Progress Of Operator Ordering Theory And Some Applications In Quantum Optics

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.