The Study Of Wavelet-fractional Fourier Transform Based On Quantum Optical Representation And Operator Theory

The symbolic method and the representation theory created by Dirac have played an important role in restating classical optical transformation from the perspective of quantum optics.In particular,the technique of integration within an ordered product(IWOP technique)is a direct "bridge" linking classical optical transformation and quantum optical operators.Through IWOP technique,we can construct quantum optical unitary operators of integral type.These unitary operators can form a one-to-one correspondence with many classical optical transformations through appropriate representations,establishing the relationship between quantum optics and classical optics.On the other hand,by virtue of IWOP technique,we can find more new optical transformations through replacing and reconstructing the quantum optical unitary operators.In this paper,using quantum optical representation and operator theory,and based on the quantum optical representation of complex wavelet transform and complex fractional Fourier transform,we recombine the integration kernels of complex wavelet transform and complex fractional Fourier transform and propose a new combined optical transform—the joint complex wavelet-fractional Fourier transform employing the IWOP technique,and the corresponding transform operator is calculated.By substituting the mother wavelet in the form of the Mexican hat,the joint complex wavelet-fractional Fourier transform of different quantum states is calculated,and the results are numerically simulated,and a three-dimensional graph is given to discuss its characteristics more intuitively.In addition,considering the fractional squeezing transform is obtained by replacing the trigonometric function of the integral kernel of the fractional Fourier transform,we then analyze the integration kernel of the wavelet transform and the fractional squeezing transform under the quantum optical representation,and propose the joint wavelet-fractional squeezing transform by the same method.Then calculated the transformation results of different quantum states,and discusses the transformation characteristics through three-dimensional graphs.Finally,the research of joint wavelet-fractional squeezing transform is extended to the entangled case.By recombining the integration kernel of complex wavelet transform and complex fractional squeezing transform,we establish the joint complex waveletfractional squeezing transform,then analyze the transformation characteristics by discussing the transformation results of different quantum states.For the above three transformations,this paper analyzes the transformation maps of different quantum states.The results show that these transformation maps show different characteristics with the changes of the selected parameters,so these transformations have certain application value in the identification of quantum states.