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Journal of the Optical Society of America A

Eigenfunctions of the complex fractional Fourier transform obtained in the context of quantum optics

Hong-Yi Fan, Liyun Hu, Ji‐Suo Wang

Published Mar 26, 2008• 14 citations
Publisher

Abstract

We employ the recently established basis (the two-variable Hermite-Gaussian function) of the generalized Bargmann space (BGBS) [Phys. Lett. A303, 311 (2002)] to study the generalized form of the fractional Fourier transform (FRFT). By using the technique of integration within an ordered product of operators and the bipartite entangled-state representations, we derive the generalized generating function of the BGBS with which the undecomposable kernel of the two-dimensional FRFT [also named complex fractional Fourier transform (CFRFT)] is obtained. This approach naturally shows that the BGBS is just the eigenfunction of the CFRFT.

DOI
10.1364/josaa.25.000974

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We employ the recently established basis (the two-variable Hermite-Gaussian function) of the generalized Bargmann space (BGBS) [Phys. Lett. A303, 311 (2002)] to study the generalized form of the fractional Fourier transform (FRFT). By using the technique of integration within an ordered product of operators and the bipartite entangled-state representations, we derive the generalized generating function of the BGBS with which the undecomposable kernel of the two-dimensional FRFT [also named complex fractional Fourier transform (CFRFT)] is obtained. This approach naturally shows that the BGBS is just the eigenfunction of the CFRFT.
Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical representation transformation and the method of integration within normal ordered product (IWOP) of operators, we find the key point for composing FrFT, and reveal the structure of FrFT. Following this procedure, a full family of generalized fractional transformations are discovered with the usual FrFT as one special case. The eigen-functions of arbitrary GFrT are derived explicitly.
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 our previous papers, the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics, and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform. The core function of the coordinate–momentum exchange operators in the addition law of fractional Fourier transform was analyzed too. In this paper, the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators (IWOP) are used to establish the entanglement fractional Fourier transform theory to the extent of quantum. A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.
Via the route of developing Dirac's symbolic method and following Dirac's assertion: "⋯ for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory", we find the generalized Fresnel operator (GFO) corresponding to the generalized Fresnel transform (GFT) in classical optics. We derive GFO's normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in quantum optics. With the aid of entangled state representation the entangled Fresnel transform is proposed; new eigenfunctions of the complex fractional Fourier transform and fractional Hankel transform are obtained; the two-variable Hermite eigenmodes of light propagation are used in studying the Talbot effect in quadratic-index media; the complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Moreover, quantum optical version of classical z-transforms is obtained on the basis of the eigenvector of creation operator. Throughout our discussions, the coherent state, squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used.