连续纠缠态表象在理论量子光学中的应用
|
摘要
Dirac的符号法是学习量子物理的人所必须习惯的“语言”,它对物理本质的深刻反映在某种程度上超越了时代,它的内涵与美仍然需要进一步的认知。有序算符内的积分技术(英文简称为IWOP技术)和连续纠缠态表象(建立在EPR量子纠缠思想上)是对量子力学Dirac符号理论的创新发展,丰富了量子力学的数理基础理论,不但进一步揭示了Dirac符号法的科学美,而且开拓了连续变量纠缠态表象在多个物理领域的新应用,本文将介绍其在理论量子光学研究中的若干应用。
一、考虑到产生和消灭算符在纠缠EPR表象中的微商对应,我们可以将描述随机过程的Fokker-Planck方程纳入到纠缠态表象意义下讨论,方程的李代数结构和物理意义可以得到很清楚的展现;一方面,利用“对相干态”(或者SU(1,1)相干态),我们指出:对相干态在纠缠态<ξ|表象中的波函数就是一类典型的Fokker-Planck微分算子的本征解。另一方面,对于某些复变量Fokker-Planck微分方程来说,其中包含的微分算子在纠缠表象中的算符对应满足SU(1,1)的李代数结构,根据算符的分解公式,我们给出了这些复变量Fokker-Planck微分方程的求解过程。
二、密度主方程是量子光学研究中的重要方法。主方程的求解方法也有多种(超算符方法、特征函数法等),这里我们提出了一个新的求解方法——纠缠态表象法。该方法能方便、简洁地将密度主方程转化为对应的普通微分方程,并能从微分方程的解中提取出约化密度算符。文中以压缩光场热库中的阻尼谐振子模型为例说明该新方法的应用情况,并且计算了密度算符的Wigner函数,具体分析了系统的退相干过程。相比于传统的求解方法,我们的纠缠态表象法简洁而有效。
三、量子力学中存在着许多变换,利用IWOP技术,可以建立经典变换与量子幺正变换之间的联系,发展表象变换理论。Dirac曾称变换理论为“我一生中最使我兴奋的一件工作”,可见表象变换理论的重要性。相干—纠缠态是一类比较特殊的表象,皆有相干和纠缠的特性,从相干—纠缠态表象下的表象变换出发,我们导出了一个新的压缩算符,并且发现该算符具有群乘的性质。而这个压缩算符在连续纠缠表象下的矩阵元则是经典Lenz-Fresnel光学变换的积分核,实现了经典Lenz-Fresnel光学变换与量子力学幺正算符之间的对应。
四、纠缠EPR表象不仅可以应用于理论量子光学研究,还可用于分析量子信息的各种理论方案和实验研究,以及玻色凝聚的相干性问题研究中。文中利用双模纠缠Wigner算符具体分析了光分束器的纠缠规则;对于相干BEC的位相特征研究,我们提出原子相干态是描述两相干BEC密度算符的忠实表象,将以往文献中提出所谓“相态”分析纳入到原子相干态中来研究相干BEC,使得问题处理方式更丰富,物理意义更加明晰。
Dirac symbolic method is a "language", which must be used by everyone for studying quantum physics. In some sense, its profoundly reflect to physical nature has beyond the era, and its connotation and beauty still needs further awareness. The technique of integration within an ordered product of operators (IWOP for short) and continuous entangled state representation (originated from the idea of entanglement by EPR) are the innovative developments to Dirac symbolic theory, enrich the basic mathematical and physical theory of quantum mechanics, and not only further reveal the scientific beauty of Dirac symbolic method, but opened up some new applications of continuous variable entangled state representation in a number of physics research areas. In this work, we will introduce its applications in theoretical quantum optics.
First, taking creation and annihilation operator mapping into differential operatorsin entangled state representation into account, we will be able to discuss the Fokker-Planck equation of a random process in the sense of entangled state represention, and show its Li algera structure and physical meaning clearly. On one hand, by virtue of "pair coherent state", We find that the wavefunction of a pair coherent state (or SU(1,1) coherent state) in the entangled state representation is just the eigenfunction of a type of Fokker-Planck differential operator. On the other hand, we point out that some complex variables Fokker-Planck differential equations, which includes a number of differential operators of SU(1,1) algebra, can be solved by using the operator decomposition formula.
Second, the reduced denity master equation (ME) is an important method in the study of quantum optics. We can employ various methods (Superoperator method, Characteristic function, etc.) to deal with ME. Here we have proposed a new method -entangled state representation method to solve ME. Using it we can conveniently and simply convert a ME into its corresponding differential equation and from solution of which we can extract the reduced density operator. In this work, we have made a damping harmonic oscillator in a squeezed vacuum bath for an example to interpret this new method and then calculated the Wigner function of the reduced density operator to specifically analyse the system's decoherence process. Comparing to the traditional method of solving ME, our method is simple and effective.
Third, there exists many transformations in quantum mechanics. By virtue of the technique of IWOP, we can construct a" bridge " between classical canonical tranformation and quantum unitary transformation, and promote a further development of Dirac's representation transformation theory. Dirac himself admired the theory of canonical transformations in quantum mechanics very much," I think that is the piece of work which has most pleased me of all the works that I've done in my lif.... The transformation theory (became) my darling." and that shows the importance of the Dirac's representation transformation theory. Coherent-entangled state is a relatively unique one which is of coherence and entanglement characteristics. Making the representation transformation to the coherent-entangled state, we have derived a new squeezing operator in a nature way and found this unitary operator can satisfy the group multiplication rule. And we have proved that this operator and the classical Lenz-Fresnel transformation are related in such a manner that the matrix element of it in the entangled state representation is just the kernel of the Lenz-Fresnel transformation and established a link between the classical Lenz-Fresnel optical transformation and its counterparts in quantum mechanics.
Fourth, we can apply the entangled state representation for not only studying theoretical quantum optics, but for the analysis of various theoretical and experimental in quantum information, as well as for deeply studying the interference between two Bose-Einstein condensates. In this work, by using the two-mode entangled Wigner operator, we have made a deep analysis to the optical beam splitter entanglement rules. In order to demonstrate the interference patterns of BECs, we have proposed that the atomic coherent state (ACS) in Schwinger bosonic realization is a faithful representation for describing the steady relative phase of interference BECs. The so-called "phase state ", which has been introduced in before literature, can be replaced by ACS. That can enrich the way of dealing with problem and clarify the physics meaning.
一、考虑到产生和消灭算符在纠缠EPR表象中的微商对应,我们可以将描述随机过程的Fokker-Planck方程纳入到纠缠态表象意义下讨论,方程的李代数结构和物理意义可以得到很清楚的展现;一方面,利用“对相干态”(或者SU(1,1)相干态),我们指出:对相干态在纠缠态<ξ|表象中的波函数就是一类典型的Fokker-Planck微分算子的本征解。另一方面,对于某些复变量Fokker-Planck微分方程来说,其中包含的微分算子在纠缠表象中的算符对应满足SU(1,1)的李代数结构,根据算符的分解公式,我们给出了这些复变量Fokker-Planck微分方程的求解过程。
二、密度主方程是量子光学研究中的重要方法。主方程的求解方法也有多种(超算符方法、特征函数法等),这里我们提出了一个新的求解方法——纠缠态表象法。该方法能方便、简洁地将密度主方程转化为对应的普通微分方程,并能从微分方程的解中提取出约化密度算符。文中以压缩光场热库中的阻尼谐振子模型为例说明该新方法的应用情况,并且计算了密度算符的Wigner函数,具体分析了系统的退相干过程。相比于传统的求解方法,我们的纠缠态表象法简洁而有效。
三、量子力学中存在着许多变换,利用IWOP技术,可以建立经典变换与量子幺正变换之间的联系,发展表象变换理论。Dirac曾称变换理论为“我一生中最使我兴奋的一件工作”,可见表象变换理论的重要性。相干—纠缠态是一类比较特殊的表象,皆有相干和纠缠的特性,从相干—纠缠态表象下的表象变换出发,我们导出了一个新的压缩算符,并且发现该算符具有群乘的性质。而这个压缩算符在连续纠缠表象下的矩阵元则是经典Lenz-Fresnel光学变换的积分核,实现了经典Lenz-Fresnel光学变换与量子力学幺正算符之间的对应。
四、纠缠EPR表象不仅可以应用于理论量子光学研究,还可用于分析量子信息的各种理论方案和实验研究,以及玻色凝聚的相干性问题研究中。文中利用双模纠缠Wigner算符具体分析了光分束器的纠缠规则;对于相干BEC的位相特征研究,我们提出原子相干态是描述两相干BEC密度算符的忠实表象,将以往文献中提出所谓“相态”分析纳入到原子相干态中来研究相干BEC,使得问题处理方式更丰富,物理意义更加明晰。
Dirac symbolic method is a "language", which must be used by everyone for studying quantum physics. In some sense, its profoundly reflect to physical nature has beyond the era, and its connotation and beauty still needs further awareness. The technique of integration within an ordered product of operators (IWOP for short) and continuous entangled state representation (originated from the idea of entanglement by EPR) are the innovative developments to Dirac symbolic theory, enrich the basic mathematical and physical theory of quantum mechanics, and not only further reveal the scientific beauty of Dirac symbolic method, but opened up some new applications of continuous variable entangled state representation in a number of physics research areas. In this work, we will introduce its applications in theoretical quantum optics.
First, taking creation and annihilation operator mapping into differential operatorsin entangled state representation into account, we will be able to discuss the Fokker-Planck equation of a random process in the sense of entangled state represention, and show its Li algera structure and physical meaning clearly. On one hand, by virtue of "pair coherent state", We find that the wavefunction of a pair coherent state (or SU(1,1) coherent state) in the entangled state representation is just the eigenfunction of a type of Fokker-Planck differential operator. On the other hand, we point out that some complex variables Fokker-Planck differential equations, which includes a number of differential operators of SU(1,1) algebra, can be solved by using the operator decomposition formula.
Second, the reduced denity master equation (ME) is an important method in the study of quantum optics. We can employ various methods (Superoperator method, Characteristic function, etc.) to deal with ME. Here we have proposed a new method -entangled state representation method to solve ME. Using it we can conveniently and simply convert a ME into its corresponding differential equation and from solution of which we can extract the reduced density operator. In this work, we have made a damping harmonic oscillator in a squeezed vacuum bath for an example to interpret this new method and then calculated the Wigner function of the reduced density operator to specifically analyse the system's decoherence process. Comparing to the traditional method of solving ME, our method is simple and effective.
Third, there exists many transformations in quantum mechanics. By virtue of the technique of IWOP, we can construct a" bridge " between classical canonical tranformation and quantum unitary transformation, and promote a further development of Dirac's representation transformation theory. Dirac himself admired the theory of canonical transformations in quantum mechanics very much," I think that is the piece of work which has most pleased me of all the works that I've done in my lif.... The transformation theory (became) my darling." and that shows the importance of the Dirac's representation transformation theory. Coherent-entangled state is a relatively unique one which is of coherence and entanglement characteristics. Making the representation transformation to the coherent-entangled state, we have derived a new squeezing operator in a nature way and found this unitary operator can satisfy the group multiplication rule. And we have proved that this operator and the classical Lenz-Fresnel transformation are related in such a manner that the matrix element of it in the entangled state representation is just the kernel of the Lenz-Fresnel transformation and established a link between the classical Lenz-Fresnel optical transformation and its counterparts in quantum mechanics.
Fourth, we can apply the entangled state representation for not only studying theoretical quantum optics, but for the analysis of various theoretical and experimental in quantum information, as well as for deeply studying the interference between two Bose-Einstein condensates. In this work, by using the two-mode entangled Wigner operator, we have made a deep analysis to the optical beam splitter entanglement rules. In order to demonstrate the interference patterns of BECs, we have proposed that the atomic coherent state (ACS) in Schwinger bosonic realization is a faithful representation for describing the steady relative phase of interference BECs. The so-called "phase state ", which has been introduced in before literature, can be replaced by ACS. That can enrich the way of dealing with problem and clarify the physics meaning.
引文
[1]P.A.M.Dirac,The Principles of Quantum Mechanics,Oxford:Clarendon Press,1930;
[2]P.A.M.Dirac,The Physicist's conception of Nature,New York:Mehra,1973;
[3]P.A.M.Dirac,Directions in Physics,New York:John Wiley,1978;
[4]P.A.M.Dirac,Recollections of an exciting area,History of 20~(th)Century Physics.New York:Academic Press,1977,109;
[5]Fan Hong-yi,H.R.Zaidi and J.R.Klauder,Phys.Rev.D(1987)1831;Fan Hong-yi 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;
[6]Hong-yi Fan,J..Opt.B:Quantum Semiclass.Opt.5(2003)R1-R17;
[7]A.Wunsche,J..Opt.B:Quantum & Semiclass.Opt.1(1999)R11;
[8]Hong-yi Fan and Hai-liang Lu,J.Opt.B,19(2005)799-856
[9]Hong-yi Fan,Hui Zou and Yue Fan,Phys.Lett.A 254(1999)137;
[10]Hong-yi Fan,Phys.Lett.A 313(2003)343;
[11]W.H.Louisell,Quantum Statistical Properties of Radiation,New York:Wiley,1973;
[12]G.Wick,Phys.Rev.,80(1950)268;
[13]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;
[14]A.Einstein,B.Podolsky and N,Rosen,Phys.Rev.,1935,47:777;
[15]E.Schrodinger,Naturwiss.23 807,(1935),translated to english in "Quantum Theory and Measurement",ed J.A.Wheeler and W.H.Zurek,Princeton University Press(1983);
[16]Fan Hong-yi and J.R.Klauder,Phys.Rev.A 49(1994)704;Hongyi Fan,Phys.Rev.A 65(2002)064102;Hong-yi Fan,Phys.Lett.A 294(2002)253;
[17]Fan Hong-yi and Fan Yue,Phys.Rev.A 54(1996)958;
[18]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:
[19]Hong-yi Fan,Nian-quan Jiang and Hai-liang Lu,Mod.Phys.Lett B,vol.16,No.30(2002)1193;
[20]Nian-quan Jiang,Phys.Rev.A,74(2006)012306;
[21]Fan Hong-yi and Lu Hai-liang,J.Phys.A:Math.Gen.37(2004)10993;Fan Hong-yi and Tang Xu-bing,J.Opt.B:Quan.& Semiclass.Opt.7(2005)5765;
[22]Fan Hong-yi,Opt.Lett.28(2003)2177;
[1]W.H.Louisell,Quantum Statistical Properties of Radiation,John Willey & Sons,1973;
[2]S.M.Ross,Stochastic Processes,John Willey & Sons,Inc.,1983;
[3]C.W.Gardiner,Handbook of Stochastic Processes,Springer,Berlin,Heidelberg,1985;
[4]H.Haken,Light Theory,Springer-Verlag,Berlin Heidelberg,1984;H.Haken,Light,Vol.1,North-Holland Publishing Company,1981;
[5]M.Orszag,Quantum Optics,Springer,New York,2000;
[6]M.O.Scully and M.S.Zubairy,Quantum Optics,Cambridge University Press,UK,1997;
[7]Hong-yi Fan and Xu-bing Tang,J.Phys.A:Math.Gen.39(2006)1-8;
[8]D.Bhaumik,K.Bhaumik and B.Dutta,J.Phys.A:Math.Gen.9(1976)1507;
[9]G.S.Agarwal,J.Opt.Soc.Am.B5(1988)1940;G.S.Agarwal,Phys.Rev.Lett.57(1986)827;
[10]M.Perelomov,Generalized Coherent States and Their Applications,Berlin:Springer,1986;
[11]J.Glauber,Phys.Rev.,131(1963)2766;
[12]C.G.Sudarshan,Phys.Rev.Lett.10(1963)277;
[13]F.Walls and G.J.Milburn,Quantum Optics,Springer-Verlag,Berlin,1994;
[14]Fan Hong-yi and Lu Hai-Liang,Commun.Theor.Phys.,46(2006)599-602;
[1]M.O.Scully and M.S.Zubairy,Quantum Optics,Cambridge University Press,UK,1997;
[2]P.L.Knight and L.Allen,Concepts of Quantum Optics,A.Wheaton & Co.Ltd.,1983;
[3]M.Sargent Ⅲ,et.al.,Laser Physics,Addison-Wesley,1974;
[4]Jan Perina,Quantum Statistics of Linear and Nonlinear Optical Phenomena,D.Reidel Publishing Company,1984;
[5]S.Haroche and J.M.Raimond,in Advances in Atomic and Molecular Physics,Vol.20,ed.D.Bates and B.Bederson(Academic Press,New York 1985);
[6]S.Haroche,in High Resolution Laser Spectroscopy,ed.K.Shimoda(Springer-Verlag,Berlin)1976;
[7]W.H.Louisell,Quantum Statistical Properties of Radiation,John Willey & Sons,1973;
[8]W.Pauli,Probleme der Modernen Physik,AronoM Sommerfeld zum 60,Geburtsage Gewidmet von Seinen Schulern,volume 1,Hirzel Verlag,1928;
[9]R.Zwanzig,Statistical Mechanics of Irreversibilutym,volume 3,New York:Interscience,1961;
[10]I.Prigogine,Non-equilibrium Statistical Mechanics.New York:Interscience,1962;
[11]G.S.Agarwal,Progress in Optics,volume 11,Amsterdam:North-Holland,1973;
[12]Luis M Ar'evalo-Aguilar and H'ector Moya-Cessa,Quantum Semiclass Opt.,10(1998)671;
[13]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;逯怀新,连续变量量子信息论的若干研究——量子变换理论的应用,中国科学技术大学博士学位论文,2003;
[14]J.Preskill,Lecture Notes for Physics 229:Quantum Information and Computation,California Institution of Technology,1998;
[15]Memorial Issue for H.Umezawa,Int.J.Mod.Phys.B,10(1996);Y.Takahashi and H.Umezawa,Collective Phenomena,2(1975)55;
[16]Hong-yi Fan and Yue Fan,Phys.Lett.A,246(1998)242;Hong-yi Fan and Hui Wang,Mod.Phys.Lett.B,14(2000)553;
[17]范洪义,量子力学表象与变换论——狄拉克符号法进展,上海科学技术出版社,上海,1997;
[1]吕乃光,傅里叶光学(第二版),机械工业出版社,2007;
[2]宋菲君、S.Jutamulia,近代光学信息处理,北京大学出版社,北京,1998;
[3]L.Mandel and E.Wolf,Optical Coherence and Quantum Optics,Cambridge,1996;
[4]E.U.Condon,Acad.Sci.USA,23(1937)158-164;H.M.Ozakatas,D.Mendlovic.J.Opt.Soc.Am.A,10(1993)2522;D.Mendlovic,H.M.Ozakatas.J.Opt.Soc.Am.A,10(1993)1875;D.Mendlovic,H.M.Ozaktas and A.W.Lohmann,Appl.Opt.33(1994)6188;M.A.Pinsky,Introduction to Fourier Analysis and Wavelets,Book/Cole,2002;
[5]V.Namias,J.Inst.Maths Applics,25(1980)241;
[6]A.C.McBride and F.H.Kerr,IMA J.Appl.Maths,39(1987)159;
[7]A.W.Lohmann,J.Opt.Soc.Am.A,10(1993)2181;
[8]A.Gerrard and N.M.Butch,Introduction to Matrix Methods in Optics,John Wiley & Sons,London,1975;
[9]A.Torre,FRFT and its applications in Optics,Progress in Optics,Vol.43,edited by E.Wolf,2003 and references therein;
[10]J.W.Goodman,Introduction to Fourier Optics,McGraw -Hill,New York,1968;
[11]Hong-yi Fan and Hai-liang Lu,Phys.Lett.A,334(2005)132;
[12]Shao-min Wang and Dao-mu Zhao,Matrix Optics,China Higher Education Press,Beijing and Springer-Verlag,Berlin Heidelberg(2000);
[13]Hong-yi Fan and Hai-liang Lu,Journal of Optics B:Quantum and Semiclassical Optics,9(2005)799-856;
[14]Hong-yi Fan,Hui Zou and Yue Fan,Phys.Lett.A 254(1999)137;
[15]Hong-yi Fan,Phys.Lett.A 313(2003)343;
[16]Hong-yi Fan,Hai-liang Lu,J.Phys.A:Math.Gen.37(2004)10993;
[17]Hong-yi Fan,Xu-bing Tang,J.Opt.B:Quantum Semielass.Opt.7(2005)s765;
[18]R.A.Campos,B.E.A.Saleh,M.C.Teich,Phys.Rev.A 50(1989)5274;
[19]C.Silberhom,et at.,Phys.Rev.Lett.86(2001)4267;
[20]范洪义,量子力学表象与变换论——狄拉克符号法进展,上海科学技术出版社,上海,1997:
[21]H.Kogelnik,Appl.Opt.4(1965)1562;B.X.Chen and B.H.Sun,Chin.Phys.Lett.20(2003)1254;
[22]Hong-yi Fan and Li-yun Hu,Chin.Phys.Lett.24:(8)(2007)2238;
[23]Hong-yi Fan and Xu-bing Tang,Phys.Lett.A 357(2006)163;
[1]C.H.Bennett,G Brassard,C.Crepeau,R.Jozsa,A.Peres and W.Wootters,Phys.Rev.Lett.70(1993)1895;
[2]Hong-yi Fan and J.R.Klauder,Phys.Rev.A 49(1994)704;
[3]Hong-yi Fan,International Journal of Modern Physics B 18:(10)(2004)1;
[4]J.W.Pan,D.Bouwmeester,H.Weinfurter and A.Zeilinger,Phys.Rev.Lett.80(1998)3891;
[5]Z.Zhao,et al.,Experimental demonstration of five-photon entanglement and open-destination teleportation,Nature,430(2004);
[6]E.Wigner,Phys.Rev.40(1932)749;
[7]Hong-yi Fan and Yu-nan Ruan,Commun.Theor.Phys.3:(3)(1984)345;2:(5)1563;Fan Hong-yi and Min Xiao,Mod.Phys.Lett.B10(1996)989.
[8]Hong-yi Fan,Phys.Lett.A 124:(6,7)(1987)303;
[9]H.J.Wu and Hong-yi Fan,Mod.Phys.Lett.B 11(1997)544;
[I0]Hong-yi Fan and Yue Fan,Mod.Phys.Lett.A 13(1998)433;
[11]范洪义,量子力学纠缠态表象及应用,上海交通大学出版社,上海,2000;
[12]K.B.Davis,M.O.Mewes,M.R.Andrews,N.J.van Druten,D.S.Durfee,D.M.Kurn and W.Ketterle,Phys.Rev.Lett.,75(1995)3996;
[13]W.J.Mullin and R.Krotkov,Am.J.Phys.74(2006)880;
[14]J.Schwinger,Quantum Theory of Angular Momentum,New York:Academic press,1965;
[15]F.T.Arrechi,E.Courtens,R.Gilmore and H.Thomas,Phys.Rev.,6(1972)2211;
[16]Hong-yi Fan,Chao Li and Zhong-hua Jiang,Phys.Lett.A,327(2004)416;
[17]L.Susskind and J.Glogower,Physics,1(1964)49;
[18]P.Carruthers and M.M.Nieto,Rev.Mod.Phys.,40(1968)411;R.Loudon and P.L.Knight,Jour.Mod.Opt.,34(1987)709;
[19]Xu-bing Tang and Hong-yi Fan,Atomic coherent state,entangled state and phase operator for studying the interference between two Bose-Einstein condensates,Commun.Theor.Phys.(in press);
[2]P.A.M.Dirac,The Physicist's conception of Nature,New York:Mehra,1973;
[3]P.A.M.Dirac,Directions in Physics,New York:John Wiley,1978;
[4]P.A.M.Dirac,Recollections of an exciting area,History of 20~(th)Century Physics.New York:Academic Press,1977,109;
[5]Fan Hong-yi,H.R.Zaidi and J.R.Klauder,Phys.Rev.D(1987)1831;Fan Hong-yi 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;
[6]Hong-yi Fan,J..Opt.B:Quantum Semiclass.Opt.5(2003)R1-R17;
[7]A.Wunsche,J..Opt.B:Quantum & Semiclass.Opt.1(1999)R11;
[8]Hong-yi Fan and Hai-liang Lu,J.Opt.B,19(2005)799-856
[9]Hong-yi Fan,Hui Zou and Yue Fan,Phys.Lett.A 254(1999)137;
[10]Hong-yi Fan,Phys.Lett.A 313(2003)343;
[11]W.H.Louisell,Quantum Statistical Properties of Radiation,New York:Wiley,1973;
[12]G.Wick,Phys.Rev.,80(1950)268;
[13]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;
[14]A.Einstein,B.Podolsky and N,Rosen,Phys.Rev.,1935,47:777;
[15]E.Schrodinger,Naturwiss.23 807,(1935),translated to english in "Quantum Theory and Measurement",ed J.A.Wheeler and W.H.Zurek,Princeton University Press(1983);
[16]Fan Hong-yi and J.R.Klauder,Phys.Rev.A 49(1994)704;Hongyi Fan,Phys.Rev.A 65(2002)064102;Hong-yi Fan,Phys.Lett.A 294(2002)253;
[17]Fan Hong-yi and Fan Yue,Phys.Rev.A 54(1996)958;
[18]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:
[19]Hong-yi Fan,Nian-quan Jiang and Hai-liang Lu,Mod.Phys.Lett B,vol.16,No.30(2002)1193;
[20]Nian-quan Jiang,Phys.Rev.A,74(2006)012306;
[21]Fan Hong-yi and Lu Hai-liang,J.Phys.A:Math.Gen.37(2004)10993;Fan Hong-yi and Tang Xu-bing,J.Opt.B:Quan.& Semiclass.Opt.7(2005)5765;
[22]Fan Hong-yi,Opt.Lett.28(2003)2177;
[1]W.H.Louisell,Quantum Statistical Properties of Radiation,John Willey & Sons,1973;
[2]S.M.Ross,Stochastic Processes,John Willey & Sons,Inc.,1983;
[3]C.W.Gardiner,Handbook of Stochastic Processes,Springer,Berlin,Heidelberg,1985;
[4]H.Haken,Light Theory,Springer-Verlag,Berlin Heidelberg,1984;H.Haken,Light,Vol.1,North-Holland Publishing Company,1981;
[5]M.Orszag,Quantum Optics,Springer,New York,2000;
[6]M.O.Scully and M.S.Zubairy,Quantum Optics,Cambridge University Press,UK,1997;
[7]Hong-yi Fan and Xu-bing Tang,J.Phys.A:Math.Gen.39(2006)1-8;
[8]D.Bhaumik,K.Bhaumik and B.Dutta,J.Phys.A:Math.Gen.9(1976)1507;
[9]G.S.Agarwal,J.Opt.Soc.Am.B5(1988)1940;G.S.Agarwal,Phys.Rev.Lett.57(1986)827;
[10]M.Perelomov,Generalized Coherent States and Their Applications,Berlin:Springer,1986;
[11]J.Glauber,Phys.Rev.,131(1963)2766;
[12]C.G.Sudarshan,Phys.Rev.Lett.10(1963)277;
[13]F.Walls and G.J.Milburn,Quantum Optics,Springer-Verlag,Berlin,1994;
[14]Fan Hong-yi and Lu Hai-Liang,Commun.Theor.Phys.,46(2006)599-602;
[1]M.O.Scully and M.S.Zubairy,Quantum Optics,Cambridge University Press,UK,1997;
[2]P.L.Knight and L.Allen,Concepts of Quantum Optics,A.Wheaton & Co.Ltd.,1983;
[3]M.Sargent Ⅲ,et.al.,Laser Physics,Addison-Wesley,1974;
[4]Jan Perina,Quantum Statistics of Linear and Nonlinear Optical Phenomena,D.Reidel Publishing Company,1984;
[5]S.Haroche and J.M.Raimond,in Advances in Atomic and Molecular Physics,Vol.20,ed.D.Bates and B.Bederson(Academic Press,New York 1985);
[6]S.Haroche,in High Resolution Laser Spectroscopy,ed.K.Shimoda(Springer-Verlag,Berlin)1976;
[7]W.H.Louisell,Quantum Statistical Properties of Radiation,John Willey & Sons,1973;
[8]W.Pauli,Probleme der Modernen Physik,AronoM Sommerfeld zum 60,Geburtsage Gewidmet von Seinen Schulern,volume 1,Hirzel Verlag,1928;
[9]R.Zwanzig,Statistical Mechanics of Irreversibilutym,volume 3,New York:Interscience,1961;
[10]I.Prigogine,Non-equilibrium Statistical Mechanics.New York:Interscience,1962;
[11]G.S.Agarwal,Progress in Optics,volume 11,Amsterdam:North-Holland,1973;
[12]Luis M Ar'evalo-Aguilar and H'ector Moya-Cessa,Quantum Semiclass Opt.,10(1998)671;
[13]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;逯怀新,连续变量量子信息论的若干研究——量子变换理论的应用,中国科学技术大学博士学位论文,2003;
[14]J.Preskill,Lecture Notes for Physics 229:Quantum Information and Computation,California Institution of Technology,1998;
[15]Memorial Issue for H.Umezawa,Int.J.Mod.Phys.B,10(1996);Y.Takahashi and H.Umezawa,Collective Phenomena,2(1975)55;
[16]Hong-yi Fan and Yue Fan,Phys.Lett.A,246(1998)242;Hong-yi Fan and Hui Wang,Mod.Phys.Lett.B,14(2000)553;
[17]范洪义,量子力学表象与变换论——狄拉克符号法进展,上海科学技术出版社,上海,1997;
[1]吕乃光,傅里叶光学(第二版),机械工业出版社,2007;
[2]宋菲君、S.Jutamulia,近代光学信息处理,北京大学出版社,北京,1998;
[3]L.Mandel and E.Wolf,Optical Coherence and Quantum Optics,Cambridge,1996;
[4]E.U.Condon,Acad.Sci.USA,23(1937)158-164;H.M.Ozakatas,D.Mendlovic.J.Opt.Soc.Am.A,10(1993)2522;D.Mendlovic,H.M.Ozakatas.J.Opt.Soc.Am.A,10(1993)1875;D.Mendlovic,H.M.Ozaktas and A.W.Lohmann,Appl.Opt.33(1994)6188;M.A.Pinsky,Introduction to Fourier Analysis and Wavelets,Book/Cole,2002;
[5]V.Namias,J.Inst.Maths Applics,25(1980)241;
[6]A.C.McBride and F.H.Kerr,IMA J.Appl.Maths,39(1987)159;
[7]A.W.Lohmann,J.Opt.Soc.Am.A,10(1993)2181;
[8]A.Gerrard and N.M.Butch,Introduction to Matrix Methods in Optics,John Wiley & Sons,London,1975;
[9]A.Torre,FRFT and its applications in Optics,Progress in Optics,Vol.43,edited by E.Wolf,2003 and references therein;
[10]J.W.Goodman,Introduction to Fourier Optics,McGraw -Hill,New York,1968;
[11]Hong-yi Fan and Hai-liang Lu,Phys.Lett.A,334(2005)132;
[12]Shao-min Wang and Dao-mu Zhao,Matrix Optics,China Higher Education Press,Beijing and Springer-Verlag,Berlin Heidelberg(2000);
[13]Hong-yi Fan and Hai-liang Lu,Journal of Optics B:Quantum and Semiclassical Optics,9(2005)799-856;
[14]Hong-yi Fan,Hui Zou and Yue Fan,Phys.Lett.A 254(1999)137;
[15]Hong-yi Fan,Phys.Lett.A 313(2003)343;
[16]Hong-yi Fan,Hai-liang Lu,J.Phys.A:Math.Gen.37(2004)10993;
[17]Hong-yi Fan,Xu-bing Tang,J.Opt.B:Quantum Semielass.Opt.7(2005)s765;
[18]R.A.Campos,B.E.A.Saleh,M.C.Teich,Phys.Rev.A 50(1989)5274;
[19]C.Silberhom,et at.,Phys.Rev.Lett.86(2001)4267;
[20]范洪义,量子力学表象与变换论——狄拉克符号法进展,上海科学技术出版社,上海,1997:
[21]H.Kogelnik,Appl.Opt.4(1965)1562;B.X.Chen and B.H.Sun,Chin.Phys.Lett.20(2003)1254;
[22]Hong-yi Fan and Li-yun Hu,Chin.Phys.Lett.24:(8)(2007)2238;
[23]Hong-yi Fan and Xu-bing Tang,Phys.Lett.A 357(2006)163;
[1]C.H.Bennett,G Brassard,C.Crepeau,R.Jozsa,A.Peres and W.Wootters,Phys.Rev.Lett.70(1993)1895;
[2]Hong-yi Fan and J.R.Klauder,Phys.Rev.A 49(1994)704;
[3]Hong-yi Fan,International Journal of Modern Physics B 18:(10)(2004)1;
[4]J.W.Pan,D.Bouwmeester,H.Weinfurter and A.Zeilinger,Phys.Rev.Lett.80(1998)3891;
[5]Z.Zhao,et al.,Experimental demonstration of five-photon entanglement and open-destination teleportation,Nature,430(2004);
[6]E.Wigner,Phys.Rev.40(1932)749;
[7]Hong-yi Fan and Yu-nan Ruan,Commun.Theor.Phys.3:(3)(1984)345;2:(5)1563;Fan Hong-yi and Min Xiao,Mod.Phys.Lett.B10(1996)989.
[8]Hong-yi Fan,Phys.Lett.A 124:(6,7)(1987)303;
[9]H.J.Wu and Hong-yi Fan,Mod.Phys.Lett.B 11(1997)544;
[I0]Hong-yi Fan and Yue Fan,Mod.Phys.Lett.A 13(1998)433;
[11]范洪义,量子力学纠缠态表象及应用,上海交通大学出版社,上海,2000;
[12]K.B.Davis,M.O.Mewes,M.R.Andrews,N.J.van Druten,D.S.Durfee,D.M.Kurn and W.Ketterle,Phys.Rev.Lett.,75(1995)3996;
[13]W.J.Mullin and R.Krotkov,Am.J.Phys.74(2006)880;
[14]J.Schwinger,Quantum Theory of Angular Momentum,New York:Academic press,1965;
[15]F.T.Arrechi,E.Courtens,R.Gilmore and H.Thomas,Phys.Rev.,6(1972)2211;
[16]Hong-yi Fan,Chao Li and Zhong-hua Jiang,Phys.Lett.A,327(2004)416;
[17]L.Susskind and J.Glogower,Physics,1(1964)49;
[18]P.Carruthers and M.M.Nieto,Rev.Mod.Phys.,40(1968)411;R.Loudon and P.L.Knight,Jour.Mod.Opt.,34(1987)709;
[19]Xu-bing Tang and Hong-yi Fan,Atomic coherent state,entangled state and phase operator for studying the interference between two Bose-Einstein condensates,Commun.Theor.Phys.(in press);