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Journal of Modern Optics

Adaption of Collins formula to fractional Fourier transform studied in the entangled state representation of quantum optics

Liyun Hu, Hong-Yi Fan

Published Aug 18, 2008• 11 citations
Publisher

Abstract

Adaption of Collins diffraction formula to the fractional Fourier transform in complex form (FFT-C) is demonstrated in entangled state |η⟩ representation of quantum optics. The additivity property of FFT-C corresponding to Collins formula has been proved by using group multiplication of the Fresnel operator and the quantum mechanical version of the Collins formula.

DOI
10.1080/09500340802130753

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Journal of Modern Optics•Oct 10, 2009•1 citation
Adaption of Collins diffraction formula to the fractional Fourier transform in complex form (FFT-C) is demonstrated in entangled state |η⟩ representation of quantum optics. The additivity property of FFT-C corresponding to Collins formula has been proved by using group multiplication of the Fresnel operator and the quantum mechanical version of the Collins formula.
Based on the correspondence between Collins diffraction formula (optical Fresnel transform) and the transformation matrix element of a three-parameters two-mode squeezing operator in the entangled state representation (Opt. Lett. 31 (2006) 2622) we further explore the relationship between output field intensity determined by the Collins formula and the input field's probability distribution along an infinitely thin phase space strip both in spacial domain and frequency domain. The entangled Wigner function is introduced for recapitulating the result.
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.
In this Letter, a new fractional entangling transformation (FrET) is proposed, which is generated in the entangled state representation by a unitary operator expfiθðab † þ a † bÞg where aðbÞ is the Bosonic annihilate operator.The operator is actually an entangled one in quantum optics and differs evidently from the separable operator, expfiθða † a þ b † bÞg, of complex fractional Fourier transformation.The additivity property is proved by employing the entangled state representation and quantum mechanical version of the FrET.As an application, the FrET of a two-mode number state is derived directly by using the quantum version of the FrET, which is related to Hermite polynomials.
Based on the correspondence between the Collins diffraction formula (optical Fresnel transform) and the transform matrix element of a three-parameter two-mode squeezing operator in the entangled state representation 1 Fan, H-Y and Lu, H-L. 2006. Opt. Lett., 31: 2622–2624. [Crossref], [PubMed] , [Google Scholar] we further explore the relationship between output field intensity determined by the Collins formula and the input field's probability distribution along an infinitely thin phase space strip, both in the spatial domain and the frequency domain. The entangled Wigner function is introduced for recapitulating the result.