在此論文中我們探討在開放系統中的量子退相干現象。我們採用費曼維儂的影響泛涵理論，分別在兩個不同的物理系統中探討量子退相干的機制與作用。首先我們考慮一個在離散譜環境影響下的量子諧振子的波包運動，探究其量子相干性消退之表現所在。我們也考量由於波包和環境交互作用所引起波包內部結構的不穩定性。該波包運動的馬可夫與非馬可夫行為之轉換與環境條件的關係也在此
工作中被指出。接著我們擴展費曼維儂的影響泛涵理論到費米子相干態表象來研究雙量子點在耦合到週遭電極的情況下的電子動力學的退相干問題。藉由最小動作量的路徑方法我們可以得到一組描述複雜馬可夫電子動力學的方程式，並從中推導一個精確的主方程，來刻劃雙量子點在週遭電極影響下的退相干行為。從中我們探討由於和電極交換電子所引起的雙兩子點內部能量結構的動態變化過程
以及奈秒尺度下電子態分布權重的複雜轉移，來加深我們對量子退相干在此典型奈米結中構的機制的了解與掌握。

In this thesis we explore decoherence in open quantum systems. We approach this problem using Feynman-Vernon’s influence functional theory in two very distinct systems. We consider first a wave packet of a particle in a harmonic trap interacting with an environment with discrete
spectra where the reduced dynamics of the wave packet is studied. Instability due to the interaction between the system and its quantum environment is investigated and Markovian to non-Markovian transition is discussed. We then extend the theory to the fermion coherent state representation to study the decoherent dynamics of electrons in a double quantum dot under the influence of electron reservoirs connected to the dots. There we derived an exact master equation for arbitrary spectral densities for dot-lead tunneling via the stationary path
equations which fully manifests the complex non-Markovian charge dynamics. Real time fluctuations of the double dot parameters are explored and decoherence are closely analyzed based on our exact solutions from Markovian to non-Markovian regimes.

[1] W. H. Zurek, Phys. Today 44 (10), 36 (1991); Rev. Mod. Phys. 75, 715 (2003).
[2] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge
Univ. Press, Cambridge, 2000).
[3] R. Brown, Phil. Mag. 4, 161 (1828).
[4] A. Einstein, Ann. Phys. (Leipzig) 17 549 (1905).
[5] P. Langevin, CR Acad. Sci. 146 530 (1908).
[6] M. Smoluchowski, Phys. Z. 13 1069 (1912).
[7] S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
[8] I. R. Seniztky, Phys. Rev. 119 670 (1960).
[9] G.W. Gardiner and P. Zoller, Quantum Noise : A Handbook of Markovian and Non-
Markovian Quantum Stochastic Methods with Applications to Quantum Optics (Springer,
Berlin, 2004).
[10] H. B. Callen, T. A. Welton, Phys. Rev. 83, 34 (1951).
[11] R. H. Koch, D. J. Van Harlingen, and J. Clarke, Phys. Rev. Lett. 45, 2132 (1980).
[12] P. Caldirola, Nuovo Cim. 18, 393 (1941).
[13] E. Kanai, Prog. Thoer. Phys. 3, 440 (1948).
[14] H. Dekker, Phys. Rev. A 16, 2126 (1977).
[15] W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).
[16] S. Nakajima, Prog. Theor. Phys. 20, 948 (1958).
[17] H. Mori, Prog. Theor. Phys. 33, 423 (1965).
[18] R. Zwanzig, J. Chem. Phys. 33, 1338 (1960).
[19] H. Haken, Rev. Mod. Phys. 47, 67 (1975).
78
[20] G. W. Ford, M. Kac and P. Mazur, J. Math. Phys. 6, 504 (1965); G. W. Ford, J. T. Lewis
and R. F. O’Connell, Phys. Rev. A 37, 4419 (1988).
[21] F. Haake, Statistical Treatment of Open Systems by Generalized Master Equation (Springer,
Berlin, 1973).
[22] L. Mandel and E. Wolf, Quantum Coherence and quantum optics (Cambridge University
Press, Cambridge, 1995).
[23] H. Risken, The Fokker Planck Equation (Springer, Berlin, 1984).
[24] A. D. Fokker, Ann. Phys. (Leipzig), 43, 310 (1915).
[25] M. Planck, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl., 325 (1917)
[26] E. P. Wigner, Phys. Rev. 40, 749 (1932).
[27] J. R. Oppenheimer, Phys. Rev. 31, 66 (1928).
[28] G. Gamow, Z. Physik 51, 204 (1928).
[29] R. W. Gurney and E. U. Condon, Nature (London), 122, 439 (1928).
[30] T. Holstein, Ann. Phys. (New York) 8, 325 (1959); D. Emin and T. Holstein,ibid 53, 439
(1969).
[31] C. P. Flynn and A. M. Stoneham, Phys. Rev. B 1, 3966 (1970).
[32] N. Takigawa and M. Abe, Phys. Rev. C 41, 2451 (1990).
[33] A.B. Balantekin and N. Takigawa, Rev. Mod. Phys. 70, 77 (1998).
[34] A. N. Cleland, J. M. Martinis, and J. Clarke, Phys. Rev. B 37, 5950 (1988).
[35] P. W. Anderson, B. I. Halperin and C. M. Varma, Phil. Mag. 25, 1 (1972).
[36] W. A. Phillips, J. Low Temp. Phys. 7, 351 (1972).
[37] J. L. Black, Glassy Metlas I, Topics in Applied Physics, Vol. 46, ed. by H.-J. G¨untherodt
and H. Beck (Springer Verlag, Berlin, 1981).
[38] J. Stockburger, U. Weiss and R. G¨orlich, Z. Phys. B 84, 457 (1991).
[39] A. J. Leggett, Directions in Condensed Matter Physics, Vol. 1, ed. by G. Grinstein and G.
Mazenko (World Scientific, Singapore, 1986).
[40] Y. Makhlin, G. Sch¨on, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001).
[41] F. Guinea, V. Hakim, and A. Muramatsu, Phys. Rev. B 32, 4410(1985).
[42] J. T. Stockburger and H. Grabert, Chem. Phys. 268, 249 (2001); Phys. Rev. Lett. 88,
170407 (2002).
79
[43] M. Grifoni and P. H¨anggi, Phys. Rep. 304, 229 (1998).
[44] F. B. Anders and A. Schiller, Phys. Rev. B 74, 245113 (2006).
[45] A. Warshel, Z.T. Chu, W.W. Parson, Science 246, 112 (1989).
[46] J. M. Martinis, S. Nam, J. Aumentado, K. M. Lang and C. Urbina, Phys. Rev. B 67,
094510 (2003).
[47] L. M. K. Vandersypen and I. L. Chuang, Rev. Mod. Phys. 76, 1037 (2005).
[48] X. Hu and S. Das Sarma, Phys. Rev. Lett. 96, 100501 (2006).
[49] G. C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D 34, 470 (1986).
[50] G. J. Milburn, Phys. Rev. A 44, 5401 (1991).
[51] R.C. Ashoori, Nature 379, 413 (1996).
[52] L.P. Kouwenhoven, T.H. Oosterkamp, M.W.S. Danoesastro, M. Eto, D.G. Austing, T.
Honda and S. Tarucha, Science 278, 1788 (1997).
[53] D.R. Stewart, D. Sprinzak, C.M. Marcus, C.I. Duruoz and J.S. Harris Jr., Science 278
1784 (1997).
[54] H. Grabert and M.H. Devoret (Ed.), Single Charge Tunneling, NATOASI Series B, vol.
294 (Plenum Press, NewYork, 1991).
[55] S.M. Reimann and M. Manninen, Rev. Mod. Phys. 74 1283 (2002).
[56] M. Tews, Annalen der Physik 13 249 (2004).
[57] E. Schr¨odinger, Naturwiss. 14, 664 (1926).
[58] R. P. Feynman and F. L. Vernon, Ann. Phys. 24, 118 (1963).
[59] A. O. Caldeira and A. J. Leggett, Physica 121A, 587 (1983).
[60] J.W. Negele and H. Orland, Quantum Many-Particle Systems (AddisonWesley, Singapore,
1988).
[61] W. H. Zurek, Phys. Rev. D 26, 1862 (1982).
[62] A. O. Caldeira and A. J. Leggett, Ann. Phys. 149, 374 (1983); 153, 445 (1984).
[63] F. Haake, and R. Reibold, Phys. Rev. A 32, 2462 (1985).
[64] H. Grabert, P. Schramm and P. -L. Ingold, Phys. Rev. Lett. 58, 1285(1987).
[65] W. G. Unruh and W. H. Zurek, Phys. Rev. D 40, 1071 (1989).
[66] B. L. Hu, J. P. Paz, and Y. H. Zhang, Phys. Rev. D 45, 2843 (1992); ibid. 47 1576 (1993).
80
[67] J. P. Paz, S. Habib and W. H. Zurek, Phys. Rev. D 47, 488 (1993).
[68] D. Braun, P. A. Braun, and F. Haake, Opt. Commun. 179, 411 (2000).
[69] G. W. Ford and R. F. O’Connell, Phys. Rev. D 64, 105020 (2001).
[70] W. T. Strunz and F. Haake, Phys. Rev. A 67, 022102 (2003); W. T. Strunz, F. Haake and
D. Braun, ibid. 67, 022101 (2003).
[71] K. Shiokawa and B. L. Hu, Phys. Rev. A 70, 062106 (2004).
[72] G. W. Ford and R. F. O’Connell, Phys. Rev. A 73, 032103 (2006).
[73] J. H. An and W. M. Zhang, Phys. Rev. A 76, 042127 (2007).
[74] C. H. Chou, T. Yu, and B. L. Hu, Phys. Rev. E 77, 011112 (2008).
[75] B. H. Bransden, and C. J. Joachen, Physics of Atoms and Molecules (Longman, London,
1983)
[76] R. Bl¨ume, R. Graham, L. Sirko, U. Smilansky, H. Walther, and K. Yamada, Phys. Rev.
Lett. 62, 341 (1991).
[77] C. Brif, H. Rabitz, A. Wallentowitz, I. A. Walmsley, Phys. Rev. A 63, 063404 (2001)
[78] M. Spanner, E. A. Shapiro and M. Ivanov, Phys. Rev. Lett. 92, 093001 (2004); E. A.
Shapiro, I. A. Walmsley, and M. Y. Ivanov, ibid. 98, 050501 (2007)
[79] S. Yoshida, C. O. Reinhold, J. Burgdorfer, W. Zhao, J. J. Mestayer, J. C. Lancaster, and
F. B. Dunning, Phys. Rev. A 75, 013414 (2007).
[80] R. F. O’Connell, Physica E19, 77 (2003).
[81] J. J. Halliwell, Phys. Rev. D 63, 085013 (2001).
[82] G. S. Engel, T. R. Calhoun, E. L. Read, T. K. Ahn, T. Mancal, Y. C. Cheng, R. E.
Blankenship, and G. R. Fleming, Nature, 446, 782 (2007); H. Lee, Y.-C. Cheng, G. R.
Fleming, Science 316, 1462 (2007).
[83] A. J. Legget, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg and W. Zwerger,
Rev. Mod. Phys. 59, 1 (1987).
[84] H. Grabert, P. Schramm, and G.-L. Ingold, Phys. Rep. 168, 115 (1988).
[85] H. Carmichael, An open systems approach to quantum optics (Springer-Verlag, New York,
1993).
[86] U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1999).
[87] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University
Press, Oxford, New York, 2002).
81
[88] J. J. Halliwell and T. Yu, Phys. Rev. D 53, 2012 (1996).
[89] K. Lindenberg and B. J. West, Phys. Rev. A30, 568 (1984); H. Callen and T. Weldon,
Phys. Rev. 83, 34 (1951); R. Kubo, Lectures in Theoretical Physics Vol. 1(Interscience, New
York, 1959), pp. 120-203.
[90] See the discussion in Ref. [62] on page 388-391, also in private communications with A. J.
Leggett.
[91] R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New
York, 1965).
[92] K. Shiokawa and R. Kapral, J. Chem. Phys. 117, 7852 (2002).
[93] W. M. Zhang, D. H. Feng and R. Gilmore, Rev. Mod. Phys. 62, 867 (1990).
[94] E. Sch¨odinger, Ber. Kgl. Akad. Wiss. Berlin, 296 (1930).
[95] J. H. An, M. Feng and W. M. Zhang, Phys. Rev. A 76, 042127 (2007).
[96] R. H. Brandenberger, Rev. Mod. Phys. 57, 1 (1985); A. Linde, Particle Physics and Infla-
tionary Cosmology (Harwood Academic, Chur, 1990) and references therein.
[97] J. P. Sethna, Phys. Rev. B 24, 698 (1981); ibid. B 25, 5050 (1982).
[98] L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, Phys. Rev. Lett. 85, 1791
(2000).
[99] L. D. Landau and E. M. Lifschitz, Statistical Physics (Pergamon, London, 1969).
[100] F. Bloch, Z. Phys. 74, 295 (1932).
[101] W. G. van der Wiel, S. DeFranceschi, J. M. Elzerman, T. Fujisawa, S. Tarucha, and L.
P. Kouwenhoven, Rev. Mod. Phys. 75, 1 (2003).
[102] T. Brandes, Phys. Rep. 408, 315 (2005).
[103] R. H. Blick and H. Lorenz, Proceedings of the IEEE International Symposium on Circuits
and Systems, II245 (2000).
[104] T. Tanamoto, Phys. Rev. A 61, 022305 (2000)
[105] L. Fedichkin and A. Fedorov, Phys. Rev. A 69, 032311 (2004); L. Fedichkin, M.
Yanchenko, and K. A. Valiev, Nanotechnology 11, 387 (2000).
[106] J. M. Elzerman, R. Hanson, J. S. Greidanus, L. H.Willems van- Beveren, S. DeFranceschi,
L. M. K. Vandersypen, S. Tarucha, and L. P. Kouwenhoven, Phys. Rev. B 67, 161308(R)
(2003)
82
[107] A. K. H¨uttel, S. Ludwig, K. Eberl, and J. P. Kotthaus, Phys. Rev. B 72, 081310(R)
(2005).
[108] T. Hayashi, T. Fujisawa, H. D. Cheong, Y. H. Jeong and Y. Hirayama, Phys. Rev. Lett.
91, 226084(2003);T. Hayashi, T. Fujisawa, H. D. Cheong, Y. H. Jeong and Y. Hirayama,
Physica E(Amsterdam) 21, 2046 (2004); T. Fujisawa, T. Hayashi and S. Sasaki, Rep. Prog.
Phys. 69, 759 (2006).
[109] J. R. Petta , A. C. Johnson, J. M. Taylor, E. A. Laird, A Yacoby, M. D. Lukin, C. M.
Marcus, M. P. Hanson, A. C. Gossard. Science 309, 1280 (2005).
[110] J. Gorman, D. G. Hasko, and D. A. Williams, Phys. Rev. Lett. 95, 090502 (2005).
[111] T. Brandes and T. Vorrath, Phys. Rev. B 66, 075341 (2002).
[112] U. Hartmann and F. K. Wilhelm, Phys. Rev. B 69, 161309(R) (2004).
[113] M. J. Storcz, U. Hartmann, S. Kohler, and F. K. Wilhelm, Phys. Rev. B 72, 235321
(2005).
[114] Z. J.Wu, K. D. Zhu, X. Z. Yuan, Y.W. Jiang, and H. Zheng, Phys. Rev. B 71, 205323
(2005).
[115] S. Vorojtsov, E. R. Mucciolo, and H. U. Baranger, Phys. Rev. B 71, 205322 (2005).
[116] V. N. Stavrou and X. Hu, Phys. Rev. B 72, 075362 (2005).
[117] M. Thorwart, J. Eckel and E. R. Micciolo, Phys. Rev. B 72, 235320 (2005).
[118] U. Hohenester, Phys. Rev. B 74, 161307(R) (2006).
[119] S. R. Woodford, A. Bringer and K. M. Indelkofer, Phys. Rev. B 76, 064306 (2007).
[120] Y. Y. Liao, Y. N. Chen, W. C. Chou, and D. S. Chuu, Phys. Rev. B 77, 033303 (2008).
[121] T.H. Stoof and Yu.V. Nazarov, Phys. Rev. B 53, 1050 (1996).
[122] S. A. Gurvitz and Ya. S. Prager, Phys. Rev. B 53, 15932 (1996).
[123] The choice of the energy reference shall not be E1 = E2 = 0 because when the two
localized states are degenerate, E1 = E2, the energy of the dots, EDQD = ((E1+E2)/
2 )(1−ρ00)+
((E1−E2)/)
2 (ρ11 − ρ22) + (ρ12 + ρ21), where the normalization condition 1 − ρ00 = ρ11 + ρ22
is used, becomes EDQD = E1+E2
2 (1 − ρ00) + (ρ12 + ρ21). When there is no electrons in
the dots, ρ00 = 1, the energy of the dots is simply zero. And the choice of E1 = E2 = 0
will cause that the energy of the dots when there is one electron in double dots, ρ11 = 1
or ρ22 = 1 is also 0. In the case where there is no charge leakage this choice of energy
reference is ok. However for the charge qubit being an open system, charge leakage shall be
considered and the choice of energy reference would matter if we look at the time evolution
of the energy of the dots. Otherwise, this choice of energy reference matters nothing as
long as the total energy configuration, the relations between the dot energy levels and the
chemical potentials in the electron reservoirs, remains the same. We can see identical time
evolutions of the density matrix at the energy references E = 0 and E = 0 with the relations
between E and μ1,2 fixed. But the dot energy EDQD given above is of course influenced by
the energy reference.
[124] D.P. DiVincenzo, Science 269, 225(1995).