2x

Equating both sides of Eq. (2.22) to constant E which is the total energy of the system; therefore, we have

2 1 d2ψ(x) − ~ + V (x)ψ(x) = E, (2.23) 2m ψ(x) d2x and 1 dT (t) i = E. (2.24) ~T (t) dt Eq. (2.23) is known as the time-independent Schr¨odingerequation. For the har-

1 2 monic potential energy V (x) = 2 mω x, Eq. (2.23) is now 2 d2ψ(x) 1 − ~ + mω2x2ψ(x) = Eψ(x), (2.25) 2m d2x 2 where ω = pk/m is the angular frequency of the oscillation. Introducing the p dimensionless variable ξ = ω/~ x, the Eq. (2.25) can be written as

d2ψ = ξ2 − K ψ, (2.26) dξ2 where K = 2E/~ω. We consider the limit, ξ → ∞, Eq. (2.26) reduces to d2ψ ≈ ξ2 ψ, (2.27) dξ2 11 which has the solution 2 2 − ξ ξ ψ(ξ) ≈ Ae 2 + Be 2 , (2.28) where A and B are arbitrary constants. With Eq. (2.28), the second term is not normalizable because it blows up when ξ → ∞. Thus, the physical solution is given by 2 − ξ ψ(ξ) = H(ξ)e 2 , (2.29) where H(ξ) is a polynomial with non-negative power. Eq. (2.29) is the solution of Eq. (2.26). Substituting this solution into Eq.(2.26), we get

H00(ξ) − 2ξH0(ξ) + (K − 1)H = 0. (2.30)

We propose that the solution is in the power series form

∞ X n H(ξ) = Cnξ , (2.31) n=0

∞ 0 X n−1 H (ξ) = nCnξ , (2.32) n=0 ∞ 00 X n H (ξ) = (n + 1) (n + 2) Cn+2ξ . (2.33) n=0 Substituting Eq. (2.31), (2.32), (2.33) into Eq. (2.30), we have

∞ X n [(n + 1) (n + 2) Cn+2 − 2nCn + (K − 1) Cn] ξ = 0. (2.34) n=0

With Eq. (2.34), the basis set {1, ξ1, ξ2, ξ3, ...., ξn} is linearly independent. This implies that

(n + 1) (n + 2) Cn+2 − 2nCn + (K − 1) Cn = 0. (2.35) which leads to the recursion formula

2n − K + 1 C = C . (2.36) n+2 (n + 1) (n + 2) n

With the Eq. (2.36), the even coeﬃcient is Ceven = P1C0 and the odd coeﬃcient is Codd = P1C1, where P1 and P2 are some constants. Then Eq. (2.31) can be rewritten in the form

H(ξ) = Heven(ξ) + Hodd(ξ). (2.37) 12

However, not all of the solutions are normalizable. If we consider for the case of very large n, Eq. (2.36) becomes P C ≈ , (2.38) n (n/2)! for some constant P . Eq. (2.31) becomes

X 1 X 1 2 H(ξ) ≈ P ξn ≈ P ξ2n ≈ P eξ . (2.39) (n/2)! n! Then H(ξ) becomes exp(ξ2) which leads to the unphysical solution. Therefore, the series must be terminated. To do this one requires that 2n − K + 1 = 0 in Eq. (2.36). The energy is given by 1 E = ω(n + ) , n = 0, 1, 2, 3, .... (2.40) ~ 2 For n = 0, there is the only one term in the series

H0(ξ) = C0 (2.41) and thus 2 − ξ ψ0(ξ) = C0e 2 . (2.42)

For n = 1, we have

H1(ξ) = C1ξ, (2.43) and hence 2 − ξ ψ1(ξ) = C1ξe 2 . (2.44)

For n = 2, we get C2 = −2C0 and C4 = 0. So,

2 H2(ξ) = C0(1 − 2ξ ), (2.45) and 2 2 − ξ ψ2(ξ) = C0(1 − 2ξ )e 2 . (2.46)

In general cases, the wave function which corresponds to each energy level is given by 2 − ξ ψn(ξ) = CnHn(ξ)e 2 , (2.47) where Hn is Hermite polynomials. We can calculate Cn from the normalization Z ∞ ∗ ψn(x)ψn(x)dx = 1, (2.48) −∞ 13

1 2 Z ∞ ~ 2 2 −ξ2 Cn Hn(ξ)e dξ = 1. (2.49) ω −∞ Using the orthogonality relation for the Hermite polynomials

Z ∞ −ξ2 √ n Hn(ξ)Hm(ξ)e dξ = π2 n!δmn, (2.50) −∞ We obtain 1 ω 4 1 Cn = √ . (2.51) π~ 2nn! Using Rodrigues formula

n 2 ∂ 2 H (ξ) = (−1)n eξ e−ξ , (2.52) n ∂ξn the wave function of harmonic oscillator is given by

1 r ω 4 1 ω − ω x2 ψn(x) = √ Hn x e 2~ . (2.53) π~ 2nn! ~ Finally, we obtain the normalised wave function for each energy level for the quantum harmonic oscillator. In the next section, we will move to the discrete-time version presented in this section.

2.4 Quantum discrete-time harmonic oscillator

We will construct the discrete-time version of the Schr¨odingerequation for quantum harmonic oscillator. We follow the construction which has been introduced in [?? ]. We begin to imagine that the continuous-time ﬂow constitutes from many tiny discrete steps. Then in quantum case, the system evolves in discrete-time steps via the unitary operator shown in Fig. 2.6

Figure 2.6 The discrete-time ﬂow.

What we have now is the time evolution of wave function

Uˆ()Ψ(x, t) = Ψ(˜˜ x, t˜) = Ψ(x(t + ), t + ), (2.54) 14 where Uˆ() is the time evolution operator and given by [?? ]

− i T (ˆp) − i V (ˆx) − i H0 Uˆ() = e ~ e ~ = e ~ , (2.55) where T (ˆp) and V (ˆx) are the kinetic and potential energies of system and H0 is modiﬁed Hamiltonian. Thep ˆ andx ˆ are the momentum and position operators and satisfy the commutation relation

[ˆx, pˆ] =x ˆpˆ − pˆxˆ = i~. (2.56)

To obtain the modiﬁed Hamiltonian, Eq. (2.55) can be expanded by using the Baker-Campbell-Hausdorﬀ series

1 i H0 = T (ˆp) + V (ˆx) − [T (ˆp),V (ˆx)] + ..... (2.57) 2 ~ and ω2 iω2 [T (ˆp),V (ˆx)] = pˆ2, xˆ2 = − ~ (ˆxpˆ +p ˆxˆ) . (2.58) 4 2 Then Eq. (2.57) can be written in the new form as

pˆ2 1 ω2 H0 = + ω2xˆ2 − (ˆxpˆ +p ˆxˆ) + ..... (2.59) 2 2 4

We choose three terms in Eq. (2.59) to be the discrete-time Hamiltonian operator since it is invariant under discrete-map, namely, Eq. (2.16) and Eq. (2.17). This mean that I˜ˆ = I,ˆ (2.60) where I is the discrete-time modiﬁed Hamiltonian operator given by

ω2 Iˆ =p ˆ2 + ω2xˆ2 − (ˆxpˆ +p ˆxˆ) . (2.61) 2

With Eq. (2.61) the discrete-time Schr¨odingerequation of quantum harmonic oscillator is

d2 d i IˆΨ (x) = − 2 + iω2 x + ω2x2 + ~ω2 Ψ (x) = E Ψ (x). (2.62) n ~ dx2 ~ dx 2 n n n

2 2 To simplify Eq. (2.62), we introduce Ψn(x) = wn(x)exp [iω x /(4~)] and substitute it into Eq. (2.62). Then we obtain

d2w (x) − 2 n + Ω2x2w (x) = E w (x). (2.63) ~ dx2 n n n 15

The angular frequency of the discrete-time version is

1 2ω2 2 Ω = ω 1 − . (2.64) 4

We use the same method from Section (2.3) to solve for the function wn(x). There- fore, the discrete-time wave function of the harmonic oscillator is given by

1 ! 4 r 2 Ω 1 Ω ( iω − Ω )x2 Ψ(x) = √ Hn x e 4~ 2~ , (2.65) π~ 2nn! ~ and its corresponding energy level is

1 E = 2 Ω n + . (2.66) n ~ 2

Under the continuum limit → 0, Eq. (2.65) and Eq. (2.66) reduce into the standard quantum harmonic oscillator as discussed in Section. (2.3).

2.5 Chapter summary

In this chapter, we start with the some necessary background on the standard La- grangian mechanics. Then, we show the idea how to construct the discrete-time Lagrangian mechanics. After that we aim to solve the wave function and energy of quantum harmonic oscillator system for both continuous and discrete time. In the next chapter, we give the idea of quantum entanglement and how to quantify the entanglement behaviour of system. CHAPTER 3 Entanglement and Entanglement Measure

In this chapter, we start with the explanation of the original idea of quantum entanglement called EPR paradox. Then, we explain about the density matrix and reduced density matrix. We also will explain about the bipartite system. The end of this chapter, we discussed the method for measure the entanglement behaviour of the bipartite system. The content of this chapter is based on the reference [15-17].

3.1 Entanglement

The Entanglement is one of the surprising properties of nature existing only the quantum realm. This property comes from the seminal paper of Einstein, Podolsky and Rosen in 1935. In this paper, they proposed a thought experiment to show that quantum mechanics is not complete to describe physical reality which later known as EPR paradox [? ]. The EPR thought experiment model is based on two assump- tions [? ]

1. Reality Principle : The properties of physical system are independent from observation and measurement.

2. Locality Principle : According to theory of relativity, if we consider two events A and B are separated by the interval: 4s2 = c2 4 t2 − 4x2 in 1 + 1 space-time diagram. For 4s2 > 0, called time-like interval, this implies that the event A and B are related to the causality connection. For 4s2 < 0, called space-like, this implies that the event A and B are not related with the causality connection. With this statement it suggests that if two systems are causally disconnected the result of any measurements perform on ﬁrst system cannot aﬀect to the result of a measurement performed on the second system. 17

Figure 3.1 The light cone structure at point A on 1+1 space time diagram.

The Fig. 3.1 shows two event connected causally. The events B and C are in the future region of event A. Then A can communicate with B and C. On the other hand, D is not in the future region of A. Then A and D cannot causally communicate.

According to Copenhagen interpretation in quantum mechanics, the wave function is completely speciﬁed by the physical system and the uncertainty principle states that two non-commuting operators of physical quantities, e.g., position and momentum cannot both be exactly measured at the same time. However, Einstein did not happy with these two statements because he stood on the reality and locality principles. To further discuss Einstein’s argument which is a magical seed for quantum entanglement, let us consider a source that produces a pair of electron and positron moving in opposite direction with initial conditions x2 − x1 = x0 and p1 + p2 = 0, where x1 and p1 are the position and momentum of the positron and x2 and p2 are the position and momentum of the electron. Then, we consider the situation that the two particles are separated with a large distance as shown in Fig. 3.2. 18

Figure 3.2 Entanglement pair of two particles is separated. The positron is given to Bob and the electron is given to Alice.

The wave function of two-particle system is represented by

∞ Z i (x1−x2+x0)p Ψ(x1, x2) = e ~ dp. (3.1) −∞

i x1p Let up(x1) = e ~ be the eigenstate of momentum of the positron. If Bob chooses to observe the momentum (p1) of positron on this the wave function in Eq. (3.1). Then, the eigenvalue equation is given by

p1up(x1) = pup(x1). (3.2)

With the result from Eq. (3.2), the momentum of the positron is p. The Eq. (3.1) can be written in form as Z ∞ Ψ(x1, x2) = φp(x2)up(x1)dp, (3.3) −∞

i (x0−x2)p where φp(x2) = e ~ is the eigenstate of momentum of the electron. According to the initial condition on momentum p1 + p2 = 0, if Bob finds that the momentum of positron is p Bob communicates through classical channel with Alice about his result. Then, Alice will definitely find that her momentum of electron is −p without measurement. The next step is that the EPR state in Eq. (3.1) can also be written in the form of

∞ ∞ Z Z i (x−x2+x0)p Ψ(x1, x2) = e ~ dp δ(x1 − x)dx −∞ −∞ Z ∞ = φx(x2)ux(x1)dx, (3.4) −∞ where ux(x1) = δ(x1 − x) and φx(x2) = δ(x − x2 + x0) are eigenstates of position. If

Alice measures position (ˆx2 ≡ x2) of her electron, the eigenvalue equation is given by

xˆ2φx(x2) = (x + x0)φx(x2). (3.5) 19

Then, Alice obtains the position of the electron x + x0. According to the initial condition x2 − x1 = x0, if Alice ﬁnd that the position of the electron is x + x0 and communicates with Bob about her result, Bob knows that the exact position of the positron is x without performing the measurement. Obviously, the quantum theory states that position and momentum do not commute, however the EPR thought experiment suggested that the non-commuting physical quantities can be simulta- neously measured. This leads to the conclusion that if quantum theory follows the reality statement, the position and momentum could exist before measuring but Bob and Alice do not know the values of the position and momentum simultane- ously from the direct measurement?. Therefore, in order to make the argument consistent with the reality principle, there must be a hidden variable. Thus, if we accept the principle local-realism, this leads to the contradiction in quantum theory because the wave function is not a complete description of the physical system.

Then, it was pointed out that the Copenhagen interpretation of quantum theory did not agree with the EPR’s locality and reality. In 1964, John Bell [? ] showed that the prediction in the quantum theory about the EPR paradox does not have the same form as that of the Einstein’s assumption or classical picture. This may suggest that the local-realism was incorrect. Then, the EPR paradox does not contradict with the quantum theory. Rather, it suggests a fundamental feature existing in quantum level called the entanglement.

3.2 Density matrix

A quantum mechanical system may be described by a wave function or a state vector in the form of linear combination of basis vectors. Suppose |ψi is the state vector. This state vector can be written in terms of the orthonormal basis vectors |ii as

X |ψi = ci|ii, (3.6) i

P 2 where i |ci| = 1. Now, suppose we have a linear Hermitian operator M whose eigenstates are {|ii} where

M|ii = mi|ii, (3.7) 20

The probability of obtaining the outcome mi when the state |ψi is measured by M is given by

2 2 Pi = |hi|ψi| = |ci| . (3.8)

In the experiment, the source producing the quantum state does not construct the pure state because of the decoherence of the system. This mixture of pure states is called mixed state. In order to describe such a quantum system one can employ the density matrix. Suppose a quantum system is consisted of a set of pure states

{|ψii}, each with probability weight pi (note that pi 6= Pi since they are classical P distribution). We would call {pi, |ψii} an ensemble of pure state with and i=1 pi = 1. Therefore, we can write the state as

n X ρ ≡ pi|ψiihψi|. (3.9) i This representation is called the density matrix or density operator. If the quantum state |ψi is pure state, Eq. (3.9) reduces to the form ρ = |ψihψ|. A density operator must have the following properties [? ] Pn 1. Unity trace : this comes from the deﬁnition that i=1 pi = 1.

2. Hermicity : we can expand each pure state |ψii in term of a set of orthonormal basis vectors |ki as M X i |ψii = ck|ki, (3.10) k=1 PM i 2 where i=1 |ck| = 1. Then the element (k, j) of the density matrix with respect to the basis set {|ki} is given by

This result represents the hermicity because after performing the hermitian conju- gate density matrix does not change. 3. Positivity : all eigenvalues of density matrix ρ cannot be negative number. This comes from the fact that every probability pi cannot be negative. 4. Trρ2 ≤ 1: Let

X m |ψmi = cj |ji. (3.11) j the square of density matrix is given by

We take trace at both sides of the square of density matrix as

2 X m ∗ n m n ∗ Tr(ρ ) = pmpn(ci ) (ci )(ck )(cl ) Tr(|kihl|), (3.12) m,n,i,k,l with the property Tr(|kihl|) = δkl. Then Eq. (3.12) becomes

2 X m ∗ n m n ∗ Tr(ρ ) = pmpn(ci ) (ci )(ck )(ck ) , (3.13) m,n,i,k or it can be written in the form

2 X 2 Tr(ρ ) = pmpn|hψm|ψni| . (3.14) m,n,i,k

Using Schwarz inequality, Eq. (3.14) becomes

2 X Tr(ρ ) ≤ pmpnhψm|ψmihψn|ψni, m,n,i,k X X ≤ pm pn, m n ≤ 1,

2 since 0 ≤ pi ≤ 1. This leads to the condition that Trρ = 1 is for pure state and Trρ2 < 1 is for mixed state. 22

3.3 Bipartite systems

From the previous subsection we explained the language to describe quantum mechanical system. We use the linear combination of all possible states for a single particle or a single mode which is related to the Hilbert space H1. In general, we can construct the quantum system in a more complicated way when it is applied to multiple particles. Therefore, the state vector lives in a more complicated space resulting from the tensor product of all Hilbert spaces related to each particle, i.e.

H = H1 ⊗ H2 ⊗ H3.... In this thesis, we focus on the two-particle system technically called the bipartite system.

The Hilbert space of bipartite system is given by

H = HA ⊗ HB, (3.15)

where HA and HB are Hilbert spaces of the ﬁrst and the second particle. So, the new bases of the new space can be constructed from tensor product of linear combination of basis on each space. Therefore, the new bases are given by

{|1i ⊗ |1i, |1i ⊗ |2i, |2i ⊗ |1i, |2i ⊗ |2i, .....}, (3.16) where {|1i, |2i, ..., |ii, ...} and {|1i, |2i, ..., |ji, ...} are orthonormal basis vectors of

HA and HB. According to these bases, a state vector can be written in the form of

X |ψiAB = cij|ii ⊗ |ji. (3.17) ij

3.4 Reduced Density Matrix

A state vector of a multi-particle system contains all information about the whole system. However, there are many cases where we are interested in only part of the system. This leads to the deﬁnition of the reduced density matrix, which is the representation of subsystem. The reduced density matrix can be found from partial trace on the density matrix of system. In this case, we use the bipartite system to show how to compute the reduced density matrix. 23

For the bipartite system, we have two particle system A and B. The reduced density matrix for subsystem A is given by

ρA ≡ TrB (ρAB) , (3.18) where ρAB is a density matrix of the bipartite system and TrB (ρAB) is the partial trace over the subsystem B. The density matrix of the bipartite system is given by

X ∗ ρA = cij(ckl) (|iihk|) ⊗ Tr (|jihl|) , i,j,k,l X ∗ = cij(ckl) (|iihk|) ⊗ (hl|ji) , i,j,k,l X ∗ = cil(ckl) (|iihk|) . i,k,l The information from the reduced density matrix shows the subtle relation between the entanglement and the mixed state. The reduced density matrix of the pure bipartite state is the mixed state only when the former state is entangled. To demonstrate this result, we use an example of a purely entangled Bell state as it is maximally entangled state of two qubits [? ]. Therefore, the density matrix of Bell state is given by |00i + |11i h00| + h11| ρAB = √ √ , 2 2 = (|00ih00| + |00ih11| + |11ih00| + |11ih11|) /2.

Then, we take the partial trace over the second system

ρA = TrB(ρ), 1 = [(|0ih0|Tr(|0ih0|) + |0ih0|Tr(|1ih1|)] + 2 1 + [|1ih1|Tr(|0ih0|) + |1ih1|Tr(|1 >< 1|)] , 2 1 = (|0ih0| + |1ih1|) . 2 From this result, the reduced density matrix of subsystem 1 is the mixed state because it is equally composed the pure state of |0i and |1i. Therefore, this method can be used to check whether the pure state is entangled or not. 24

3.5 Entanglement entropy

In the previous subsection, we discuss about the density matrix and use density matrix to calculate the reduced density matrix of the bipartite system. For pure state, the reduced density matrix can be used to identify whether the system is in entangled state or not. However, we can say nothing about the magnitude of the entanglement. In this subsection we use the density matrix to compute the entropy called Von Neumann entropy and linear entropy because both entropies can quantify the entanglement behaviour of the system. Suppose that we have ensemble of states

{ρ1 = |ψ1ihψ1|, ρ2 = |ψ2ihψ2|, ρ3 = |ψ3ihψ3|, ....., ρn = |ψnihψn|} which are related to a set of probability {p1, p2, p3, ...., pn}. The density matrix can be written in the form (see section 3.2) n n X X ρ = pi|ψiihψi| = piρi, (3.20) i=1 i where all of the states belong to the Hilbert space of dimension n. We now deﬁne the von Neumann entropy as

n X S(ρ) = −Tr(ρlnρ), ⇔ S(ρ) = − pilnpi. (3.21) i=1

With Eq. (3.21), we observe that when the minimum entropy Smin(ρ) = 0, the state is pure. This implies that the density matrix have only one eigenvalue. On the other hand, if p1 = p2 = p3..... = pn = 1/n the entropy is maximum: Smax(ρ) = lnn.

Let us now consider the Bell state, from the previous subsection. With the Bell state, we use von Neumann entropy to measure the entanglement of the bipartite system. According to the measurement, we observe that the entanglement between the subsystems is related to the entropy computed from the reduced density matrix. With this example, one can show that the entropies of the both subsystems are equal, leading to the entanglement of the both subsystems are the same. Therefore, we now introduce the Bell state (maximally entangled state) as

1 |Ψi = √ (|0i|1i + |1i|0i). (3.22) 2 25

And, the density matrix is given by 1 ρ = |ΨihΨ| = [(|0ih0| ⊗ |1ih1|) + (|0ih1| ⊗ |1ih0|)] AB 2 1 + [(|1ih0| ⊗ |0ih1|) + (|1ih1| ⊗ |0ih0|)] . 2

With this density matrix, The reduced density matrix of subsystem 1 is

ρA = TrBρAB, 1 = (|0ih1| + |1ih1|), 2 so we get the diagonal matrix which contains the same eigenvalues. This suggests that the maximum entropy. To claim this statement, one ﬁnds the von Neumann entropy of the subsystem 1 as

SA = −Tr(ρAlnρA) 1 1 1 1 = − log ( ) − log ( ) 2 2 2 2 2 2 = ln2

= 0.63.

In general case, the entropy of system is between 0 and 1, namely, 0 ≤ S ≤ lnn. With the result from the example, the entropy is equal to lnn. In this case, the result suggests that the magnitude of the entanglement is at its maximum value. On the other hand, the von Neumann entropy is zero for a separated state. Furthermore, if we ﬁnd the partial trace over the ﬁrst system, we get the reduced density matrix of subsystem 2, which is the same as that mentioned earlier.

There is another way to measure the entanglement called the linear entropy. This is a simply version of von Neumann entropy. With the Eq. (3.21) and the expansion of ln(x) = (x − 1) + (x − 1)2/2 + ....., the von Neumann entropy becomes

S = −T r(ρlnρ) = T r ρ ρ − 1 + (ρ − 1)2/2 + .....

2 2 ' −T r(ρ − ρ) = 1 − T rρ ≡ SL

The linear-entropy is bounded between 0 and 1: 0 ≤ SL ≤ 1. If the linear-entropy is one the particles are in the maximally entangled state. If the linear-entropy is zero the quantum correlation between the two particles vanishes. 26

3.6 Chapter summary

In the present chapter, we start with the idea of the entanglement from the EPR paradox and then we discuss about representation of quantum system via the density matrix and use the density matrix to compute the entropy in order to quantify the entanglement. In the next chapter, we will discuss about the coupled harmonic oscillators of discrete-time and continuous-rime for both classical and quantum system. CHAPTER 4 Ground state entanglement for coupled harmonic oscillators

In this chapter, w e explain about the setting of a system of coupled harmonic oscillators and then w e construct the Euler-Lagrange equation and Hamilton equation in the continuous-time regime to ﬁnd the equation of motion with the normal modes of motion. Then w e are interested in investigating the behaviour of the system on both continuous and discrete time. W e aim to study the quantum behaviour on discrete-time ﬂow and compare the results of the behaviours in discrete and continuous time. T o study the entanglement behaviour, w e use a linear-entropy as a test to quantify the entanglement between the ground state of the quantum continuous- time and discrete-time coupled harmonic oscillators. A t the end of this chapter, the entanglement behaviour of system in both time scales is discussed.

4.1 Continuous-time coupled harmonic oscillators

The system is composed of t w o identical particles with a unit mass and three springs. Two particles are attached to one another b y a spring with a spring constant σ. Each particle is attached to the w a l l through the springs, with spring constant k as shown in Fig. 4.1.

Figure 4.1 The local coupled harmonics oscillators.

The Hamiltonian of the system is given b y p2 p2 1 1 1 H(p , p , x , x ) = 1 + 2 + kx2 + kx2 + σ(x − x )2, (4.1) 1 2 2 2 2 2 21 22 2 1 2 where the ﬁrst t w o terms represents the kinetic energy for each particle. The third and the fourth terms are the interaction with the environment. And the last term 28 is the interaction between the particles themselves.

Figure 4.2 The contour plots of the potential function (a) σ = 0, k 6= 0 , (b) σ = 0.3, k = 0.1.

In Fig. 4.2 (a) the interaction between both particles vanishes (σ = 0), whereas both of them couple with the environment (k 6= 0). In this case, the potential terms in Eq. (4.1) reduces to a paraboloid surface giving rise to circular contours. On the other hand, when the coupling between both particles is non zero, the term

1 2 2 σ(x1 − x2) couples the variables x1 and x2 causing the stretching and tilting of the circular contours. This transforms the circular contours into the tilted ellipses.

To decouple x1 and x2, a new set of coordinates are now introduced such that

x1 + x2 X1 = √ , (4.2) 2 x1 − x2 X2 = √ , (4.3) 2 where X1 and X2 are a new set of coordinates. With this new coordinates, the Hamiltonian (4.1) becomes P 2 ω2X2 P 2 ω2X2 H(P ,P ,X ,X ) = 1 + 1 1 + 2 + 2 2 , (4.4) 1 2 1 2 2 2 2 2 where p1 + p2 P1 = √ , (4.5) 2 p1 − p2 P2 = √ , (4.6) 2 √ √ are the momenta. The ω1 = k and ω2 = k + 2σ are two normal modes of the motion. The advantage of the Hamiltonian (4.4) is that the centre of mass coordinates and the relative coordinates are decoupled. 29

4.2 Euler-Lagrange equation and equation of motion of continuous time

In this section, we start with writing the Lagrangian of coupled harmonic oscillators, which is given by

X˙ 2 X˙ 2 ω2X2 ω2X2 L(X , X˙ ,X , X˙ ) = 1 + 2 − 1 1 − 2 2 . (4.7) 1 1 2 2 2 2 2 2

Then the Euler-Lagrange equation of the coupled harmonic oscillators come from variational principle. With the Eq. (4.7), the Euler-Lagrange equation is given by

∂L d ∂L − = 0, (4.8) ˙ ∂X2 dt ∂X2 and ∂L d ∂L − = 0. (4.9) ˙ ∂X1 dt ∂X1 On the other hand, the Hamilton’s equation for coupled harmonic oscillators are given by

∂H ˙ = X1 = P1, ∂P1 ∂H ˙ 2 = −P1 = ω1X1, (4.10) ∂X1 and

∂H ˙ = X2 = P2, ∂P2 ∂H ˙ 2 = −P2 = ω2X2. (4.11) ∂X2 Obviously, the results from Eq. (4.10), Eq. (4.11) and Eq. (4.9), (4.8) lead to the equation of motion of the coupled harmonic oscillators. Therefore, the equation of motion of this system is given by

¨ 2 X1 + ω1X1 = 0, (4.12)

¨ 2 X2 + ω2X2 = 0. (4.13)

The solution of Eq. (4.12) and Eq. (4.13) are

X1 = x1 + x2 = C1 sin(ω1t + φ1), (4.14)

X2 = x1 − x2 = C2 sin(ω2t + φ2), (4.15) 30

where C1,C2, φ1 and φ2 are arbitrary constants that are determined by the four initial conditions, x1(0), x2(0), x˙ 1(0), x˙ 2(0). We see that x1(t) can be obtained by adding Eq. (4.14) and Eq. (4.15), while x2(t) can be obtained by subtracting Eq. (4.14) and Eq. (4.15) giving

x1(t) = C1 sin(ω1t + φ1) + C2 sin(ω2t + φ2), (4.16)

x2(t) = C2 sin(ω1t + φ1) − C2 sin(ω2t + φ2), (4.17)

With the Eq. (4.16) and Eq. (4.17), if we chose C2 = 0 then we have

x1 = x2 = C1 sin(ω1t + φ1), (4.18)

so both particles move coherently in the same direction with the frequency ω1 ( The middle spring is never stretched ). On the other hand, if we chose C1 = 0 then we have

x1 = −x2 = C2 sin(ω2t + φ2), (4.19) the two particles move incoherently in the opposite directions, both inward and outward with frequency ω2 because the middle spring is stretched or compressed, see Fig. 4.3

Figure 4.3 Two normal modes of motion. (a) Mode 1 is the center of mass motion. (b) Mode 2 is the relative motion.

The solution in the Eq. (4.18) and Eq. (4.19) represent the motion of system. For each mode of motion is related to the normal frequency ω1 and ω2. The next section, we discuss about the discrete map coming form the discrete-time Lagrangian and discrete-time Euler Lagrange equation. 31

4.3 Lagrangian equation and Equation of motion of Discrete time

In this section, we introduce the Lagrangian of discrete-time system and construct the discrete Euler-Lagrange equation. In order to solve for the equation of motion of the coupled harmonic oscillators, we substitute the discrete-time Lagrangian function to the discrete-time Euler-Lagrange equation. Under the continuum limit → 0, the continuous-time equation of motion is recovered.

We introduce the discrete-time Lagrangian

(X˜ − X )2 ω2X2 (X˜ − X )2 ω2X2 L(X˜ , X˜ ,X ,X ) = 1 1 − 1 1 + 2 2 − 2 2 , (4.20) 1 2 1 2 22 2 22 2 where X1(t) and X2(t) are positions of the particles at the current time t and ˜ ˜ X1 ≡ X1(t + ), X2 ≡ X2(t + ) are the shift in positions in discrete-time direction. With the discrete Euler equations, we have

∂L ∂L0 0 = + , (4.21) ˜ ˜ ∂X2 ∂X2 and ∂L ∂L0 0 = + , (4.22) ˜ ˜ ∂X1 ∂X1 0 0 ˜ ˜ ˜ ˜ where L ≡ L (X1, X2, X1, X2). The equation of motion in discrete-time is given by

X˜ − 2X˜ + X 2 2 2 + ω2X = 0, (4.23) 2 2 2

X˜ − 2X˜ + X 1 1 1 + ω2X = 0. (4.24) 2 1 1 Under the continuum limit → 0, Eq. (4.23) and Eq. (4.24) yield back the standard equations of the motion of harmonic oscillators. On the other hand, we introduce the discrete-time Hamiltonian as

˜ ˜ ˜2 2 2 ˜2 2 2 H(P1, P2,X1,X2) = P1 + ω1X1 /2 + P2 + ω2X2 /2 , (4.25)

where P1(t), P2(t) and X1(t), X2(t) are the discrete-time momenta and positions ˜ of the particles at a given time t. The shifted momenta are P1 = P1(t + ) and 32

˜ P2 = P2(t + ), where is the discrete-time step. The discrete-time Hamilton equations are

∂H X˜ − X = 1 1 = P˜ , ˜ 1 ∂P1 ˜ ! ∂H P1 − P1 2 = − = ω1X1, (4.26) ∂X1 and

∂H X˜ − X = 2 2 = P˜ , ˜ 2 ∂P2 ˜ ! ∂H P2 − P2 2 = − = ω2X2, (4.27) ∂X2

The pair of Eq. (4.26) and Eq. (4.27) can be written in the form of

˜ 2 2 ˜ 2 X1 = (1 − ω1 )X1 + P1 , and P1 = −ω1X1 + P1 , (4.28) and ˜ 2 2 ˜ 2 X2 = (1 − ω2 )X2 + P2 , and P2 = −ω2X2 + P2 . (4.29)

Eq. (4.28) and Eq. (4.29) are called discrete maps. In the next section, we study the continuous-time system of quantum coupled harmonic oscillators.

4.4 Continuous-time quantum coupled harmonic oscillators

In this section, we study the quantum system of the coupled harmonic oscillators. We begin with the Schr¨odingerequation of the coupled harmonic oscillators. Then, we use the wave function from the Schr¨odingerequation to calculate the probability density of the system.

Using Eq. (4.4), we can promote the Hamiltonian of the system into the operators. Therefore, the continuous-time Hamiltonian of the system can be written as ˆ ˆ ˆ ˆ2 2 2 ˆ2 2 2 H(P1, P2,X1,X2) = P1 + ω1X1 /2 + P2 + ω2X2 /2. (4.30)

On the other hand, the Hamiltonian in Eq. (4.30) can be separated into two parts

ˆ ˆ ˆ H = H1 + H2, (4.31) 33

ˆ ˆ where H1 and H2 are the Hamiltonians of the ﬁrst particle and the second particle, respectively. The Schr¨odingerequation of the system can be written in the form of eigenvalue equation

ˆ ˆ ˆ H(P1, P2,X1,X2)Ψ(X1,X2) = EΨ(X1,X2), (4.32)

where E = E1 + E2 is total energy of system. The wave function of system can be separated as Ψ(X1,X2) = ϕ(X1)φ(X2). Since the Hamiltonian operator is consisted of two parts, we may write Eq. (4.32)

ˆ H1ϕ(X1) = E1ϕ(X1), (4.33)

ˆ H2φ(X2) = E2φ(X2). (4.34)

Writing Eq. (4.33) in diﬀerential form, we have

2 2 2 −~ d ϕ(X1) ω1 2 2 + X1 ϕ(X1) = E1ϕ(X1). (4.35) 2 dX1 2 With the Eq. (4.35) has the same form as that of the quantum oscillator in the Chapter 2. Thus we can use the same method to ﬁnd the wave function and the energy of system. The wave function of centre of mass is given by

1 r ω1 4 1 ω1 − ω1 X2 ϕn(X1) = √ Hn X1 e 2~ 1 , (4.36) π~ 2nn! ~ with the corresponding energy of

1 E = ω (n + ), n = 0, 1, 2, 3, .... (4.37) 1 ~ 1 2

By the same token Eq. (4.34) can be solved and the associated wave function is given by 1 r ω2 4 1 ω2 − ω2 X2 φm(X2) = √ Hm X2 e 2~ 2 , (4.38) π~ 2mm! ~ and the corresponding energy to the wave function is

1 E = ω (m + ), m = 0, 1, 2, 3, .... (4.39) 2 ~ 2 2

Therefore, the total wave function of system is given by

1 1 r r ω1 4 ω2 4 1 1 ω1 ω2 Ψnm(X1,X2) = √ √ Hn X1 Hm X2 π~ π~ 2nn! 2mm! ~ ~ h ω1 2i h ω2 2i × exp − X1 exp − X2 . (4.40) 2~ 2~ 34

The probability density for each energy level is given by

1 1 r r 2 ω1 2 ω2 2 1 1 2 ω1 2 ω2 |Ψnm| = Hn X1 Hm X2 π~ π~ 2nn! 2mm! ~ ~ −ω1 2 −ω2 2 × exp X1 exp X2 . (4.41) ~ ~ Fig. 4.4 shows the probability contours for the ground state and the ﬁrst excited state

Figure 4.4 Contours of the probability density for the ground state and ﬁrst excited states.

The energy of system is 1 1 E = ω (n + ) + ω (m + ), (4.42) ~ 1 2 ~ 2 2 where n, m are energy levels. In the next section we study about the discrete-time quantum case of the coupled harmonic oscillators.

4.5 Discrete-time quantum coupled harmonic oscillators

In this section, we study the discrete-time system mentioned in Section. 4.4. The standard method is to start by writing the discrete-time Schr¨odinger equation. In this present work, we may start by considering the time evolution of the wave function ˆ ˜ ˜ ˜ U()Ψ(X1,X2, t) = Ψ(X1, X2, t˜), (4.43) where ˜ ˜ ˜ Ψ(X1, X2, t˜) = Ψ(X1(t + ),X2(t + ), t + ), (4.44) 35 and

i i − T (P1,P2) − V (X1,X2) Uˆ() = e ~ e ~ . (4.45) when T (P1,P2) and V (X1,X2) in Eq. (4.45) are the kinetic and potential energies for centre of mass and relative coordinates, the time evolution operator can be possibly written in the form ˆ ˆ ˆ U() = UX1 ()UX2 (), (4.46) where

i i ˆ − T (P1) − V (X1) UX1 () = e ~ e ~ , (4.47)

i i ˆ − T (P2) − V (X2) UX2 () = e ~ e ~ . (4.48) Then Eq. (4.43) becomes

˜ ˜ ˜ ˜ ˜ ˆ ˆ ˜ ˜ ˜ ˜ ϕ˜X1 (X1, t)φX2 (X2, t) = UX1 ()ϕX1 (X1, t)UX2 ()φX2 (X2, t) = Ψ(X1, X2, t). (4.49)

The time evolution operator in Eq. (4.47) and Eq. (4.48) may be written as

i i i 0 − T (P1) − V (X1) − H ˆ ~ X1 UX1 () = e ~ e ~ = e , (4.50)

i i i 0 − T (P2) − V (X2) − H ˆ ~ X2 UX2 () = e ~ e ~ = e , (4.51) we expand term exp [−iT1/~] and exp [−iV1/~] in equation (4.50) with respect to discrete step " 2 3 # i −i 1 −i 2 1 −i 3 exp − T1 = 1 + T1 + T1 + T1 + ... , (4.52) ~ ~ 2! ~ 3! ~ and " 2 3 # i −i 1 −i 2 1 −i 3 exp − V1 = 1 + V1 + V1 + V1 + ... . (4.53) ~ ~ 2! ~ 3! ~ Inserting Eq. (4.52) and (4.53) into Eq. (4.50) and (4.51) and using the Baker- CampbellHausdorﬀ formula [? ], the modiﬁed Hamiltonian becomes " 2 3 # i 0 −i 1 −i 2 1 −i 3 exp − H1 = 1 + T1 + T1 + T1 + ... ~ ~ 2! ~ 3! ~ " 2 3 # −i 1 −i 2 1 −i 3 × 1 + V1 + V1 + V1 + ... ~ 2! ~ 3! ~ 2 2 3 1 −i 1 −i 2 1 −i 3 = 1 + T1V1 + T1 + V1 + 2! ~ 2! ~ 3! ~ 3 3 4 1 −i 2 1 −i 2 1 −i 3 + T1V1 + T1 V1 + T1V1 + 2! ~ 2! ~ 3! ~ 4 1 −i 2 2 + T1 V1 + ...... (4.54) 2!2! ~ 36

Taking the natural logarithm on both sides of Eq. (4.54), we get

−iH0 1 = ln(1 + A), (4.55) ~ where A contains all terms on the right-hand-side of Eq. (4.54), excluding 1. We expand the right hand side of Eq. (4.55) by using Taylor’s series

0 2 2 2 −iH1 i i −i 1 −i 1 −i = − V1 − T1 + T1V1 − V1T1 − T1V1 + ..... ~ ~ ~ ~ 2! ~ 2! ~ (4.56) Then,

0 2 2 −iH1 i i 1 −i 1 −i = − V1 − T1 + T1V1 − V1T1 + ..... (4.57) ~ ~ ~ 2! ~ 2! ~ We will get 0 1 −i H1 = V1 + T1 + [T1V1 − V1T1] + ..... (4.58) 2! ~

From the relation [T1,V1] = T1V1 − V1T1 and then 0 1 −i H1 = T1 + V1 + [T1,V1] + ..... (4.59) 2! ~

ˆ2 2 2 Using T1 = P1 /2 and V1 = ω1X1 /2, we obtain " # Pˆ2 ω2X2 ω2 h i [T ,V ] = 1 , 1 1 = 1 Pˆ2,X2 . (4.60) 1 1 2 2 4 1 1

Using the relations h i h i h i AˆB,ˆ Cˆ = Aˆ B,ˆ Cˆ + A,ˆ Cˆ B.ˆ (4.61) h i h i h i A,ˆ BˆCˆ = Bˆ A,ˆ Cˆ + A,ˆ Bˆ C.ˆ (4.62)

Eq. (4.60) can be rearranged such that

ω2 h i h i [T ,V ] = 1 Pˆ Pˆ , Xˆ 2 + Pˆ , Xˆ 2 Pˆ 1 1 4 1 1 1 1 1 1 ω2 h i h i h i h i = 1 Pˆ Xˆ Pˆ , Xˆ + Pˆ , Xˆ Xˆ + Xˆ Pˆ , Xˆ + Pˆ , Xˆ Xˆ Pˆ 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ω2 h i h i h i h i = 1 Pˆ Xˆ Pˆ , Xˆ + Pˆ Pˆ , Xˆ Xˆ + Xˆ Pˆ , Xˆ Pˆ + Pˆ , Xˆ Xˆ Pˆ 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ω2 h i = 1 Pˆ , Xˆ Pˆ Xˆ + Xˆ Pˆ 2 1 1 1 1 1 1 −iω2 = 1~ Pˆ Xˆ + Xˆ Pˆ . (4.63) 2 1 1 1 1 37

Therefore, Eq. (4.59) becomes 1 −ω2 H0 = Pˆ2 + ω2Xˆ 2 + 1 Pˆ Xˆ + Xˆ Pˆ + ...... (4.64) 1 2 1 1 1 2 1 1 1 1 From Eq. (4.64), the new Hamiltonian is given by ω2 I = Pˆ2 + ω2Xˆ 2 − 1 Pˆ Xˆ + Xˆ Pˆ , (4.65) 1 1 1 1 2 1 1 1 1 since is small so that the higher order terms in Eq. (4.64) can be neglected and this new Hamiltonian I is invariant under the mapping Eq. (4.28):

˜ I1 = I1. (4.66)

Eq. (4.65) is then treated as the modiﬁed Hamiltonian. Therefore, the discrete-time Schr¨odingerequation is given by

2 ˆ 2 d 2 d 2 2 i~ 2 I1ϕ(X1) = −~ 2 + iω1~X1 + ω1X1 + ω1 ϕ(X1) = E1ϕ(X1). (4.67) dX1 dX1 2

2 2 Introducing an auxiliary wave function as ϕ(X1) = w(X1)exp [iω1X1/(4~)], Eq. (4.67) reduces to that of the quantum oscillator. We obtain

2 2 d w(X1) 2 2 −~ 2 + Ω1X1 w(X1) = E1w(X1), (4.68) dX1

where 1 2ω2 2 Ω = ω 1 − 1 , (4.69) 1 1 4 which is the angular frequency of the ﬁrst mode Eq. (4.68) has the same mathematical form as that of Eq. (4.35). Therefore, the eigenfunction of Eq. (4.68) is given by 1 ! 4 r Ω1 1 Ω1 − Ω1 X2 w(X1) = √ Hn X1 e 2~ 1 . (4.70) π~ 2nn! ~ Substituting Eq. (4.70) back into the auxiliary wave function, this results in

1 2 r ! iω Ω Ω 4 1 Ω 1 − 1 X2 1 1 4~ 2~ 1 ϕ(X1) = √ Hn X1 e . (4.71) π~ 2nn! ~

Eq. (4.71) is the discrete-time wave function of the ﬁrst normal mode. Using the same method to calculate the discrete-time wave function of the second normal mode, we obtain the discrete-time wave function as

1 2 r ! iω Ω Ω 4 1 Ω 2 − 2 X2 2 2 4~ 2~ 2 φX2 = √ Hm X2 e . (4.72) π~ 2mm! ~ 38

With Eq. (4.71) and the Eq. (4.72), the full discrete-time wave function of the quantum coupled harmonic oscillators can be written as

1 1 ! ! 4 4 r r Ω1 Ω2 1 1 Ω1 Ω2 Ψnm = √ √ Hn X1 Hm X2 π~ π~ 2nn! 2mm! ~ ~ 2 2 iω1 Ω1 2 iω2 Ω2 2 × exp − X1 × exp − X2 . (4.73) 4~ 2~ 4~ 2~ The total energy is

1 1 E = 2 Ω n + + 2 Ω m + , (4.74) nm ~ 1 2 ~ 2 2 where n, m = 0, 1, 2, 3, .... and the discrete-time probability density is given by

1 1 2 2 r r 2 Ω1 Ω2 1 1 2 Ω1 2 Ω2 |Ψnm| = Hn( X1)Hm( X2) π~ π~ 2nn! 2mm! ~ ~ Ω1 2 Ω2 2 × exp − X1 exp − X2 . (4.75) ~ ~ According to Fig. 4.5, the probability contours for the ground state and the ﬁrst excited state are shown.

Figure 4.5 Contours of the discrete-time probability density for the ground state and ﬁrst excited states. 39

According to Fig. 4.5, the probability of the discrete-time wave function is a little bit broader than that of the continuous-time wave function in Fig. 4.4. This results p from the fact that both the Hermite Polynomials Hy( Ωi/~Xi) and the exponential 2 terms exp [−ΩiXi /(2~)] contain the discrete-time parameter . From this result, we may interpret the effect of the discrete-time parameter that in the same point on the probability density contour. In the discrete-time case we have the possibly to find the particle which different form the continuous-time case. In the next section, we using the discrete-time wave function to compute the linear entropy.

4.6 Linear-entropy for continuous-time coupled harmonic oscillators

We use linear entropy to quantify the entanglement behaviour of coupled harmonic oscillators system. In this section, we show the method to compute the linear entropy in continuous-time case. We use the continuous-time wave function from the Eq. (4.40).

1 1 r r ω1 4 ω2 4 1 1 ω1 ω2 Ψnm(X1,X2) = √ √ Hn X1 Hm X2 π~ π~ 2nn! 2mm! ~ ~ h ω1 2i h ω2 2i × exp − X1 exp − X2 . (4.76) 2~ 2~

Therefore, the ground state wave function in coordinate x1 and x2 is

1 1 2 2 ω1 4 ω2 4 − ω1 (x1+x2) − ω2 (x1−x2) Ψ00(X1,X2) = e 2~ 2 e 2~ 2 . (4.77) π~ π~ From this wave function, we can construct the density matrix:

0 0 ∗ 0 0 ρ(x1, x2; x1, x2) = Ψ00(x1, x2)Ψ00(x1, x2). (4.78)

This is the pure state in the space of x1 and x2. If we do not make any measurements 0 in x1 space, we can construct the density matrix ρ(x2, x2) by integrating over x1: Z 0 0 ρ(x2, x2) = ρ(x1, x2; x1, x2)dx1, (4.79)

1 2 γ − β γ 2 02 β 0 − (x +x ) x2x = e 2~ 2 2 e ~ 2 . (4.80) π~ 40

1 1 where γ = 4 (ω1 + ω2) + ω1ω2/ω1 + ω2 and β = 4 (ω1 + ω2) − ω1ω2/ω1 + ω2. We then compute Tr(ρ2) to ﬁnd the linear entropy S. The result is Z 2 0 0 0 Tr(ρ ) = ρ(x2, x2)ρ(x2, x2)dx2dx2, (4.81) γ − β = , (4.82) pγ2 − β2 and the continuous-time linear entropy is deﬁned as

γ − β S = 1 − Tr(ρ2) = 1 − . (4.83) pγ2 − β2

In the next section, we calculate the linear entropy of the discrete-time case.

4.7 Linear-entropy for discrete-time coupled harmonic oscillators

In this section, we use the linear entropy to quantify the entanglement behaviour of the coupled harmonic oscillators system on discrete-time case. We start with the discrete-time wave function of quantum coupled harmonic oscillators which is given by

1 1 ! ! 4 4 r r Ω1 Ω2 1 1 Ω1 Ω2 Ψnm = √ √ Hn X1 Hm X2 π~ π~ 2nn! 2mm! ~ ~ 2 2 iω1 Ω1 2 iω2 Ω2 2 × exp − X1 × exp − X2 . (4.84) 4~ 2~ 4~ 2~

With Eq. (4.84), the ground state wave function in coordinate x1 and x2 is

1 1 4 4 iω2 2 iω2 2 Ω1 Ω2 ( 1 − Ω1 ) (x1+x2) +( 2 − Ω2 ) (x1−x2) Ψ00 = e 4~ 2~ 2 4~ 2~ 2 . (4.85) π~ π~ The density matrix is deﬁne as

0 0 ∗ 0 0 ρ(x1, x2; x1, x2) = Ψ00(x1, x2)Ψ00(x1, x2), 1 (Ω Ω ) 2 1 2 02 1 0 0 1 2 − (Ω1+Ω2)(x +x ) − (Ω1−Ω2)2(x1x2+x x ) = e 4~ 1 1 e 4~ 1 2 π~ 1 2 02 i 2 2 2 02 i 2 2 0 0 − (Ω1+Ω2)(x +x ) (ω +ω )(x −x ) (ω −ω )2(x1x2−x x ) × e 4~ 2 2 e 8~ 1 2 1 1 e 8~ 1 2 1 2

i (ω2+ω2)(x2−x02) × e 8~ 1 2 2 2 . (4.86) 41

Without proof, it should be noted that Eq. (4.86) is pure density matrix. Then, we

0 can construct the discrete-time density matrix ρ(x2, x2) as Z 0 0 ρ(x2, x2) = ρ(x1, x2; x1, x2)dx1,

1 (Ω Ω ) 2 1 2 02 i 2 2 2 02 1 2 − (Ω1+Ω2)(x +x ) (ω +ω )(x −x ) = e 4~ 2 2 e 8~ 1 2 2 2 π ∞ ~ Z 1 2 1 0 i 2 2 0 − (Ω1+Ω2)2x − (Ω1−Ω2)2(x2+x )x1 (ω −ω )2(x2−x )x1 × e 4~ 1 e 4~ 2 e 8~ 1 2 2 dx1. −∞ (4.87)

We consider the integration term of Eq. (4.87) by exploiting the Gaussian integral and then the discrete-time density matrix is given by

1 ∗ ∗ 2 ∗ ∗ (Ω −Ω ) (γ − β ) −γ 2 02 β 0 i (ω2+ω2)− i 1 2 (ω2−ω2) (x2−x02) 0 2 (x2+x2 )+ (x2x2) 8 1 2 8 (Ω +Ω ) 1 2 2 2 ρ(x2, x2) = √ e ~ ~ e ~ ~ 1 2 , π~ (4.88) 2 2 2 2 2 2 2 2 ∗ 1 Ω1Ω2 (ω1 −ω2 ) ∗ 1 Ω1Ω2 (ω1 −ω2 ) where γ = (Ω1 +Ω2)+ + and β = (Ω1 +Ω2)− + . 4 Ω1+Ω2 16(Ω1+Ω2) 4 Ω1+Ω2 16(Ω1+Ω2) In this case, we calculate Tr(ρ2) with Eq. (4.81). Therefore, Z 2 0 0 0 Tr(ρ ) = ρ(x2, x2)ρ(x2, x2)dx2dx2, (4.89)

∞ ∞ ∗ ∗ ∗ ∗ Z Z (γ − β ) γ 2 02 2β 0 − (x2+x2 )+ (x2x2) 0 = e ~ ~ dx2dx2, (4.90) −∞ −∞ π~ γ∗ − β∗ = . (4.91) p(γ∗)2 − (β∗)2

The linear entropy of discrete-time system is given by

γ∗ − β∗ SL = 1 − . (4.92) p(γ∗)2 − (β∗)2

Then, the results from Eq.(4.83) and Eq. (4.92) are used the physical meaning which is discussed to interpret in the next section.

4.8 Result and discussion

We study a quantum property called entanglement for the two coupled harmonic oscillators through the linear entropy. From the section 4.6 and 4.7, we get the linear entropy of coupled harmonics oscillators in Eq. (4.83) and Eq. (4.92). Therefore, we 42

Figure 4.6 The relation between the linear entropy and the interaction (σ) with diﬀerent amount of the discrete-time scale and the external interaction (k).

using both linear-entropy in continuous-time and discrete-time to plotting as shown in figure. According to Fig. 4.6, in the continuous time(solid lines), the entanglement of the system at the ground state increases as the interaction between particles σ increases, while the interaction with environment k is fixed. The system approaches to the maximally entangled state SL → 1 as the parameter σ approaches to infinity implying that the oscillation mode Ω1 (the center of mass motion) significantly dominates over the oscillation mode Ω2 (the relative motion). We also find that when the parameter k increases, the entanglement will rise more slowly with the increasing value of the parameter σ. This means that the oscillation mode Ω2 becomes more significant with increasing k which then makes the oscillation mode

Ω1 more difficult to overcome the oscillation mode Ω2. In the case that the parameter k is infinitely large, the entanglement of the system is extremely suppressed due to the domination of the oscillation mode Ω2. We may now say that less relative motion (the oscillation mode 2) of the system implies more entanglement. In the discrete time case, the entanglement of the system behaves almost the same with the continuous time case. Except that we cannot freely vary the values of the parameter σ and the parameter k since there are the cut-off conditions coming from

2 the fact that both Ω1 and Ω2 must be positive values. This implies that 0 ≤ ω2 < 2 2 4/ since ω2 ≥ ω1. In terms of σ, this will give the inequality 0 ≤ σ < 2/ − k/2 which also implies that 0 ≤ k < 4/2. Both k and σ cannot satisfy their respective 43 upper bounds (k = 4/2 and σ = 2/2 − k/2) because that will cause the wave function (4.84) to vanish which means the state does not exist (implying that the motion of the system cannot be in any oscillation modes). If k > 4/2 (which implies

2 σ > 2/ − k/2) the oscillation frequencies Ω1 and Ω2 will become imaginary and the wave function is now not well define. This is the reason that k ≥ 4/2 and σ ≥ 2/2 − k/2 present unphysical situations and have to be excluded from the our consideration. In the physical situations, if we fix the value of the parameter k, the entanglement of the system will increase as the parameter σ increases and the entanglement will only asymptotically approach 1, but never reaches 1, before the parameter σ gets to the cut-off point σ = 2/2 − k/2. Increasing the value of the parameter k will suppress the entanglement of the system like those in the continuous time case.

4.9 Chapter summary

In this chapter, we have already shown the set up of coupled harmonic oscillators system and identify the mode of motion of system. To shown the method to dis- cretize equation from continuous-time to discrete-time, we introduce discrete-time Lagrangian and Hamiltonian of coupled harmonic oscillators which lead to discrete maps. Then, we study the behaviour of the system for both time scales and focus on the entanglement behaviour of the system. The entanglement behaviour is quantify by the linear entropy. At the end of this chapter the entanglement behaviour of the system is discussed. CHAPTER 5 CONCLUDING REMARKS

We can conclude the results from two different points of view. Firstly, from the first assumption in this thesis, if we take the continuous time constituted from the discrete steps, we find that the discrete time flow affects the system behaviours leading to the extra feature, namely, the broadening of the probability contour and the cut-off condition for the ground state entanglement entropy. The cut-off condition represents the relation between the discrete-time step , the interaction between two subsystem σ and strength of the harmonic potential k. The interaction strength σ between two subsystems is bounded by a function of k and k is in turn bounded by a function of discrete step . This behaviour is completely different from the continuous-time scale, where = 0 all of these features are washed away under the continuum limit → 0.

Secondly, we can look at these results from the way the system is sampled, assum- ing that time itself is fundamentally continuous whereas the discreteness arises from the experimental samplings of the positions and velocities of the system at a given frequency determined by 1/. In this case the diﬀerence in the linear entropy for each value of is due to the diﬀerence in the sampling rate itself. We also discovered that the bounds for are not actually physical but it is due to the rate at which the system is sampled which is equal to the Nyquist sampling rate of the system. Thus the reason that the observation becomes unphysical beyond the bounded region is because the sampling rate is less than the Nyquist frequency, which can potentially make the results of the observation become distorted.

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NAME Mr. Watcharanon Kantayasakun

DATE OF BIRTH 7 August 1993

EDUCATIONAL RECORD

HIGH SCHOOL High School Graduation Surawittayakarn School, 2011

BACHELOR’S DEGREE Bachelor of Science (Applied Physics) King Mongkut’s University of Technology Thonburi, 2015

MASTER’S DEGREE Master of Science (Physics) King Mongkut’s University of Technology Thonburi, 2017

PUBLICATION Kantayasakun, W., Yoo-Kong, S., Deesuwan, T., Tanasittikosol, M. and Liewrian, W., 2017, “Ground State Entanglement Entropy for Discrete-time Two Coupled Harmonic Oscillators”, Accepted to Publish in IOP Conference Series