The Progress of Optics Signal Analysis in Quantum Mechanics and Quantum Tomography
|School||University of Science and Technology of China|
|Keywords||operator ordering theory the IWOP technique the Weyl order theIWWOP technique the Wigner operator quantum tomography theRadon transform the Wavelet transform|
In quantum mechanics, the physical observables are all represented by the Hermitian operators. These are generally not commutative between them. Therefore, when the function of classical mechanics promote to the operators of quantum mechanics, the form of corresponding is not only one. For the different operator ordering forms, or for the different corresponding rules, their corresponding operators are different. So when people correspond the function of classical mechanics to the operators of quantum mechanics, they will have a problem that how to order the operators. There are some definite operator ordering forms, such as the normal ordering form, the anti-normal ordering form and the Weyl ordering form. In addition, the Weyl ordering form has a remarkable property, i.e.,it has similar invariance when the Weyl ordered operators transform under similar transformations. To our knowledge, there are two main approaches to handle the operator ordering problems. They are the Lie algebra method and Louisell’s differential operation method via the coherent state representation. However, these two methods are not very efficient in the processes which calculate the complex quantum operator ordering problems. In order to solve this problem, Prof.Fan firstly proposed the technique of integration within an ordered product of operators(IWOP). The IWOP technique generalizes the Newton-Leibniz rule to the integrations over the ket-bra operators in quantum mechanics and builds a bridge between classical mechanics and quantum mechanics.Based on the operator ordering theory, we introduced the analysis methods of classical optics to th quantum optics by the IWOP technique. The analysis methods involved the Tomography technique, the Radon transform and the Wavelet transform. These efforts are very useful to study the optical transformations in quantum mechanics and enrich the Dirac’s symbolic method and the transformation theory.The whole thesis is divided into six chapters, arranged in details as follows:In Chapter.1, based on the IWOP technique, we resurvey the preliminary quantum representations such as the coordinate representation, the momentum representation, the particle number representation and the coherent representation. And then, we introduced some new quantum representations such as the entangled state representation, the coordinate-momentum intermediate representation, the entangled coherent representation, the three-mode entangled state representation and the three-mode entangled coherent representation. They have clear physical meaning.In Chapter.2, we introduced the Weyl ordering form and the Wigner operator, Weyl ordering operator formula, the technique of integration within the Weyl ordering product of operators and the order-invariance of Weyl ordered operators under similar transformations. Using the IWOP technique, we derived the form of the Wigner operator in different basic quantum representations. At last, we derived the form of the Wigner operator in different new quantum representations we introduced in Chapter.1.In Chapter.3, at first we introduced a dimension reduced optical analysis method which is the Tomography technique. The basis of its mathematical is the Radon transform. And then we presented the mathematical formulation of Radon transforms in different dimensions. Then we promote them to the quantum field, and derived the Radon transformation of Wigner operators in both single-mode and two-mode by the coordinate-momentum intermediate representation and the entangled intermediate representation.In Chapter.4, we did more detailed research on the Radon transformation of Wigner operators and calculated the Radon transformation of Wigner operators with various optical processes. We come to a conclusion that the alteration of Radon transformation of signal’s Wigner function through these optical processes can be ascribed to the variation of Radon transformation parameters. This method simplify the calculation processes and can be promote the Radon transformation to the quantum optics. At last, we did the same works in two-mode situation by the entangled intermediate representation.In Chapter.5, we introduced a dimension enhanced optical analysis method which is the Wavelet transformation. In the process when we promote the Wavelet transform to the quantum field, we find a general formula of qualified continuum mother wavelets---High-order Mexican Hat wavelets by the Admissible condition. And then, we make comparisons among different wavelet transforms computed with our new mother wavelets and the classical Mexican Hat wavelet by several special optic pulses and find the character of the High-order Mexican Hat wavelets.In the last chapter, we summarized the research works of the above and some possible future research directions in these fields are given.