Based on the newly developed coherent-entangled state representation,we propose the so-called Fresnel-Weyl complementary transformation operator.The new operator plays the roles of both Fresnel transformation（for（a 1 a 2）/√ 2） and the Weyl transformation（for（a 1 ＋ a 2）/√ 2）.Physically,（a 1 a 2）/√ 2 and（a 1 ＋ a 2）/√ 2 could be a symmetric beamsplitter＇s two output fields for the incoming fields a 1 and a 2.We show that the two transformations are concisely expressed in the coherent-entangled state representation as a projective operator in the integration form.
In terms of the coherent state evolution in phase space,we present a quantum mechanical version of the classical Liouville theorem.The evolution of the coherent state from ｜z〉to｜sz-rz＊〉 corresponds to the motion from a point z（q,p） to another point sz-rz＊ with ｜s｜2-｜r｜2=1.The evolution is governed by the so-called Fresnel operator U（s,r） that was recently proposed in quantum optics theory,which classically corresponds to the matrix optics law and the optical Fresnel transformation,and obeys group product rules.In other words,we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space,which seems to be a combination of quantum statistics and quantum optics.
By using the two-mode Fresnel operator we derive a multiplication rule of two-dimensional （2D） Collins diffraction formula, the inverse of 2D Collins diffraction integration can also be conveniently derived in this way in the context of quantum optics theory.
By virtue of the entangled state representation （Hong-Yi Fan and J R Klauder 1994 Phys. Rev. A 49 704） and the two-mode squeezing operator＇s natural representation （Hong-Yi Fan and Yue Fan 1996 Phys. Rev. A 54 958） we propose the squeeze-swapping mechanism which can generate quantum entanglement and new squeezed states of continuum variables.
