Constructing New State Vectors In Quantum Light Field And Analysing Their Properties With IWOP Theory

Schrodinger’s wave mechanics representation, Heisenberg’s matrix mechanic-s representation (these two representation are summarized by Dirac as symbolic method) and Feyman’s routing integration are three equivalent expressions about quantum mechanics. Quantum phase space theory is another one and has been widely used in statistical physics, quantum optics and nonlinear physics for its outstanding advantage. The so-called phase space expression is to correspond quantum operators to classical coordinate-momentum function in q-p phase s-pace (such as Weyl correspondence rule) to deduce Wigner function of quantum state and constract the time evolutionary equation similar to Schrodinger’s wave equation of Wigner function. The establishment of the corresponding relationship between classical mechanics and quantum mechanics is one of the main content of the research about quantum phase space theory and also a basic starting point to build quantum phase space theory. Fortunately, the technique of integration within an ordered product (IWOP) of operators just provides a new bridge con-necting classic changes and quantum operator for us and a natural transition shortcut to find a unitary operator in process of realizing the classic transforma-tion to quantum unitary transformation. IWOP technology can not only realize the integration of Dirac ket-bra type operator and pioneers a new development direction for Newton-Leibniz integration, but also exploit the potential applica-tions of Dirac symbolic method and representation theory. Quantum states in optical fields and their properties have been a topic of great interest in quantum optics and quantum information, moreover constructing new quantum states in optical fields is a hotspot. At present, the basic approach for constructing new quantum states is making use of the principle of superposition in quantum me-chanics, for example, some superposition states of Fock states, coherent states, squeezed states, displaced Fock states and so on. Another approach is that acting an operator on a reference state, for example, the usual squeezed state can be generated by operating the squeezing operator on a coherent state. We make full use of IWOP technology and regulate quantum states in optical fields by using the second method, so we construct some new quantum states and research their properties. This thesis is organized as follows:In Chap.1, to begin with, the technique of integration within ordered product of operators (IWOP), first proposed by Prof. Fan Hong-yi, is briefly reviewed, and complete expression of pure gaussian integral form of quantum mechanics related representation is also derived. Based on this technique fundamental quantum states in optical field are understanded and described from new point of view, such as Fock state, coherent state and thermal state.In Chap.2, in view of the non-gaussian compression states with nonclassicality having potential applications in the field of quantum information, we introduce a new type of photon-subtracted and photon-added queezed coherent state on the basis of a new type of queezed coherent state, and derive their normalized coefficients and related some statistical distribution functions analytically and graphically. The nonclassicality of these states are discussed in terms of the in-tense oscillation of photon-number distribution and the negativity of the Wigner function.In Chap.3, by using un-normalized coherent state expression of Fock state and transformation skills from Weyl ordering to normal ordering of operators, we derive the PND of single-mode and two-mode squeezed chaotic state analytically. Being worth to be mentioned, the PND of single-mode and two-mode squeezed chaotic state are a Legendre polynomial and a Jacobi polynomial. Corresponding characters of Jacobi polynomials and Legendre polynomials are deduced in view of physical theory, which embodies the mathematical beauty of quantum physics fully.In Chap.4, by repeatedly operating the combination of Bosonic creation and annihilation operators on the coherent state, we propose a generalized photon- modulated coherent state whose normalization factor is related to single-variable Hermite polynomials. We also investigate two important statistical distribution function (e.g. PND and WF) and discuss its nonclassicality in terms of the nega-tivity of the Wigner function.In Chap.5, we investigate the time evolution of a quantum state in the laser channel by virtue of the thermo entangled state representation. This result help us to calculate the evolving result of the photon-added coherent state governed by the master equation describing the laser process and derive the normal ordered form of the density operator in the laser channal. In addition, we not only calculate its WF analytically which is a form of the Laguerre-Gaussian function, but also show the time evolving result on the different parameters graphically.In Chap.6, by virtue of the coherent state representation and the operator ordering method we find a new approach for transiting Hermite polynomials to La-guerre polynomials and derive some expansion properties of laguerre polynomials operator.In Chap.7, by virtue of the linear combination of photon creation operator and annihilation operator we derive some new identities about Hermite polyno-mial operators and Laguerre polynomial operators and the definition of Laguerre polynomial with the help of polynomial operators. But beyond that, we expand the contents of chapter6and derive a new approach for transiting Hermite poly-nomials to associated Laguerre polynomials.