由相干态诱导出的非经典量子态统计属性研究
|
摘要
在量子光学和量子信息学中，光场的非经典性质一直以来都是一个备受关注的研究课题.一般地，光场的非经典性通过一些具体的量子统计特点体现出来的，如亚Poissonian光子统计、反聚束、压缩以及Wigner函数的部分负值分布等.近些年来，人们普遍认识到量子力学态叠加原理是能够使得量子态呈现各种非经典效应的基本原理.因此，基于态叠加原理，人们构造出了许多具有显著非经典效应的量子态.比如，两个相干态的对称或反对称叠加态(称之为奇偶相干态)具有压缩和反聚束等非经典效应.另一种产生新量子态的方法是将某些算符作用到相应的初始态上，这里的初始态原则上可以是任意的量子态，例如真空态、相干态或者热态.比如，压缩态是通过将压缩算符作用在真空态或者相干态上产生.光子增加相干态是通过将光子产生算符a连续作用到相干态上而得到等等.研究非经典态的一个重要且行之有效的途径，就是尽可能多的在量子力学框架内构造出一些量子态，进而研究它们的统计属性，从而发现可能具有的非经典效应.因此，从理论上构造出一些新的量子态并研究它们的统计性质具有重要的实际意义.我们知道，一个量子态的Wigner函数包含了该量子态在整个相空的全部信息，因此量子态的演化可以用它的Wigner函数来描述.另外，由于Wigner函数不是严格的概率分布函数而是具有准概率分布函数的性质，因而可正可负.对于经典态或者准经典态如典型的相干态，其Wigner函数值总是非负的，因此Wigner函数展现出负值是量子态非经典特征的体现.本论文主要介绍了我们在量子调控相干态和压缩相干态等方面的一些研究进展.利用相干态表象下Wigner算符以及算符正规乘积内的积分技术，重构和得出了一些量子态的Wigner函数，并根据这些Wigner函数在相空间中随复变量的变化关系，详细地讨论了这些量子态所展现出来的非经典性质.最后，利用热场动力学理论和热纠缠态表象推导出的密度算符和Wigner函数在振幅衰减通道和热通道中的演化公式，讨论了这些量子态的退相干效应.本文的主要研究工作包括以下五个方面：
1.一类综合了增光子压缩真空态和减光子压缩真空态的光子调制压缩真空态的理论构造及其非经典特性研究.首先由产生算符a和湮灭算符a的相干叠加构造成的调制算符作用于压缩真空态，从而构造出了光子调制压缩真空态，进而利用有序算符内的积分技术和新发现的Legendre多项式的新形式得出了归一化系数.接着推导出了Wigner函数和与压缩Schro¨dinger猫态的保真度.
2.光子增加相干态在热通道中的统计属性研究.利用由热纠缠态表象推导出的密度算符在热通道中的演化公式和相干态表象下的Wigner算符以及正规乘积内的积分技术，得出了光子增加相干态在热通道中的Wigner函数.另外，借助于坐标—动量中介表象，得出了光子增加相干态在热通道中的Tomogram函数.分析研究表明：当演化时间趋向于无穷大时，光子增加相干态在热通道中的密度算符、Wigner函数和Tomogram函数均退化成了热态的密度算符、Wigner函数和Tomogram函数.
3.光子增加扣除相干态及其统计性质.首先对构造出的光子增加扣除相干态进行了归一化，接着推导出了光子增加扣除相干态的光子数分布、Mandel Q-参数、P-函数、Q-函数、Wigner函数以及与原始相干态的保真度.然后利用由热纠缠态表象推导出的Wigner函数在振幅衰减通道中演化公式，讨论了该态的退相干效应.
4.讨论了光子扣除和光子增加压缩相干态的统计属性.借助于算符Weyl编序下的相似变换不变性和正规乘积内的积分技术，得出了压缩相干态正规乘积形式的密度算符.在此基础上，通过将湮灭算符和产生算符分别作用于压缩相干态，得出了光子扣除和光子增加压缩相干态.进一步得出了归一化系数、Mandel Q-参数、光子数分布、压缩度以及Wigner函数等的解析表达式，并讨论了光子扣除和光子增加压缩相干态的非经典性和非高斯性.借助于密度算符和Wigner函数在热通道中演化公式讨论了其退相干过程.
5.研究了由增强压缩热态诱导出的非经典非高斯量子态的统计属性及退相干效应.通过重复对增强压缩热态进行增光子操作得出了光子增加增强压缩热态并进一步导出了归一化系数的解析表达式.通过导出Mandel Q-参数、光子数分布、Wigner函数的解析表达式讨论了其非经典非高斯行为；讨论了两类不同压缩参数对非经典性的影响.最后研究了与原始增强压缩热态的保真度以及在热通道中的退相干过程.
本论文的内容章节安排如下：第一章简要介绍非经典量子态和相干态的理论基础，列举了若干产生非经典态的方法和衡量非经典性的有效判据，介绍了相干态的定义和性质.第二章主要介绍有序算符内的积分技术基本理论，并用该积分技术从新的角度导出常用的量子力学表象、量子态的Wigner函数、以及密度算符和Wigner函数在各种通道中随时间的演化公式，为后续章节内容的研究做准备.第三章介绍了光子调制压缩真空态，一种综合了增光子压缩真空态和减光子压缩真空态的新的量子态.第四章讨论了光子增加相干态和光子增加扣除相干态的统计属性.第五章介绍了由压缩相干态诱导出的非经典非高斯态.第六章介绍了增光子增强压缩热态的统计属性.我们相信以上对这些量子态的研究能一定程度上丰富量子态调控和量子态工程理论，具有较高的学术参考价值和一定的实用意义.
The nonclassicality of optical fields has been a hot topic in the development of quantum op-tics and quantum information processing. Usually, the nonclassicality manifests itself in specificproperties of quantum statistics, such as sub-Poissonian photon statistics, antibunching, squeez-ing in one of the quadratures, and partial negative Wigner function, etc. In recent years，peopleextensively know that it is the principle of superposition in quantum mechanics that produce var-ious kinds of non-classical effects of quantum states. Therefore, people construct many quantumstates exhibiting remarkable nonclassical effects on the basis of the principle. For example, thesuperposed states of two coherent states have squeezing and anti-bunching effect. Another wayto constructing new quantum states is employing the associated operators on the original states，where the original states can be arbitrary states in principle, e.g., vacuum states, coherent states,and thermal states, etc. For example, the squeezed states can be generated by operating the squeez-ing operator on vacuum states or coherent states; the photon-added coherent states is constructedby employing the photon creation operator a on coherent states repeatedly, etc. An importantand effective approach to studying nonclassical states is to constructing quantum states as manyas possible in the framework of quantum mechanics, and then find out new nonclassical effects bystudying their quantum statistical properties. Therefore, it is of practical meaning to constructingnew quantum states theoretically and investigating their nonclassical properties. We know that theWigner function of a quantum state possesses all information of the quantum state in the wholephase space, so the evolution of quantum states can be described by Wigner function. However,Wigner function is not probability distribution function but quasi-probability distribution function,whose value can be positive or negative. For the classical or quasi-classical states (e.g., coherentstates), their Wigner function is always non-negative. Thus the partial negativity of Wigner func-tion is indeed a good indication of the highly nonclassical character of quantum states. In this paperwe mainly introduce the research progress in the manipulation of coherent states and squeezed co-herent states. Using the Wigner operator in coherent states representation and the technique ofintegration within an ordered product of operators, we reconstruct and obtain the Wigner functionsfor these quantum states. And then in term of the variations of the Wigner function with respectto complex variables in phase space, we discuss their nonclassical properties in detail. Finally, by virtue of the evolution formula of Wigner function in amplitude damping channel and thermalchannel derived by the theory of thermal field dynamics and thermal entanglement states represen-tation, we discuss the decoherence effect of these quantum states. The main works are summarizedas five parts:
1. The construction and investigation of photon manipulation squeezed vacuum states thatintegrate photon-added and photon-subtracted squeezed vacuum states. Firstly, by applying theoperation of the coherent superposition of photon addition operator a and subtraction operator aon squeezed vacuum states, we obtain the so-called photon manipulation squeezed vacuum states.And then, we derive the normalization constant by virtue of the technique of integration within anordered product and the newly found expression of Legendre polynomials. And also derive thephoton-number distribution, the Wigner function, and the fidelity between squeezed Schro¨dingercat states.
2. The investigation on statistical properties of photon-added coherent states in thermal chan-nel. We obtain the evolution of density operator in thermal channel obtained by thermal entan-glement states representation and the evolution of Wigner function in thermal channel by usingthe Wigner operator in coherent states representation. Additionally, based on the coordinate—momentum intermediate representation, we obtain the tomogram function of photon-added coher-ent states in thermal channel. Indicated by the analysis and investigation is that the density operator,Wigner function, and tomogram function of photon-added coherent states in thermal channel shallreduce to the density operator, Wigner function, and tomogram function of thermal states when theevolution time tends to infinity.
3. The study on statistical properties of photon-added-then-subtracted coherent states. Bytaking advantage of the operator ordering theory in quantum optics and the technique of inte-gration within an ordered product, the analytical expressions of normalization constant, photon-number distribution, Mandel’s Q-parameter, P-function, Q-function, Wigner function, and fidelitybetween photon-added-then-subtracted coherent states and initial coherent states are derived, andits nonclassicality is discussed in detail accordingly. In addition, using the time evolution of Wignerfunction in amplitude damping channel obtained by thermal entanglement states representation, thedecoherence effect of photon-added-then-subtracted coherent states in amplitude damping channelis studied.
4. The investigation on statistical properties of photon-subtracted and photon-added squeezedcoherent states. Employing the invariance of Weyl ordered operator under similar transformationand the technique of integration within an ordered product, we derive the normally ordered formof density operator for squeezed coherent states. On the basis of the normally ordered densityoperator, by repeatedly subtracting from and adding photons to squeezed coherent states, we the-oretically construct so-called photon-subtracted and photon-added squeezed coherent states. By analytical calculating Mandel’s Q-parameter, quadratures squeezing, photon-number distribution,and Wigner function, etc. we study the nonclassicality of photon-subtracted and photon-addedsqueezed coherent states. On the other hand, the decoherence process of photon-subtracted andphoton-added squeezed coherent states in thermal channel is studied through the time evolution ofdensity operator and Wigner function, respectively.
5. The photon-added squeezing enhanced thermal states is introduced theoretically by re-peatedly applying photon creation operator on the squeezing enhanced thermal states. And thenby virtue of the normally ordered form density operator for enhanced squeezing thermal states,we investigate the statistical properties on account of the analytical expressions of normalizationconstant, Mandel’s Q-parameter, photon-number distribution, Wigner function, and fidelity be-tween photon-added squeezing enhanced thermal states and squeezing enhanced thermal states. Inaddition, the decoherence process of photon-added squeezing enhanced thermal states in thermalchannel is also included.
The structure of this dissertation is arranged as follows:
In Chap.1, we briefly introduce the basic theories of nonclassical quantum states and coherentstates, including some way that can generate nonclassical states and some effective criteria thatcan measure nonclassicality, and the definition and properties of coherent states is also reviewed.In Chap.2, we introduce the technique of integration within an ordered product. And based onthis technique some fundamental representations in quantum mechanics, Wigner function, timeevolution of density operator and Wigner function in several channels are derived. And it is thefoundation for further investigation in the following chapters. In Chap.3, we study a kind of statesthat integrate photon-added and photon-subtracted squeezed vacuum states—photon modulationsqueezed vacuum states. In Chap.4, we study the statistical properties of photon-added coherentstates and photon-added-then-subtracted coherent states. In Chap.5, we obtain the nonclassicaland non-Gaussian states associated with squeezed coherent states after nonclassical non-Gaussianoperation, including photon addition and photon subtraction. In Chap.6, we obtain the nonclassicaland non-Gaussian states associated with squeezing enhanced thermal states. We believe that theinvestigation of these quantum states, to some extent, can further enrich the quantum states ma-nipulation and quantum states engineering theory, possessing high academic value and practicalsignificance.
1.一类综合了增光子压缩真空态和减光子压缩真空态的光子调制压缩真空态的理论构造及其非经典特性研究.首先由产生算符a和湮灭算符a的相干叠加构造成的调制算符作用于压缩真空态，从而构造出了光子调制压缩真空态，进而利用有序算符内的积分技术和新发现的Legendre多项式的新形式得出了归一化系数.接着推导出了Wigner函数和与压缩Schro¨dinger猫态的保真度.
2.光子增加相干态在热通道中的统计属性研究.利用由热纠缠态表象推导出的密度算符在热通道中的演化公式和相干态表象下的Wigner算符以及正规乘积内的积分技术，得出了光子增加相干态在热通道中的Wigner函数.另外，借助于坐标—动量中介表象，得出了光子增加相干态在热通道中的Tomogram函数.分析研究表明：当演化时间趋向于无穷大时，光子增加相干态在热通道中的密度算符、Wigner函数和Tomogram函数均退化成了热态的密度算符、Wigner函数和Tomogram函数.
3.光子增加扣除相干态及其统计性质.首先对构造出的光子增加扣除相干态进行了归一化，接着推导出了光子增加扣除相干态的光子数分布、Mandel Q-参数、P-函数、Q-函数、Wigner函数以及与原始相干态的保真度.然后利用由热纠缠态表象推导出的Wigner函数在振幅衰减通道中演化公式，讨论了该态的退相干效应.
4.讨论了光子扣除和光子增加压缩相干态的统计属性.借助于算符Weyl编序下的相似变换不变性和正规乘积内的积分技术，得出了压缩相干态正规乘积形式的密度算符.在此基础上，通过将湮灭算符和产生算符分别作用于压缩相干态，得出了光子扣除和光子增加压缩相干态.进一步得出了归一化系数、Mandel Q-参数、光子数分布、压缩度以及Wigner函数等的解析表达式，并讨论了光子扣除和光子增加压缩相干态的非经典性和非高斯性.借助于密度算符和Wigner函数在热通道中演化公式讨论了其退相干过程.
5.研究了由增强压缩热态诱导出的非经典非高斯量子态的统计属性及退相干效应.通过重复对增强压缩热态进行增光子操作得出了光子增加增强压缩热态并进一步导出了归一化系数的解析表达式.通过导出Mandel Q-参数、光子数分布、Wigner函数的解析表达式讨论了其非经典非高斯行为；讨论了两类不同压缩参数对非经典性的影响.最后研究了与原始增强压缩热态的保真度以及在热通道中的退相干过程.
本论文的内容章节安排如下：第一章简要介绍非经典量子态和相干态的理论基础，列举了若干产生非经典态的方法和衡量非经典性的有效判据，介绍了相干态的定义和性质.第二章主要介绍有序算符内的积分技术基本理论，并用该积分技术从新的角度导出常用的量子力学表象、量子态的Wigner函数、以及密度算符和Wigner函数在各种通道中随时间的演化公式，为后续章节内容的研究做准备.第三章介绍了光子调制压缩真空态，一种综合了增光子压缩真空态和减光子压缩真空态的新的量子态.第四章讨论了光子增加相干态和光子增加扣除相干态的统计属性.第五章介绍了由压缩相干态诱导出的非经典非高斯态.第六章介绍了增光子增强压缩热态的统计属性.我们相信以上对这些量子态的研究能一定程度上丰富量子态调控和量子态工程理论，具有较高的学术参考价值和一定的实用意义.
The nonclassicality of optical fields has been a hot topic in the development of quantum op-tics and quantum information processing. Usually, the nonclassicality manifests itself in specificproperties of quantum statistics, such as sub-Poissonian photon statistics, antibunching, squeez-ing in one of the quadratures, and partial negative Wigner function, etc. In recent years，peopleextensively know that it is the principle of superposition in quantum mechanics that produce var-ious kinds of non-classical effects of quantum states. Therefore, people construct many quantumstates exhibiting remarkable nonclassical effects on the basis of the principle. For example, thesuperposed states of two coherent states have squeezing and anti-bunching effect. Another wayto constructing new quantum states is employing the associated operators on the original states，where the original states can be arbitrary states in principle, e.g., vacuum states, coherent states,and thermal states, etc. For example, the squeezed states can be generated by operating the squeez-ing operator on vacuum states or coherent states; the photon-added coherent states is constructedby employing the photon creation operator a on coherent states repeatedly, etc. An importantand effective approach to studying nonclassical states is to constructing quantum states as manyas possible in the framework of quantum mechanics, and then find out new nonclassical effects bystudying their quantum statistical properties. Therefore, it is of practical meaning to constructingnew quantum states theoretically and investigating their nonclassical properties. We know that theWigner function of a quantum state possesses all information of the quantum state in the wholephase space, so the evolution of quantum states can be described by Wigner function. However,Wigner function is not probability distribution function but quasi-probability distribution function,whose value can be positive or negative. For the classical or quasi-classical states (e.g., coherentstates), their Wigner function is always non-negative. Thus the partial negativity of Wigner func-tion is indeed a good indication of the highly nonclassical character of quantum states. In this paperwe mainly introduce the research progress in the manipulation of coherent states and squeezed co-herent states. Using the Wigner operator in coherent states representation and the technique ofintegration within an ordered product of operators, we reconstruct and obtain the Wigner functionsfor these quantum states. And then in term of the variations of the Wigner function with respectto complex variables in phase space, we discuss their nonclassical properties in detail. Finally, by virtue of the evolution formula of Wigner function in amplitude damping channel and thermalchannel derived by the theory of thermal field dynamics and thermal entanglement states represen-tation, we discuss the decoherence effect of these quantum states. The main works are summarizedas five parts:
1. The construction and investigation of photon manipulation squeezed vacuum states thatintegrate photon-added and photon-subtracted squeezed vacuum states. Firstly, by applying theoperation of the coherent superposition of photon addition operator a and subtraction operator aon squeezed vacuum states, we obtain the so-called photon manipulation squeezed vacuum states.And then, we derive the normalization constant by virtue of the technique of integration within anordered product and the newly found expression of Legendre polynomials. And also derive thephoton-number distribution, the Wigner function, and the fidelity between squeezed Schro¨dingercat states.
2. The investigation on statistical properties of photon-added coherent states in thermal chan-nel. We obtain the evolution of density operator in thermal channel obtained by thermal entan-glement states representation and the evolution of Wigner function in thermal channel by usingthe Wigner operator in coherent states representation. Additionally, based on the coordinate—momentum intermediate representation, we obtain the tomogram function of photon-added coher-ent states in thermal channel. Indicated by the analysis and investigation is that the density operator,Wigner function, and tomogram function of photon-added coherent states in thermal channel shallreduce to the density operator, Wigner function, and tomogram function of thermal states when theevolution time tends to infinity.
3. The study on statistical properties of photon-added-then-subtracted coherent states. Bytaking advantage of the operator ordering theory in quantum optics and the technique of inte-gration within an ordered product, the analytical expressions of normalization constant, photon-number distribution, Mandel’s Q-parameter, P-function, Q-function, Wigner function, and fidelitybetween photon-added-then-subtracted coherent states and initial coherent states are derived, andits nonclassicality is discussed in detail accordingly. In addition, using the time evolution of Wignerfunction in amplitude damping channel obtained by thermal entanglement states representation, thedecoherence effect of photon-added-then-subtracted coherent states in amplitude damping channelis studied.
4. The investigation on statistical properties of photon-subtracted and photon-added squeezedcoherent states. Employing the invariance of Weyl ordered operator under similar transformationand the technique of integration within an ordered product, we derive the normally ordered formof density operator for squeezed coherent states. On the basis of the normally ordered densityoperator, by repeatedly subtracting from and adding photons to squeezed coherent states, we the-oretically construct so-called photon-subtracted and photon-added squeezed coherent states. By analytical calculating Mandel’s Q-parameter, quadratures squeezing, photon-number distribution,and Wigner function, etc. we study the nonclassicality of photon-subtracted and photon-addedsqueezed coherent states. On the other hand, the decoherence process of photon-subtracted andphoton-added squeezed coherent states in thermal channel is studied through the time evolution ofdensity operator and Wigner function, respectively.
5. The photon-added squeezing enhanced thermal states is introduced theoretically by re-peatedly applying photon creation operator on the squeezing enhanced thermal states. And thenby virtue of the normally ordered form density operator for enhanced squeezing thermal states,we investigate the statistical properties on account of the analytical expressions of normalizationconstant, Mandel’s Q-parameter, photon-number distribution, Wigner function, and fidelity be-tween photon-added squeezing enhanced thermal states and squeezing enhanced thermal states. Inaddition, the decoherence process of photon-added squeezing enhanced thermal states in thermalchannel is also included.
The structure of this dissertation is arranged as follows:
In Chap.1, we briefly introduce the basic theories of nonclassical quantum states and coherentstates, including some way that can generate nonclassical states and some effective criteria thatcan measure nonclassicality, and the definition and properties of coherent states is also reviewed.In Chap.2, we introduce the technique of integration within an ordered product. And based onthis technique some fundamental representations in quantum mechanics, Wigner function, timeevolution of density operator and Wigner function in several channels are derived. And it is thefoundation for further investigation in the following chapters. In Chap.3, we study a kind of statesthat integrate photon-added and photon-subtracted squeezed vacuum states—photon modulationsqueezed vacuum states. In Chap.4, we study the statistical properties of photon-added coherentstates and photon-added-then-subtracted coherent states. In Chap.5, we obtain the nonclassicaland non-Gaussian states associated with squeezed coherent states after nonclassical non-Gaussianoperation, including photon addition and photon subtraction. In Chap.6, we obtain the nonclassicaland non-Gaussian states associated with squeezing enhanced thermal states. We believe that theinvestigation of these quantum states, to some extent, can further enrich the quantum states ma-nipulation and quantum states engineering theory, possessing high academic value and practicalsignificance.
引文
[1] Knight P L and Allen L. Concept of quantum optics [M]. Oxford: Pergamon Press,54–69,1983.
[2] Marek P, Jeong H, and Kim M S. Generating ‘squeezed’ superpositions of coherent statesusing photon addition and subtraction [J]. Phys. Rev. A,78,063811,2008.
[3] Lee S Y, Ji S W, Kim H J, and Nha H. Enhancing quantum entanglement for continuousvariables by coherent superposition of photon subtraction and addition [J]. Phys. Rev. A,84,012302,2011.
[4] Agarwal G S and Tara K. Nonclassical properties of states generated by the excitations on acoherent state [J]. Phys. Rev. A,43,492–497,1991.
[5] Zhang Z and Fan H. Properties of states generated by excitations on a squeezed vacuumstate [J]. Phys. Lett. A,165,14–18,1992.
[6] Kis Z, Adam P, and Janszky J. Properties of states generated by excitations on the amplitudesqueezed states [J]. Phys. Lett. A,188,16–20,1994.
[7] Xin Z Z, Duan Y B, Zhang H M, Hirayama M, and Matumoto K I. Excited two-photoncoherent state of the radiation field [J]. J. Phys. B: At. Mol. Opt. Phys.,29,4493–4506,1996.
[8] Man’ko V I and Wu¨nsche A. Properties of squeezed-state excitations [J]. Quantum Semi-class. Opt.,9,381–409,1997.
[9] Xin Z Z, Duan Y B, Zhang H M, Qian W J, Hirayama M, and Matumoto K I. Excitedeven and odd coherent states of the radiation field [J]. J. Phys. B: At. Mol. Opt. Phys,29,2597–2606,1996.
[10] Dodonov V V, Korennoy Y A, Man’ko V I, and Moukhin Y A. Nonclassical properties ofstates generated by the excitation of even/odd coherent states of light [J]. Quantum Semi-class. Opt.,8,413–427,1996.
[11] Neergaard-Nielsen J S, Melholt Nielsen B, Hettich C, M mer K, and Polzik E S. Generationof a superposition of odd photon number states for quantum information networks [J]. Phys.Rev. Lett.,97,083604,2006．
[12] Ourjoumtsev A, Tualle-Brouri R, Laurat J, and Grangier Ph. Generating optical Schro¨dingerkittens for quantum information processing [J]. Science,312,83–86,2006．
[13] Wakui K, Takahashi H, Fumsawa A, and Sasaki M. Photon subtracted squeezed states gen-erated with periodically poled KTiOPO4[J]. Opt．Express,15,3568–3574,2007．
[14] Dakna M, Anhut T, Opatmy T, Knoll L, and Welsch D-G. Generating Schrodinger-cat-likestates by means of conditional measurements on a beam splitter [J]. Phys. Rev. A,55,3184–3194,1997．
[15] Glancy S and de Vasconcelos H M. Methods for producing optical coherent state superpo-sitions [J]. J. Opt. Soc. Am. B,25,712–733,2008.
[16] Ourjoumtsev A, Dantan A, Tualle-Brouri R, and Grangier Ph. Increasing entanglement be-tween Gaussian states by coherent photon subtraction [J]. Phys. Rev. Lett.,98，030502,2007.
[17] Browne D E, Eisert J, Scheel S, and Plenio M B. Driving non-Gaussian to Gaussian stateswith linear optics [J]. Phys. Rev. A,67,062320,2003.
[18] Nha H and Carmichael H J. Proposed test of quantum nonlocality for continuous variables[J]. Phys. Rev. Lett.,93,020401,2004.
[19] Garc′a-Pat′ron R, Fiura′sˇek J, Cerf N J, Wenger J, Tualle-Brouri R, and Grangier Ph. Proposalfor a loophole-free Bell test using homodyne detection [J]. Phys. Rev. Lett.,93,130409,2004.
[20] Bartlett S D and Sanders B C. Universal continuous-variable quantum computation: Re-quirement of optical nonlinearity for photon counting [J]. Phys. Rev. A,65,042304,2002.
[21] Short R and Mandel L. Observation of sub-poissonian photon statistics [J]. Phys. Rev. Lett.,51,384–387,1983.
[22] Kimble H J, Dagenais M, and Mandel L. Photon antibunching in resonance fluorescence [J].Phys. Rev. Lett.,39,691–695,1977.
[23] Stoler D. Equivalence classes of minimum uncertainty packets [J]. Phys. Rev. D,1,3217–3219,1970.
[24] Stoler D. Equivalence classes of minimum uncertainty packets II [J]. Phys. Rev. D,4,1925–1926,1971.
[25] Yuen H P. Two-photon coherent states of the radiation field [J]. Phys. Rev. A,13,2226–2243,1976.
[26] Mandel L. Sub-Poissonian photon statistics in resonance fluorescence [J]. Opt. Lett.,4,205–207,1979.
[27] Spaelter S, Burk M, Stroessner U, Sizmann A, and Leuchs G. Propagation of quantumproperties of sub-picosecond solitons in a fiber [J]. Opt. Express,2,77–83,1998.
[28] Caves C M. Quantum-mechanical noise in an interferometer [J]. Phys. Rev. D,23,1693–1708,1981.
[29] Slusher R E, Hollberg L W, Yurke B, Mertz J C, and Valley J F. Observation of squeezedstates generated by four-wave mixing in an optical cavity [J]. Phys. Rev. Lett.,55,2409–2412,1985.
[30] Machida S, Yamamoto Y, and Itaya Y. Observation of amplitude squeezing in a constant-current–driven semiconductor laser [J]. Phys. Rev. Lett.,58,1000–1003,1987.
[31] Raizen M G, Orozco L A, Xiao Min, Boyd T L, and Kimble H J. Squeezed-state generationby the normal modes of a coupled system [J]. Phys. Rev. Lett.,59,198–201,1987.
[32] Hillery M. Squeezing of the square of the field amplitude in second harmonic generation[J]. Opt. Commun.,62,135–138,1987.
[33] Hong C K and Mandel L. Higher-order squeezing of a quantum field [J]. Phys. Rev. Lett.,54,323–325,1985.
[34] Hong C K and Mandel L. Generation of higher-order squeezing of quantum electromagneticfields [J]. Phys. Rev. A，32,974–982,1985.
[35] Zhang Z M, Xu L, Chai J L, and Li F L, A new kind of higher-order squeezing of radiationfield [J]. Phys. Lett. A，150,27–30，1990.
[36] Du S D and Gong C D. Squeezing of the kth power of the field amplitude [J]. Phys. Lett. A,168,296–300,1992.
[37] Lee J, Kim J, and Nha H. Demonstrating higher-order nonclassical effects by photon-addedclassical states: realistic schemes [J]. J. Opt. Soc. Am. B,26,1363–1369,2009.
[38] Kenfack A and Zyczkowski K. Negativity of the Wigner function as an indicator of nonclas-sicality [J]. J. Opt. B,6,396–404,2004.
[39] Lee C T. Measure of the nonclassicality of nonclassical states [J]. Phys. Rev. A,44, R2775–R2778,1991.
[40] Sudarshan E C G. Equivalence of semiclassical and quantum mechanical descriptions ofstatistical light beams [J]. Phys. Rev. Lett.,10,277–279,1963.
[41] Scully M O and Zubairy M S. Quantum optics [M].(Cambridge University,1997).
[42] Hillery M. Nonclassical distance in quantum optics [J]. Phys. Rev. A,35,725–732,1987.
[43] Asbo′th J K, Calsamiglia J, and Ritsch H. Computable measure of nonclassicality for light[J]. Phys. Rev. Lett.,94,173602,2005.
[44] Wei U. Quantum dissipative system [M].(Singapore: Word Scientific，1999).
[45] Gell-Mann M and Hartle J B. Classical equations for quantum systems [J]. Phys. Rev. D,47,3345–3382,1993.
[46] Walls D E and Milbum G J. Quantum optics [M].(Springer-Verlag, Berlin,1994).
[47] Orszag M. Quanturn optics [M].(Springer Press,2000).
[48] Elze H T. Quantum decoherence, entropy and thermalization in strong interactions at highenergy.(I). Noisy and dissipative vacuum effects in toy models [J]. Nucl．Phys. B,436,213–261,1995.
[49] Zurek W H. Environment-induced superselection rules [J]. Phys. Rev. D,26,1862–1880,1982.
[50] Unruh W G. Maintaining coherence in quantum computers [J]. Phys. Rev. A,51,992–997,1995.
[51] Bouwmeester D, Ekert A, and Zeilinger A. The physics of quantum information [M].(Springer-Verlag, Berlin,2000).
[52] Bennett C H and DiVincenzo D P. Quantum information and computation [J]. Nature,404,247–255,2000.
[53] Klauder J R and Skagerstam B S. Coherent states: applications in physics and mathematicalphysics [M]. World Scientific,1985.
[54] Klauder J R and Sudarshan E C G. Fundamentals of quantum optics [M]. New York: Ben-jamin,1968.
[55] Perelomov A M. Generalized coherent states and their applications [M]. Berlin: Springer,1986.
[56] Zhang W M. Coherent States: Theory and some applications [J]. Rev. Mod. Phys.,62,867–927,1990.
[57] Onofri E. A note on coherent state representations of Lie groups [J]. J. Math. Phys.,16,1087–1089,1975.
[58] Jurcˇo B,Sˇtovicˇek P. Coherent states for quantum compact groups [J]. Comm. Math. Phys.,182,221–251,1996.
[59] Scherk J. An introduction to the theory of dual models and strings [J]. Rev. Mod. Phys.,47,123–164,1975.
[60] Davies M J and Heller E J. Multidimensional wave functions from classical trajectories [J].J. Chem. Phys.,75,3919–3924,1981.
[61] Schro¨dinger E. Quantization as a problem of eigenvalue [J]. Naturwissenschaften,14,664–666,1926.
[62] Glauber R J. The quantum theory of optical coherence [J]. Phys. Rev.,130,2529–2539,1963.
[63] Glauber R J. Coherent and incoherent states of radiation field [J]. Phys. Rev.,131,2766–2788,1963．
[64] Sivakumar S. Studies on nonlinear coherent states [J]. J. Opt B：Quantum Semiclass. Opt.,2, R61–R75,2000.
[65] Loundon R. The quantum theory of light [M]. Oxford: Clarendon Press,1983.
[66] Louisell W H. Quantum statstical properities of radiation [M]. New York: Wiley1973.
[67] Dirac P A M. The principles of quantum mechanics [M]. Oxford: Clarendon Press,1930.
[68] Fan H Y. Operator ordering in quantum optics theory and the development of Dirac’s sym-bolic method [J]. J. Opt. B,5, R147–R163,2003.
[69] Wigner E. On the quantum correction for thermodynamic equilibrium [J]. Phys. Rev.,40,749–759,1932.
[70] Fan H Y. Newton–Leibniz integration for ket–bra operators in quantum mechanics (IV)–Integrations within Weyl ordered product of operators and their applications [J]. Ann. Phys.,323,500–526,2008.
[71] Mehta C L. Diagonal coherent-state representation of quantum operators [J]. Phys. Rev.Lett.,18,752–754,1967.
[72] Fan H Y. New antinormal ordering expansion for density operators [J]. Phys. Lett. A,161,1–4,1991.
[73] Weyl H. Quantenmechanik und gruppentheorie [J]. Z. Phys.,46,1–46,1927.
[74] Fan H Y. Weyl ordering quantum mechanical operators by virtue of the IWWP technique[J]. J. Phys. A: Math. Gen.,25,3443–3447,1992.
[75] Hu L Y and Fan H Y. Wavelet transformation and Wigner-Husimi distribution function [J].Int J Theor. Phys.,48,1539–1544,2009.
[76] Einstein A, Podolsky B, and Rosen N. Can quantum-mechanical description of physicalreality be considered complete [J]. Phys. Rev.,47,777–780,1935.
[77] Fan H Y, Zaidi H R, and Klauder J R. New approach for calculating the normally orderedform of squeeze operators [J]. Phys. Rev. D,35,1831–1834,1987.
[78] Fan H Y and Klaude J R. Eigenvectors of two particles’ relative position and total momen-tum [J]. Phys. Rev. A,49,704–707,1994.
[79] Fan H Y. Time evolution of the Wigner function in the entangled-state representation [J].Phys. Rev. A,65,064102,2002.
[80] Fan H Y. Application of EPR entangled state representation in quantum teleportation ofcontinuous variables [J]. Phys. Lett. A,294,253–257,2002.
[81] Fan H Y and Fan Y. Representations of two-mode squeezing transformations [J]. Phys. Rev.A,54,958–960,1996.
[82] Gardiner C and Zoller P. Quantum noise [M].(Springer Berlin，2000).
[83] Agarwal G S. Brownian motion of a quantum oscillator [J]. Phys. Rev. A,4,739–747,1971.
[84] Daniel D J and Milburn G J. Destruction of quantum coherence in a nonlinear oscillator viaattenuation and amplification [J]. Phys. Rev. A,39,4628–4640,1989.
[85] Kim M S and Buzek V. Schro¨dinger-cat states at finite temperature: Influence of a finite-temperature heat bath on quantum interferences [J]. Phys. Rev. A,46,4239–4251,1992.
[86] Fan H Y and Fan Y. New representation of thermal states in thermal field dynamics [J].Phys. Lett. A,246,242–246,1998.
[87] Wang C C and Fan H Y. Evolution of a thermo vacuum state in a single-mode amplitudedissipative channel [J]. Chin. Phys. Lett.,27,110302-1–110302-3,2010.
[88] Fan H Y. Weyl ordering expansion of power product of coordinate and momentum operators[J]. Commun. Theor. Phys.,54,439–442,2010.
[89] Hu L Y and Fan H Y. Two-mode squeezed number state as a two-variable Hermite-polynomial excitation on the squeezed vacuum [J]. J. Mod. Opt.,55,2011–2024,2008.
[90] Fan H Y and Tang X B. Wavefunction of a pair coherent state in the entangled state repre-sentation as an eigenfunction of a type of Fokker–Planck differential operator [J]. J. Phys.A,39,9831–9838,2006.
[91] Magnus W, Oberhettinger F, and Soni R P. Formulas and theorem for the special functionsof mathematical physics [M].3rd ed.(Springer-Verlag,1966).
[92] Fan H Y and Hu L Y. Infinite-dimensional Kraus operators for describing amplitude-damping channel and laser process [J]. Opt. Comm.,282,932–935,2009.
[93] Takahashi Y and Umezawa H. Thermo field dynamics [J]. Int. J. Mod. Phys. B,10,1755–1805,1996.
[94] Puri R R. Mathematical methods of quantum optics [M].(Springer-Verlag,2001), AppendixA.
[95] Fan H Y and Lu H L. Wave-function transformations by general SU(1,1) single-modesqueezing and analogy to Fresnel transformations in wave optics [J]. Opt. Commun.,258,51–58,2006.
[96] Fan H Y and Lu H L.2-mode Fresnel operator and entangled Fresnel transform [J]. Phys.Lett. A,334,132–1392005.
[97] Biswas A and Agarwal G S. Nonclassicality and decoherence of photon-subtracted squeezedstates [J]. Phys. Rev. A,75,032104，2007.
[98] Olivares S and Paris M G A. Squeezed Fock state by inconclusive photon subtraction [J]. J.Opt. B: Quantum Semiclass. Opt.,7, S616,2005.
[99] Olivares S and Paris M G A. Photon subtracted states and enhancement of nonlocality in thepresence of noise [J]. J. Opt. B: Quantum Semiclass. Opt.,7, S392,2005.
[100] Fan H Y, Meng X G, and Wang J S. New form of Legendre polynomials obtained by virtueof excited squeezed state and IWOP technique in quantum optics [J]. Commun. Theor. Phys.46,845–848,2006.
[101] Dodonov V V. Nonclassical states in quantum optics: a squeezed review of the first75years[J]. J. Opt. B,4, R1–R33,2002.
[102] Hu L Y and Fan H Y. Nonclassicality of photon-added squeezed vacuum and its decoherencein thermal environment [J]. J. Mod. Opt.,57,1344–1354,2010.
[103] Lee S Y and Nha H. Quantum state engineering by a coherent superposition of photonsubtraction and addition [J]. Phys. Rev. A,82,053812,2010.
[104] Hu L Y and Fan H Y. Statistical properties of photon-subtracted squeezed vacuum in thermalenvironment [J]. J. Opt. Soc. Am. B,25,1955-1964,2008.
[105] Xu X X, Yuan H C, and Fan H Y. Decoherence of photon-subtracted squeezed vacuum statein dissipative channel [J]. Chinese Phys. B,20,024203,2011.
[106] Schleich W P. Quantum optics in phase space [M]. Wiley-VCH, Berlin,2001.
[107] Li H M and Yuan H C. Simple approach to deriving some operator ordering formulas inquantum optics [J]. Int J Theor Phys,49,2121–2130,2010.
[108] Sugita A and Aiba H. Second moment of the Husimi distribution as a measure of complexityof quantum states [J]. Phys. Rev. E65,036205,2002.
[109] Ourjoumtsev A, Jeong Hyunseok, Tualle-Brouri R, and Grangier P. Generation of opticalSchro¨dinger cats from photon number states [J]. Nature448,784–786,2007.
[110] Ralph T C, Gilchrist A, Milburn G J, Munro W J, and Glancy S. Quantum computation withoptical coherent states [J]. Phys. Rev. A,68,042319,2003.
[111] Jeong H, Kim M S, and Lee J. Quantum-information processing for a coherent superpositionstate via a mixedentangled coherent channel [J]. Phys. Rev. A,64,052308,2001.
[112] Munro W J, Nemoto K, Milburn G J, and Braunstein S L. Weak-force detection with super-posed coherent states [J]. Phys. Rev. A,66,023819,2002.
[113] You W L, Li Y W, and Gu S J Fidelity, dynamic structure factor, and susceptibility in criticalphenomena [J]. Phy. Rev. E76,022101,2007.
[114] Bogolyubov N N. Lectures on quantum statistics [M]. New York: Gordon and Breach,1987.
[115] Zavatta A, Viciani S, and Bellini M. Quantum-to-classical transition with single-photon-added coherent states of light [J]. Science,306,660–662,2004.
[116] Zavatta A, Viciani S, and Bellini M. Single-photon excitation of a coherent state: catchingthe elementary step of stimulated light emission [J]. Phys. Rev. A,72,023820,2005.
[117] Meng X G, Wang Z, Fan H Y, and Wang J S. Squeezed number state and squeezed thermalstate: decoherence analysis and nonclassical properties in the laser process [J]. J. Opt. Soc.Am. B,29,1835–1843,2012.
[118] Vogel K and Risken H. Determination of quasiprobability distributions in terms of proba-bility distributions for the rotated quadrature phase [J]. Phys. Rev. A,40,2847–2849,1989.
[119] Smithey D T, Beck M, Raymer M G, and Faridani A. Measurement of the Wigner distribu-tion and the density matrix of a light mode using optical homodyne tomography: applicationto squeezed states and the vacuum [J]. Phys. Rev. Lett.,70,1244–1247,1993.
[120] Fan H Y and Chen H L. Two-parameter Radon transformation of the Wigner operator andits inverse [J]. Chin. Phys. Lett.,18,850–853,2001.
[121] Fan H Y and Li G C. Radon transformation of the Wigner operator for two-mode correlatedsystem in generalized entangled state representation [J]. Mod. Phys. Lett. A,15,499–507,2000.
[122] Kim M S. Recent developments in photon-level operations on travelling light fields [J]. J.Phys. B,41,133001,2008.
[123] Lee S Y, Park J, Ji S W, Ooi C H R, and Lee H W. Nonclassicality generated by photonannihilation-then-creation and creation-then-annihilation operations [J]. J. Opt. Soc. Am. B,26,1532–1537,2009.
[124] Yang Y and Li F L. Nonclassicality of photon-subtracted and photon-added-then-subtractedGaussian states [J]. J. Opt. Soc. Am. B,26,830–835,2009.
[125] Fan H Y and Fan Y. New bosonic operator ordering identities gained by the entangled staterepresentation and two-variable Hermite polynomials [J]. Commun. Theor. Phys.,38,297–300,2002.
[126] Hu L Y and Fan H Y. Infinite-dimensional Kraus operators for describing amplitude-damping channel and laser process [J]. Opt. Commun.,282,932–935,2009.
[127] Jozsat R. Fidelity for mixed quantum states [J]. J. Mod. Opt.,41,2315–2323,1994.
[128] Yuen H P. Two-photon coherent states of the radiation field [J]. Phys. Rev. A,13,2226–2243,1976.
[129] Yuen H P and Shapiro J H. Optical communication with two-photon coherent states–Part I:Quantum-state propagation and quantum-noise [J]. IEEE Trans. Inform. Theory,24,657–668,1978.
[130] Shapiro J H, Yuen H P, and Machado Mata J A. Optical communication with two-photoncoherent states–Part II: Photoemissive detection and structured receiver performance [J].IEEE Trans. Inform. Theory,25,179–192,1979.
[131] Goda K, Miyakawa O, Mikhailov E E, Saraf S, Adhikari R, Mckenzie K, Ward R, Vass S,Weinstein A J, and Mavalvala N. A quantum-enhanced prototype gravitational-wave detec-tor [J]. Nature Phys.,4,472–476,2008.
[132] Vahlbruch H, Chelkowski S, Hage B, Franzen A, Danzmann K, and Schnabel R. Demon-stration of a squeezed-light-enhanced power-and signal-recycled Michelson interferometer[J]. Phys. Rev. Lett.,95,211102-1–211102-4,2005.
[133] Mandel L and Wolf E. Optical coherence and quantum optics [M].(Cambridge: CambridgeUniversity Press),1995.
[134] Schleich W and Wheeler J A. Oscillations in photon distribution of squeezed states andinterference in phase space [J]. Nature,326,574–577,1987.
[135] Schleich W and Wheeler J A. Oscillations in photon distribution of squeezed states [J]. J.Opt. Soc. Am. B,4,1715–1722,1987.
[136] Vahlbruch H, Mehmet M, Chelkowski S, Hage B, Franzen A, Lastzka N, Gober S, Danz-mann K, and Schnabel R. Observation of squeezed light with10-dB quantum-noise reduc-tion [J]. Phys. Rev. Lett.,100,033602,2008.
[137] Mehmet M, Vahlbruch H, Lastzka N, Danzmann K, and Schnabel R. Observation ofsqueezed states with strong photon-number oscillations [J]. Phys. Rev. A,81,013814,2010.
[138] Braunstein S L and Kimble H J. Dense coding for continuous variables [J]. Phys. Rev. A,61,042302,2000.
[139] Takeno Y, Yukawa M, Yonezawa H, and Furusawa A. Observation of9dB quadraturesqueezing with improvement of phase stability in homodyne measurement [J]. Opt. Express,15,4321,2007.
[140] Hsu M T L, Hetet G, Glockl O, Longdell J J, Buchler B C, Bachor H A, and Lam P K.Quantum study of information delay in electromagnetically induced transparency [J]. Phys.Rev. Lett.,97,183601,2006.
[141] Fan H Y, Yuan H C, Xu X X, and Jiang N Q. New approach for obtaining the squeezing-enhanced state and its Wigner function by virtue of the Weyl–Wigner quantization scheme[J]. Phys. Scr.,83,015403,2011.
[142] Wang S and Fan H Y. Statistical properties of the squeezing-enhanced thermal state [J]. J.Opt. Soc. Am. B,29,15–22,2012.
[143] Agarwal G S and Tara K. Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics, Phys. Rev. A,46,4851992.
[144] Chizhov A V, Knoll L, and Welsch D G. Continuous-variable quantum teleportation throughlossy channels [J]. Phy. Rev. A,65,022310,2002.
[2] Marek P, Jeong H, and Kim M S. Generating ‘squeezed’ superpositions of coherent statesusing photon addition and subtraction [J]. Phys. Rev. A,78,063811,2008.
[3] Lee S Y, Ji S W, Kim H J, and Nha H. Enhancing quantum entanglement for continuousvariables by coherent superposition of photon subtraction and addition [J]. Phys. Rev. A,84,012302,2011.
[4] Agarwal G S and Tara K. Nonclassical properties of states generated by the excitations on acoherent state [J]. Phys. Rev. A,43,492–497,1991.
[5] Zhang Z and Fan H. Properties of states generated by excitations on a squeezed vacuumstate [J]. Phys. Lett. A,165,14–18,1992.
[6] Kis Z, Adam P, and Janszky J. Properties of states generated by excitations on the amplitudesqueezed states [J]. Phys. Lett. A,188,16–20,1994.
[7] Xin Z Z, Duan Y B, Zhang H M, Hirayama M, and Matumoto K I. Excited two-photoncoherent state of the radiation field [J]. J. Phys. B: At. Mol. Opt. Phys.,29,4493–4506,1996.
[8] Man’ko V I and Wu¨nsche A. Properties of squeezed-state excitations [J]. Quantum Semi-class. Opt.,9,381–409,1997.
[9] Xin Z Z, Duan Y B, Zhang H M, Qian W J, Hirayama M, and Matumoto K I. Excitedeven and odd coherent states of the radiation field [J]. J. Phys. B: At. Mol. Opt. Phys,29,2597–2606,1996.
[10] Dodonov V V, Korennoy Y A, Man’ko V I, and Moukhin Y A. Nonclassical properties ofstates generated by the excitation of even/odd coherent states of light [J]. Quantum Semi-class. Opt.,8,413–427,1996.
[11] Neergaard-Nielsen J S, Melholt Nielsen B, Hettich C, M mer K, and Polzik E S. Generationof a superposition of odd photon number states for quantum information networks [J]. Phys.Rev. Lett.,97,083604,2006．
[12] Ourjoumtsev A, Tualle-Brouri R, Laurat J, and Grangier Ph. Generating optical Schro¨dingerkittens for quantum information processing [J]. Science,312,83–86,2006．
[13] Wakui K, Takahashi H, Fumsawa A, and Sasaki M. Photon subtracted squeezed states gen-erated with periodically poled KTiOPO4[J]. Opt．Express,15,3568–3574,2007．
[14] Dakna M, Anhut T, Opatmy T, Knoll L, and Welsch D-G. Generating Schrodinger-cat-likestates by means of conditional measurements on a beam splitter [J]. Phys. Rev. A,55,3184–3194,1997．
[15] Glancy S and de Vasconcelos H M. Methods for producing optical coherent state superpo-sitions [J]. J. Opt. Soc. Am. B,25,712–733,2008.
[16] Ourjoumtsev A, Dantan A, Tualle-Brouri R, and Grangier Ph. Increasing entanglement be-tween Gaussian states by coherent photon subtraction [J]. Phys. Rev. Lett.,98，030502,2007.
[17] Browne D E, Eisert J, Scheel S, and Plenio M B. Driving non-Gaussian to Gaussian stateswith linear optics [J]. Phys. Rev. A,67,062320,2003.
[18] Nha H and Carmichael H J. Proposed test of quantum nonlocality for continuous variables[J]. Phys. Rev. Lett.,93,020401,2004.
[19] Garc′a-Pat′ron R, Fiura′sˇek J, Cerf N J, Wenger J, Tualle-Brouri R, and Grangier Ph. Proposalfor a loophole-free Bell test using homodyne detection [J]. Phys. Rev. Lett.,93,130409,2004.
[20] Bartlett S D and Sanders B C. Universal continuous-variable quantum computation: Re-quirement of optical nonlinearity for photon counting [J]. Phys. Rev. A,65,042304,2002.
[21] Short R and Mandel L. Observation of sub-poissonian photon statistics [J]. Phys. Rev. Lett.,51,384–387,1983.
[22] Kimble H J, Dagenais M, and Mandel L. Photon antibunching in resonance fluorescence [J].Phys. Rev. Lett.,39,691–695,1977.
[23] Stoler D. Equivalence classes of minimum uncertainty packets [J]. Phys. Rev. D,1,3217–3219,1970.
[24] Stoler D. Equivalence classes of minimum uncertainty packets II [J]. Phys. Rev. D,4,1925–1926,1971.
[25] Yuen H P. Two-photon coherent states of the radiation field [J]. Phys. Rev. A,13,2226–2243,1976.
[26] Mandel L. Sub-Poissonian photon statistics in resonance fluorescence [J]. Opt. Lett.,4,205–207,1979.
[27] Spaelter S, Burk M, Stroessner U, Sizmann A, and Leuchs G. Propagation of quantumproperties of sub-picosecond solitons in a fiber [J]. Opt. Express,2,77–83,1998.
[28] Caves C M. Quantum-mechanical noise in an interferometer [J]. Phys. Rev. D,23,1693–1708,1981.
[29] Slusher R E, Hollberg L W, Yurke B, Mertz J C, and Valley J F. Observation of squeezedstates generated by four-wave mixing in an optical cavity [J]. Phys. Rev. Lett.,55,2409–2412,1985.
[30] Machida S, Yamamoto Y, and Itaya Y. Observation of amplitude squeezing in a constant-current–driven semiconductor laser [J]. Phys. Rev. Lett.,58,1000–1003,1987.
[31] Raizen M G, Orozco L A, Xiao Min, Boyd T L, and Kimble H J. Squeezed-state generationby the normal modes of a coupled system [J]. Phys. Rev. Lett.,59,198–201,1987.
[32] Hillery M. Squeezing of the square of the field amplitude in second harmonic generation[J]. Opt. Commun.,62,135–138,1987.
[33] Hong C K and Mandel L. Higher-order squeezing of a quantum field [J]. Phys. Rev. Lett.,54,323–325,1985.
[34] Hong C K and Mandel L. Generation of higher-order squeezing of quantum electromagneticfields [J]. Phys. Rev. A，32,974–982,1985.
[35] Zhang Z M, Xu L, Chai J L, and Li F L, A new kind of higher-order squeezing of radiationfield [J]. Phys. Lett. A，150,27–30，1990.
[36] Du S D and Gong C D. Squeezing of the kth power of the field amplitude [J]. Phys. Lett. A,168,296–300,1992.
[37] Lee J, Kim J, and Nha H. Demonstrating higher-order nonclassical effects by photon-addedclassical states: realistic schemes [J]. J. Opt. Soc. Am. B,26,1363–1369,2009.
[38] Kenfack A and Zyczkowski K. Negativity of the Wigner function as an indicator of nonclas-sicality [J]. J. Opt. B,6,396–404,2004.
[39] Lee C T. Measure of the nonclassicality of nonclassical states [J]. Phys. Rev. A,44, R2775–R2778,1991.
[40] Sudarshan E C G. Equivalence of semiclassical and quantum mechanical descriptions ofstatistical light beams [J]. Phys. Rev. Lett.,10,277–279,1963.
[41] Scully M O and Zubairy M S. Quantum optics [M].(Cambridge University,1997).
[42] Hillery M. Nonclassical distance in quantum optics [J]. Phys. Rev. A,35,725–732,1987.
[43] Asbo′th J K, Calsamiglia J, and Ritsch H. Computable measure of nonclassicality for light[J]. Phys. Rev. Lett.,94,173602,2005.
[44] Wei U. Quantum dissipative system [M].(Singapore: Word Scientific，1999).
[45] Gell-Mann M and Hartle J B. Classical equations for quantum systems [J]. Phys. Rev. D,47,3345–3382,1993.
[46] Walls D E and Milbum G J. Quantum optics [M].(Springer-Verlag, Berlin,1994).
[47] Orszag M. Quanturn optics [M].(Springer Press,2000).
[48] Elze H T. Quantum decoherence, entropy and thermalization in strong interactions at highenergy.(I). Noisy and dissipative vacuum effects in toy models [J]. Nucl．Phys. B,436,213–261,1995.
[49] Zurek W H. Environment-induced superselection rules [J]. Phys. Rev. D,26,1862–1880,1982.
[50] Unruh W G. Maintaining coherence in quantum computers [J]. Phys. Rev. A,51,992–997,1995.
[51] Bouwmeester D, Ekert A, and Zeilinger A. The physics of quantum information [M].(Springer-Verlag, Berlin,2000).
[52] Bennett C H and DiVincenzo D P. Quantum information and computation [J]. Nature,404,247–255,2000.
[53] Klauder J R and Skagerstam B S. Coherent states: applications in physics and mathematicalphysics [M]. World Scientific,1985.
[54] Klauder J R and Sudarshan E C G. Fundamentals of quantum optics [M]. New York: Ben-jamin,1968.
[55] Perelomov A M. Generalized coherent states and their applications [M]. Berlin: Springer,1986.
[56] Zhang W M. Coherent States: Theory and some applications [J]. Rev. Mod. Phys.,62,867–927,1990.
[57] Onofri E. A note on coherent state representations of Lie groups [J]. J. Math. Phys.,16,1087–1089,1975.
[58] Jurcˇo B,Sˇtovicˇek P. Coherent states for quantum compact groups [J]. Comm. Math. Phys.,182,221–251,1996.
[59] Scherk J. An introduction to the theory of dual models and strings [J]. Rev. Mod. Phys.,47,123–164,1975.
[60] Davies M J and Heller E J. Multidimensional wave functions from classical trajectories [J].J. Chem. Phys.,75,3919–3924,1981.
[61] Schro¨dinger E. Quantization as a problem of eigenvalue [J]. Naturwissenschaften,14,664–666,1926.
[62] Glauber R J. The quantum theory of optical coherence [J]. Phys. Rev.,130,2529–2539,1963.
[63] Glauber R J. Coherent and incoherent states of radiation field [J]. Phys. Rev.,131,2766–2788,1963．
[64] Sivakumar S. Studies on nonlinear coherent states [J]. J. Opt B：Quantum Semiclass. Opt.,2, R61–R75,2000.
[65] Loundon R. The quantum theory of light [M]. Oxford: Clarendon Press,1983.
[66] Louisell W H. Quantum statstical properities of radiation [M]. New York: Wiley1973.
[67] Dirac P A M. The principles of quantum mechanics [M]. Oxford: Clarendon Press,1930.
[68] Fan H Y. Operator ordering in quantum optics theory and the development of Dirac’s sym-bolic method [J]. J. Opt. B,5, R147–R163,2003.
[69] Wigner E. On the quantum correction for thermodynamic equilibrium [J]. Phys. Rev.,40,749–759,1932.
[70] Fan H Y. Newton–Leibniz integration for ket–bra operators in quantum mechanics (IV)–Integrations within Weyl ordered product of operators and their applications [J]. Ann. Phys.,323,500–526,2008.
[71] Mehta C L. Diagonal coherent-state representation of quantum operators [J]. Phys. Rev.Lett.,18,752–754,1967.
[72] Fan H Y. New antinormal ordering expansion for density operators [J]. Phys. Lett. A,161,1–4,1991.
[73] Weyl H. Quantenmechanik und gruppentheorie [J]. Z. Phys.,46,1–46,1927.
[74] Fan H Y. Weyl ordering quantum mechanical operators by virtue of the IWWP technique[J]. J. Phys. A: Math. Gen.,25,3443–3447,1992.
[75] Hu L Y and Fan H Y. Wavelet transformation and Wigner-Husimi distribution function [J].Int J Theor. Phys.,48,1539–1544,2009.
[76] Einstein A, Podolsky B, and Rosen N. Can quantum-mechanical description of physicalreality be considered complete [J]. Phys. Rev.,47,777–780,1935.
[77] Fan H Y, Zaidi H R, and Klauder J R. New approach for calculating the normally orderedform of squeeze operators [J]. Phys. Rev. D,35,1831–1834,1987.
[78] Fan H Y and Klaude J R. Eigenvectors of two particles’ relative position and total momen-tum [J]. Phys. Rev. A,49,704–707,1994.
[79] Fan H Y. Time evolution of the Wigner function in the entangled-state representation [J].Phys. Rev. A,65,064102,2002.
[80] Fan H Y. Application of EPR entangled state representation in quantum teleportation ofcontinuous variables [J]. Phys. Lett. A,294,253–257,2002.
[81] Fan H Y and Fan Y. Representations of two-mode squeezing transformations [J]. Phys. Rev.A,54,958–960,1996.
[82] Gardiner C and Zoller P. Quantum noise [M].(Springer Berlin，2000).
[83] Agarwal G S. Brownian motion of a quantum oscillator [J]. Phys. Rev. A,4,739–747,1971.
[84] Daniel D J and Milburn G J. Destruction of quantum coherence in a nonlinear oscillator viaattenuation and amplification [J]. Phys. Rev. A,39,4628–4640,1989.
[85] Kim M S and Buzek V. Schro¨dinger-cat states at finite temperature: Influence of a finite-temperature heat bath on quantum interferences [J]. Phys. Rev. A,46,4239–4251,1992.
[86] Fan H Y and Fan Y. New representation of thermal states in thermal field dynamics [J].Phys. Lett. A,246,242–246,1998.
[87] Wang C C and Fan H Y. Evolution of a thermo vacuum state in a single-mode amplitudedissipative channel [J]. Chin. Phys. Lett.,27,110302-1–110302-3,2010.
[88] Fan H Y. Weyl ordering expansion of power product of coordinate and momentum operators[J]. Commun. Theor. Phys.,54,439–442,2010.
[89] Hu L Y and Fan H Y. Two-mode squeezed number state as a two-variable Hermite-polynomial excitation on the squeezed vacuum [J]. J. Mod. Opt.,55,2011–2024,2008.
[90] Fan H Y and Tang X B. Wavefunction of a pair coherent state in the entangled state repre-sentation as an eigenfunction of a type of Fokker–Planck differential operator [J]. J. Phys.A,39,9831–9838,2006.
[91] Magnus W, Oberhettinger F, and Soni R P. Formulas and theorem for the special functionsof mathematical physics [M].3rd ed.(Springer-Verlag,1966).
[92] Fan H Y and Hu L Y. Infinite-dimensional Kraus operators for describing amplitude-damping channel and laser process [J]. Opt. Comm.,282,932–935,2009.
[93] Takahashi Y and Umezawa H. Thermo field dynamics [J]. Int. J. Mod. Phys. B,10,1755–1805,1996.
[94] Puri R R. Mathematical methods of quantum optics [M].(Springer-Verlag,2001), AppendixA.
[95] Fan H Y and Lu H L. Wave-function transformations by general SU(1,1) single-modesqueezing and analogy to Fresnel transformations in wave optics [J]. Opt. Commun.,258,51–58,2006.
[96] Fan H Y and Lu H L.2-mode Fresnel operator and entangled Fresnel transform [J]. Phys.Lett. A,334,132–1392005.
[97] Biswas A and Agarwal G S. Nonclassicality and decoherence of photon-subtracted squeezedstates [J]. Phys. Rev. A,75,032104，2007.
[98] Olivares S and Paris M G A. Squeezed Fock state by inconclusive photon subtraction [J]. J.Opt. B: Quantum Semiclass. Opt.,7, S616,2005.
[99] Olivares S and Paris M G A. Photon subtracted states and enhancement of nonlocality in thepresence of noise [J]. J. Opt. B: Quantum Semiclass. Opt.,7, S392,2005.
[100] Fan H Y, Meng X G, and Wang J S. New form of Legendre polynomials obtained by virtueof excited squeezed state and IWOP technique in quantum optics [J]. Commun. Theor. Phys.46,845–848,2006.
[101] Dodonov V V. Nonclassical states in quantum optics: a squeezed review of the first75years[J]. J. Opt. B,4, R1–R33,2002.
[102] Hu L Y and Fan H Y. Nonclassicality of photon-added squeezed vacuum and its decoherencein thermal environment [J]. J. Mod. Opt.,57,1344–1354,2010.
[103] Lee S Y and Nha H. Quantum state engineering by a coherent superposition of photonsubtraction and addition [J]. Phys. Rev. A,82,053812,2010.
[104] Hu L Y and Fan H Y. Statistical properties of photon-subtracted squeezed vacuum in thermalenvironment [J]. J. Opt. Soc. Am. B,25,1955-1964,2008.
[105] Xu X X, Yuan H C, and Fan H Y. Decoherence of photon-subtracted squeezed vacuum statein dissipative channel [J]. Chinese Phys. B,20,024203,2011.
[106] Schleich W P. Quantum optics in phase space [M]. Wiley-VCH, Berlin,2001.
[107] Li H M and Yuan H C. Simple approach to deriving some operator ordering formulas inquantum optics [J]. Int J Theor Phys,49,2121–2130,2010.
[108] Sugita A and Aiba H. Second moment of the Husimi distribution as a measure of complexityof quantum states [J]. Phys. Rev. E65,036205,2002.
[109] Ourjoumtsev A, Jeong Hyunseok, Tualle-Brouri R, and Grangier P. Generation of opticalSchro¨dinger cats from photon number states [J]. Nature448,784–786,2007.
[110] Ralph T C, Gilchrist A, Milburn G J, Munro W J, and Glancy S. Quantum computation withoptical coherent states [J]. Phys. Rev. A,68,042319,2003.
[111] Jeong H, Kim M S, and Lee J. Quantum-information processing for a coherent superpositionstate via a mixedentangled coherent channel [J]. Phys. Rev. A,64,052308,2001.
[112] Munro W J, Nemoto K, Milburn G J, and Braunstein S L. Weak-force detection with super-posed coherent states [J]. Phys. Rev. A,66,023819,2002.
[113] You W L, Li Y W, and Gu S J Fidelity, dynamic structure factor, and susceptibility in criticalphenomena [J]. Phy. Rev. E76,022101,2007.
[114] Bogolyubov N N. Lectures on quantum statistics [M]. New York: Gordon and Breach,1987.
[115] Zavatta A, Viciani S, and Bellini M. Quantum-to-classical transition with single-photon-added coherent states of light [J]. Science,306,660–662,2004.
[116] Zavatta A, Viciani S, and Bellini M. Single-photon excitation of a coherent state: catchingthe elementary step of stimulated light emission [J]. Phys. Rev. A,72,023820,2005.
[117] Meng X G, Wang Z, Fan H Y, and Wang J S. Squeezed number state and squeezed thermalstate: decoherence analysis and nonclassical properties in the laser process [J]. J. Opt. Soc.Am. B,29,1835–1843,2012.
[118] Vogel K and Risken H. Determination of quasiprobability distributions in terms of proba-bility distributions for the rotated quadrature phase [J]. Phys. Rev. A,40,2847–2849,1989.
[119] Smithey D T, Beck M, Raymer M G, and Faridani A. Measurement of the Wigner distribu-tion and the density matrix of a light mode using optical homodyne tomography: applicationto squeezed states and the vacuum [J]. Phys. Rev. Lett.,70,1244–1247,1993.
[120] Fan H Y and Chen H L. Two-parameter Radon transformation of the Wigner operator andits inverse [J]. Chin. Phys. Lett.,18,850–853,2001.
[121] Fan H Y and Li G C. Radon transformation of the Wigner operator for two-mode correlatedsystem in generalized entangled state representation [J]. Mod. Phys. Lett. A,15,499–507,2000.
[122] Kim M S. Recent developments in photon-level operations on travelling light fields [J]. J.Phys. B,41,133001,2008.
[123] Lee S Y, Park J, Ji S W, Ooi C H R, and Lee H W. Nonclassicality generated by photonannihilation-then-creation and creation-then-annihilation operations [J]. J. Opt. Soc. Am. B,26,1532–1537,2009.
[124] Yang Y and Li F L. Nonclassicality of photon-subtracted and photon-added-then-subtractedGaussian states [J]. J. Opt. Soc. Am. B,26,830–835,2009.
[125] Fan H Y and Fan Y. New bosonic operator ordering identities gained by the entangled staterepresentation and two-variable Hermite polynomials [J]. Commun. Theor. Phys.,38,297–300,2002.
[126] Hu L Y and Fan H Y. Infinite-dimensional Kraus operators for describing amplitude-damping channel and laser process [J]. Opt. Commun.,282,932–935,2009.
[127] Jozsat R. Fidelity for mixed quantum states [J]. J. Mod. Opt.,41,2315–2323,1994.
[128] Yuen H P. Two-photon coherent states of the radiation field [J]. Phys. Rev. A,13,2226–2243,1976.
[129] Yuen H P and Shapiro J H. Optical communication with two-photon coherent states–Part I:Quantum-state propagation and quantum-noise [J]. IEEE Trans. Inform. Theory,24,657–668,1978.
[130] Shapiro J H, Yuen H P, and Machado Mata J A. Optical communication with two-photoncoherent states–Part II: Photoemissive detection and structured receiver performance [J].IEEE Trans. Inform. Theory,25,179–192,1979.
[131] Goda K, Miyakawa O, Mikhailov E E, Saraf S, Adhikari R, Mckenzie K, Ward R, Vass S,Weinstein A J, and Mavalvala N. A quantum-enhanced prototype gravitational-wave detec-tor [J]. Nature Phys.,4,472–476,2008.
[132] Vahlbruch H, Chelkowski S, Hage B, Franzen A, Danzmann K, and Schnabel R. Demon-stration of a squeezed-light-enhanced power-and signal-recycled Michelson interferometer[J]. Phys. Rev. Lett.,95,211102-1–211102-4,2005.
[133] Mandel L and Wolf E. Optical coherence and quantum optics [M].(Cambridge: CambridgeUniversity Press),1995.
[134] Schleich W and Wheeler J A. Oscillations in photon distribution of squeezed states andinterference in phase space [J]. Nature,326,574–577,1987.
[135] Schleich W and Wheeler J A. Oscillations in photon distribution of squeezed states [J]. J.Opt. Soc. Am. B,4,1715–1722,1987.
[136] Vahlbruch H, Mehmet M, Chelkowski S, Hage B, Franzen A, Lastzka N, Gober S, Danz-mann K, and Schnabel R. Observation of squeezed light with10-dB quantum-noise reduc-tion [J]. Phys. Rev. Lett.,100,033602,2008.
[137] Mehmet M, Vahlbruch H, Lastzka N, Danzmann K, and Schnabel R. Observation ofsqueezed states with strong photon-number oscillations [J]. Phys. Rev. A,81,013814,2010.
[138] Braunstein S L and Kimble H J. Dense coding for continuous variables [J]. Phys. Rev. A,61,042302,2000.
[139] Takeno Y, Yukawa M, Yonezawa H, and Furusawa A. Observation of9dB quadraturesqueezing with improvement of phase stability in homodyne measurement [J]. Opt. Express,15,4321,2007.
[140] Hsu M T L, Hetet G, Glockl O, Longdell J J, Buchler B C, Bachor H A, and Lam P K.Quantum study of information delay in electromagnetically induced transparency [J]. Phys.Rev. Lett.,97,183601,2006.
[141] Fan H Y, Yuan H C, Xu X X, and Jiang N Q. New approach for obtaining the squeezing-enhanced state and its Wigner function by virtue of the Weyl–Wigner quantization scheme[J]. Phys. Scr.,83,015403,2011.
[142] Wang S and Fan H Y. Statistical properties of the squeezing-enhanced thermal state [J]. J.Opt. Soc. Am. B,29,15–22,2012.
[143] Agarwal G S and Tara K. Nonclassical character of states exhibiting no squeezing or sub-Poissonian statistics, Phys. Rev. A,46,4851992.
[144] Chizhov A V, Knoll L, and Welsch D G. Continuous-variable quantum teleportation throughlossy channels [J]. Phy. Rev. A,65,022310,2002.