Symplectic-dilation mixed wavelet transform and its correspondence in quantum optics

TL;DR

This paper introduces the symplectic-dilation mixed wavelet transform (SDWT), combining real-variable dilation and complex-variable symplectic transforms, with applications in quantum optics and classical optics correspondence.

Contribution

It develops the SDWT, establishes its mathematical properties, and links it to quantum and classical optical transforms using entangled-coherent state representation.

Findings

SDWT possesses Parseval theorem and inversion formula.

Derived quantum transform operator corresponding to SDWT.

Established classical optics counterpart as lens-Fresnel mixed transform.

Abstract

The symplectic wavelet transformation [Opt. Lett. 31 (2006) 3432], which is related to quantum optical Fresnel transform, is developed to the symplectic-dilation mixed wavelet transform (SDWT). The SDWT involves both a real-variable dilation-transform and a complex-variable symplectic transform, and possesses well-behaved properties such as Parseval theorem and inversion formula. The entangled-coherent state representation (ESCR) not only underlies the SDWT, but also helps to derive the corresponding quantum transform operator whose counterpart in classical optics is the lens-Fresnel mixed transform [Phys. Lett. A 357 (2006) 163].