International Journal of Theoretical Physics

, Volume 54, Issue 4, pp 1283–1291 | Cite as

From Weyl Expansion of Operator to Ordering of Operator and its New Applications

  • Fang Jia
  • Hao-Liang Zhang
  • Li-Yun Hu
  • Hong-Yi Fan


By using the IWOP technique, Weyl expansion of operator is derived. Based on this, three new ordering formulas of operators are presented, which have been applied to calculate Q-P and P-Q ordering of some operators. As other applications of Weyl expansion, the formula of photon counting is also easily obtained, which are related to Wigner and Q-function. In particular, noticing the Weyl ordering of displacement operator is itself, Weyl ordering invariance under similarity transformations is conveniently proved, instead of using of Wigner operator.


Weyl expansion Operator ordering Photon counting Weyl ordering invariance 



Project supported by the National Natural Science Foundation of China (Grant No. 11264018), the Natural Science Foundation of Jiangxi Province of China (Grant No. 20132BAB212006), and the Research Foundation of the Education Department of Jiangxi Province of China (no GJJ14274) as well as Degree and postgraduate education teaching reform project of jiangxi province(No. JXYJG-2013-027).


  1. 1.
    Scully, M.O., Zubairy, M.S.: Quantum Optics. Cambridge (1997)Google Scholar
  2. 2.
    Fan, H.Y., Wang, T.T: Int. J. Theor. Phys. 48, 441 (2009)CrossRefMATHGoogle Scholar
  3. 3.
    Fan, H.Y., Yuan, H.C., Jiang, N.Q.: Sci. China: Phys. Mech. Astron. 53, 1626 (2010)ADSGoogle Scholar
  4. 4.
    Fan, H.Y.: Chin. Phys. B 19, 050303 (2010)Google Scholar
  5. 5.
    Lee, H.W.: Phys. Rep. 259, 147 (1995)CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Fan, H.Y., Hu, L.Y., Yuan, H.C.: Chin. Phys. B 19, 060305 (2010)CrossRefADSGoogle Scholar
  7. 7.
    Fan, H.Y.: Sci. China: Phys. Mech. Astron. 55, 762 (2012)CrossRefADSGoogle Scholar
  8. 8.
    Hu, L.Y., Zhang, H.L., Jia, F., Tao, X.Y.: Chin. Phys. B 22, 120301 (2013)CrossRefGoogle Scholar
  9. 9.
    Hu, L.Y., Liu, S.Y., Zheng, K.M., Jia, F., Fan, H.Y.: Int. J. Theor. Phys. 53, 380 (2014)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Marian, P., Marian, T.A.: Phys. Rev. A 74, 042306 (2006)CrossRefADSGoogle Scholar
  11. 11.
    Braunstein, S.L., Kimble, H.J.: Phys. Rev. Lett. 80, 869 (1998)CrossRefADSGoogle Scholar
  12. 12.
    Fan, H.Y.: Ann. Phys. 323, 500 (2008)CrossRefADSMATHGoogle Scholar
  13. 13.
    Fan, H.Y.: J. Opt. B Quantum Semiclass Opt. 5, R147 (2003)CrossRefADSGoogle Scholar
  14. 14.
    Fan, H.Y.: Representation and Transformation Theory in Quantum Mechanics (Shanghai: Shanghai Scientific and Technical Publisher, 1997) (in Chinese) p. 27.Google Scholar
  15. 15.
    Hu, L.Y., Xu, X.X., Guo, Q., Fan, H.Y.: Opt. Commun. 283, 5074 (2010)CrossRefADSGoogle Scholar
  16. 16.
    Loudon, R.: The Quantum Theory of Light. Oxford University Press, Oxford (1983)Google Scholar
  17. 17.
    Orszag, M.: Quantum Optics. Springer, Berlin (2000)CrossRefMATHGoogle Scholar
  18. 18.
    Kelley, P.L., Kleiner, W.H.: Phys. Rev. 136, 316 (1964)CrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Scully, M.O., Lamb, W.E.: Phys. Rev. 179, 368 (1969)CrossRefADSGoogle Scholar
  20. 20.
    Mollow, B.R.: Phys. Rev. 168, 1896 (1968)CrossRefADSGoogle Scholar
  21. 21.
    Fan, H.Y., Hu, L.Y.: Opt. Lett. 33, 443 (2008)CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Fang Jia
    • 1
    • 2
  • Hao-Liang Zhang
    • 2
  • Li-Yun Hu
    • 2
  • Hong-Yi Fan
    • 1
  1. 1.Department of Material Science and EngineeringUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Key Laboratory of Optoelectronic and TelecommunicationJiangxi Normal UniversityNanchangChina

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