1. ICO-21 2008 Quantum optical methods in classical optics: Optical realizations of quantum systems Héctor Moya-Cessa Instituto Nacional de Astrofísica, Optica y Electrónica Tonantzintla, Pue MEXXICO

2. ICO-21 2008 Talk will cover Optical ralization of a quantum invariant Optical realization of a quantum beam splitter Wigner function to evaluate some divergent series

3. ICO-21 2008 Time dependent harmonic oscillator (classical) Ermakov-Lewis Invariant Ermakov equation Lewis, PRL (1967). Optical ralization of a quantum invariant M. Fernández Guasti (Metropolitan University of Mexico) S. Chávez Cerda (INAOE) V. Arrizon (INAOE)

4. ICO-21 2008 Squeezing & displacement Translates into QM as H. Moya-Cessa and M. Fernández Guasti PHYSICS LETTERS A 311, 1 (2003). G is an invariant provided its derivative is zero

5. ICO-21 2008 Time dependence multiplies only one operator

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8. Praxial wave equation Suponemos ahora dos medios GRIN pegados GRaded INdex referring to an optical material with refractive index in the form of a parabolic curve, decreasing from the center towards the cladding.

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10. Units such that =1

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13. Arfken

14. ICO-21 2008 Optical realization of a quantum beam splitter R. Mar Sarao (INAOE)

15. ICO-21 2008 Splitting in a 50:50 beam splitter R.A. Campos, B.E.A. Saleh, and M. C. Teich, Phys. Rev. A 40, 1371 (1989).

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17. Again paraxial wave equation Consider the copropagation of two beams, probe and signal, in a Kerr medium. The probe beam produces the index of refraction If the probe beam has a Gaussian profile, Is astigmatic and slightly tilted,a term xy is produced S. Chávez-Cerda, J.R.Moya-Cessa, and H. Moya-Cessa, J. of the Opt. Soc. of Am. B 24, 404-407 (2007).

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19. Wigner function as a tool to evaluate divergente series Roberto de Jesús León (INAOE) E. Martí Panameño (Puebla University) H. Moya-Cessa and P.L. Knight, Phys. Rev. A 48, 2479-2481 (1993). Glauber displacement operator

20. ICO-21 2008 Abel: "Divergent series are on the whole devil's work, and it is a shame that one dares to found any proof on them. One can get out of them what one wants if one uses them, and it is they which have made so much unhappiness and so many paradoxes. Can one think of anything more appalling than to say that where m is a positive number. Here's something to laugh at, friends."

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23. Conclusions It was shown the optical realization of the Lewis-Ermakov invariant and of the “quantum” beam splitter The Wigner function was used to evaluate some divergent series.