The Study Of Fractional Squeezing Transformation And Its Related Properties By Virtue Of Quantum Mechanical Representation And Operator Theory

Representation theory not only has a broad application in many areas of physics,but also provides a good platform for studying the issue of other subjects.Electing the proper representation to solve the specific problem can let us save a lot of work.It tells us that we should look for more representations to simplify the study of science.And by virtue of the integration technique within ordered product of opertators,we have found a series of new quantum optics representations,including the intermediary entangled state representation,the thermal entangled state representation,etc.At the same time,by means of projecting the symplectic transformation of the classical phase space into quantum Hilbert space through the coherent state representation,we built a series of integral type unitary operator of quantum mechanics.By proper representation,such as coordinate representation,momentum representation,entangled state representation,the unitary operator can form one-to-one correspondence relationship with many classical optical transformation,including the fractional squeezing transform,fractional Hankel transform and generalized Fresnel transform and so on.On the other hand,from the perspective of the optical development history,every kind of optical devices always corresponds to a kind of the discovered optical transformation,such as the lens group is the transform device of the fractional Fourier transform,etc.On the contrary,once a new optical transformation were found,we hope some optical device can realize it.So the introduction of the new transformation make quantum optics and the classical optical theory more and more colorful.Therefore,in this article,we will use the representation and operator theory of quantum mechanics,to do a series of research on the fractional Fourier transform and the complex fractional Fourier transform,including the quantum mechanical representation expression of fractional Fourier transform,additivity,eigenmode and convolution theorem.On this basis,through the variable substitution for the integral kernel of the fractional Fourier transform,which uses hyperbolic function instead of trigonometric function,we proposed a new kind of integral transform.Studies haveshown that this kind of transformation is just the fractional squeezing transform.Like the fractional Fourier transform,it also satisfies additivity.We also discussed the application of the fractional squeezing transform,its convolution theorem,and its relationship with the Wigner transform.Due to the wide application of the quantum entanglement,in this article,we have also generalized the above discussion to two particles entangled case.