非高斯量子态及其非经典性质的研究
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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,光场非经典性质通过一些具体的量子统计特点体现出来的,如反聚束、亚泊松光子统计、光场的压缩特性及Wigner函数的部分负分布等等。其中,由于Wigner函数具有准概率分布函数的性质而不是严格的概率分布函数,故可正可负。对于准经典态(如典型的相干态),其Wigner函数值总是非负,而Wigner函数取负值则是量子态具有非经典特征。为了寻找新的具有非经典性质的量子态,已有不少作者提出了产生各种非经典态的方案。最常见的一种方法就是利用量子力学中的态叠加原理,例如利用Fock态、相干态、压缩态、平移数态等量子态的叠加态。另一种方法就是将一个算符作用在一个参考态上,比如说,通常的压缩态可以通过压缩算符作用在相干态上产生。近年来,对某态通过光子增加或光子扣除也可以诱导出具有非经典性质的态(如光子增加相干态)。实际上,上述所涉及到的态大多数不是高斯量子态。所谓高斯量子态,是指其相空间Wigner分布函数具有高斯形式的量子态。但是,随着实验技术的发展,实验和理论物理学家也在尝试利用非高斯态作为信息源。发现它们在量子计算机中的应用,同时可以改善隐形传输、克隆和存储等。因此,非高斯量子态的研究引起了人们的注意。本文主要针对若干高斯量子态进行一些非高斯性操作,产生一些新的非高斯量子态,并展现出其非经典特征。其主要内容包括:
一、简要介绍一些量子光学的理论基础。简要回顾范洪义教授提出的有序算符内的积分技术(IWOP技术)的基本理论,并用该技术从新的角度导出常用的量子力学表象。特别给出了用IWOP技术和技巧方便地求出量子态的相空间分布函数,如Wigner函数、Husimi函数和Tomogram函数等。介绍了两个新的光子计数公式。另外还列举了描述量子态非经典属性的其它表征方法。
二、采用一种新的途径推导出几种复杂系统所对应的广义热真空态,即部分求迹再结合IWOP技术。于是对于复杂系统中,力学量的系综平均的计算也就转为求纯态下广义热真空态的期望值,这不但给计算带来了很大的方便,而且丰富了热场动力学理论。这里,利用所求出的复杂系统相应的热真空态很方便求解出该系统的内能及其分布。
三、研究了对热场进行非高斯操作所诱导出几种非高斯态及其相关的一些非经典性质,包括光子增加或扣除热态、光子调制热态。在求出这些非高斯态的归一化系数之后,采用IWOP技术和Weyl编序在相似变换下的不变性求出光子增加或扣除热态的Wigner函数解析表达式。重点讨论了光子增加热态在热环境下的退相干的统计性质,如光子数分布与Wigner函数的时间演化,发现了Wigner函数的负部随时间的增加逐渐消失,并在介观RLC电路量子化方案中分析这些非高斯态的涨落问题。其次,我们还研究了光子调制热态,它是通过产生和湮灭算符的线性组合算符连续作用在热态上产生的,并讨论了该量子态的性质,如保真情况、准几率分布函数、光子计数分布以及Tomogram函数。
四、研究了由单模压缩态衍生出的非高斯量子态及其非经典性质。首先介绍了相干态表象下的压缩算符及其对应的广义压缩粒子数态,分析了其非经典性质,讨论该量子态在耗散通道中的退相干问题,得到了Wigner分布函数演化的解析表达式,清楚地了解有关参数对Wigner分布函数的影响。其次讨论光子先增后减压缩真空态,实际上它是两种压缩粒子态的叠加,并利用Hilbert-Schmidt距离来度量其非高斯性,给出了腔QED的产生方案。最后研究光子增加和扣除压缩真空态,得到其归一化系数与Wigner函数解析式,并分别与压缩猫态(叠加相干态)的保真度的进行比较,结果发现任意光子扣除(或增加)压缩真空态都可以产生一个高保真度的压缩猫态。
五、研究了由双模压缩态衍生出的非高斯量子态及其非经典性质。首先讨论了单-双模连续压缩算符及其压缩态,并研究了该态的压缩效应,关联函数、反聚束效应以及粒子数分布等统计属性。特别是,利用算符的Weyl编序下相似不变性质,解析地导出了连-续单双模压缩真空态的Wigner函数表达式。重点是将光子扣除单模压缩真空态的情况推广到光子扣除双模压缩真空态(一个非高斯态),并通过IWOP技术讨论光子扣除双模压缩态的非经典性质。结果表明,其归一化系数是一个关于压缩参数的Jacobi多项式,其Wigner函数与双变量厄密多项式有关。
六、讨论了光子扣除(增加)压缩热态及其非经典性质。对于任意数目光子扣除压缩热态,导出了归一化系数的解析表达式——一个关于压缩参数以及热态平均光子数的Legendre多项式;也导出它们的几个准概率分布函数的解析表达式。解析导出了压缩热态与光子扣除压缩热态间的保真度,发现该保真度随光子扣除数和压缩参数的增加而单调减少。最后还讨论了它们的退相干问题。
Nonclassicality of optical fields has been a topic of great interest in quantum optics andquantum information processing. Usually, the nonclassicality manifests itself in specificproperties of quantum statistics, such as the antibunching, sub-Poissonian photon statistics,squeezing in one of the quadratures of the field, partial negative Wigner distribution, etc.Among them, the Wigner function is the quasi-probability distribution function, whose valuecan be positive or negative. For the quasi-classical state (such as a typical coherent state), itsWigner function is always non-negative. Thus, the partial negativity of Wigner function isindeed a good indication of the highly nonclassical character of quantum states. In order toobtain new and non-classical quantum states, many researchers have proposed a variety ofschemes. The basic approach for constructing non-classical quantum states is making use ofthe principle of superposition in quantum mechanics, for example, some superposition statesof Fock states, coherent states, squeezed states, displaced Fock states and so on. Anotherapproach is that acting an operator on a reference state, for example, the usual squeezedstates can be generated by operating the squeezing operator on the coherent states. In recentyears, it is found that subtracting photons from or adding photons to quantum states canobtain some non-classical states such as photon-added coherent states. In fact, these statesabove are not involved in the Gaussian quantum states. The so-called Gaussian quantumstate is defined as state with a Gaussian Wigner function. However, with the developmentof experimental techniques, experimental and theoretical physicists are trying to use non-Gaussian state as the source of information. The most notable example is certainly their usefor an optical quantum computer, alongside their employment for improving teleportation,cloning, and storage. Therefore, considerable attention has been paid to a class of non-Gaussian quantum states. In this thesis, some new non-Gaussian quantum states, exhibitingtheir non-classical features, can be obtained from a number of Gaussian quantum states afteroperating non-Gaussian operation. This thesis is organized as follows:
1. We brie?y introduce some basis theories of quantum optics. To begin with, thetechnique of integration within ordered product of operators (IWOP), first proposed by Prof. Fan, is brie?y reviewed, and based on this technique fundamental representations in quantummechanics are derived from new point of view. Especially we give the IWOP technique andskills to easily deduce several distribution functions in the phase space, such as the Wignerfunction, Husimi function and Tomogram function and so on. At the same time, two newphoton counting formulas are presented. It also lists other methods describing non-classicalproperties of quantum states.
2. We employ a new way, the partial trace method and the IWOP technique, to derivegeneralized thermal vacuum states corresponding to some complex systems, namely, obtain-ing the corresponding pure state of a mixed state in the extended Hilbert space. So for acomplex system, the ensemble average of turns into calculating the pure state’s expectation,this is not only convenient for calculations, but also develops and enriches the theory of ther-mal field dynamics. In addition, it is very convenient for solving the internal energy and itsdistribution in the complex system system by using its corresponding thermal vacuum state.
3. We obtain some non-Gaussian states associated with thermal field after non-Gaussianoperation, including photon-added, photon-subtracted, and photon-modulated, and studytheir non-classical properties. After deriving normalization coefficients on these non-Gaussian states, we obtain their explicit expressions of the Wigner functions, by adoptingthe IWOP technique and the Weyl ordered operators’invariance under similar transforma-tions. Next, we focus on investigating the decoherence of statistical properties for photon-added thermal field in the thermal environment, such as photon number distribution and theWigner function with time evolution. It shows that the partial negativity of Wigner func-tion is gradually disappear with the increase of time. We discuss the ?uctuation problemsof these non-Gaussian states in quantum mesoscopic RLC circuit. Secondly, we also studythe photon-modulation thermal state, generated by operating the linear combination of thecreation and annihilation operators on the thermal state, and discuss its non-classical prop-erties such as fidelity, quasi-probability distribution function, photon counting distribution,and Tomogram function.
4. We study non-Gaussian states evolved from single-mode squeezed states and theirnon-classical properties. At first, we introduce the generalized squeezing operator in thecoherent state representation and the corresponding squeezed Fock state, analyze its non-classical characteristics. Its analytical expression of the Wigner function with time evolu-tion is obtained in the dissipative channel. Then, we discuss photon-added-then-subtractedsqueezed vacuum state, which is actually the superposition of two kind squeezed Fock states,evaluate its non-Gaussianity by using Hilbert-Schmidt distance measure and present the gen- eration scheme through the cavity QED. Finally, we propose photon-added and photon-subtracted squeezed vacuum states, deduce their normalization coefficients and Wignerfunctions, and deduce the analytical expressions of the fidelity between photon subtracted(or added) squeezed vacuum state and squeezed cat state. For the same number photon-subtraction as photon-addition, a squeezed cat state with a lower fidelity yet higher amplitudecan be generated by the case of photon-addition. In this sense, although photons addition op-erator are more difficult than that of photons subtraction in experiment, photons addition canalso be a powerful tool to generate a cat state with large amplitude.
5. We study non-Gaussian states evolved from two-mode squeezed states and their non-classical properties. Firstly we discuss the single- and two-mode successive squeezed oper-ator and its squeezed states. Some statistical properties such as squeezing effect, correlationfunction, anti-bunching effect, and photon number distribution are investigated. In partic-ular, we analytically derive its Wigner function by virtue of the Weyl ordered operators’invariance under similar transformations. Moreover, we focus on investigating the statisticalproperties of photon subtractions from a two-mode squeezed vacuum state, which is also anon-Gaussian state, and its non-classical properties with the help of the IWOP technique. Itis found that the normalization of this state is the Jacobi polynomial of the squeezing pa-rameter, A compact expression for the Wigner function is related to two-variable Hermitepolynomial.
6. We investigate nonclassical properties of the field states generated by subtracting(adding) any number photon from the squeezed thermal state. It is found that the normaliza-tion factor of photon-subtracted squeezed thermal state is a Legendre polynomial of squeez-ing parameter and average photon number of thermal state. Their expressions of severalquasi-probability distributions are derived analytically. Additionally, we devote to calcu-lating the fidelity between the photon-subtracted squeezed thermal state and the squeezedthermal state. It is shown that the fidelity decreases monotonously with the increment ofboth photon-subtraction number and the squeezing parameter. Finally, we also discuss theirdecoherence.
一、简要介绍一些量子光学的理论基础。简要回顾范洪义教授提出的有序算符内的积分技术(IWOP技术)的基本理论,并用该技术从新的角度导出常用的量子力学表象。特别给出了用IWOP技术和技巧方便地求出量子态的相空间分布函数,如Wigner函数、Husimi函数和Tomogram函数等。介绍了两个新的光子计数公式。另外还列举了描述量子态非经典属性的其它表征方法。
二、采用一种新的途径推导出几种复杂系统所对应的广义热真空态,即部分求迹再结合IWOP技术。于是对于复杂系统中,力学量的系综平均的计算也就转为求纯态下广义热真空态的期望值,这不但给计算带来了很大的方便,而且丰富了热场动力学理论。这里,利用所求出的复杂系统相应的热真空态很方便求解出该系统的内能及其分布。
三、研究了对热场进行非高斯操作所诱导出几种非高斯态及其相关的一些非经典性质,包括光子增加或扣除热态、光子调制热态。在求出这些非高斯态的归一化系数之后,采用IWOP技术和Weyl编序在相似变换下的不变性求出光子增加或扣除热态的Wigner函数解析表达式。重点讨论了光子增加热态在热环境下的退相干的统计性质,如光子数分布与Wigner函数的时间演化,发现了Wigner函数的负部随时间的增加逐渐消失,并在介观RLC电路量子化方案中分析这些非高斯态的涨落问题。其次,我们还研究了光子调制热态,它是通过产生和湮灭算符的线性组合算符连续作用在热态上产生的,并讨论了该量子态的性质,如保真情况、准几率分布函数、光子计数分布以及Tomogram函数。
四、研究了由单模压缩态衍生出的非高斯量子态及其非经典性质。首先介绍了相干态表象下的压缩算符及其对应的广义压缩粒子数态,分析了其非经典性质,讨论该量子态在耗散通道中的退相干问题,得到了Wigner分布函数演化的解析表达式,清楚地了解有关参数对Wigner分布函数的影响。其次讨论光子先增后减压缩真空态,实际上它是两种压缩粒子态的叠加,并利用Hilbert-Schmidt距离来度量其非高斯性,给出了腔QED的产生方案。最后研究光子增加和扣除压缩真空态,得到其归一化系数与Wigner函数解析式,并分别与压缩猫态(叠加相干态)的保真度的进行比较,结果发现任意光子扣除(或增加)压缩真空态都可以产生一个高保真度的压缩猫态。
五、研究了由双模压缩态衍生出的非高斯量子态及其非经典性质。首先讨论了单-双模连续压缩算符及其压缩态,并研究了该态的压缩效应,关联函数、反聚束效应以及粒子数分布等统计属性。特别是,利用算符的Weyl编序下相似不变性质,解析地导出了连-续单双模压缩真空态的Wigner函数表达式。重点是将光子扣除单模压缩真空态的情况推广到光子扣除双模压缩真空态(一个非高斯态),并通过IWOP技术讨论光子扣除双模压缩态的非经典性质。结果表明,其归一化系数是一个关于压缩参数的Jacobi多项式,其Wigner函数与双变量厄密多项式有关。
六、讨论了光子扣除(增加)压缩热态及其非经典性质。对于任意数目光子扣除压缩热态,导出了归一化系数的解析表达式——一个关于压缩参数以及热态平均光子数的Legendre多项式;也导出它们的几个准概率分布函数的解析表达式。解析导出了压缩热态与光子扣除压缩热态间的保真度,发现该保真度随光子扣除数和压缩参数的增加而单调减少。最后还讨论了它们的退相干问题。
Nonclassicality of optical fields has been a topic of great interest in quantum optics andquantum information processing. Usually, the nonclassicality manifests itself in specificproperties of quantum statistics, such as the antibunching, sub-Poissonian photon statistics,squeezing in one of the quadratures of the field, partial negative Wigner distribution, etc.Among them, the Wigner function is the quasi-probability distribution function, whose valuecan be positive or negative. For the quasi-classical state (such as a typical coherent state), itsWigner function is always non-negative. Thus, the partial negativity of Wigner function isindeed a good indication of the highly nonclassical character of quantum states. In order toobtain new and non-classical quantum states, many researchers have proposed a variety ofschemes. The basic approach for constructing non-classical quantum states is making use ofthe principle of superposition in quantum mechanics, for example, some superposition statesof Fock states, coherent states, squeezed states, displaced Fock states and so on. Anotherapproach is that acting an operator on a reference state, for example, the usual squeezedstates can be generated by operating the squeezing operator on the coherent states. In recentyears, it is found that subtracting photons from or adding photons to quantum states canobtain some non-classical states such as photon-added coherent states. In fact, these statesabove are not involved in the Gaussian quantum states. The so-called Gaussian quantumstate is defined as state with a Gaussian Wigner function. However, with the developmentof experimental techniques, experimental and theoretical physicists are trying to use non-Gaussian state as the source of information. The most notable example is certainly their usefor an optical quantum computer, alongside their employment for improving teleportation,cloning, and storage. Therefore, considerable attention has been paid to a class of non-Gaussian quantum states. In this thesis, some new non-Gaussian quantum states, exhibitingtheir non-classical features, can be obtained from a number of Gaussian quantum states afteroperating non-Gaussian operation. This thesis is organized as follows:
1. We brie?y introduce some basis theories of quantum optics. To begin with, thetechnique of integration within ordered product of operators (IWOP), first proposed by Prof. Fan, is brie?y reviewed, and based on this technique fundamental representations in quantummechanics are derived from new point of view. Especially we give the IWOP technique andskills to easily deduce several distribution functions in the phase space, such as the Wignerfunction, Husimi function and Tomogram function and so on. At the same time, two newphoton counting formulas are presented. It also lists other methods describing non-classicalproperties of quantum states.
2. We employ a new way, the partial trace method and the IWOP technique, to derivegeneralized thermal vacuum states corresponding to some complex systems, namely, obtain-ing the corresponding pure state of a mixed state in the extended Hilbert space. So for acomplex system, the ensemble average of turns into calculating the pure state’s expectation,this is not only convenient for calculations, but also develops and enriches the theory of ther-mal field dynamics. In addition, it is very convenient for solving the internal energy and itsdistribution in the complex system system by using its corresponding thermal vacuum state.
3. We obtain some non-Gaussian states associated with thermal field after non-Gaussianoperation, including photon-added, photon-subtracted, and photon-modulated, and studytheir non-classical properties. After deriving normalization coefficients on these non-Gaussian states, we obtain their explicit expressions of the Wigner functions, by adoptingthe IWOP technique and the Weyl ordered operators’invariance under similar transforma-tions. Next, we focus on investigating the decoherence of statistical properties for photon-added thermal field in the thermal environment, such as photon number distribution and theWigner function with time evolution. It shows that the partial negativity of Wigner func-tion is gradually disappear with the increase of time. We discuss the ?uctuation problemsof these non-Gaussian states in quantum mesoscopic RLC circuit. Secondly, we also studythe photon-modulation thermal state, generated by operating the linear combination of thecreation and annihilation operators on the thermal state, and discuss its non-classical prop-erties such as fidelity, quasi-probability distribution function, photon counting distribution,and Tomogram function.
4. We study non-Gaussian states evolved from single-mode squeezed states and theirnon-classical properties. At first, we introduce the generalized squeezing operator in thecoherent state representation and the corresponding squeezed Fock state, analyze its non-classical characteristics. Its analytical expression of the Wigner function with time evolu-tion is obtained in the dissipative channel. Then, we discuss photon-added-then-subtractedsqueezed vacuum state, which is actually the superposition of two kind squeezed Fock states,evaluate its non-Gaussianity by using Hilbert-Schmidt distance measure and present the gen- eration scheme through the cavity QED. Finally, we propose photon-added and photon-subtracted squeezed vacuum states, deduce their normalization coefficients and Wignerfunctions, and deduce the analytical expressions of the fidelity between photon subtracted(or added) squeezed vacuum state and squeezed cat state. For the same number photon-subtraction as photon-addition, a squeezed cat state with a lower fidelity yet higher amplitudecan be generated by the case of photon-addition. In this sense, although photons addition op-erator are more difficult than that of photons subtraction in experiment, photons addition canalso be a powerful tool to generate a cat state with large amplitude.
5. We study non-Gaussian states evolved from two-mode squeezed states and their non-classical properties. Firstly we discuss the single- and two-mode successive squeezed oper-ator and its squeezed states. Some statistical properties such as squeezing effect, correlationfunction, anti-bunching effect, and photon number distribution are investigated. In partic-ular, we analytically derive its Wigner function by virtue of the Weyl ordered operators’invariance under similar transformations. Moreover, we focus on investigating the statisticalproperties of photon subtractions from a two-mode squeezed vacuum state, which is also anon-Gaussian state, and its non-classical properties with the help of the IWOP technique. Itis found that the normalization of this state is the Jacobi polynomial of the squeezing pa-rameter, A compact expression for the Wigner function is related to two-variable Hermitepolynomial.
6. We investigate nonclassical properties of the field states generated by subtracting(adding) any number photon from the squeezed thermal state. It is found that the normaliza-tion factor of photon-subtracted squeezed thermal state is a Legendre polynomial of squeez-ing parameter and average photon number of thermal state. Their expressions of severalquasi-probability distributions are derived analytically. Additionally, we devote to calcu-lating the fidelity between the photon-subtracted squeezed thermal state and the squeezedthermal state. It is shown that the fidelity decreases monotonously with the increment ofboth photon-subtraction number and the squeezing parameter. Finally, we also discuss theirdecoherence.
引文
[1] Barbieri M, Spagnolo N, Genoni M G, etc. Phys. Rev. A 82 (2010) 063833.
[2] Furusawa A, S?ensen J L, Braunstein S L, etc. Science 282 (1998) 706.
[3] Andersen U L, Josse V, Leuchs G. Phys. Rev. Lett. 94 (2005) 240503.
[4] Lund A. Ralph T C, Haselgrove H L. Phys. Rev. Lett. 100 (2008) 030503.
[5] Agarwal G S, Tara K. Phys. Rev. A 43 (1991) 492.
[6] Zavatta A, Viciani S, Bellini M. Science 306 (2004) 660.
[7] Zavatta A, Viciani S, Bellini M. Phys. Rev. A 72 (2005) 023820.
[8] Fan H Y, Zaidi H R. Phys. Rev. A 37 (1988) 2985; Fan H Y, Wunsche A. J. Opt. B: Quantum &Semiclass. Opt. 2 (2000) 464.
[9] Fan H Y, Lu H L, Fan Y. Ann. Phys. 321 (2006) 480.
[10] Fan H Y. Phys. Lett. A 161 (1991) 1.
[11] Weyl H. Z. Phys. 46 (1927) 1.
[12] Louisell W H. Quantum Statistical Properties of Radiation. Wiley, New York, 1973.
[13] Wick G. Phys. Rev. 80 (1950) 268.
[14]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.
[15] Fan H Y. J. Phys. A 25 (1992) 3443.
[16] Fan H Y. Mod. Phys. Lett. A 15 (2000) 2297.
[17] Fan H Y. Ann. Phys. 323 (2008) 500.
[18] Klauder J R, Skargerstam B S. Coherent States. World Scientific, Singapore, 1985.
[19] Glauber R J. Phys. Rev. 130 (1963) 2529; 131 (1963) 2766.
[20] Wigner E. Phys. Rev. 40 (1932) 749; O’Connell R F, Wigner E P. Phys. Lett. A 83 (1981) 145.
[21] Schleich W P. Quantum Optics in Phase Space. Wiley-VCH, Birlin, 2001.
[22] Feynman R P. Statistical Physics. Benjamin W A, 1972.
[23] Hillery M, O’Connell R F, Scully M O, Wigner E P. Phys. Rep. 106 (1984) 121; Balazs N L,Jennings B K. Phys. Rep. 104 (1984) 347.
[24] Vogel K, Risken H. Phys. Rev. A 40 (1989) 2847.
[25] Husimi K. Proc. Phys.-Math. Soc. Jpn. 22 (1940) 264.
[26] Guo Q. Int. J. Theor. Phys. 46 (2007) 3135; Guo Q, Fan H Y. Int. J. Theor. Phys. 47 (2008) 3234.
[27] Fan H Y and Yu G C. Mod. Phys. Lett. A 15 (2000) 2297.
[28] Kelly P L, Kleiner W H. Phys. Rev. 136 (1964) 316.
[29] Loudon R. The Quantum Theory of Light. Oxford University Press, 1983.
[30] Fan H Y, Hu L Y. Opt. Lett. 33 (2008) 443.
[31] Mandel L. Opt. Lett. 4 (1979) 205.
[32] Lee C T. Phys. Rev. A 44 (1991) R2775.
[33] Kenfack A, Zyczkowski K. J. Opt. B: Quantum Semiclass. Opt. 6 (2004) 396.
[34] Hillery M. Phys. Rev. A 35 (1987) 725.
[35] Asboth J K, Calsamiglia J, Ritsch H. Phys. Rev. Lett. 94 (2005) 173602.
[36] Takahashi Y, Umezawa H. Thermo field dynamics. Collective Phenomena 2 (1975) 55; MemorialIssue for H. Umezawa, Int. J. Mod. Phys. B 10 (1996) 1563.
[37]范洪义,量子力学纠缠态表象及应用,上海交通大学出版社,2001.
[38]范洪义,从量子力学到量子光学,上海交通大学出版社,2005.
[39]范洪义,胡利云,光学变换-从量子到经典,上海交通大学出版社,2010.
[40]汪志诚,热力学统计物理,高等教育出版社,2008.
[41] Hu L Y, Fan H Y. Chin. Phys. Lett. 26 (2009) 090307.
[42] Barnett S M, Radmore P M. Methods in Theoretical Quantum Optics. Oxford: Clarendon press,1997.
[43] Hu L Y, Fan H Y. Commun. Theor. Phys. 51 (2009) 321.
[44] Fan H Y, Xu X X. J. Math. Phys. 50 (2009) 063303.
[45] Fan H Y, Xu X X, Hu L Y. Chin. Phys. B 19 (2010) 060502.
[46] Pathria R K. Statistical Mechannics. Singapore: World Scientific, 2001.
[47] Hu L Y, Fan H Y. Mod. Phys. Lett. A 24 (2009) 2263.
[48] Fan H Y, Jiang N Q. Phys. Scr. 78 (2008) 045402.
[49] Fan H Y, Chen H L. Chin. Phys. Lett. 18 (2001) 850.
[50] Wang J S, Sun C Y. Int. J. Theor. Phys. 37 (1998) 1213.
[51] Louisell W H. Quantum Statistical Properties of Radiation.John Wiley, New York,1973.– 127–
[52] Xu X X, Yuan H C, Hu L Y, Fan H Y. Thermal ?uctuation and thermodynamics of mesoscopic RLCcircuit at finite temperature calculated by using generalized thermal vacuum state (to be submitted)
[53] Mandel L, Wolf E. Optical Coherence and Quantum Optics. Cambridge: Cambridge UniversityPress, 1995.
[54] Fan H Y, Hu L Y. Commun. Theor. Phys. 51 (2009) 506.
[55] Xue X X, Fan H Y, Hu L Y, Yuan H C. Physica A 390 (2011) 18.
[56] Parigi V, Zavatta A, Kim M S, Bellini M. Science 317 (2007) 1890.
[57] Boyd R W, Chan K W, O’Sullivan M N. Science 317 (2007) 1874.
[58] Wenger J, Tualle-Brouri R, Grangier P. Phys. Rev. Lett. 92 (2004) 153601.
[59] Zavatta A, Parigi V, Bellini M. Phys. Rev. A 75 (2007) 052106.
[60] Ourjoumtsev A, Dantan A, Tualle-Brouri R, Grangier P. Phys. Rev. Lett. 98 (2007) 030502.
[61] Ourjoumtsev A, Dantan A, Tualle-Brouri R, Grangier P. Phys. Rev. Lett. 96 (2006) 213601.
[62] Biswas A, Agarwal G S. Phys. Rev. A 75 (2007) 032104.
[63] Hu L Y, Fan H Y. J. Opt. Soc. Am. B 25 (2008) 1955.
[64] Hu L Y, Fan H Y. Mod. Phys. Lett. A 24 (2009) 2263.
[65] Wang S J, Xu X X, Ma S J. Int. J. Quant. Inf. (2011) in Press.
[66] Fan H Y, J. Phys. A 25 (1992) 3443; Fan H Y, Ann. Phys. 323 (2008) 1502.
[67] Fan H Y, Fan Y. Mod. Phys. Lett. A 13 (1998) 433.
[68] Fan H Y, Fan Y. Int. J. Mod. Phys. A 17 (2002) 701.
[69] Gardiner C and Zoller P. Quantum Noise. Springer, Berlin, 2000.
[70] Fan H Y, Hu L Y. Mod. Phys. Lett. B 22 (2008) 2435.
[71] Hu L Y, Fan H Y. Opt. Commun. 282 (2009) 4379.
[72] Fan H Y, Hu L Y. Opt. Commun. 281 (2008) 5571.
[73] Vogel K and Risken H Phys. Rev. A 40 (1989) 2827.
[74] Xu X X, Hu L Y, Fan H Y, Modern Physics Letters B 25 (2011) 31.
[75] Xu X X, Yuan H C, Hu L Y, Fan H Y. Statistical properties of generalized photon-modulatedthermal state. (2011) (to be submitted).
[76] Rainville E D. Special Function. New York: MacMillan 1960.
[77] Klauder J R, Skargerstam B S. Coherent states. Singapore, World Scientific, 1985;Glauber R J.Phys. Rev. 130 (1963) 2529; Phys. Rev. 131 (1963) 2766.
[78] Fan H Y, Meng X G, Wang J S. Commun. Theor. phys. 46 (2006) 845.
[79] Genoni M G, Paris M G A, Banaszek K. Phys. Rev. A 76 (2007) 042327; Phys. Rev. A 78 (2008)060303(R).
[80] Scully M O, Zubairy M S. Quantum Optics. Cambridge university Press, 1998.
[81] Sudarshan E C G. Phys. Rev. lett. 10 (1963) 277.
[82] Methta C L. Phys. Rev. lett. 18 (1967) 752.
[83] Mundarain D F, Stephany J. J. Phys. A: Math. Gen. 37 (2004) 3869.
[84] Kelly P L, Kleiner W H. Phys. Rev. 136 (1964) 316.
[85] Orszag M. Quantum Optics. Berlin, Springer, 2000.
[86] Loudon R, Knight P L. J. Mod. Opt. 34 (1987) 709; Buzek V. J. Mod. Opt. 37 (1990) 303; DodonovV V. J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1.
[87] Yuen H P. Phys. Rev. A 13 (1976) 2226.
[88] Walls D. F. Nature 306 (1983) 141.
[89] Schleich P. W. Quantum Optics in Phase Space. Berlin, Wiley, 2001.
[90] Bogolyubov N N. Lectures on Quantum Statistics (Vol.1). New York, Gordon and Breach, 1987.
[91] Fan H Y, Chen J H. Commun. Theor. Phys. 38 (2002) 147.
[92] Olivares S, Paris M G A. J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S616.
[93] Hu L Y, Fan H Y. Opt. Commu. 282 (2009) 4379.
[94] Genoni M G, Paris M G A, Banaszek K. Phys. Rev. A 76 (2007) 042327.
[95] Genoni M G, Paris M G A, Banaszek K. Phys. Rev. A 78 (2008) 060303(R).
[96] Ivan J S, Kumar M S, Simon R. arXiv: 0812.2800v1(quant-ph).
[97] Gulfam Q A, Islam R, Ikram M. J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 165502.
[98] Ramsey N.F. Molecular Beams, Oxford University Press, New York, 1985.
[99] Einstein A, Podolsky B, N. Rosen. Phys. Rev. 47 (1935) 777.
[100] Selleri F. Quantum Mechanics Versus Local Realism: The Einstein-Podolsky-Rosen Paradox. NewYork, Plenum Press, 1988.
[101] Nielsen M A, Chuang I L. Quantum Computation and Quantum Information. Cambridge UniversityPress 2000.
[102] Bouwmeester D. The Physics of Quantum Information. Berlin, Springer 2000.
[103] Fan H Y, Yu G C. Phys. Rev. A 65 (2002) 033829.
[104] Fan H Y, Klauder J R. Phys. Rev. A 49 (1994) 704.
[105] Mandel L, Wolf E. Optical Coherence and Quantum Optics. Cambridge University Press 1995.
[106] Loudon R, Knight P L. J. Mod. Opt. 34 (1987) 709.
[107] Dodonov V V. J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1.
[108] Furusawa A, Soensen J L, Braunstein S L, Fuchs C A, Kimble H J, Polzik E S. Science 282 (1998)706.
[109] Opatrny T, Kurizki G, Welsch D G. Phys. Rev. A 61 (2000) 032302.
[110] Olivares S, Paris M G A, Bonifacio R. Phys. Rev. A 67 (2003) 032314.
[111] Kitagawa A, Takeoka M, Sasaki M, Che?es A. Phys. Rev. A 73 (2006) 042310.
[112]徐学翔,袁洪春,胡利云,范洪义.单双模连续压缩真空态及其量子统计性质.物理学报.(2011)待投.
[113] Schrodinger E. Naturwiss. 23 (1935) 807.
[114] Bennett C H, Brassard G,Crepeau C, Jozsa R. Phys. Rev. Lett. 70 (1993) 1895.
[115] Fan H Y, Klauder J. R. Phys. Rev. A 49 (1994) 704; Fan H Y, Phys. Rev. A 65 (2002) 064102; FanH Y, Phys. Lett. A 294 (2002) 253;
[116] Fan H Y, Fan Y. Phys. Rev. A 54 (1996) 958;
[117] Preskill J. Quantum Information and Computation. Pasadena, CA: California Institute of Technol-ogy, 1998.
[118] Buzek V. J. Mod. Opt. 37 (1990) 303.
[119] Hu L Y, Fan H Y. J. Phys. A: Math. Gen. 39 (2006) 14133.
[120] Jiang N Q. Phys. Rev. A 74 (2006) 012306.
[121] Hu L Y, Fan H Y. J. Phys. B: At. Mol. Opt. Phys. 40 (2007) 2099.
[122] Fan H Y. Opt. Lett. 28 (2003) 2177.
[123] Hu LY, Xu X X, Fan H Y, J. Opt. Soc. Am. B, 27 (2010) 286.
[124] Fan H Y, Xu Z H. Phys. Rev. A 50 (1994) 2921.
[125] Magnus W. Formulas and theorems for the special functions of mathematical physics. Springer,1996.
[126] Zhang Z X, Fan H Y. Phys. Lett. A 174 (1993) 206.
[127] Bronstein I N, Semendiajew K A. Traschenbuch der Mathematik. Leipzig, 1959.
[128] Lee C T. Phys. Rev. A 41 (1990) 1569.– 130–
[129]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.
[130] Hu LY, Xu X X, Guo Q, Fan H Y. Opt. Commun. 283 (2010) 5074.
[131] Aspachs M, Calsamiglia J. Phys. Rev. A 79 (2009) 033834.
[132] Marian P, Marian T A. Phys. Rev. A 47 (1993) 4487.
[133] Marian P, Marian T A. Phys. Rev. A 47 (1993) 4474.
[134] Fiurasek J, Cerf N. J. Phys. Rev. Lett. 93 (2004) 063601.
[135] Wenger J, Fiurasek J, Tualle-Brouri R, Cerf N J, Grangier P. Phys. Rev. A 70 (2004) 053812.
[136] Hu L Y, Xu X X, Wang Z S, Xu X F. Phys. Rev. A 82 (2010) 043842.
[137] Kim M S. J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 133001.
[138] Kitagawa A, Takeoka M, Sasaki M, Che?es A. Phys. Rev. A 73 (2006) 042310.
[139] Fiurasek J, Cerf N J. Phys. Rev. A 72 (2005) 033822.
[140] Orszag M. Quantum Optics. Springer-Verlag, 2000.
[141] Loudon R. The Quantum Theory of Light, 2nd ed. Oxford U. Press, 1983.
[142] Sudarshan E C G. Phys. Rev. Lett. 10 (1963) 277.
[143] Fan H Y. Phys. Lett. A 124 (1987) 303.
[144] Hu L Y, Fan H Y. Chin. Phys. B 18, 4657 (2009).
[145] Genoni M G, Paris M G A, Banaszek K. Phys. Rev. A 78 (2008) 060303(R).
[146] Genoni M G, Paris M G A, Banaszek K. Phys. Rev. A 76 (2007) 042327.
[147] Fan H Y. Commun. Theor. Phys. 42 (2004) 339.
[2] Furusawa A, S?ensen J L, Braunstein S L, etc. Science 282 (1998) 706.
[3] Andersen U L, Josse V, Leuchs G. Phys. Rev. Lett. 94 (2005) 240503.
[4] Lund A. Ralph T C, Haselgrove H L. Phys. Rev. Lett. 100 (2008) 030503.
[5] Agarwal G S, Tara K. Phys. Rev. A 43 (1991) 492.
[6] Zavatta A, Viciani S, Bellini M. Science 306 (2004) 660.
[7] Zavatta A, Viciani S, Bellini M. Phys. Rev. A 72 (2005) 023820.
[8] Fan H Y, Zaidi H R. Phys. Rev. A 37 (1988) 2985; Fan H Y, Wunsche A. J. Opt. B: Quantum &Semiclass. Opt. 2 (2000) 464.
[9] Fan H Y, Lu H L, Fan Y. Ann. Phys. 321 (2006) 480.
[10] Fan H Y. Phys. Lett. A 161 (1991) 1.
[11] Weyl H. Z. Phys. 46 (1927) 1.
[12] Louisell W H. Quantum Statistical Properties of Radiation. Wiley, New York, 1973.
[13] Wick G. Phys. Rev. 80 (1950) 268.
[14]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.
[15] Fan H Y. J. Phys. A 25 (1992) 3443.
[16] Fan H Y. Mod. Phys. Lett. A 15 (2000) 2297.
[17] Fan H Y. Ann. Phys. 323 (2008) 500.
[18] Klauder J R, Skargerstam B S. Coherent States. World Scientific, Singapore, 1985.
[19] Glauber R J. Phys. Rev. 130 (1963) 2529; 131 (1963) 2766.
[20] Wigner E. Phys. Rev. 40 (1932) 749; O’Connell R F, Wigner E P. Phys. Lett. A 83 (1981) 145.
[21] Schleich W P. Quantum Optics in Phase Space. Wiley-VCH, Birlin, 2001.
[22] Feynman R P. Statistical Physics. Benjamin W A, 1972.
[23] Hillery M, O’Connell R F, Scully M O, Wigner E P. Phys. Rep. 106 (1984) 121; Balazs N L,Jennings B K. Phys. Rep. 104 (1984) 347.
[24] Vogel K, Risken H. Phys. Rev. A 40 (1989) 2847.
[25] Husimi K. Proc. Phys.-Math. Soc. Jpn. 22 (1940) 264.
[26] Guo Q. Int. J. Theor. Phys. 46 (2007) 3135; Guo Q, Fan H Y. Int. J. Theor. Phys. 47 (2008) 3234.
[27] Fan H Y and Yu G C. Mod. Phys. Lett. A 15 (2000) 2297.
[28] Kelly P L, Kleiner W H. Phys. Rev. 136 (1964) 316.
[29] Loudon R. The Quantum Theory of Light. Oxford University Press, 1983.
[30] Fan H Y, Hu L Y. Opt. Lett. 33 (2008) 443.
[31] Mandel L. Opt. Lett. 4 (1979) 205.
[32] Lee C T. Phys. Rev. A 44 (1991) R2775.
[33] Kenfack A, Zyczkowski K. J. Opt. B: Quantum Semiclass. Opt. 6 (2004) 396.
[34] Hillery M. Phys. Rev. A 35 (1987) 725.
[35] Asboth J K, Calsamiglia J, Ritsch H. Phys. Rev. Lett. 94 (2005) 173602.
[36] Takahashi Y, Umezawa H. Thermo field dynamics. Collective Phenomena 2 (1975) 55; MemorialIssue for H. Umezawa, Int. J. Mod. Phys. B 10 (1996) 1563.
[37]范洪义,量子力学纠缠态表象及应用,上海交通大学出版社,2001.
[38]范洪义,从量子力学到量子光学,上海交通大学出版社,2005.
[39]范洪义,胡利云,光学变换-从量子到经典,上海交通大学出版社,2010.
[40]汪志诚,热力学统计物理,高等教育出版社,2008.
[41] Hu L Y, Fan H Y. Chin. Phys. Lett. 26 (2009) 090307.
[42] Barnett S M, Radmore P M. Methods in Theoretical Quantum Optics. Oxford: Clarendon press,1997.
[43] Hu L Y, Fan H Y. Commun. Theor. Phys. 51 (2009) 321.
[44] Fan H Y, Xu X X. J. Math. Phys. 50 (2009) 063303.
[45] Fan H Y, Xu X X, Hu L Y. Chin. Phys. B 19 (2010) 060502.
[46] Pathria R K. Statistical Mechannics. Singapore: World Scientific, 2001.
[47] Hu L Y, Fan H Y. Mod. Phys. Lett. A 24 (2009) 2263.
[48] Fan H Y, Jiang N Q. Phys. Scr. 78 (2008) 045402.
[49] Fan H Y, Chen H L. Chin. Phys. Lett. 18 (2001) 850.
[50] Wang J S, Sun C Y. Int. J. Theor. Phys. 37 (1998) 1213.
[51] Louisell W H. Quantum Statistical Properties of Radiation.John Wiley, New York,1973.– 127–
[52] Xu X X, Yuan H C, Hu L Y, Fan H Y. Thermal ?uctuation and thermodynamics of mesoscopic RLCcircuit at finite temperature calculated by using generalized thermal vacuum state (to be submitted)
[53] Mandel L, Wolf E. Optical Coherence and Quantum Optics. Cambridge: Cambridge UniversityPress, 1995.
[54] Fan H Y, Hu L Y. Commun. Theor. Phys. 51 (2009) 506.
[55] Xue X X, Fan H Y, Hu L Y, Yuan H C. Physica A 390 (2011) 18.
[56] Parigi V, Zavatta A, Kim M S, Bellini M. Science 317 (2007) 1890.
[57] Boyd R W, Chan K W, O’Sullivan M N. Science 317 (2007) 1874.
[58] Wenger J, Tualle-Brouri R, Grangier P. Phys. Rev. Lett. 92 (2004) 153601.
[59] Zavatta A, Parigi V, Bellini M. Phys. Rev. A 75 (2007) 052106.
[60] Ourjoumtsev A, Dantan A, Tualle-Brouri R, Grangier P. Phys. Rev. Lett. 98 (2007) 030502.
[61] Ourjoumtsev A, Dantan A, Tualle-Brouri R, Grangier P. Phys. Rev. Lett. 96 (2006) 213601.
[62] Biswas A, Agarwal G S. Phys. Rev. A 75 (2007) 032104.
[63] Hu L Y, Fan H Y. J. Opt. Soc. Am. B 25 (2008) 1955.
[64] Hu L Y, Fan H Y. Mod. Phys. Lett. A 24 (2009) 2263.
[65] Wang S J, Xu X X, Ma S J. Int. J. Quant. Inf. (2011) in Press.
[66] Fan H Y, J. Phys. A 25 (1992) 3443; Fan H Y, Ann. Phys. 323 (2008) 1502.
[67] Fan H Y, Fan Y. Mod. Phys. Lett. A 13 (1998) 433.
[68] Fan H Y, Fan Y. Int. J. Mod. Phys. A 17 (2002) 701.
[69] Gardiner C and Zoller P. Quantum Noise. Springer, Berlin, 2000.
[70] Fan H Y, Hu L Y. Mod. Phys. Lett. B 22 (2008) 2435.
[71] Hu L Y, Fan H Y. Opt. Commun. 282 (2009) 4379.
[72] Fan H Y, Hu L Y. Opt. Commun. 281 (2008) 5571.
[73] Vogel K and Risken H Phys. Rev. A 40 (1989) 2827.
[74] Xu X X, Hu L Y, Fan H Y, Modern Physics Letters B 25 (2011) 31.
[75] Xu X X, Yuan H C, Hu L Y, Fan H Y. Statistical properties of generalized photon-modulatedthermal state. (2011) (to be submitted).
[76] Rainville E D. Special Function. New York: MacMillan 1960.
[77] Klauder J R, Skargerstam B S. Coherent states. Singapore, World Scientific, 1985;Glauber R J.Phys. Rev. 130 (1963) 2529; Phys. Rev. 131 (1963) 2766.
[78] Fan H Y, Meng X G, Wang J S. Commun. Theor. phys. 46 (2006) 845.
[79] Genoni M G, Paris M G A, Banaszek K. Phys. Rev. A 76 (2007) 042327; Phys. Rev. A 78 (2008)060303(R).
[80] Scully M O, Zubairy M S. Quantum Optics. Cambridge university Press, 1998.
[81] Sudarshan E C G. Phys. Rev. lett. 10 (1963) 277.
[82] Methta C L. Phys. Rev. lett. 18 (1967) 752.
[83] Mundarain D F, Stephany J. J. Phys. A: Math. Gen. 37 (2004) 3869.
[84] Kelly P L, Kleiner W H. Phys. Rev. 136 (1964) 316.
[85] Orszag M. Quantum Optics. Berlin, Springer, 2000.
[86] Loudon R, Knight P L. J. Mod. Opt. 34 (1987) 709; Buzek V. J. Mod. Opt. 37 (1990) 303; DodonovV V. J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1.
[87] Yuen H P. Phys. Rev. A 13 (1976) 2226.
[88] Walls D. F. Nature 306 (1983) 141.
[89] Schleich P. W. Quantum Optics in Phase Space. Berlin, Wiley, 2001.
[90] Bogolyubov N N. Lectures on Quantum Statistics (Vol.1). New York, Gordon and Breach, 1987.
[91] Fan H Y, Chen J H. Commun. Theor. Phys. 38 (2002) 147.
[92] Olivares S, Paris M G A. J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S616.
[93] Hu L Y, Fan H Y. Opt. Commu. 282 (2009) 4379.
[94] Genoni M G, Paris M G A, Banaszek K. Phys. Rev. A 76 (2007) 042327.
[95] Genoni M G, Paris M G A, Banaszek K. Phys. Rev. A 78 (2008) 060303(R).
[96] Ivan J S, Kumar M S, Simon R. arXiv: 0812.2800v1(quant-ph).
[97] Gulfam Q A, Islam R, Ikram M. J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 165502.
[98] Ramsey N.F. Molecular Beams, Oxford University Press, New York, 1985.
[99] Einstein A, Podolsky B, N. Rosen. Phys. Rev. 47 (1935) 777.
[100] Selleri F. Quantum Mechanics Versus Local Realism: The Einstein-Podolsky-Rosen Paradox. NewYork, Plenum Press, 1988.
[101] Nielsen M A, Chuang I L. Quantum Computation and Quantum Information. Cambridge UniversityPress 2000.
[102] Bouwmeester D. The Physics of Quantum Information. Berlin, Springer 2000.
[103] Fan H Y, Yu G C. Phys. Rev. A 65 (2002) 033829.
[104] Fan H Y, Klauder J R. Phys. Rev. A 49 (1994) 704.
[105] Mandel L, Wolf E. Optical Coherence and Quantum Optics. Cambridge University Press 1995.
[106] Loudon R, Knight P L. J. Mod. Opt. 34 (1987) 709.
[107] Dodonov V V. J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1.
[108] Furusawa A, Soensen J L, Braunstein S L, Fuchs C A, Kimble H J, Polzik E S. Science 282 (1998)706.
[109] Opatrny T, Kurizki G, Welsch D G. Phys. Rev. A 61 (2000) 032302.
[110] Olivares S, Paris M G A, Bonifacio R. Phys. Rev. A 67 (2003) 032314.
[111] Kitagawa A, Takeoka M, Sasaki M, Che?es A. Phys. Rev. A 73 (2006) 042310.
[112]徐学翔,袁洪春,胡利云,范洪义.单双模连续压缩真空态及其量子统计性质.物理学报.(2011)待投.
[113] Schrodinger E. Naturwiss. 23 (1935) 807.
[114] Bennett C H, Brassard G,Crepeau C, Jozsa R. Phys. Rev. Lett. 70 (1993) 1895.
[115] Fan H Y, Klauder J. R. Phys. Rev. A 49 (1994) 704; Fan H Y, Phys. Rev. A 65 (2002) 064102; FanH Y, Phys. Lett. A 294 (2002) 253;
[116] Fan H Y, Fan Y. Phys. Rev. A 54 (1996) 958;
[117] Preskill J. Quantum Information and Computation. Pasadena, CA: California Institute of Technol-ogy, 1998.
[118] Buzek V. J. Mod. Opt. 37 (1990) 303.
[119] Hu L Y, Fan H Y. J. Phys. A: Math. Gen. 39 (2006) 14133.
[120] Jiang N Q. Phys. Rev. A 74 (2006) 012306.
[121] Hu L Y, Fan H Y. J. Phys. B: At. Mol. Opt. Phys. 40 (2007) 2099.
[122] Fan H Y. Opt. Lett. 28 (2003) 2177.
[123] Hu LY, Xu X X, Fan H Y, J. Opt. Soc. Am. B, 27 (2010) 286.
[124] Fan H Y, Xu Z H. Phys. Rev. A 50 (1994) 2921.
[125] Magnus W. Formulas and theorems for the special functions of mathematical physics. Springer,1996.
[126] Zhang Z X, Fan H Y. Phys. Lett. A 174 (1993) 206.
[127] Bronstein I N, Semendiajew K A. Traschenbuch der Mathematik. Leipzig, 1959.
[128] Lee C T. Phys. Rev. A 41 (1990) 1569.– 130–
[129]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.
[130] Hu LY, Xu X X, Guo Q, Fan H Y. Opt. Commun. 283 (2010) 5074.
[131] Aspachs M, Calsamiglia J. Phys. Rev. A 79 (2009) 033834.
[132] Marian P, Marian T A. Phys. Rev. A 47 (1993) 4487.
[133] Marian P, Marian T A. Phys. Rev. A 47 (1993) 4474.
[134] Fiurasek J, Cerf N. J. Phys. Rev. Lett. 93 (2004) 063601.
[135] Wenger J, Fiurasek J, Tualle-Brouri R, Cerf N J, Grangier P. Phys. Rev. A 70 (2004) 053812.
[136] Hu L Y, Xu X X, Wang Z S, Xu X F. Phys. Rev. A 82 (2010) 043842.
[137] Kim M S. J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 133001.
[138] Kitagawa A, Takeoka M, Sasaki M, Che?es A. Phys. Rev. A 73 (2006) 042310.
[139] Fiurasek J, Cerf N J. Phys. Rev. A 72 (2005) 033822.
[140] Orszag M. Quantum Optics. Springer-Verlag, 2000.
[141] Loudon R. The Quantum Theory of Light, 2nd ed. Oxford U. Press, 1983.
[142] Sudarshan E C G. Phys. Rev. Lett. 10 (1963) 277.
[143] Fan H Y. Phys. Lett. A 124 (1987) 303.
[144] Hu L Y, Fan H Y. Chin. Phys. B 18, 4657 (2009).
[145] Genoni M G, Paris M G A, Banaszek K. Phys. Rev. A 78 (2008) 060303(R).
[146] Genoni M G, Paris M G A, Banaszek K. Phys. Rev. A 76 (2007) 042327.
[147] Fan H Y. Commun. Theor. Phys. 42 (2004) 339.