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Wavelet Transformation and Wigner-Husimi Distribution Function

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Abstract

We find that the optical wavelet transformation can be used to study the Husimi distribution function in phase space theory of quantum optics. We prove that the Husimi distribution function of a quantum state |ψ〉 is just the modulus square of the wavelet transform of \(e^{-x^{2}/2}\) with ψ(x) being the mother wavelet up to a Gaussian function. Thus a convenient approach for calculating various Husimi distribution functions of miscellaneous quantum states is presented.

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References

  1. Wigner, E.: Phys. Rev. 40, 749 (1932)

    Article  MATH  ADS  Google Scholar 

  2. O’Connell, R.F., Wigner, E.P.: Phys. Lett. A 83, 145 (1981)

    Article  ADS  Google Scholar 

  3. Hillery, M., O’Connell, R.F., Scully, M.O., Wigner, E.P.: Phys. Rep. 106, 121 (1984)

    Article  ADS  Google Scholar 

  4. Vogel, K., Risken, H.: Phys. Rev. A 40, 2847 (1989)

    Article  ADS  Google Scholar 

  5. Agawal, G.S., Wolf, E.: Phys. Rev. D 2, 2161 (1972)

    Article  ADS  Google Scholar 

  6. Agawal, G.S., Wolf, E.: Phys. Rev. D 2, 2187 (1972)

    Article  ADS  Google Scholar 

  7. Agawal, G.S., Wolf, E.: Phys. Rev. D 2, 2206 (1972)

    Article  ADS  Google Scholar 

  8. Bužek, V., Knight, P.L.: Prog. Opt. 34, 1 (1995)

    Article  Google Scholar 

  9. Bužek, V., Keitel, C.H., Knight, P.L.: Phys. Rev. A 51, 2575 (1995)

    Article  ADS  Google Scholar 

  10. Dodonov, V.V., Man’ko, V.I.: Theory of Nonclassical States of Light. Taylor & Francis, New York (2003)

    Google Scholar 

  11. Husimi, K.: Proc. Phys. Math. Soc. Jpn. 22, 264 (1940)

    MATH  Google Scholar 

  12. Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

    MATH  Google Scholar 

  13. Kaiser, G.: A Friendly Guide to Wavelets. Birkhäuser, Basel (1994)

    MATH  Google Scholar 

  14. Chui, C.K.: An Introduction to Wavelets. Academic Press, San Diego (1992)

    MATH  Google Scholar 

  15. Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford U. Press, London (1958)

    MATH  Google Scholar 

  16. Orszag, M.: Quantum Optics. Springer, Berlin (2000)

    MATH  Google Scholar 

  17. Walls, D.F., Milburn, G.J.: Quantum Optics. Springer, Berlin (1994)

    MATH  Google Scholar 

  18. Schleich, W.P.: Quantum Optics in Phase Space. Wiley-VCH, Berlin (2001)

    Book  MATH  Google Scholar 

  19. Fan, H.-Y., Lu, H.-L., Fan, Y.: Ann. Phys. 321, 480 (2006)

    Article  MATH  ADS  Google Scholar 

  20. Hu, L., Fan, H.: Int. J. Theor. Phys. 47, 1058 (2008)

    Article  MATH  Google Scholar 

  21. Wünsche, A.: J. Opt. B, Quantum Semiclass. Opt. 1, R11 (1999)

    Article  Google Scholar 

  22. Fan, H.-Y.: J. Opt. B, Quantum Semiclass. Opt. 5, R147 (2003)

    Article  ADS  Google Scholar 

  23. Fan, H.-Y., Lu, H.-L.: Opt. Lett. 31, 407 (2006)

    Article  ADS  Google Scholar 

  24. Walls, D.F.: Nature 324, 210 (1986)

    Article  ADS  Google Scholar 

  25. Loudon, R., Knight, P.L.: J. Mod. Opt. 34, 709 (1987)

    Article  MATH  ADS  Google Scholar 

  26. Dodonov, V.V.: J. Opt. B, Quantum Semiclass. Opt. 4, R1–R33 (2002)

    Article  ADS  Google Scholar 

  27. Mehta, C.L.: Phys. Rev. Lett. 18, 752 (1967)

    Article  ADS  Google Scholar 

  28. Klauder, J.R., Skargerstam, B.S.: Coherent States. World Scientific, Singapore (1985)

    MATH  Google Scholar 

  29. Glauber, R.J.: Phys. Rev. 130, 2529 (1963)

    Article  ADS  Google Scholar 

  30. Weyl, H.: Z. Phys. 46, 1 (1927)

    ADS  Google Scholar 

  31. Fan, H.-Y.: Ann. Phys. (2007). doi:10.1016/j.aop.2007.06.003

    Google Scholar 

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Authors and Affiliations

  1. College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, 330022, China

    Li-Yun Hu

  2. Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, China

    Hong-Yi Fan

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  1. Li-Yun Hu
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  2. Hong-Yi Fan
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Correspondence to Li-Yun Hu.

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Hu, LY., Fan, HY. Wavelet Transformation and Wigner-Husimi Distribution Function. Int J Theor Phys 48, 1539–1544 (2009). https://doi.org/10.1007/s10773-009-0008-z

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  • DOI: https://doi.org/10.1007/s10773-009-0008-z

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