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College Physics

The quantum mechanical correspondence for classical Fresnel operator

Fan Hong‐Yi

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

The quantum mechanical correspondence for the classical Fresnel transformation is introduced,which is quantum Fresnel operator corresponding to the symplectic motion of the representing point for coherent state in phase space.

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In terms of the coherent state evolution in phase space, we present a quantum mechanical version of the classical Liouville theorem. The evolution of the coherent state from | z 〉 to | sz − rz *〉 corresponds to the motion from a point z ( q , p ) to another point sz − rz * with | s | 2 − | r | 2 = 1. The evolution is governed by the so-called Fresnel operator U ( s , r ) that was recently proposed in quantum optics theory, which classically corresponds to the matrix optics law and the optical Fresnel transformation, and obeys group product rules. In other words, we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space, which seems to be a combination of quantum statistics and quantum optics.
By virtue of the new technique of performing integration over Dirac’s ket-bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel-Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond 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 the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed 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”.
Corresponding to the Fresnel transform there exists a unitary operator in quantum optics theory, which could be known the Fresnel operator (FO). We show that the multiplication rule of the FO naturally leads to the quantum optical ABCD law. The canonical operator methods as mapping of ray-transfer ABCD matrix is explicitly shown by the normally ordered expansion of the FO through the coherent state representation and the technique of integration within an ordered product of operators. We show that time evolution of the damping oscillator embodies the quantum optical ABCD law.
The symplectic wavelet transform, which is related to the 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 the Parseval theorem and the inversion formula. The entangled-coherent state representation 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.
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.