Theory Of Entangled Fourier Transformation And Fractional Squeezing Transformation For Optical Imaging

Optical imaging system as one of the most important information processing systems in optics,by using the linear transformation theory and spectrum analysis technology and propagation characteristics of light to transfer the information such as the structure,grayscale and color of objects.Developing the theory of optical information propagation and transformation,extending the imaging range of optical system and improving the imaging accuracy,have become a very important front topic in modern optics.For example,The lens is the most basic device in the geometric optical system,and its imaging theory corresponds to the Fourier transform.Another example,The theory of fractional Fourier transform proposed in recent years can be applied to the propagation of light in optical fibers,and it is also a bridge between the theory of optical diffraction and the theory of Wigner distribution function about light field.Therefore,in order to develop more optical application fields,we urgently need to enrich and expand the theory of integral transformation.We develop the traditional Fourier optical transformations(such as Fourier transformation and Fourier fractional transformation and Fresnel transformation,etc)to the case involving entanglement,we propose entangled Fourier integral transformation(EFIT)and fractional squeezing transformation,which provides experimental physicists with new image-making mechanism.This motivation came from the idea:how to introduce entanglement to optical transforms since entanglement is now dominating many fields of physics.For example,it seeks to transform the functional image of the product between two independent polynomials xmyn into the functional image of two-variable Hermite polynomial(which may be achieved by designing new lens combinations),corresponding to the ongoing research on quantum entanglement.Since the basis vector of the function space about two-body entangled state of continuous variables is two-variable Hermite polynomial Hm?n(x,y),which is a new complete and orthogonal basis in function space,and the transformation of two independent polynomials xmyn into Hm?n(x,y)is a classical entanglement transformation,which will be widely used in quantum entanglement theory.Instead of the traditional approach,we will take the path of quantum optics to classical optics.The research content of this paper mainly includes the following three parts:1.In order to entangle the optical image function to be transformed,we have proposed the entangled Fourier integral transform(EFIT),which has the property of keeping modulus-invariant,and has its inverse transformation.Then we applied this transform into the operator function of quantum mechanics.With the help of the integration of within an ordered product of operator,we studied the EFIT of Wigner operator and found that the EFIT of a classical function is just only related to its Weyl-corresponding operator's matrix element in coordinate—momentum representation,which helps to find other new optical transformations,such as fractional squeezing transformation.As a by-product,we also established new relations about operators're-ordering,and it can convert P-Q ordering and Q-P ordering into Weyl ordering respectively.2.The work of the second step is extending the work of the first step to the two-mode case,we have proposed a new complex entangled Fourier integration transform which can establish a new relationship between a two-mode operator's matrix element in the entangled state representation and its Wigner function.This complex integration keeps modulus invariant and therefore invertible.Based on this and the Weyl-Wigner correspondence theory,we find a two-mode operator which is responsible for complex fractional squeezing transformation.The entangled state representation and the Weyl ordering form of the two-mode Wigner operator is fully used in our derivation which brings convenience.The achievements of these two steps o have used the theory of I WOP,which is a self-contained system and shows a series.It is the result of the combination of quantum optics and classical optics.3.Expand on the basis of the first two parts of the work,by using the P representation of the density operator,the diffusion equation of the quantum density operator is derived.Furthermore,by introducing the Weyl ordering of quantum operators and the corresponding Weyl quantization scheme,the equation describing the quantum diffusion channel is derived,and the evolution of the Wigner operator in the quantum channel is given.The evolution law of Wigner operator from point source function to Gaussian function at t time is shown,which is concise and physically clear.On this basis,the evolution of coherent states through quantum diffusion channels is discussed.