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arXiv:quant-ph/0508066 · Optics Letters

General formula for finding mother wavelets by virtue of Dirac's representation theory and the coherent state

Hong-Yi Fan, Lu Hai-Liang

Published Jan 1, 2006• 22 citations
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Abstract

The admissibility condition of a mother wavelet is explored in the context of quantum optics theory. By virtue of Dirac's representation theory and the coherent state property we derive a general formula for finding qualified mother wavelets. A comparison between a wavelet transform computed with the newly found mother wavelet and one computed with a Mexican hat wavelet is presented.

DOI
10.1364/ol.31.000407

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The admissibility condition of a mother wavelet is explored in the context of quantum optics theory. By virtue of Dirac's representation theory and the coherent state property we derive a general formula for finding qualified mother wavelets. A comparison between a wavelet transform computed with the newly found mother wavelet and one computed with a Mexican hat wavelet is presented.
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.
For the efficient identification of quantum states, we propose the notion of linear canonical wavelet transform in the framework of quantum mechanics. Using the machinery of Dirac representation theory and integration within an ordered product of operators, we recast the linear canonical wavelet transform to a matrix element of the squeezing–displacing operator U(μ,s)KM between analyzing vector ⟨ψ| and two-mode quantum state vector |f⟩ to be transformed. We also derive the inner product relation and inversion formula for the linear canonical wavelet transform in the realm of quantum mechanics. Lastly, we present an explicit example for the lucid implementation of linear canonical wavelet transform in identifying the quantum states.
In this paper, we propose the so-called continuous Fresnel-wavelet combinatorial transform which means that the mother wavelet undergoes the Fresnel transformation. This motivation can let the mother-wavelet-state itself vary from |〉 to F†r,s|〉, except for variation within the family of dilations and translations. The Parseval's equality, admissibility condition and inverse transform of this continuous Fresnel-wavelet combinatorial transform are analysed. By taking certain parameters and using the admissibility condition of this continuous Fresnel-wavelet combinatorial transform, we obtain some mother wavelets. A comparison between the newly found mother wavelets is presented.