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Modern Physics Letters A

Induced entangled state representations for generating fractional Fourier–Hankel transform

Cui-hong Lv, Su-Qing Zhang, Hong-Yi Fan

Published Oct 30, 2018• 0 citations
Publisher

Abstract

Based on the fact that the quantum mechanical version of Hankel transform kernel (the Bessel function) is just the transform between [Formula: see text] and [Formula: see text], two induced entangled state representations, and working with them, we derive fractional Fourier–Hankel transformation (FrFHT) caused by the operator [Formula: see text], where [Formula: see text] is named the core operator and is essential to the fractional transformation. The fractional property (additive rule) of the FrFHT can be explicitly proved.

DOI
10.1142/s0217732318502115

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Based on the fact that the quantum mechanical version of Hankel transform kernel (the Bessel function) is just the transform between [Formula: see text] and [Formula: see text], two induced entangled state representations, and working with them, we derive fractional Fourier–Hankel transformation (FrFHT) caused by the operator [Formula: see text], where [Formula: see text] is named the core operator and is essential to the fractional transformation. The fractional property (additive rule) of the FrFHT can be explicitly proved.
Based on the fact that the quantum mechanical version of Hankel transform kernel (the Bessel function) is just the transform between | q , r 〉 and ( s , r ′|, two induced entangled state representations are given, and working with them we derive fractional squeezing-Hankel transform (FrSHT) caused by the operator e − i α ( a 1 † a 2 † + a 1 a 2 ) e − i π a 2 † a 2 , which is an entangled fractional squeezing transform operator. The additive rule of the FrSHT can be explicitly proved.
By a quantum mechanical analysis of the additive rule F α[F β[f]]=F α+β[f], which the fractional Fourier transformation (FrFT) F α[f] should satisfy, we reveal that the position-momentum mutual-transformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation (CFrFT), which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.
In this Letter, a new fractional entangling transformation(Fr ET) is proposed, which is generated in the entangled state representation by a unitary operator expfiθeabt abTg where aebT is the Bosonic annihilate operator. The operator is actually an entangled one in quantum optics and differs evidently from the separable operator, expfiθeaa t bbTg, of complex fractional Fourier transformation. The additivity property is proved by employing the entangled state representation and quantum mechanical version of the Fr ET. As an application, the Fr ET of a two-mode number state is derived directly by using the quantum version of the Fr ET, which is related to Hermite polynomials.