The Quantum Optical Correspondence Of Classical Optical Transformations And Their Applications

With the rapid development of quantum mechanic in the beginning of the twentieth century, researchers devoted themselves to the continuous study of kinds of questions in quantum mechanic. Among these research, including the search of the quantum corresponding of classical optical transformations. Although the classical optical transformations have been fruitfulness up to now, some questions have not been solved commendably, and that quantum optics was proposed, lots of similarities in classical and quantum optics were found in a way. Therefore, finding the quantum corresponding of classical optical transformations and developing the quantum or classical optics have becoming an inevitable trend.Firstly, based on the fractional Fourier transformation (FrFT) and complex fractional Fourier transformation (CFrFT), a new kind of fractional transform (the FrET) was proposed successfully by replacing the integration kernel with a new one.Employing the IWOP technique and the entangled state representation, the unitary operator corresponding to the FrET is just a entangling operator and is evidently different from that in the complex FrFT. The additivity property, an important fractional feature, was easily proved by using the matrix element expression and the completeness relation of entangled state representation in the framework of quantum mechanics. It should point out that not only found the quantum mechanical correspondence (unitary operator) of the classical optical transform, but also developed the classical one from the viewpoint of quantum optics. That is to say, it is possible that some other transforms can be presented by using different quantum mechanical unitary operators or representations. In these derivation, the IWOP technique plays an important role, many problems become easier with the help of IWOP.Next, based on the quantum Hadmard transformation, the quantum fractional Hadmard transformation (FHT) with continuous variables was introduced. It found that the corresponding fractional Hadmard transformation operator (FHTO) could be decomposed into a single-mode fractional operator and two single-mode squeezing operators. The FHT was also introduced by using the bipartite entangled state representation, the corresponding FHTO could be decomposed into two single-mode fractional operators and two two-mode squeezing operators. Because of the development of quantum mechanic representation, the FHT under three-mode entangled state |p) was proposed further. The FHTO could be decomposed into three single-mode fractional operators and two three-mode squeezing operators. It was easily obtained that these FHTOs satisfy the additivity under certain condition by using above decomposition forms. These results are different from FrFT. For any quantum state |f) , the measurement result for the transformed quantum state by continuous state|x)(|?> or|?>) just corresponds to a squeezing FrFT.In search of the quantum corresponding of classical optical transformations has important meaning to develop quantum mechanic representations and obtain more and more quantum unitary operators. It can help not only know and understand the classical optical transformations, but also the development of the classical optical transformations.