量子退相干的纠缠态表象方法论
|
摘要
退相干是现代量子信息与量子计算中普遍存在的本质问题,对于研究它是当今物理界的一个重点和难点。难在目前国际上尚无一个处理连续变量量子态的退相干的行之有效的理论。尽管存在所谓的超算符方法,但它有很多局限性。本文提出一个崭新的处理量子退相干的理论,即引入热纠缠态表象处理开放量子系统的退相干问题,我们的处理方法不仅方便有效,而且能丰富和发展量子的算符和表示理论。这是因为,一方面,IWOP技术将原本只适用于可对易的牛顿—莱布尼兹积分公式推广到对量子力学非对易量子算符的运算,从而建立了联系q数与c数的一座桥梁;另一方面,热纠缠态表象的引入,本身就体现了量子系统与外界热环境的纠缠本性,在热纠缠态表象中讨论退相干问题,能较直接地反映其本质,从而有明显的成效。其优点在于(1)能方便有效地将密度算符主方程转化为c数方程;(2)能解析求解密度算符,并给出其Kraus算符和表示;(3)结合有序算符内的积分技术可以证明Kraus算符的归一化;(4)可方便地导出Wigner、光子数分布等函数的时间演化公式。因此,这套方法有望被广泛地采用。本文主要内容如下:
一、基于IWOP技术和热场动力学方法,通过引入一个连续变量的热纠缠态表象|η>来处理开放系统的量子退相干问题。利用|η表象可以方便简洁地将密度算符(ρ)主方程转化成一个形式上较传统c数方程更简洁的关于函数<η|ρ>的c数方程,即将密度算符的求解问题转换为求解关于<η|ρ>的微分方程。为此我们还建立了热纠缠态表象表示与Wigner函数、密度算符以及正P表示等之间的关系及相应的逆关系。此外,利用<η|ρ>还可将任意算符在系统ρ下的平均值的计算转化为热场动力学框架下的矩阵元,为计算平均值提供便利。
二、进一步应用|η>表象方法求解了多种常见量子退相干模型下的密度算符主方程,如振幅阻尼模型,压缩热库谐振子模型,位相扩散(阻尼)模型和一些广义的位相扩散模型,低级近似下的激光模型(包括了有限温度下的情况),有Kerr介质存在时的振幅阻尼模型以及参量下转化过程等,从而展示了时间演化过程中不同量子通道所具有的丰富特征(如衰减、放大、相扩散等),导出了密度算符的解析解——即首次求解给出了若干密度算符的算符和表示(无限求和表示)和Kraus算符,并证明了其归一化条件(这在以往的文献中尚未见报导)。这些结果不但能够帮助我们以一种直观的方式抓住退相干的内在本质(量子纠缠),而且为求解其他一些退相干模型提供方便。此外,在量子隐形传态的描述中,将离散形式的算符和表示进一步推广到连续(积分)形式的算符和表示的情况。从而实现了Kraus算符理论由有限维算符和表示到无限维算符和表示、连续算符和表示的丰富与发展。
三、利用算符的反正规乘积表示和Weyl表示导出了两个新的光子计数公式,其中一个通过密度算符的Wigner函数表示,另一个通过密度算符的Q函数表示。基于热纠缠态表象,建立了若干退相干模型下量子系统Wigner函数的时间演化公式,即初态的Wigner函数与任意时刻的Wigner函数之间的关系。结合新光子计数公式中光子数分布与Wigner函数之间的关系,导出了计算退相干模型下光子数分布的新公式。这些公式将给研究退相干量子系统的非经典特性带来很大的方便。
四、基于IWOP技术和以上方法,对于任意数的光子增加或扣除量子态(如热环境中的光子增加相干态、光子扣除单模(双模)压缩态),通过导出有关函数的解析表达式深入讨论它们在退相干模型中的演化问题,包括态的归一化系数、Wigner函数、Husimi函数、光子分布以及Tomography等等。这些解析结果不仅有助于清楚地理解耗散库对量子态的影响,而且能为实验测量结果提供有价值的参考。
Decoherence is a universal essential problem in modern quantum information and quan-tum computation. It is significant and difficult to study this problem in modern physical field.Its difficulty is from the absent of an effective theory to deal with decoherence of quantumstate with continuum variables. Although there is a super-operator method, it has much in-sufficiency. In this thesis, we propose a new theory to deal with quantum decoherence, i.e.,by introducing a thermo entangled state representation. Our method is not only convenientand effective, but also can enrich and develop the sum-representation theory of Kraus opera-tor. The reasons are that, on one hand, the IWOP technique generalizes the Newton-Leibnizrule to the integrations over the operators in quantum mechanics, which is usually not com-mutative, thus it creates a bridge between classical mechanics and quantum mechanics; onthe other hand, the thermo entangled state representation itself exhibits the entanglement be-tween quantum system and environment so that the essence of decoherence can be open outdirectly by using this new representation. The special merits are (1) converting convenientlythe master equation (ME) of density operator to c-number equation; (2) solving analyticallyMS and obtaining the Kraus operator sum representation of density operator; (3) provingeffectively the normalization of Kraus operator by using the IWOP technique; (4) derivingthe time evolution of Wigner function and photon distribution in a concise process. Thus ourtheory may be adapted extensively. Our main contents are as follows:
一. Based on the IWOP technique and the thermo dynamics theory, we introduce a newthermo entangled state representation with continuous variables to study the decoherence ofopen quantum system. By using |η>, one can convert conveniently the master equationof density operatorρinto c-number (differential) equation about function <η|ρ>, whichis more concise than that in traditional form. For solving <η|ρ>, we have also set upthe relation between the thermo entangled state representation and Wigner function, densityoperator and positive P-representation, and vice versa. In addition, using <η|ρ> we put thecalculation for any operator’s average value in density operator into a matrix element in theframe of thermo dynamics theory, which is more convenient.
二. Using |η>, we further have analytically solved some familiar MEs of quantum op-tical systems, such as amplitude damping, (generalized) phase diffusion, squeezing sensitivereservoirs, laser theory in the lowest-order approximation (involving ME at finite tempera-ture), amplitude damping with a Kerr medium and parametric down conversion. For the firsttime, we have derived the infinite sum representation of density operators and proved the nor-malization of Kraus operators by virtue of the IWOP technique. These results can provide uswith an intuitionistic way to grasp the essence of decoherence (quantum entanglement), butwith conveniences for solving other decoherence models. In addition, we have generalizedthe discrete sum representation of density operators to the continuum sum representation inquantum teleportation. This further develops the Kraus operator sum representation form thediscrete finite sum case to the discrete infinite sum case, then to the continuum sum case.
三.Employing the antinormal ordering form of product of operators and Weyl repre-sentation, we have derived two new photon-count formulas, which are related to Wignerfunction and Q-representation of density operator, respectively. Further, for some decoher-ence models, we have derived the evolution formula of Wigner function by using |η> andseveral new formulas of photon distribution from the relation between photon distributionand Wigner function. These formulas will provide us with convenience for studying thenonclassicality of quantum system.
四.By virtue of the IWOP technique and these method above, we have discussed fullyand analytically the evolution of some classical quantum states (photon-added coherent state,single-mode and two-mode photon-subtracted squeezed state) in environment, such as nor-malization factor, Wigner function, Husimi function, photon distribution and quantum to-mography. These analytical results can help us to not only understand clearly the dissipativereservoir’s effect to quantum state, but also provide a valuable reference for the experimentaltest.
一、基于IWOP技术和热场动力学方法,通过引入一个连续变量的热纠缠态表象|η>来处理开放系统的量子退相干问题。利用|η表象可以方便简洁地将密度算符(ρ)主方程转化成一个形式上较传统c数方程更简洁的关于函数<η|ρ>的c数方程,即将密度算符的求解问题转换为求解关于<η|ρ>的微分方程。为此我们还建立了热纠缠态表象表示与Wigner函数、密度算符以及正P表示等之间的关系及相应的逆关系。此外,利用<η|ρ>还可将任意算符在系统ρ下的平均值的计算转化为热场动力学框架下的矩阵元,为计算平均值提供便利。
二、进一步应用|η>表象方法求解了多种常见量子退相干模型下的密度算符主方程,如振幅阻尼模型,压缩热库谐振子模型,位相扩散(阻尼)模型和一些广义的位相扩散模型,低级近似下的激光模型(包括了有限温度下的情况),有Kerr介质存在时的振幅阻尼模型以及参量下转化过程等,从而展示了时间演化过程中不同量子通道所具有的丰富特征(如衰减、放大、相扩散等),导出了密度算符的解析解——即首次求解给出了若干密度算符的算符和表示(无限求和表示)和Kraus算符,并证明了其归一化条件(这在以往的文献中尚未见报导)。这些结果不但能够帮助我们以一种直观的方式抓住退相干的内在本质(量子纠缠),而且为求解其他一些退相干模型提供方便。此外,在量子隐形传态的描述中,将离散形式的算符和表示进一步推广到连续(积分)形式的算符和表示的情况。从而实现了Kraus算符理论由有限维算符和表示到无限维算符和表示、连续算符和表示的丰富与发展。
三、利用算符的反正规乘积表示和Weyl表示导出了两个新的光子计数公式,其中一个通过密度算符的Wigner函数表示,另一个通过密度算符的Q函数表示。基于热纠缠态表象,建立了若干退相干模型下量子系统Wigner函数的时间演化公式,即初态的Wigner函数与任意时刻的Wigner函数之间的关系。结合新光子计数公式中光子数分布与Wigner函数之间的关系,导出了计算退相干模型下光子数分布的新公式。这些公式将给研究退相干量子系统的非经典特性带来很大的方便。
四、基于IWOP技术和以上方法,对于任意数的光子增加或扣除量子态(如热环境中的光子增加相干态、光子扣除单模(双模)压缩态),通过导出有关函数的解析表达式深入讨论它们在退相干模型中的演化问题,包括态的归一化系数、Wigner函数、Husimi函数、光子分布以及Tomography等等。这些解析结果不仅有助于清楚地理解耗散库对量子态的影响,而且能为实验测量结果提供有价值的参考。
Decoherence is a universal essential problem in modern quantum information and quan-tum computation. It is significant and difficult to study this problem in modern physical field.Its difficulty is from the absent of an effective theory to deal with decoherence of quantumstate with continuum variables. Although there is a super-operator method, it has much in-sufficiency. In this thesis, we propose a new theory to deal with quantum decoherence, i.e.,by introducing a thermo entangled state representation. Our method is not only convenientand effective, but also can enrich and develop the sum-representation theory of Kraus opera-tor. The reasons are that, on one hand, the IWOP technique generalizes the Newton-Leibnizrule to the integrations over the operators in quantum mechanics, which is usually not com-mutative, thus it creates a bridge between classical mechanics and quantum mechanics; onthe other hand, the thermo entangled state representation itself exhibits the entanglement be-tween quantum system and environment so that the essence of decoherence can be open outdirectly by using this new representation. The special merits are (1) converting convenientlythe master equation (ME) of density operator to c-number equation; (2) solving analyticallyMS and obtaining the Kraus operator sum representation of density operator; (3) provingeffectively the normalization of Kraus operator by using the IWOP technique; (4) derivingthe time evolution of Wigner function and photon distribution in a concise process. Thus ourtheory may be adapted extensively. Our main contents are as follows:
一. Based on the IWOP technique and the thermo dynamics theory, we introduce a newthermo entangled state representation with continuous variables to study the decoherence ofopen quantum system. By using |η>, one can convert conveniently the master equationof density operatorρinto c-number (differential) equation about function <η|ρ>, whichis more concise than that in traditional form. For solving <η|ρ>, we have also set upthe relation between the thermo entangled state representation and Wigner function, densityoperator and positive P-representation, and vice versa. In addition, using <η|ρ> we put thecalculation for any operator’s average value in density operator into a matrix element in theframe of thermo dynamics theory, which is more convenient.
二. Using |η>, we further have analytically solved some familiar MEs of quantum op-tical systems, such as amplitude damping, (generalized) phase diffusion, squeezing sensitivereservoirs, laser theory in the lowest-order approximation (involving ME at finite tempera-ture), amplitude damping with a Kerr medium and parametric down conversion. For the firsttime, we have derived the infinite sum representation of density operators and proved the nor-malization of Kraus operators by virtue of the IWOP technique. These results can provide uswith an intuitionistic way to grasp the essence of decoherence (quantum entanglement), butwith conveniences for solving other decoherence models. In addition, we have generalizedthe discrete sum representation of density operators to the continuum sum representation inquantum teleportation. This further develops the Kraus operator sum representation form thediscrete finite sum case to the discrete infinite sum case, then to the continuum sum case.
三.Employing the antinormal ordering form of product of operators and Weyl repre-sentation, we have derived two new photon-count formulas, which are related to Wignerfunction and Q-representation of density operator, respectively. Further, for some decoher-ence models, we have derived the evolution formula of Wigner function by using |η> andseveral new formulas of photon distribution from the relation between photon distributionand Wigner function. These formulas will provide us with convenience for studying thenonclassicality of quantum system.
四.By virtue of the IWOP technique and these method above, we have discussed fullyand analytically the evolution of some classical quantum states (photon-added coherent state,single-mode and two-mode photon-subtracted squeezed state) in environment, such as nor-malization factor, Wigner function, Husimi function, photon distribution and quantum to-mography. These analytical results can help us to not only understand clearly the dissipativereservoir’s effect to quantum state, but also provide a valuable reference for the experimentaltest.
引文
[1] U. Wei, Quantum Dissipative System (Singapore: Word Scientific, 1999)
[2] M. Gell-Mann and J. B. Hartle, Phys. Rev. D 47 (1993) 3345
[3] D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994);M. Orszag, Quan-tum Optics (Springer Press, 2000)
[4] H. T. Elze, Nucl. Phys. B 436 (1995) 213
[5] W. H. Zurek, Phys. Rev. D 26 (1982) 1862
[6] W. G. Unruh, Phys. Rev. A 51 (1995) 992
[7] D. Bouwmeester, A. Ekert and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag,Berlin, 2000)
[8] C. H. Bennett and D. P. DiVincenzo, Nature (London) 404 (2000) 247
[9] W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973)
[10] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, UK, 1997)
[11] P. L. Knight and L. Allen, Concepts of Quantum Optics (A. Wheaton & Co.Ltd., 1983)
[12] M. Sargent III, et. al., Laser Physics (Addison-Wesley, 1974)
[13] Jan Perina, Quantum Statistics of Linear and Nonlinear Optical Phenomena, (D. Reidel PublishingCompany, 1984)
[14] S. Haroche and J. M. Raimond, in Advances in Atomic and Molecular Physics, Vol. 20, ed. D. Batesand B. Bederson (Academic Press, New York, 1985)
[15] S. Haroche, in High Resolution Laser Spectroscopy, ed. K. Shimoda (Springer-Verlag, Berlin,1976)
[16] A. Mufti, H. A. Schmitt, A. B. Balantekin and M. Sargent III, J. Opt. Soc. Am. B 10 (1993) 2100;S. Chatuverdi and V. Srinivasan, J. Mod. Opt. 38 (1991) 777
[17] H. F. Arnoldus, J. Opt. Soc. Am. B 13 (1996) 1099; M. Ban, Phys. Rev. A 47 (1993) 5093; J. Math.Phys. 33 (1992) 3213; G. M. d’Ariano, Phys. Lett. A 187 (1994) 231
[18] X. X. Yi, C. Li, and J. C. Su, Phys. Rev. A 62 (2000) 013819
[19] L. M. Are‘valo-Aguilar and H. Moya-Cessa, Quantum Semiclassic. Opt. 10 (1998) 671
[20] J. Preskill, Lecture Notes for Physics 229: Quantum Information and Computation, (CaliforniaInstitution of Technology, 1998)
[21] F. Haake, Z. Phys. 223 (1969) 353; 223 (1969) 364171
[22] F. Haake, Statistical Treament of Open Systems by Generalized Master Equations, in SpringerTracts in Modern Physics, Vol. 66 (Springer-Verlag, Berlin, 1973) pp. 98-168
[23] W. Pauli, Probleme der Modernen Physik, Aronold Sommerfeld zum 60, Geburtsage Gewidmetvon Seinen Schulern, volume 1, (Hirzel Verlag, 1928)
[24] R. Zwanzig, Statistical Mechanics of Irreversibilutym, volume 3, (New York: Interscience, 1961)
[25] I. Prigogine, Non-equilibrium Statistical Mechanics (New York: Interscience, 1962)
[26] G. S. Agarwal, Progress in Optics, volume 11, (Amsterdam: North-Holland, 1973)
[27] H. X. Lu, J. Yang, Y. D. Zhang and Z. B. Chen, Phys. Rev. A 67 (2003) 024101; H. X. Lu, Z. B.Chen and Y. D. Zhang, Mod. Phys. Lett. B 16 (2002) 595.
[28] P. A. M. Dirac, The Principles of Quantum Mechanics, (Oxford: Clarendon Press, 1930).
[29] P. A. M. Dirac, The Physicist’s conception of Nature, (New York: Mehra, 1973).
[30] P. A. M. Dirac, Directions in Physics, (New York: John Wiley, 1978)
[31] P. A. M. Dirac, Recollections of an exciting area, History of 20th Century Physics, (New York:Academic Press, 1977, 109).
[32] Hong-yi Fan, H. R. Zaidi and J. R. Klauder, Phys. Rev. D 35 (1987) 1831; Hong-yi Fan and H. R.Zaidi, Phys. Rev. A 37 (1988) 2985; Hong-yi Fan and J. R. Klauder, J. Phys. A 21 (1988) L715;Hong-yi Fan and Hui Zou, Phys. Lett. A 252 (1999) 281; Hong-yi Fan and A. Wunsche, J. Opt. B:Quantum & Semiclass. Opt. 2 (2000) 464.
[33] Hong-yi Fan, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) R147; H. Y. Fan, H. L. Lu and Y. Fan,Ann. Phys. 321 (2006) 480.
[34] A. Wunsche, J. Opt. B: Quantum & Semiclass. Opt. 1 (1999) R11.
[35] Hong-yi Fan, Phys. Lett. A 161 (1991) 1.
[36] H. Weyl, Z. Phys. 46 (1927) 1.
[37] G. Wick, Phys. Rev., 80 (1950) 268.
[38]范洪义,量子力学表象与变换论—狄拉克符号法进展,(上海:上海科学技术出版社,1997).
[39] R. J. Glauber, Phys. Rev. 130 (1963) 2529; 131 (1963) 2766.
[40] J. R. Klauder and B.S. Skargerstam, Coherent States, (World Scientific, Singapore, 1985).
[41] R. Loudon, and P.L. Knight, J. Mod. Opt. 34 (1987) 709; V. Buzˇek, J. Mod. Opt. 37 (1990) 303;Dodonov V V, J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1; See e.g., G. M. D’Ariano, M. G.Rassetti, J. Katriel and A. I. Solomon, Squeezed and Nonclassical Light, ed P Tombesi and E RPike (New York: Plenum 1989) p301.
[42] H. P. Yuen, Phys. Rev. A 13 (1976) 2226.
[43] D. F. Walls, Nature 306 (1983) 141.
[44] P. W. Schleich, Quantum Optics in Phase Space (Berlin: Wiley) 2001.
[45] N. N. Bogolyubov, Lectures on Quantum Statistics (Vol.1), (New York: Gordon and Breach, 1987).
[46] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47 (1935) 777.
[47] E. Schrodinger, Naturwiss. 23 (1935) 807, translated to english in“Quantum Theory and Measure-ment”, ed J. A. Wheeler and W. H. Zurek, Princeton University Press (1983).
[48] C. H. Bennett, G. Brassard,C. Crepeau, R. Jozsa, etc., Phys. Rev. Lett. 70 (1993) 1895.
[49] Hong-yi Fan and J. R. Klauder, Phys. Rev. A 49 (1994) 704; Hong-yi Fan, Phys. Rev. A 65 (2002)064102; Hong-yi Fan, Phys. Lett. A 294 (2002) 253;
[50] Hong-yi Fan and Yue Fan, Phys. Rev. A 54 (1996) 958;
[51] J. Preskill, Quantum Information and Computation (Lecture Notes for Physics vol 229) (Pasadena,CA: California Institute of Technology 1998).
[52]范洪义,量子力学纠缠态表象及应用,上海:上海交通大学出版社,2001.
[53] Li-yun Hu and Hong-yi Fan, J. Phys. A: Math. Gen. 39 (2006) 14133; Hong-yi Fan, Nian-quanJiang and Hai-liang Lu, Mod. Phys. Lett B 16 (2002) 1193.
[54] Nian-quan Jiang, Phys. Rev. A 74 (2006) 012306;
[55] Li-yun Hu and Hong-yi Fan, J. Phys. B: At. Mol. Opt. Phys. 40 (2007) 2099; Li-yun Hu and Hong-yi Fan, Int. J. Mod. Phys. A, (2009) to appear; Hong-yi Fan and Lu Hai-liang, J. Phys. A: Math.Gen. 37 (2004) 10993.
[56] Hong-yi Fan, Opt. Lett. 28 (2003) 2177.
[57] E. Wigner, Phys. Rev. 40 (1932) 749; R. F. O’Connell and E. P. Wigner, Phys. Lett. A 83 (1981)145.
[58] R. P. Feynman, Statistical Physics, W. A. Benjamin, 1972.
[59] See e.g., M. Hillery, R. F. O’Connell, M. O. Scully and E. P. Wigner, Phys. Rep. 106 (1984) 121;N. L. Balazs and B. K. Jennings, Phys. Rep. 104 (1984) 347.
[60] K. Vogel and H. Risken, Phys. Rev. A 40 (1989) 2847.
[61] Hong-yi Fan and Hai-ling Cheng, Commun. Theor. Phys. 36 (2001) 651.
[62] Hong-yi Fan and Li-yun Hu, Commun. Theor. Phys. 51 (2009) 321.
[63] Hong-yi Fan and Li-yun Hu, Commun. Theor. Phys. (2009) in press.
[64] Hong-yi Fan and Yue Fan, Commun. Theor. Phys. 38 (2002) 403; 40 (2003) 645.
[65] J. Schwigner, Quantum Theory of Angular Momentum, (New York, Academic Press, 1965).
[66] Y. Takahashi and H. Umezawa, Collecive Phenomena 2 (1975) 55.
[67] Memorial Issue for H. Umezawa, Int. J. Mod. Phys. B 10 (1996) 1695 memorial issue and refer-ences therein.
[68] H. Umezawa, Advanced Field Theory-Micro, Macro, and Thermal Physics (AIP 1993).
[69] G. S. Agarwal, R. Simon, Opt. Commun. 92 (1992) 105; H. Y. Fan, N. Q. Jiang and H. L. Lu, Opt.Commun. 234 (2004) 277.
[70] H. Y. Fan and Y. Fan, Phys. Lett. A 246 (1998) 242.
[71] H. Y. Fan and J. R. Klauder, Phys. Rev. A 49 (1994) 704.
[72] H. Y. Fan and Y. Fan, Phys. Rev. A 54 (1996) 958.
[73] C. Gardiner and P. Zoller, Quantum Noise, Springer Berlin, (2000).
[74] H. Risken, in: The Fokker–Planck Equation: Methods of Solutions and Applications (Springer,Berlin, 1984).
[75] C.W. Gardiner, in: Handbook of Stochastic Methods, 2nd ed., (Springer, Heidelberg, 1990).
[76]唐绪兵,博士学位论文,中国科学技术大学,2008.
[77] H. Y. Fan and Y. Fan, J. Phys. A 35 (2002) 6873.
[78] H. Y. Fan and H. L. Lu, Mod. Phys. Lett. B, 21 (2007) 183.
[79] R. R. Puri, Mathematical Methods of Quantum Optics Springer-Verlag, Berlin, (2001), AppendixA.
[80] H. Y. Fan, Ann. Phys. 323 (2008) 1502; H. Y. Fan and H. Q. Li, Chin. Phys. Lett. 24 (2007) 3322.
[81] H. Y. Fan, Ann. Phys. 323 (2008) 500.
[82] H. Weyl, Z. Phys. 46 (1927) 1.
[83] H. Y. Fan, H. R. Zaidi, Phys. Lett. A 124 (1987) 303; H. Y. Fan, Phys. Rev. A, 65 (2002) 064102.
[84] P. D. Drummond and C. W. Gardiner, J. Phys. A 13 (1980) 2353.
[85] H. Y. Fan and Min Xiao, Phys. Lett. A 244 (1998) 256.
[86] H. Y. Fan, Phys. Lett. A 161 (1991) 1.
[87] H. Y. Fan, Z. W. Liu and T. N. Ruan, Commun. Theor. Phys. 3 (1984) 175; H. Y. Fan and A.Wu¨nsche, Eur. Phys. J. D 15 (2001) 405.
[88] X. F. Xu, Commun. Theor. Phys. 45 (2006) 332.
[89] H. Y. Fan and X. Min, Phys. Lett. A 219 (1996) 175.
[90] H. Y. Fan and X. L. Xu, Commun. Theor. Phys. 48 (2007) 67; L. Fu and H. Y. Fan, Commun.Theor. Phys. 49 (2008) 1461.
[91] N. L. Balazs and B. K. Jennings, Phys. Rep. 104 (1984) 347; W. Schleich, Quantum Optics, NewYork: Wiley, (2001); T. Tatarsky, Sov. Phys.-Usp. 26 (1983) 311.
[92] H. Y. Fan, Commun. Theor. Phys. 37 (2002) 289.
[93] Magnus W, Oberhettinger F and Soni R P, Formulas and Theorems for the Special Functions ofMathematical Physics, 3rd Ed., Springer Verlag, 1966.
[94] X. L. Xu, H. Q. Li and H. Y. Fan, Commun. Theor. Phys. 49 (2008) 1159;H. Y. Fan and J. S. Wang, Commun. Theor. Phys. 47 (2007) 611; H. Y. Fan and W. B. Gao, Com-mun. Theor. Phys. 45 (2006) 617.
[95] L. Y. Hu, N. R. Zhou and H. Y. Fan, Commun. Theor. Phys. 48 (2007) 629; L. Y. Hu and N. R.Zhou, Commun. Theor. Phys. 47 (2007) 805.
[96] Hong-yi Fan, Representation and Transformation Theory in Quantum Mechanics, (Shanghai Sci-entific & Technical, Shanghai, 1997) (in Chinese).
[97] E. Scho¨dinger, Naturwissenschaften 23 (1935) 807.
[98] Hong-yi Fan and Li-yun Hu, Chin. Phys. B (2009) in press.
[99] M. Ueda and N. Imoto, Phys. Rev. A 41 (1990) 3891;M. Ueda, Phys. Rev. A 41 (1990) 3845.
[100] C. Gardiner and P. Zoller, Quantum Noise (Springer Berlin, 2000).
[101] G. S. Agarwal, Phys. Rev. A, 4, (1971) 739.
[102] D. J. Daniel and G. J. Milburn, Phys. Rev. A, 39 (1989) 4628.
[103] M. S. Kim and V. Buzek, Phys. Rev. A 46 (1992) 4239.
[104] H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-PlanckEquations, Springer-Verlag, Berlin, 1999; H. J. Carmichael, Statistical Methods in Quantum Optics2: Non-Classical Fields, (Springer-Verlag, Berlin, 2008).
[105] Hong-yi Fan, Commun. Theor. Phys. 42 (2004) 339.
[106] A. Wu¨nsche, J. Computational and Appl. Math. 133 (2001) 665.
[107] A. Wu¨nsche, J . Phys. A: Math. and Gen. 33 (2000) 1603.
[108] H. X. Lu, J Yang, Y. D. Zhang and Z. B. Chen, Phys. Rev. A. 67 (2003) 024101.
[109] He′ctor Moya-Cessa, Phys. Rep. 432 (2006) 1.
[110] M. S. Kim and V. Buzek, Phys. Rev. A 47 (1993) 610.
[111] S. J. D. Phoenix, Phys. Rev. A 41 (1990) 5132.
[112] C. M. Caves, Phys. Rev. D 26, (1982) 1817; G. J. Milburn, J. Phys. A 17, (1984) 737; C.M. Cavesand B. L. Schumacher, Phys. Rev. A 31, (1985) 3068; M. A. Dupertuis and S. Stenholm, J. Opt.Soc. Am. B 4, (1987) 1102; M. A. Dupertuis, S. M. Barnett and S. Stenholm, ibid. 4, (1987) 1124;
[113] V Bizek, P. L. Knight and I. K. Kudryavtsev, Phys. Rev. A 44, (1991) 1931; A. K. Ekert and P. L.Knight, Phys. Rev. A 42, (1990) 487; T. A. B. Kennedy and D. F. Walls, ibid. 37, (1988) 152.
[114] X.L Xu and H.Y. Fan, Chin. Phys. B (2008) to appear.
[115] Hong-yi Fan, Phys. Lett. A 313 (2003) 343.
[116] Hong-yi Fan and Cao Li, Phys. Lett. A 325 (2004) 188.
[117] Hong-yi Fan, Hui Zou and Yue Fan, Phys. Lett. A 254 (1999) 137.
[118] G. J. Milburn and C. A. Kolmes, Phys. Rev. Lett. 56 (1986) 2237.
[119] S. Chaturvedi, V. Srinivasan and G. S. Agarwal, J. Phys. A. Math. Gen 32 (1999) 1909.
[120] S. Chaturvedi and V. Srinivasan, Phys. Rev. A 43 (1991) 4054; J. Mod. Opt. 38 (1991) 777.
[121] G. S. Agarwal, Phys. Rev. A 61 (1999) 013807.
[122] J.G. Peixoto and de Faria, Eur. Phys. D. 42 (2007) 153 (arXiv:quant-ph/0512060v2).
[123] Hong-yi Fan, Xu-bing Tang and Li-yun Hu, Commun. Theor. Phys. 2008 to appear.
[124] C. H. Bennett, G. Brassard, C. Creau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70(1993) 1895; L. Vaidman, Phys. Rev. A. 49 (1994) 1473.
[125] Li-yun Hu and Hong-yi Fan, Commun. Theor. Phys. 50 (2008) 879
[126] K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theroy. Lecture Notesin Physics, Vol. 190. (Springer-Verlag, Berlin, 1983).
[127] E. B. Davies, Quantum Theory of Open Systems (New York: Academic, 1976).
[128] M. Ozawa, J. Math. Phys. 25 (1984) 79.
[129] H. Y. Fan and X. L. Xu, Commun. Theor. Phys. 48 (2007) 67.
[130] Hong-yi Fan and Li-yun Hu, Commun. Theor. Phys. 2008 to appear
[131] P. L. Kelley and W. H. Kleiner, Phys. Rev. 136 (1964) 316.
[132] L. Mandel, E. C. G. Sudarshan and E. Wolf, Proc. Phys. Soc. (London) 84 (1964) 435.
[133] M. O. Scully and W. E. Lamb, Jr. Quantum Theory of an Optical Maser. III. Theory of Photoelec-tron Counting Statistics, Phys. Rev. 179 (1969) 368.
[134] B. R. Mollow, Quantum Theory of Field Attenuation, Phys. Rev. 168 (1968) 1896.
[135] R. Loudon, The Quantum Theory of Light, Oxford University Press, 2nd, 1983.
[136] E. C. G. Sudarshan, Phys. Rev. Lett. 10 (1963) 277.
[137] H.-Y. Fan, Ann. Phys. (2007), doi: 10.1016/j.aop.2007.08.009. (see Eq. (45))
[138] H.-Y. Fan, J. Phys. A, 25 (1992) 3443.
[139] H. Weyl, The Classical Groups, Princeton University Press, 1953.
[140] G. S. Agarwal and G. Adam, Phys. Rev. A 39 (1989) 6259.
[141] D. Bouwmeester et al., The Physics of Quantum Information (Springer, Berlin, 2000).
[142] H. J. Kimble et al., Phys. Rev. Lett. 39 (1977) 691.
[143] R. Short and L. Mandel, Phys. Rev. Lett. 51 (1983) 384.
[144] G. S. Agarwal, K. Tara, Phys. Rev. A 43 (1991) 492.
[145] S. Sivakumar, J. Opt. B: Quantum Semiclass. Opt. 2 (2000) R61; S. Sivakumar, J. Phys. A: Math.Gen. 32 (1999) 2441.
[146] V V. Dodonov, M. A. Marchiolli, Y. A. Korennoy, V. I. Man’ko, Y. A. Moukhin, Phys. Rev. A 58(1998) 4087.
[147] A. Zavatta, S. Viciani, and M. Bellini, Science 306 (2004) 660.
[148] A. Zavatta, S. Viciani, and M. Bellini, Phys. Rev. A 72 (2005) 023820.
[149] K. Husimi, Proc. Phys. Math. Soc. Jpn. 22 (1940) 264.
[150] H. Y. Fan and Y. L. Yang, Phys. Lett. A, 353 (2006) 439.
[151] M. O. Scully and W. E. Lamb, Jr. 1969 Phys. Rev. 179 368.
[152] B. R. Mollow, 1968 Phys. Rev. 168 1896.
[153] D. T. Smithey et al., Phys. Rev. Lett. 70 (1993) 1244.
[154] H. Y. Fan and H. L. Chen, Chin. Phys. Lett. 18 (2001) 850.
[155] L. Y. Hu and H. Y. Fan, J. Mod. Opt. 55 (2008) 2011-2024.
[156] A. Vourdas, Phys.Rev.A 39 (1989) 206; A. Vourdas, Phys.Rev.A 37 (1988) 3890.
[157] J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. M?lmer, and E. S. Polzik, Phys. Rev.Lett. 97 (2006) 083604.
[158] A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, Ph. Grangier, Science 312 (2006) 83.
[159] K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, Opt. Express 15 (2007) 3568.
[160] M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, Phys. Rev. A 55 (1997) 3184.
[161] S. Glancy and H. M. de Vasconcelos, J. Opt. Soc. Am. B 25 (2008) 712.
[162] A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, Phys. Rev. Lett. 98 (2007) 030502.
[163] D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, Phys. Rev. A 67 (2003) 062320.
[164] H. Nha and H. J. Carmichael, Phys. Rev. Lett. 93 (2004) 020401.
[165] R. Garc′?a-Patro′n, J. Fiura′sˇek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, Phys. Rev.Lett. 93 (2004) 130409.
[166] S. D. Bartlett and B. C. Sanders, Phys. Rev. A 65 (2002) 042304.
[167] S. Olivares and Matteo G. A. Paris, J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S616.
[168] S. Olivares and Matteo G. A. Paris, J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S392.
[169] A. Biswas and G. S. Agarwal, Phys. Rev. A 75 (2007) 032104.
[170] H. Y. Fan and Vander J. Linde, J. Phys. A 24 (1989) 2529.
[171] F. De Martini, Phys. Rev. Lett. 81 (1998) 2842.
[172] M. S. Kim, E. Park, P. L. Knight and H. Jeong, Phys. Rev. A 71 (2005) 043805.
[173] H.Y. Fan, X.G. Meng and J.S. Wang, Commun. Theor. Phys. 46 (2006) 845.
[174] Hong-Yi Fan and Yue Fan, Commun. Theor. Phys. 38 (2002) 297.
[175] X. X. Zhang and Hong-yi Fan. Phys. Lett. A 165 (1992) 14.
[176] I. S. Gradshteyn and L. M. Ryzhik, Tables of Integration Series and Products (Academic Press,New York, 1980).
[177] A. P. Prudnikov, Yu. A. Bychkov, and O. I. Marychev, Integrals and Series: Volume 2: SpecialFunctions (Nauka, Moscow, 1983), subsection 5.12.2.
[178] A. Furusawa, J. L. S?rensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble and E. S. Polzik, Science282 (1998) 706.
[179] T. Opatrny′, G. Kurizki and D-G Welsch, Phys. Rev. A 61 (2000) 032302.
[180] S. Olivares, M. G. A. Paris and R. Bonifacio, Phys. Rev. A 67 (2003) 032314.
[181] A. Kitagawa, M. Takeoka, M. Sasaki and A. Che?es, Phys. Rev. A 73 (2006) 042310.
[182] R. Garc′?a-Patro′n, J. Fiura′sˇek, N. J. Cerf, J. Wenger, R. Tualle-Brouri and Ph. Granger, Phys. Rev.Lett. 93 (2004) 130409; R. Garc′?a-Patro′n, J. Fiura′sˇek and N. J. Cerf, Phys. Rev. A 71 (2005)022105.
[183] For a recent PHD tutorial paper, see M. S. Kim, J. Phys. B. At. Mol. Opt. Phys. 41 (2008) 133001.
[184] Hong-yi Fan and Zhihua Xu, Phys. Rev. A 50 (1994) 2921; Hong-yi Fan, Phys. Rev. A 41 (1990)1526.
[185] W. Magnus et al., Formulas and theorems for the special functions of mathematicalphysics(Springer, 1996).
[186] Z. X. Zhang and Hong-yi Fan, Phys. Lett. A 174 (1993) 206.
[187] I. N. Bronstein, K. A. Semendiajew, Traschenbuch der Mathematik, (B. G. Teubner Verlagsgesell-shaft, Leipzig 1959).
[188] C. T. Lee, Phys. Rev. A 41 (1990) 1569.
[189]范洪义,唐绪兵,量子力学数理基础进展,合肥:中国科学技术大学出版社,2008.
[190] Tohya Hiroshima, Phys. Rev. A 63 (2001) 022305.
[191] H. Jeong, J. Lee and M. S. Kim, Phys. Rev. A 61 (2000) 052101; J. Lee, M. S. Kim and H. Jeong,Phys. Rev. A 62 (2000) 032305.
[192] M. S. Kim and N. Imoto, Phys. Rev. A 52 (1995) 2401.
[193] C. T. Lee, Phys. Rev. A 44 (1991) R2775.
[194] J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Phys. Rev. Lett. 23 (1969) 880; J. F. Clauserand M. A. Horne, Phys. Rev. D. 10 (1974) 526.
[195] K. Banaszek and K. Wo′dkiewicz, Phys. Rev. A. 58 (1998) 4345; Phys. Rev. Lett. 82 (1999) 2009.
[196] A. O. Caldeira and A. J. Leggett, Physica A 121 (1983) 587.
[197] For a review article, see A. Serafini, M. G. A. Paris, F. Illuminati and S. De Siena, J. Opt. B:Quantum Semiclass. Opt. 7 (2005) R19.
[2] M. Gell-Mann and J. B. Hartle, Phys. Rev. D 47 (1993) 3345
[3] D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994);M. Orszag, Quan-tum Optics (Springer Press, 2000)
[4] H. T. Elze, Nucl. Phys. B 436 (1995) 213
[5] W. H. Zurek, Phys. Rev. D 26 (1982) 1862
[6] W. G. Unruh, Phys. Rev. A 51 (1995) 992
[7] D. Bouwmeester, A. Ekert and A. Zeilinger, The Physics of Quantum Information (Springer-Verlag,Berlin, 2000)
[8] C. H. Bennett and D. P. DiVincenzo, Nature (London) 404 (2000) 247
[9] W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973)
[10] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, UK, 1997)
[11] P. L. Knight and L. Allen, Concepts of Quantum Optics (A. Wheaton & Co.Ltd., 1983)
[12] M. Sargent III, et. al., Laser Physics (Addison-Wesley, 1974)
[13] Jan Perina, Quantum Statistics of Linear and Nonlinear Optical Phenomena, (D. Reidel PublishingCompany, 1984)
[14] S. Haroche and J. M. Raimond, in Advances in Atomic and Molecular Physics, Vol. 20, ed. D. Batesand B. Bederson (Academic Press, New York, 1985)
[15] S. Haroche, in High Resolution Laser Spectroscopy, ed. K. Shimoda (Springer-Verlag, Berlin,1976)
[16] A. Mufti, H. A. Schmitt, A. B. Balantekin and M. Sargent III, J. Opt. Soc. Am. B 10 (1993) 2100;S. Chatuverdi and V. Srinivasan, J. Mod. Opt. 38 (1991) 777
[17] H. F. Arnoldus, J. Opt. Soc. Am. B 13 (1996) 1099; M. Ban, Phys. Rev. A 47 (1993) 5093; J. Math.Phys. 33 (1992) 3213; G. M. d’Ariano, Phys. Lett. A 187 (1994) 231
[18] X. X. Yi, C. Li, and J. C. Su, Phys. Rev. A 62 (2000) 013819
[19] L. M. Are‘valo-Aguilar and H. Moya-Cessa, Quantum Semiclassic. Opt. 10 (1998) 671
[20] J. Preskill, Lecture Notes for Physics 229: Quantum Information and Computation, (CaliforniaInstitution of Technology, 1998)
[21] F. Haake, Z. Phys. 223 (1969) 353; 223 (1969) 364171
[22] F. Haake, Statistical Treament of Open Systems by Generalized Master Equations, in SpringerTracts in Modern Physics, Vol. 66 (Springer-Verlag, Berlin, 1973) pp. 98-168
[23] W. Pauli, Probleme der Modernen Physik, Aronold Sommerfeld zum 60, Geburtsage Gewidmetvon Seinen Schulern, volume 1, (Hirzel Verlag, 1928)
[24] R. Zwanzig, Statistical Mechanics of Irreversibilutym, volume 3, (New York: Interscience, 1961)
[25] I. Prigogine, Non-equilibrium Statistical Mechanics (New York: Interscience, 1962)
[26] G. S. Agarwal, Progress in Optics, volume 11, (Amsterdam: North-Holland, 1973)
[27] H. X. Lu, J. Yang, Y. D. Zhang and Z. B. Chen, Phys. Rev. A 67 (2003) 024101; H. X. Lu, Z. B.Chen and Y. D. Zhang, Mod. Phys. Lett. B 16 (2002) 595.
[28] P. A. M. Dirac, The Principles of Quantum Mechanics, (Oxford: Clarendon Press, 1930).
[29] P. A. M. Dirac, The Physicist’s conception of Nature, (New York: Mehra, 1973).
[30] P. A. M. Dirac, Directions in Physics, (New York: John Wiley, 1978)
[31] P. A. M. Dirac, Recollections of an exciting area, History of 20th Century Physics, (New York:Academic Press, 1977, 109).
[32] Hong-yi Fan, H. R. Zaidi and J. R. Klauder, Phys. Rev. D 35 (1987) 1831; Hong-yi Fan and H. R.Zaidi, Phys. Rev. A 37 (1988) 2985; Hong-yi Fan and J. R. Klauder, J. Phys. A 21 (1988) L715;Hong-yi Fan and Hui Zou, Phys. Lett. A 252 (1999) 281; Hong-yi Fan and A. Wunsche, J. Opt. B:Quantum & Semiclass. Opt. 2 (2000) 464.
[33] Hong-yi Fan, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) R147; H. Y. Fan, H. L. Lu and Y. Fan,Ann. Phys. 321 (2006) 480.
[34] A. Wunsche, J. Opt. B: Quantum & Semiclass. Opt. 1 (1999) R11.
[35] Hong-yi Fan, Phys. Lett. A 161 (1991) 1.
[36] H. Weyl, Z. Phys. 46 (1927) 1.
[37] G. Wick, Phys. Rev., 80 (1950) 268.
[38]范洪义,量子力学表象与变换论—狄拉克符号法进展,(上海:上海科学技术出版社,1997).
[39] R. J. Glauber, Phys. Rev. 130 (1963) 2529; 131 (1963) 2766.
[40] J. R. Klauder and B.S. Skargerstam, Coherent States, (World Scientific, Singapore, 1985).
[41] R. Loudon, and P.L. Knight, J. Mod. Opt. 34 (1987) 709; V. Buzˇek, J. Mod. Opt. 37 (1990) 303;Dodonov V V, J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1; See e.g., G. M. D’Ariano, M. G.Rassetti, J. Katriel and A. I. Solomon, Squeezed and Nonclassical Light, ed P Tombesi and E RPike (New York: Plenum 1989) p301.
[42] H. P. Yuen, Phys. Rev. A 13 (1976) 2226.
[43] D. F. Walls, Nature 306 (1983) 141.
[44] P. W. Schleich, Quantum Optics in Phase Space (Berlin: Wiley) 2001.
[45] N. N. Bogolyubov, Lectures on Quantum Statistics (Vol.1), (New York: Gordon and Breach, 1987).
[46] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. 47 (1935) 777.
[47] E. Schrodinger, Naturwiss. 23 (1935) 807, translated to english in“Quantum Theory and Measure-ment”, ed J. A. Wheeler and W. H. Zurek, Princeton University Press (1983).
[48] C. H. Bennett, G. Brassard,C. Crepeau, R. Jozsa, etc., Phys. Rev. Lett. 70 (1993) 1895.
[49] Hong-yi Fan and J. R. Klauder, Phys. Rev. A 49 (1994) 704; Hong-yi Fan, Phys. Rev. A 65 (2002)064102; Hong-yi Fan, Phys. Lett. A 294 (2002) 253;
[50] Hong-yi Fan and Yue Fan, Phys. Rev. A 54 (1996) 958;
[51] J. Preskill, Quantum Information and Computation (Lecture Notes for Physics vol 229) (Pasadena,CA: California Institute of Technology 1998).
[52]范洪义,量子力学纠缠态表象及应用,上海:上海交通大学出版社,2001.
[53] Li-yun Hu and Hong-yi Fan, J. Phys. A: Math. Gen. 39 (2006) 14133; Hong-yi Fan, Nian-quanJiang and Hai-liang Lu, Mod. Phys. Lett B 16 (2002) 1193.
[54] Nian-quan Jiang, Phys. Rev. A 74 (2006) 012306;
[55] Li-yun Hu and Hong-yi Fan, J. Phys. B: At. Mol. Opt. Phys. 40 (2007) 2099; Li-yun Hu and Hong-yi Fan, Int. J. Mod. Phys. A, (2009) to appear; Hong-yi Fan and Lu Hai-liang, J. Phys. A: Math.Gen. 37 (2004) 10993.
[56] Hong-yi Fan, Opt. Lett. 28 (2003) 2177.
[57] E. Wigner, Phys. Rev. 40 (1932) 749; R. F. O’Connell and E. P. Wigner, Phys. Lett. A 83 (1981)145.
[58] R. P. Feynman, Statistical Physics, W. A. Benjamin, 1972.
[59] See e.g., M. Hillery, R. F. O’Connell, M. O. Scully and E. P. Wigner, Phys. Rep. 106 (1984) 121;N. L. Balazs and B. K. Jennings, Phys. Rep. 104 (1984) 347.
[60] K. Vogel and H. Risken, Phys. Rev. A 40 (1989) 2847.
[61] Hong-yi Fan and Hai-ling Cheng, Commun. Theor. Phys. 36 (2001) 651.
[62] Hong-yi Fan and Li-yun Hu, Commun. Theor. Phys. 51 (2009) 321.
[63] Hong-yi Fan and Li-yun Hu, Commun. Theor. Phys. (2009) in press.
[64] Hong-yi Fan and Yue Fan, Commun. Theor. Phys. 38 (2002) 403; 40 (2003) 645.
[65] J. Schwigner, Quantum Theory of Angular Momentum, (New York, Academic Press, 1965).
[66] Y. Takahashi and H. Umezawa, Collecive Phenomena 2 (1975) 55.
[67] Memorial Issue for H. Umezawa, Int. J. Mod. Phys. B 10 (1996) 1695 memorial issue and refer-ences therein.
[68] H. Umezawa, Advanced Field Theory-Micro, Macro, and Thermal Physics (AIP 1993).
[69] G. S. Agarwal, R. Simon, Opt. Commun. 92 (1992) 105; H. Y. Fan, N. Q. Jiang and H. L. Lu, Opt.Commun. 234 (2004) 277.
[70] H. Y. Fan and Y. Fan, Phys. Lett. A 246 (1998) 242.
[71] H. Y. Fan and J. R. Klauder, Phys. Rev. A 49 (1994) 704.
[72] H. Y. Fan and Y. Fan, Phys. Rev. A 54 (1996) 958.
[73] C. Gardiner and P. Zoller, Quantum Noise, Springer Berlin, (2000).
[74] H. Risken, in: The Fokker–Planck Equation: Methods of Solutions and Applications (Springer,Berlin, 1984).
[75] C.W. Gardiner, in: Handbook of Stochastic Methods, 2nd ed., (Springer, Heidelberg, 1990).
[76]唐绪兵,博士学位论文,中国科学技术大学,2008.
[77] H. Y. Fan and Y. Fan, J. Phys. A 35 (2002) 6873.
[78] H. Y. Fan and H. L. Lu, Mod. Phys. Lett. B, 21 (2007) 183.
[79] R. R. Puri, Mathematical Methods of Quantum Optics Springer-Verlag, Berlin, (2001), AppendixA.
[80] H. Y. Fan, Ann. Phys. 323 (2008) 1502; H. Y. Fan and H. Q. Li, Chin. Phys. Lett. 24 (2007) 3322.
[81] H. Y. Fan, Ann. Phys. 323 (2008) 500.
[82] H. Weyl, Z. Phys. 46 (1927) 1.
[83] H. Y. Fan, H. R. Zaidi, Phys. Lett. A 124 (1987) 303; H. Y. Fan, Phys. Rev. A, 65 (2002) 064102.
[84] P. D. Drummond and C. W. Gardiner, J. Phys. A 13 (1980) 2353.
[85] H. Y. Fan and Min Xiao, Phys. Lett. A 244 (1998) 256.
[86] H. Y. Fan, Phys. Lett. A 161 (1991) 1.
[87] H. Y. Fan, Z. W. Liu and T. N. Ruan, Commun. Theor. Phys. 3 (1984) 175; H. Y. Fan and A.Wu¨nsche, Eur. Phys. J. D 15 (2001) 405.
[88] X. F. Xu, Commun. Theor. Phys. 45 (2006) 332.
[89] H. Y. Fan and X. Min, Phys. Lett. A 219 (1996) 175.
[90] H. Y. Fan and X. L. Xu, Commun. Theor. Phys. 48 (2007) 67; L. Fu and H. Y. Fan, Commun.Theor. Phys. 49 (2008) 1461.
[91] N. L. Balazs and B. K. Jennings, Phys. Rep. 104 (1984) 347; W. Schleich, Quantum Optics, NewYork: Wiley, (2001); T. Tatarsky, Sov. Phys.-Usp. 26 (1983) 311.
[92] H. Y. Fan, Commun. Theor. Phys. 37 (2002) 289.
[93] Magnus W, Oberhettinger F and Soni R P, Formulas and Theorems for the Special Functions ofMathematical Physics, 3rd Ed., Springer Verlag, 1966.
[94] X. L. Xu, H. Q. Li and H. Y. Fan, Commun. Theor. Phys. 49 (2008) 1159;H. Y. Fan and J. S. Wang, Commun. Theor. Phys. 47 (2007) 611; H. Y. Fan and W. B. Gao, Com-mun. Theor. Phys. 45 (2006) 617.
[95] L. Y. Hu, N. R. Zhou and H. Y. Fan, Commun. Theor. Phys. 48 (2007) 629; L. Y. Hu and N. R.Zhou, Commun. Theor. Phys. 47 (2007) 805.
[96] Hong-yi Fan, Representation and Transformation Theory in Quantum Mechanics, (Shanghai Sci-entific & Technical, Shanghai, 1997) (in Chinese).
[97] E. Scho¨dinger, Naturwissenschaften 23 (1935) 807.
[98] Hong-yi Fan and Li-yun Hu, Chin. Phys. B (2009) in press.
[99] M. Ueda and N. Imoto, Phys. Rev. A 41 (1990) 3891;M. Ueda, Phys. Rev. A 41 (1990) 3845.
[100] C. Gardiner and P. Zoller, Quantum Noise (Springer Berlin, 2000).
[101] G. S. Agarwal, Phys. Rev. A, 4, (1971) 739.
[102] D. J. Daniel and G. J. Milburn, Phys. Rev. A, 39 (1989) 4628.
[103] M. S. Kim and V. Buzek, Phys. Rev. A 46 (1992) 4239.
[104] H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-PlanckEquations, Springer-Verlag, Berlin, 1999; H. J. Carmichael, Statistical Methods in Quantum Optics2: Non-Classical Fields, (Springer-Verlag, Berlin, 2008).
[105] Hong-yi Fan, Commun. Theor. Phys. 42 (2004) 339.
[106] A. Wu¨nsche, J. Computational and Appl. Math. 133 (2001) 665.
[107] A. Wu¨nsche, J . Phys. A: Math. and Gen. 33 (2000) 1603.
[108] H. X. Lu, J Yang, Y. D. Zhang and Z. B. Chen, Phys. Rev. A. 67 (2003) 024101.
[109] He′ctor Moya-Cessa, Phys. Rep. 432 (2006) 1.
[110] M. S. Kim and V. Buzek, Phys. Rev. A 47 (1993) 610.
[111] S. J. D. Phoenix, Phys. Rev. A 41 (1990) 5132.
[112] C. M. Caves, Phys. Rev. D 26, (1982) 1817; G. J. Milburn, J. Phys. A 17, (1984) 737; C.M. Cavesand B. L. Schumacher, Phys. Rev. A 31, (1985) 3068; M. A. Dupertuis and S. Stenholm, J. Opt.Soc. Am. B 4, (1987) 1102; M. A. Dupertuis, S. M. Barnett and S. Stenholm, ibid. 4, (1987) 1124;
[113] V Bizek, P. L. Knight and I. K. Kudryavtsev, Phys. Rev. A 44, (1991) 1931; A. K. Ekert and P. L.Knight, Phys. Rev. A 42, (1990) 487; T. A. B. Kennedy and D. F. Walls, ibid. 37, (1988) 152.
[114] X.L Xu and H.Y. Fan, Chin. Phys. B (2008) to appear.
[115] Hong-yi Fan, Phys. Lett. A 313 (2003) 343.
[116] Hong-yi Fan and Cao Li, Phys. Lett. A 325 (2004) 188.
[117] Hong-yi Fan, Hui Zou and Yue Fan, Phys. Lett. A 254 (1999) 137.
[118] G. J. Milburn and C. A. Kolmes, Phys. Rev. Lett. 56 (1986) 2237.
[119] S. Chaturvedi, V. Srinivasan and G. S. Agarwal, J. Phys. A. Math. Gen 32 (1999) 1909.
[120] S. Chaturvedi and V. Srinivasan, Phys. Rev. A 43 (1991) 4054; J. Mod. Opt. 38 (1991) 777.
[121] G. S. Agarwal, Phys. Rev. A 61 (1999) 013807.
[122] J.G. Peixoto and de Faria, Eur. Phys. D. 42 (2007) 153 (arXiv:quant-ph/0512060v2).
[123] Hong-yi Fan, Xu-bing Tang and Li-yun Hu, Commun. Theor. Phys. 2008 to appear.
[124] C. H. Bennett, G. Brassard, C. Creau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70(1993) 1895; L. Vaidman, Phys. Rev. A. 49 (1994) 1473.
[125] Li-yun Hu and Hong-yi Fan, Commun. Theor. Phys. 50 (2008) 879
[126] K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theroy. Lecture Notesin Physics, Vol. 190. (Springer-Verlag, Berlin, 1983).
[127] E. B. Davies, Quantum Theory of Open Systems (New York: Academic, 1976).
[128] M. Ozawa, J. Math. Phys. 25 (1984) 79.
[129] H. Y. Fan and X. L. Xu, Commun. Theor. Phys. 48 (2007) 67.
[130] Hong-yi Fan and Li-yun Hu, Commun. Theor. Phys. 2008 to appear
[131] P. L. Kelley and W. H. Kleiner, Phys. Rev. 136 (1964) 316.
[132] L. Mandel, E. C. G. Sudarshan and E. Wolf, Proc. Phys. Soc. (London) 84 (1964) 435.
[133] M. O. Scully and W. E. Lamb, Jr. Quantum Theory of an Optical Maser. III. Theory of Photoelec-tron Counting Statistics, Phys. Rev. 179 (1969) 368.
[134] B. R. Mollow, Quantum Theory of Field Attenuation, Phys. Rev. 168 (1968) 1896.
[135] R. Loudon, The Quantum Theory of Light, Oxford University Press, 2nd, 1983.
[136] E. C. G. Sudarshan, Phys. Rev. Lett. 10 (1963) 277.
[137] H.-Y. Fan, Ann. Phys. (2007), doi: 10.1016/j.aop.2007.08.009. (see Eq. (45))
[138] H.-Y. Fan, J. Phys. A, 25 (1992) 3443.
[139] H. Weyl, The Classical Groups, Princeton University Press, 1953.
[140] G. S. Agarwal and G. Adam, Phys. Rev. A 39 (1989) 6259.
[141] D. Bouwmeester et al., The Physics of Quantum Information (Springer, Berlin, 2000).
[142] H. J. Kimble et al., Phys. Rev. Lett. 39 (1977) 691.
[143] R. Short and L. Mandel, Phys. Rev. Lett. 51 (1983) 384.
[144] G. S. Agarwal, K. Tara, Phys. Rev. A 43 (1991) 492.
[145] S. Sivakumar, J. Opt. B: Quantum Semiclass. Opt. 2 (2000) R61; S. Sivakumar, J. Phys. A: Math.Gen. 32 (1999) 2441.
[146] V V. Dodonov, M. A. Marchiolli, Y. A. Korennoy, V. I. Man’ko, Y. A. Moukhin, Phys. Rev. A 58(1998) 4087.
[147] A. Zavatta, S. Viciani, and M. Bellini, Science 306 (2004) 660.
[148] A. Zavatta, S. Viciani, and M. Bellini, Phys. Rev. A 72 (2005) 023820.
[149] K. Husimi, Proc. Phys. Math. Soc. Jpn. 22 (1940) 264.
[150] H. Y. Fan and Y. L. Yang, Phys. Lett. A, 353 (2006) 439.
[151] M. O. Scully and W. E. Lamb, Jr. 1969 Phys. Rev. 179 368.
[152] B. R. Mollow, 1968 Phys. Rev. 168 1896.
[153] D. T. Smithey et al., Phys. Rev. Lett. 70 (1993) 1244.
[154] H. Y. Fan and H. L. Chen, Chin. Phys. Lett. 18 (2001) 850.
[155] L. Y. Hu and H. Y. Fan, J. Mod. Opt. 55 (2008) 2011-2024.
[156] A. Vourdas, Phys.Rev.A 39 (1989) 206; A. Vourdas, Phys.Rev.A 37 (1988) 3890.
[157] J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. M?lmer, and E. S. Polzik, Phys. Rev.Lett. 97 (2006) 083604.
[158] A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, Ph. Grangier, Science 312 (2006) 83.
[159] K. Wakui, H. Takahashi, A. Furusawa, and M. Sasaki, Opt. Express 15 (2007) 3568.
[160] M. Dakna, T. Anhut, T. Opatrny, L. Knoll, and D.-G. Welsch, Phys. Rev. A 55 (1997) 3184.
[161] S. Glancy and H. M. de Vasconcelos, J. Opt. Soc. Am. B 25 (2008) 712.
[162] A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, and Ph. Grangier, Phys. Rev. Lett. 98 (2007) 030502.
[163] D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, Phys. Rev. A 67 (2003) 062320.
[164] H. Nha and H. J. Carmichael, Phys. Rev. Lett. 93 (2004) 020401.
[165] R. Garc′?a-Patro′n, J. Fiura′sˇek, N. J. Cerf, J. Wenger, R. Tualle-Brouri, and P. Grangier, Phys. Rev.Lett. 93 (2004) 130409.
[166] S. D. Bartlett and B. C. Sanders, Phys. Rev. A 65 (2002) 042304.
[167] S. Olivares and Matteo G. A. Paris, J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S616.
[168] S. Olivares and Matteo G. A. Paris, J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S392.
[169] A. Biswas and G. S. Agarwal, Phys. Rev. A 75 (2007) 032104.
[170] H. Y. Fan and Vander J. Linde, J. Phys. A 24 (1989) 2529.
[171] F. De Martini, Phys. Rev. Lett. 81 (1998) 2842.
[172] M. S. Kim, E. Park, P. L. Knight and H. Jeong, Phys. Rev. A 71 (2005) 043805.
[173] H.Y. Fan, X.G. Meng and J.S. Wang, Commun. Theor. Phys. 46 (2006) 845.
[174] Hong-Yi Fan and Yue Fan, Commun. Theor. Phys. 38 (2002) 297.
[175] X. X. Zhang and Hong-yi Fan. Phys. Lett. A 165 (1992) 14.
[176] I. S. Gradshteyn and L. M. Ryzhik, Tables of Integration Series and Products (Academic Press,New York, 1980).
[177] A. P. Prudnikov, Yu. A. Bychkov, and O. I. Marychev, Integrals and Series: Volume 2: SpecialFunctions (Nauka, Moscow, 1983), subsection 5.12.2.
[178] A. Furusawa, J. L. S?rensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble and E. S. Polzik, Science282 (1998) 706.
[179] T. Opatrny′, G. Kurizki and D-G Welsch, Phys. Rev. A 61 (2000) 032302.
[180] S. Olivares, M. G. A. Paris and R. Bonifacio, Phys. Rev. A 67 (2003) 032314.
[181] A. Kitagawa, M. Takeoka, M. Sasaki and A. Che?es, Phys. Rev. A 73 (2006) 042310.
[182] R. Garc′?a-Patro′n, J. Fiura′sˇek, N. J. Cerf, J. Wenger, R. Tualle-Brouri and Ph. Granger, Phys. Rev.Lett. 93 (2004) 130409; R. Garc′?a-Patro′n, J. Fiura′sˇek and N. J. Cerf, Phys. Rev. A 71 (2005)022105.
[183] For a recent PHD tutorial paper, see M. S. Kim, J. Phys. B. At. Mol. Opt. Phys. 41 (2008) 133001.
[184] Hong-yi Fan and Zhihua Xu, Phys. Rev. A 50 (1994) 2921; Hong-yi Fan, Phys. Rev. A 41 (1990)1526.
[185] W. Magnus et al., Formulas and theorems for the special functions of mathematicalphysics(Springer, 1996).
[186] Z. X. Zhang and Hong-yi Fan, Phys. Lett. A 174 (1993) 206.
[187] I. N. Bronstein, K. A. Semendiajew, Traschenbuch der Mathematik, (B. G. Teubner Verlagsgesell-shaft, Leipzig 1959).
[188] C. T. Lee, Phys. Rev. A 41 (1990) 1569.
[189]范洪义,唐绪兵,量子力学数理基础进展,合肥:中国科学技术大学出版社,2008.
[190] Tohya Hiroshima, Phys. Rev. A 63 (2001) 022305.
[191] H. Jeong, J. Lee and M. S. Kim, Phys. Rev. A 61 (2000) 052101; J. Lee, M. S. Kim and H. Jeong,Phys. Rev. A 62 (2000) 032305.
[192] M. S. Kim and N. Imoto, Phys. Rev. A 52 (1995) 2401.
[193] C. T. Lee, Phys. Rev. A 44 (1991) R2775.
[194] J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Phys. Rev. Lett. 23 (1969) 880; J. F. Clauserand M. A. Horne, Phys. Rev. D. 10 (1974) 526.
[195] K. Banaszek and K. Wo′dkiewicz, Phys. Rev. A. 58 (1998) 4345; Phys. Rev. Lett. 82 (1999) 2009.
[196] A. O. Caldeira and A. J. Leggett, Physica A 121 (1983) 587.
[197] For a review article, see A. Serafini, M. G. A. Paris, F. Illuminati and S. De Siena, J. Opt. B:Quantum Semiclass. Opt. 7 (2005) R19.