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Chinese Physics B

The Fresnel—Weyl complementary transformation

Chuanmei Xie, Hong-Yi Fan

Published Oct 1, 2012• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1056/21/10/100302

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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.
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.
Based on the newly-developed coherent-entangled state representation we propose the so-called Fresnel–Hadamard complementary transformation, which is unitary. The new unitary operator plays the role of both Fresnel transformation for and Hadamard transformation for respectively. Physically, and could be a symmetric beamsplitter's two output fields. We show that the two transformations are concisely expressed in the coherent-entangled state representation as a projective operator in integration form.
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.
By combining the beam splitter and the Fresnel transform, a protocol is proposed to generate a new entangled state representation, called the intermediate coherent-entangled state (ICES) representation. The properties, such as eigenvalue equation, completeness relation and orthogonal relation, are investigated. The conjugate state representation of the ICES and the Schmidt decomposing of the ICES are also discussed. As applications, a new squeezing operator and some operator identities by using the ICES are obtained.
We find a new complex 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 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 are fully used in our derivation which brings convenience.