International Journal of Theoretical Physics

, Volume 55, Issue 7, pp 3164–3172 | Cite as

Expectation Value Theorem for Thermo Vacuum States of Optical Chaotic Field and Negative-Binomial Field

  • Zhi-Long Wan
  • Hong-Yi Fan


For the density operator (mixed state) describing chaotic light and negative-binomial field there exist the corresponding thermal vacuum state (pure state) in the real-fictitious space. Using the method of integration within ordered product of operators we find the expectation value theorem in these two thermo vacuum states respectively. The thermal average theorem of translation operator is also deduced. Application of the new thermo vacuum state in calculating photon number disturibution and fluctuation and thermal average is presented.


Expectation value theorem Chaotic field Negative binomial optical field Thermo vacuum state Integration method within ordered product of operators 



This work is supported by the National Natural Science Foundation of China(Grant Nos. 11175113 and 11574295).


  1. 1.
    Hu, L.Y., Fan, H.Y.: Wigner functions of thermo number state, photon subtracted and added thermo vacuum state at finite temperature. Mod. Phys. Lett. A 24(28), 2263–2274 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Fan, H.Y., Jiang, N.Q.: Thermo wigner operator in thermo field dynamics: its introduction and application. Phys. Scr. 78(4), 2517–2530 (2008)CrossRefMATHGoogle Scholar
  3. 3.
    Wu, H.J., Fan, H.Y.: Two-mode wigner operator in 〈η| representation. Mod. Phys. Lett. B 11(13), 549–554 (1997)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Takahashi, Y., Umezawa, H.: Thermo field dynamics. Collective Phenomena 2, 55–88 (1975)MathSciNetMATHGoogle Scholar
  5. 5.
    Takahashi, Y., Umezawa, H.: Thermo field dynamics. Int. J. Mod. Phys. B 10(13), 1755–1805 (1996). [Reprint]ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hu, L.Y., Fan, H.Y.: Infinite operator-sum representation of density operator for a dissipative cavity with Kerr medium derived by virtue of entangled state representation. Int. J. Theor. Phys. 48(12), 3396–3402 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hu, L.Y., Fan, H.Y.: Two-mode squeezed number state as a two-variable Hermite-polynomial excitation on the squeezed vacuum. J. Mod. Opt. 55(13), 2011–2024 (2008)CrossRefMATHGoogle Scholar
  8. 8.
    Fan, H.Y., Zhan, D.H.: New generating function formulae of even- and odd-Hermite polynomials obtained and applied in the context of quantum optics. Chin. Phys. B 23(6), 060301 (2014)ADSCrossRefGoogle Scholar
  9. 9.
    Hu, L.Y., Fan, H.Y.: Adaption of Collins formula to fractional Fourier transform studied in the entangled state representation of quantum optics. J. Mod. Opt. 55(15), 2429–2437 (2008)CrossRefMATHGoogle Scholar
  10. 10.
    Fan, H.Y., Fan, Y.: Coherent state representation for generalized bosonic bogolyubov transformation. Mod. Phys. Lett. B 11(26), 1157–1160 (1997)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Fan, H.Y., Wang, H.: New applicatinos of 〈η| representation in thermal field statistics. Mod. Phys. Lett. B 14(15), 553–562 (2000)ADSCrossRefGoogle Scholar
  12. 12.
    Wan, Z.L., Fan, H.Y.: Thermo-vacuum state in a negative binomial optical field and its application. Acta Phys. Sin. 64(10), 190302 (2015)Google Scholar
  13. 13.
    Gradsbteyn I.S., Ryzbik L.M.: Table of Integrals, Series, and Products, 6thed. World Scientific, Singapore (2000)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Mathematical and Chemical IndustryChangzhou Institute of TechnologyChangzhouChina
  2. 2.Department of Material Science and EngineeringUniversity of Science and Technology of ChinaHeifeiChina

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