臺灣博碩士論文加值系統

本論文分別把兩態系統（在磁場中的自旋-1/2粒子）和簡諧振子（電磁場）當作環境，來探討環境如何影響一個兩態系統。我們找出兩態系統跟環境所容許的量子狀態，再找到這種狀態下兩態系統的約化密度矩陣，經由Von Neumann熵，最後得到可以代表環境的等效溫度。等效溫度的值受到環境的能級與環境和兩態系統耦合強度有關，而等效溫度和兩態系統跟環境的糾纏程度有密切的關係。
最後，我們探討單頻電磁場環境對自旋-1/2粒子的幾何相位的影響。我們得到在單頻電磁場環境中，自旋-1/2粒子的幾何相位會受到環境和自旋粒子耦合強度及環境能級影響。當耦合強度越強，幾何相位的大小就越小。

In this investigation, we model the environment either as a two-level system (TLS, spinor in a magnetic field) or an oscillator, and study the influence of environment on a TLS. By finding out quantum eigen-state of the system (TLS + environment), we calculate the reduced density matrix of the TLS. Finally, we define the effective temperature of the environment by comparing the von Neumann entropy with thermodynamic entropy. The effective temperature is related to the energy split of the environment and the coupling constant. For these two kinds of environment, we can define the degree of entanglement by the effective temperature.
Finally, we regard an oscillator as the environment and consider the influence of environment on the Berry phase of a TLS. The Berry phase still has some relationship with the energy split of the environment and the coupling constant. When coupling is stronger, the Berry phase is smaller.

內文
第一章 前言…………………………………………………………………………..1
第二章 原理簡介
2-1 密度矩陣（Density matrix）及熵（Entropy）…………………………3
2-2 配分函數（Partition function）…………………………………………8
2-3 幾何相位（Berry phase）..…………………………………………….11
2-4 傑尼-卡彌哈米爾頓（Jaynes-Cummings Hamiltonian）….…………..12
2-5 路徑積分（Feynman’s Path integral）…………………………………14
第三章 自旋-1/2和自旋-1/2的耦合
3-1 等效溫度…………………………………………………………….…20
3-2 糾纏程度（Degree of entanglement）…………………………..……...29
第四章 自旋-1/2和諧振子的耦合
4-1 等效溫度……………………………………………………………….34
4-2 糾纏程度……………………………………………………………….44
第五章 電磁場中的自旋粒子
5-1 電磁場對自旋-1/2粒子幾何相位的影響……………………………..46
5-2 自旋-1/2粒子在電磁場裡的動力學…………………………………..52
第六章 結論…………………………………………………………………………56
附錄…………………………………………………………………………………..58
參考資料……………………………………………………………………………..62

[1] R.P. Feynman, Found. Phys. 16, 507 (1986)
[2] D.P. Divincenzo and P.W. Shor, Phys. Rev. Lett. 77, 3260(1996); E. Knill and R. Laflamme, Phys. Rev. A 55, 900(1997)
[3] Y. Makhlin, G. Schon, and A. Shnirman, Rev. of Mod. Phys. 73, 357(2001)
[4] C. Monroe, D.M. Meekhof, B.E. King, W.M. Itano, and D.J. Wineland, Phys. Rev. Lett. 75, 4714(1995)
[5] Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, and H.J. Kimble, Phys. Rev. Lett. 75, 4710(1995)
[6] N.A. Gershenfeld and I.L. Chang, Science 275, 350 (1997)
[7] M. K. Vandersypen et al., Nature 414, 883(2001)
[8] D.G. Cory, M.D. Price, W. Mass, E. Knill, R. Laflamme, W.H. Zurek, T.F. Havel, and S.S. Somaroo, Phys. Rev. Lett. 81, 2152(1998)
[9] Ch. Miniatura et al., Phys. Rev. Lett. 69, 261(1992) ; A.G. Wagh et al., Phys. Rev. Lett. 78, 755(1997); C.L. Webb et al., Phys Rev. A 60, R1783(1999)
[10] R.S. Whitney et al., arXiv:cond-mat/0107359(2001); K.M. Fonseca et al., arXiv:quant-ph/0003049(2002); A. Carollo et al., arXiv:quant-ph/0202128(2002)
[11] C. Anastopoulos, Phys. Rev. D 59, 045001(1999)
[12] J. Von Neumann, Die Mathematische Grundlagen der Quanten-mechanik（Springer, Berlin, 1932）; J. Von Neumann, Mathematical Foundation of Quantum Mechanics（Princeton University Press, Princeton）
[13] M. V. Berry, Proc. Roy. Soc. A 392, 45 (1984)
[14] A. Shapere and F. Wilczek eds., Geometric phases in physics (World Scientific, Singapore, 1989).
[15] J.E. Bayfield, Quantum Evolution, p.61-63
[16] R. Shankar, Principle of Quantum Mechanics 2ed(1994)
[17] R.P. Feynman, Quantum Mechanics and Path Integrals, Chap3
[18] N. Nagaosa, Field Theory of Condensed Material Systems, p.47-50
[19] R. Kubo, Principles of Statistical Mechanics, p.38-41
[20] A. Einstein, B. Podolsky, N. Rosen, Physical Review 47, 777 (1935)
[21] Schrödinger, E. Naturwissenschaften 23 (1935)
[22] J. Bell, "On the Einstein Podolsky Rosen paradox" , Physics I #3, 195 (1964);曾謹言, 量子力學 卷II 第三版
[23]A.F. Abouraddy, B.E.A. Saleh+, A.V. Sergienko, and M.C. Teich, arXiv:quant-ph/0109081
[24] H.J. Bernstein, Phys. Rev. Lett. 18, 1102(1967); Y. Aharonov and L. Susskind, Phys. Rev. 158, 1237(1967)
[25] A.G. Wagh et al., Phys. Rev. Lett. A 146, 369(1990)
[26] A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg and W. Zwerger, Rev. Mod. Phys. 59, 1(1987)