The Study Of Fractional Squeezing-hankel Transform And Its Related Properties Based On Quantum Optical Entangled State Representation

Throughout the development of optics,we know that each optical transform corresponds to a group of optical equipment,for example,the optical equipment corresponding to the fractional Fourier transform is a group of lenses,otherwise,the discovery of new optical transform will promote the development of new optical equipment.Therefore,it is of great significance to propose more and more new optical transforms for the continuous enrichment and development of optical theory.With the continuous development of quantum theory,it is found that the classical optical transform can find its representation of quantum mechanics.Many problems that cannot be solved by classical optics can be explained by quantum theory.Among them,quantum optical representation theory can replace a complex abstract quantity with a simple representation,thus simplifying the research of the problem.The technique of integration within an ordered product(IWOP)of operators is an integration method suitable for operator calculation in the quantum field.It not only solves the problem of the operators are not commutative,but also is a useful tool to find new representations,and establishes the connection between the classical optical transform and the quantum optical unitary operators.The main content of this article is based on the original optical transform,using quantum optical representation theory and IWOP technique to construct new optical transforms.Based on the relationship between fractional squeezing transform and single-mode Fresnel transform,this paper further discusses the relationship between entangled fractional squeezing transform and two-mode Fresnel transform,and proposes a method to prove the additivity property of the entangled fractional squeezing transform according to the multiplication rule of the two-mode Fresnel transform.Secondly,based on the representation of complex fractional Fourier transform and Hankel transform in quantum mechanics,we use representation theory and IWOP technique to recombine their integral kernel,and get a new integral kernel through integral calculation,so as to construct a new transform called fractional Fourier-Hankel transform,and study its related properties.The fractional FourierHankel transform of quantum states is calculated,and the result is simulated.By analyzing the change characteristics of the graph,it is found that the transform plays a certain role in the recognition of quantum states.Compared with the results of the corresponding complex fractional Fourier transform,it is found that the fractional Fourier-Hankel transform has more advantages in quantum state recognition.In addition,the advantage of the transform in engineering calculation is found by solving differential equations.Finally,the above research content is extended to the squeezing transform,that is,the fractional squeezing-Hankel transform is obtained by reorganizing the integral kernel of the entangled fractional squeezing transform and Hankel transform.Its linear and additivity properties are proved,and its related applications are discussed.The transform results of quantum states are also discussed in the form of three-dimensional graphs.It is found that the transform can also recognize quantum states.Compared with the result of entangled fractional squeezing transform,it is found that the advantage of fractional squeezing-Hankel transform is more obvious.