By introducing the entangled state representation and the two-mode Fresnel operator we provide quantum mechanical version for Bessel beam's classical propagation in ABCD optical system. This provides the opportunity of studying various classical Fresnel transformations in the context of quantum optics.
Enlightened by the algorithmic isomorphism between the rotation of the Wigner distribution function (WDF) and the αth fractional Fourier transform, we show that the optical Fresnel transform performed on the input through an ABCD system makes the output naturally adapting to the associated Wigner transform, i.e. there exists algorithmic isomorphism between ABCD transformation of the WDF and the optical Fresnel transform. We prove this adaption in the context of operator language. Both the single-mode and the two-mode Fresnel operators as the image of classical Fresnel transform are introduced in our discussions, while the two-mode Wigner operator in the entangled state representation is introduced for fitting the two-mode Fresnel operator.
We develop the mathematical properties of the ABCD -transformation from the classical and quantum mechanical points of view. First, we list the four types of generating function which generate the ABCD -transformation in classical mechanics. Second, we introduce the unitary operator of the ABCD -transformation. Next, we calculate the normal ordering of this unitary operator and derive the kernels in coordinate-momentum phase space. The kernels are comprised of the generating functions, which generate the ABCD -transformation in classical mechanics. This reveals a new correspondence between classical and quantum mechanics. As an application of these kernels, we show the propagation of coherent and Gaussian beams in the context of quantum optics which corresponds to the ABCD -transformation in matrix optics.
Corresponding to the Fresnel transform there exists a unitary operator in quantum optics theory, which could be known the Fresnel operator (FO). We show that the multiplication rule of the FO naturally leads to the quantum optical ABCD law. The canonical operator methods as mapping of ray-transfer ABCD matrix is explicitly shown by the normally ordered expansion of the FO through the coherent state representation and the technique of integration within an ordered product of operators. We show that time evolution of the damping oscillator embodies the quantum optical ABCD law.