量子光学中算符编序理论进展及其应用
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摘要
在量子力学、量子光学与量子场论中物理可观察量用厄米算符来表示的,它们之间一般来说是不对易的,人们总是面临着算符排序问题,所以这成了人们感兴趣的研究课题之一.算符的排序不同,它们对应的经典函数也不同,这是量子相空间理论基础.常见的算符排序形式有正规乘积、反正规乘积与Weyl编序.当一密度算符排成正规乘积后,它的相干态矩阵元与Q函数就立刻得到;而一旦知道它的反正规乘积,就便知道相应P函数.尤其,算符Weyl编序对路径积分理论和量子统计特别有用,一旦知道它的Weyl编序形式就知道Weyl经典对应.它具有一个特殊的性质,即Weyl编序在相似变换下具有有序不变性,这为实际的运算带来很大的方便.目前,在现有文献中主要有两种方法解决算符排序问题:李代数方法与相干态表象微分操作方法.但是这两种方法不适用于一些复杂算符排序.本文利用范洪义教授所提出的有序算符内的积分技术(简称为IWOP技术)来处理算符排序问题. IWOP技术就是将原本只适用于可对易函数的牛顿-莱布尼兹积分推广到对量子力学Dirac符号法ket-bra积分型非对易量子算符的运算,从而建立了联系q数与c数的一座桥梁.在本文中算符编序理论的发展主要体现以下四个方面: 1)利用导出的新表象并结合IWOP技术导出新的算符恒等式,如纠缠态表象、坐标-动量中介表象、纠缠相干态表象等. 2)利用IWOP技术发展量子相空间的新积分变换,并用来导出其它一些复杂算符函数的Weyl编序形式,如坐标-动量算符函数、动量-坐标算符函数以及多模组合算符函数等. 3)基于算符恒等式与IWOP技术简捷地导出很多新的积分公式,而不需要真正进行积分运算.另外,不同的算符编序方案对同一个积分的结果表现形式不同,它们之间的相互转换直接导出新的积分公式. 4)将IWOP技术发展到算符s编序内的积分技术(简称为IWSOP技术),导出了任意密度算符的s-编序展开公式,同时也导出了带s-参数的新算符公式.这些不仅进一步发展与完善了算符编序理论,也丰富了Dirac的符号法和变换理论,对量子光学中非经典态的研究十分有用.其具体内容包括:
一、分别介绍了正规乘积内积分技术与反正规乘积内积分技术,并用IWOP技术重新审视量子力学基本表象,从数理统计中的正态分布可以更加直捷地导出原有的表象以及给出了一些多模指数算符的正规乘积与反正规乘积展开形式.这些都展示了IWOP技术给出经典变换过渡到量子幺正变换算符的捷径,可以发现很多新的幺正算符,找到很多新的有用的量子力学表象.也介绍了Weyl编序内的积分技术、任意算符的Weyl编序展开公式以及Weyl编序在相似变换下的有序不变性.
二、根据量子相空间的积分变换(范氏变换),推广到纠缠Wigner算符的新积分变换.它有良好的变换特性,具有可逆保迹性.利用该积分变换导出了一些复杂算符的Weyl编序形式.这不仅发展了变换理论,而且丰富了量子相空间理论.
三、基于Hermite多项式(单变量与双变量)的产生函数直接导出算符恒等式,并根据这些算符等式易导出有关Hermite多项式的递推公式与一些复杂的算符等式.结合IWOP技术与量子力学表象的完备性导出若干新的数学积分公式,而不需要真正进行积分运算.另外,交替使用正规乘积、反正规乘积与Weyl编序的积分技术导出若干新的算符公式.这开辟了一条新的途径直接通过量子力学的排序算符导出一些复杂的数学积分公式.
四、把IWOP技术发展到IWSOP技术(玻色情况与费米情况).它统一了正规乘积积分技术(s = 1)、反正规乘积积分技术(s = ?1)以及Weyl编序内积分技术(s = 0),进而导出了任意密度算符的s-编序展开公式,完善了带s-参数的量子化方案与算符编序理论.并结合IWSOP技术导出了s-编序算符的恒等式、量子力学基本表象的s-编序展开式以及Wigner算符的s-编序展开形式.
五、导出了量子光学中光子计数新公式.一方面根据原有量子光学中光子计数公式,导出了密度算符P表示的光子计数公式并利用该公式计算出相干态、混沌光场、平移混沌光场等光场的光子计数分布;另一方面基于IWSOP技术导出带s参数的光子计数公式,当s取不同值时,就有不同形式的光子计数公式.根据双变量Hermite多项式的递推公式直捷地导出最小不确定热态,还利用Weyl编序在相似变换下具有有序不变性导出了最小不确定热态的Wigner函数解析表达式.
六、利用前面导出的算符恒等式很容易导出了Agarwal等人所提出的激发相干态(光子增加相干态)的归一化系数.在此基础上,扩展到广义激发相干态、激发Bell型纠缠相干态与单模激发GHZ型纠缠相干态,并导出了相应的归一化系数.此外,分别研究了它们的光子计数分布、Wigner函数、纠缠Concurrence、Bell不等式的违背等非经典特征并提出了这些态的物理实现方案.利用IWOP技术与多粒子纠缠态表象构造出多模广义的SU(1,1)压缩算符和压缩态,并研究其高阶压缩、Winger函数、Bell不等式的违背情况等非经典性质.
In quantum mechanics, quantum optics and quantum field theory, physical observablesare all represented by Hermitian operators, and these operators are not generally commuta-tive between them. People always face with the operator ordering problem, thus this becomesone of very significant and interesting research topics. For the different operator orderingforms, their corresponding classical function are different, which is the theoretical basis ofphase space quantum mechanics. There are some definite operator orderings, such as normalordering, antinormal ordering, and Weyl ordering. When the density operator is in orderingits matrix element in the coherent state and Q function are directly obtained. After convertingthe density operators into their antinormal ordering forms, their Glaubler P-representationcan be directly written down in the coherent state representation. In particular, among themthe Weyl ordering of operators is the direct result of Weyl quantization recipe and is veryuseful to path integral theory and quantum statistics. As long as the Weyl product form ofoperator are derived, its classical Weyl correspondence is known. In addition, the Weyl or-dering has a remarkable property, i.e., the order-invariance of Weyl ordered operators undersimilar transformations, which is of great convenience to deal with some real problems. Toour knowledge, in most literature two main approaches are available to handle operator order-ing problems, including the Lie algebra method and Louisell’s differential operation methodvia the coherent state representation. However, these two methods seem not very efficient inplaying their advantage to the complex quantum operator ordering problems. Here, we adoptthe technique of integration within an ordered product of operators (IWOP), first proposedby Prof. Fan, to deal with the operator ordering problem, and further develop the operator or-dering theory. The IWOP technique generalizes the Newton-Leibniz rule to the integrationsover the ket-bra operators in quantum mechanics, and it creates a bridge between classicalmechanics and quantum mechanics. In this thesis, the development of the operator orderingtheory is mainly embodied in four aspects: 1) Using the derived new quantum representationand the IWOP technique, we have derived some new operator identities, such as the entan-gled state representation, coordinate-momentum intermediate representation, and entangledcoherent state representation and so on. 2) By virtue of the IWOP technique we have de- veloped the new integral transformation in quantum phase space, and based on this integraltransformation the Weyl ordering forms about some other complicated operator functionsare obtained, such as coordinate-momentum operator function, momentum-coordinate op-erator function and multi-mode combinations’operator function. 3) According to operatoridentities and the IWOP technique we have quickly and easily exported a lot of new integralformula, without really performing the integrations in the ordinary way. Additionally for thesame integrated result, we obtain its different form by using the different operator ordering.These forms may convert each other, which directly derive some new integral formulas 4)The IWOP technique is evolved into the technique of integration within s-ordered productof operators (IWSOP). Employing the IWSOP technique we deduce the s-ordered expan-sion formula of density operator, which is applicable to deriving the s-ordered expansionsof some operators. These not only further develop and improve the operator ordering the-ory, which are very useful to study the nonclassical state in quantum optics, but also enrichDirac’s symbolic method and the transformation theory. The whole thesis is arranged indetail as follows:
1. We introduce the technique of integration within the normally and antinormally or-dered product of operators, respectively. Based on the IWOP technique we re-survey thepreliminary quantum representations and neatly derive them from the normal distribution inmathematical statistics. Moreover, normal ordering and antinormal ordering forms of somemultimode exponential operators are given as well. It is found that the IWOP techniqueaccomplish the transitions from classical transformations to quantum mechanical unitaryoperators, and further reveals the beauty and elegance of Dirac’s symbolic method and trans-formation theory. By virtue of the IWOP technique you can find not only many new unitaryoperator, but also many new and useful quantum representation. In addition, we presentthe technique of integration within the Weyl ordering product of operators, Weyl orderingoperator formula, and the order-invariance of Weyl ordered operators under similar transfor-mations.
2. Based on Fan’s integration transformation, we find a new two-fold complex integra-tion transformation about the entangled Wigner operator in phase space quantum mechanics,which is invertible and obeys Parseval theorem. In this way, some complicated operator or-dering problems can be solved and the contents of phase space quantum mechanics can beenriched.
3. Using the generating function of Hermite polynomials including single-variable andtwo-variable immediately yields a lot of operator identities about Hermite polynomials. Based on these results, we easily deduce the well-known recurrence relations of Hermitepolynomials and some other more complicated operator identities. In addition, by alternatelyusing the technique of integration within normal, antinormal, and Weyl ordering of operatorswe deduce some new integration formulas. This may open a new route of directly derivingsome complicated mathematical integration formulas by virtue of the quantum mechanicaloperator ordering technique.
4. The IWOP technique is evolved into the technique of integration within s-orderedproduct of operators (IWSOP) including Bose operators and Fermi operators, which unifiesthe technique of integration within normal ordered(s = 1), antinormally ordered(s = ?1)and Weyl ordered(s = 0) product of operators. Then we deduce the s-ordered expansion for-mula of density operator, which is applicable to deriving the s-ordered expansions of someoperators. This enriches and develops quantization scheme and the operator ordering theory.With the help of the IWSOP technique, the quantum-mechanical fundamental representa-tions can be recast into s-ordering operator expansions and the s-ordered form of the usualWigner operator is derived.
5. Some new photocount formulas is presented in quantum optics. On the one hand,based on the original photon counting distribution formula we derive a new quantum me-chanical photon counting distribution formula related to density operator’s P-representation,which brings convenience to photocount’s calculation of some optical fields such as coherentstate, chaotic light field, displaced chaotic field. On the other hand, employing the s-orderedoperator expansion formula and the IWSOP technique, we obtain another new photocountformula for the general parameter s. The different form of photocount formulas is acquiredfor the different value of s. We use a new and simply approach to deriving the thermo-minimum uncertainty states via the relation of two-variable Hermite polynomials. In addi-tion, the compact expression for its corresponding Wigner function is obtained analyticallyby using the Weyl-ordered invariance under the similar transformations, which seems a newresult.
6. According to the derived operator identities above, it is very easy to obtain the nor-malization factor of excited (photon-added) proposed by Agarwal et al. Based on this state,we present some other excited coherent state such as generalized excited coherent state,excited Bell-type entangled coherent state, and single-mode excited GHZ-type entangledcoherent states and derive their corresponding normalization factor. In addition, their non-classical characteristics are analytically investigated such as photon counting distribution,Wigner function, Concurrence of entanglement, Bell inequality violation etc. and the phys- ical realization of these states are proposed as well. Finally, by constructing a generalizedmulti-partite entangled state representation and introducing the ket-bra integral in this repre-sentation, we use the IWOP technique to find a new set of generalized bosonic realization ofthe generators of the SU(1, 1) algebra, which can compose a generalized multi-mode squeez-ing operator. Furthermore, we construct some squeezed states using a generalized multi-mode squeezing operator, and we examine their non-classical properties such the higher-order squeezing, and Wigner function, the violation of the Bell inequality and so on.
一、分别介绍了正规乘积内积分技术与反正规乘积内积分技术,并用IWOP技术重新审视量子力学基本表象,从数理统计中的正态分布可以更加直捷地导出原有的表象以及给出了一些多模指数算符的正规乘积与反正规乘积展开形式.这些都展示了IWOP技术给出经典变换过渡到量子幺正变换算符的捷径,可以发现很多新的幺正算符,找到很多新的有用的量子力学表象.也介绍了Weyl编序内的积分技术、任意算符的Weyl编序展开公式以及Weyl编序在相似变换下的有序不变性.
二、根据量子相空间的积分变换(范氏变换),推广到纠缠Wigner算符的新积分变换.它有良好的变换特性,具有可逆保迹性.利用该积分变换导出了一些复杂算符的Weyl编序形式.这不仅发展了变换理论,而且丰富了量子相空间理论.
三、基于Hermite多项式(单变量与双变量)的产生函数直接导出算符恒等式,并根据这些算符等式易导出有关Hermite多项式的递推公式与一些复杂的算符等式.结合IWOP技术与量子力学表象的完备性导出若干新的数学积分公式,而不需要真正进行积分运算.另外,交替使用正规乘积、反正规乘积与Weyl编序的积分技术导出若干新的算符公式.这开辟了一条新的途径直接通过量子力学的排序算符导出一些复杂的数学积分公式.
四、把IWOP技术发展到IWSOP技术(玻色情况与费米情况).它统一了正规乘积积分技术(s = 1)、反正规乘积积分技术(s = ?1)以及Weyl编序内积分技术(s = 0),进而导出了任意密度算符的s-编序展开公式,完善了带s-参数的量子化方案与算符编序理论.并结合IWSOP技术导出了s-编序算符的恒等式、量子力学基本表象的s-编序展开式以及Wigner算符的s-编序展开形式.
五、导出了量子光学中光子计数新公式.一方面根据原有量子光学中光子计数公式,导出了密度算符P表示的光子计数公式并利用该公式计算出相干态、混沌光场、平移混沌光场等光场的光子计数分布;另一方面基于IWSOP技术导出带s参数的光子计数公式,当s取不同值时,就有不同形式的光子计数公式.根据双变量Hermite多项式的递推公式直捷地导出最小不确定热态,还利用Weyl编序在相似变换下具有有序不变性导出了最小不确定热态的Wigner函数解析表达式.
六、利用前面导出的算符恒等式很容易导出了Agarwal等人所提出的激发相干态(光子增加相干态)的归一化系数.在此基础上,扩展到广义激发相干态、激发Bell型纠缠相干态与单模激发GHZ型纠缠相干态,并导出了相应的归一化系数.此外,分别研究了它们的光子计数分布、Wigner函数、纠缠Concurrence、Bell不等式的违背等非经典特征并提出了这些态的物理实现方案.利用IWOP技术与多粒子纠缠态表象构造出多模广义的SU(1,1)压缩算符和压缩态,并研究其高阶压缩、Winger函数、Bell不等式的违背情况等非经典性质.
In quantum mechanics, quantum optics and quantum field theory, physical observablesare all represented by Hermitian operators, and these operators are not generally commuta-tive between them. People always face with the operator ordering problem, thus this becomesone of very significant and interesting research topics. For the different operator orderingforms, their corresponding classical function are different, which is the theoretical basis ofphase space quantum mechanics. There are some definite operator orderings, such as normalordering, antinormal ordering, and Weyl ordering. When the density operator is in orderingits matrix element in the coherent state and Q function are directly obtained. After convertingthe density operators into their antinormal ordering forms, their Glaubler P-representationcan be directly written down in the coherent state representation. In particular, among themthe Weyl ordering of operators is the direct result of Weyl quantization recipe and is veryuseful to path integral theory and quantum statistics. As long as the Weyl product form ofoperator are derived, its classical Weyl correspondence is known. In addition, the Weyl or-dering has a remarkable property, i.e., the order-invariance of Weyl ordered operators undersimilar transformations, which is of great convenience to deal with some real problems. Toour knowledge, in most literature two main approaches are available to handle operator order-ing problems, including the Lie algebra method and Louisell’s differential operation methodvia the coherent state representation. However, these two methods seem not very efficient inplaying their advantage to the complex quantum operator ordering problems. Here, we adoptthe technique of integration within an ordered product of operators (IWOP), first proposedby Prof. Fan, to deal with the operator ordering problem, and further develop the operator or-dering theory. The IWOP technique generalizes the Newton-Leibniz rule to the integrationsover the ket-bra operators in quantum mechanics, and it creates a bridge between classicalmechanics and quantum mechanics. In this thesis, the development of the operator orderingtheory is mainly embodied in four aspects: 1) Using the derived new quantum representationand the IWOP technique, we have derived some new operator identities, such as the entan-gled state representation, coordinate-momentum intermediate representation, and entangledcoherent state representation and so on. 2) By virtue of the IWOP technique we have de- veloped the new integral transformation in quantum phase space, and based on this integraltransformation the Weyl ordering forms about some other complicated operator functionsare obtained, such as coordinate-momentum operator function, momentum-coordinate op-erator function and multi-mode combinations’operator function. 3) According to operatoridentities and the IWOP technique we have quickly and easily exported a lot of new integralformula, without really performing the integrations in the ordinary way. Additionally for thesame integrated result, we obtain its different form by using the different operator ordering.These forms may convert each other, which directly derive some new integral formulas 4)The IWOP technique is evolved into the technique of integration within s-ordered productof operators (IWSOP). Employing the IWSOP technique we deduce the s-ordered expan-sion formula of density operator, which is applicable to deriving the s-ordered expansionsof some operators. These not only further develop and improve the operator ordering the-ory, which are very useful to study the nonclassical state in quantum optics, but also enrichDirac’s symbolic method and the transformation theory. The whole thesis is arranged indetail as follows:
1. We introduce the technique of integration within the normally and antinormally or-dered product of operators, respectively. Based on the IWOP technique we re-survey thepreliminary quantum representations and neatly derive them from the normal distribution inmathematical statistics. Moreover, normal ordering and antinormal ordering forms of somemultimode exponential operators are given as well. It is found that the IWOP techniqueaccomplish the transitions from classical transformations to quantum mechanical unitaryoperators, and further reveals the beauty and elegance of Dirac’s symbolic method and trans-formation theory. By virtue of the IWOP technique you can find not only many new unitaryoperator, but also many new and useful quantum representation. In addition, we presentthe technique of integration within the Weyl ordering product of operators, Weyl orderingoperator formula, and the order-invariance of Weyl ordered operators under similar transfor-mations.
2. Based on Fan’s integration transformation, we find a new two-fold complex integra-tion transformation about the entangled Wigner operator in phase space quantum mechanics,which is invertible and obeys Parseval theorem. In this way, some complicated operator or-dering problems can be solved and the contents of phase space quantum mechanics can beenriched.
3. Using the generating function of Hermite polynomials including single-variable andtwo-variable immediately yields a lot of operator identities about Hermite polynomials. Based on these results, we easily deduce the well-known recurrence relations of Hermitepolynomials and some other more complicated operator identities. In addition, by alternatelyusing the technique of integration within normal, antinormal, and Weyl ordering of operatorswe deduce some new integration formulas. This may open a new route of directly derivingsome complicated mathematical integration formulas by virtue of the quantum mechanicaloperator ordering technique.
4. The IWOP technique is evolved into the technique of integration within s-orderedproduct of operators (IWSOP) including Bose operators and Fermi operators, which unifiesthe technique of integration within normal ordered(s = 1), antinormally ordered(s = ?1)and Weyl ordered(s = 0) product of operators. Then we deduce the s-ordered expansion for-mula of density operator, which is applicable to deriving the s-ordered expansions of someoperators. This enriches and develops quantization scheme and the operator ordering theory.With the help of the IWSOP technique, the quantum-mechanical fundamental representa-tions can be recast into s-ordering operator expansions and the s-ordered form of the usualWigner operator is derived.
5. Some new photocount formulas is presented in quantum optics. On the one hand,based on the original photon counting distribution formula we derive a new quantum me-chanical photon counting distribution formula related to density operator’s P-representation,which brings convenience to photocount’s calculation of some optical fields such as coherentstate, chaotic light field, displaced chaotic field. On the other hand, employing the s-orderedoperator expansion formula and the IWSOP technique, we obtain another new photocountformula for the general parameter s. The different form of photocount formulas is acquiredfor the different value of s. We use a new and simply approach to deriving the thermo-minimum uncertainty states via the relation of two-variable Hermite polynomials. In addi-tion, the compact expression for its corresponding Wigner function is obtained analyticallyby using the Weyl-ordered invariance under the similar transformations, which seems a newresult.
6. According to the derived operator identities above, it is very easy to obtain the nor-malization factor of excited (photon-added) proposed by Agarwal et al. Based on this state,we present some other excited coherent state such as generalized excited coherent state,excited Bell-type entangled coherent state, and single-mode excited GHZ-type entangledcoherent states and derive their corresponding normalization factor. In addition, their non-classical characteristics are analytically investigated such as photon counting distribution,Wigner function, Concurrence of entanglement, Bell inequality violation etc. and the phys- ical realization of these states are proposed as well. Finally, by constructing a generalizedmulti-partite entangled state representation and introducing the ket-bra integral in this repre-sentation, we use the IWOP technique to find a new set of generalized bosonic realization ofthe generators of the SU(1, 1) algebra, which can compose a generalized multi-mode squeez-ing operator. Furthermore, we construct some squeezed states using a generalized multi-mode squeezing operator, and we examine their non-classical properties such the higher-order squeezing, and Wigner function, the violation of the Bell inequality and so on.
引文
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[2] Dirac PAM. Principles of quantum mechanics (4ed) [M]. Oxford: Claredon Press, 1958.
[3] Fan HY. Representation and Transformation Theory in Quantum Mechanics——Progress ofDirac’s symbolic method (in Chinese) [M]. Shanghai: Shanghai Scientific & Technical Pub-lishers, 1997.
[4] Fan HY. Entangled State Representations in Quantum Mechanics and Their Applications (in Chi-nese) [M]. Shanghai: Shanghai Jiao Tong University Press, 2001.
[5] Fan HY. From Quantum Mechanics to Quantum Optics——Development of the MathematicalPhysics (in Chinese) [M]. Shanghai: Shanghai Jiao Tong University Press, 2005.
[6] Fan HY, Tang XB. Progress of mathematics and physics’s foundation in quantum mechanics (inChinese) [M]. Hefei: University of Science and Technology of China Press, 2008.
[7] Fan HY, Hu LY. Optical transformation: from quantum optics to classical optics (in Chinese) [M].Shanghai: Shanghai Jiao Tong University Press, 2010.
[8] Fan HY, Lu HL, Fan Y. Newton-Leibniz integration for ket-bra operators in quantum mechanicsand derivation of entangled state representations [J]. Ann. Phys., 2006, 321(2): 480-494.
[9] Fan HY. Newton-Leibniz integration for ket-bra operators (II) - application in deriving densityoperator and generalized partition function formula [J]. Ann. Phys., 2007, 322(4): 866-885.
[10] Fan HY. Newton-Leibniz integration for ket-bra operators (III) - application in fermionic quantumstatistics [J]. Ann. Phys., 2007, 322(4): 886-902.
[11] Fan HY. Newton-Leibniz integration for ket-bra operators in quantum mechanics (IV) - Integrationswithin Weyl ordered product of operators and their applications [J]. Ann. Phys., 2008, 323(2): 500-526.
[12] Fan HY. Newton-Leibniz integration for ket-bra operators in quantum mechanics (V) - Deriv-ing normally ordered bivariate-normal-distribution form of density operators and developing theirphase space formalism [J]. Ann. Phys., 2008, 323(6): 1502-1528.
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