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Acta Physica Sinica

Ridgelet transform for quantum optical states

Hai-jun Yu, Zhong Guo-Bao, MA Jian-guo, Ren Gang

Published Jan 1, 2013• 1 citation
Publisher

Abstract

We derive a useful formula for the normal ordering of two-mode operator by the technique of integration within an ordered product (IWOP) of operators. Furthermore, we also calculate the Radon transformations for the coherent state, special squeezed coherent state, intermediary entangled state in the view of the quantum mechanics. Based on the results above, the Mexican hat's mother wavelet function is selected to analyse Ridgelet transformations of these three quantum optical states.

DOI
10.7498/aps.62.144203

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We derive a useful formula for the normal ordering of two-mode operator by the technique of integration within an ordered product (IWOP) of operators. Furthermore, we also calculate the Radon transformations for the coherent state, special squeezed coherent state, intermediary entangled state in the view of the quantum mechanics. Based on the results above, the Mexican hat's mother wavelet function is selected to analyse Ridgelet transformations of these three quantum optical states.
Inspired by the wavelet transform in quantum mechanics, we define the new Ridgelet transform for quantum optics by rewriting the classic Ridgelet transform via the two-mode coordinate representation in Fock space. Furthermore, we give the explicit form of the asymmetric operator's integral and derive two useful formulas for the normal ordering of the two-mode operator with the help of the technique of integration within an ordered product (IWOP) of operators. By choosing the two-variable Mexican hat's mother wavelet function, we analyse the Ridgelet transforms of the coherent state, special squeezed coherent state, intermediary entangled state on the basis of the theories we have mentioned.
By the newly developed technique of integration within an ordered product (IWOP) of 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 can not 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. 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 to 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 IWOP technique are fully used. All these confirms 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".
In the context of quantum mechanics we employ the technique of integration within an ordered product of operators to recast the classical wavelet transform to a squeezing-displacing transform between the mother wavelet vector and the state vector to be transformed. In this way we propose the wavelet-transform spectrum for quantum optical states. For some typical states we obtain numerical results which imply that the spectrum can be used to recognize a variety of quantum optical states, and the inverse wavelet transform has the possibility to play a role in quantum state engineering.