Entangled photon states in consecutive nonlinear optical interactions


Quantum theory of two consecutive light-wave parametric interactions with aliquant frequencies produced by a common pump wave in a crystal is developed. Using the differentiation method, the unitary evolution operator of the system is reduced to an ordered form that allows the calculation of the field state and the statistical characteristics of interacting waves. It is shown that, for the initial vacuum field state, the created photons obey the super-Poisson statistics at the interacting frequencies and are in a multiparticle entangled state.

This is a preview of subscription content, log in to check access.


  1. 1.

    L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, Cambridge, 1995; Nauka, Moscow, 2000).

    Google Scholar 

  2. 2.

    M. D. Reid and P. D. Drummond, Phys. Rev. Lett. 60, 2731 (1988); M. D. Reid, Phys. Rev. A 40, 913 (1989).

    Article  ADS  Google Scholar 

  3. 3.

    N. Korolkova, G. Leuchs, R. Loudon, et al., Phys. Rev. A 65, 052306 (2002); R. Schnabel, W. P. Bowen, N. Treps, et al., Phys. Rev. A 67, 012316 (2003).

  4. 4.

    The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation, Ed. by D. Bouwmeester, A. K. Ekert, and A. Zeilinger (Springer, Berlin, 2000; Postmarket, Moscow, 2002).

    Google Scholar 

  5. 5.

    D. N. Klyshko, Photons and Nonlinear Optics (Nauka, Moscow, 1980).

    Google Scholar 

  6. 6.

    J. S. Bell, Physics (Long Island City, N.Y.) 1, 195 (1964).

    Google Scholar 

  7. 7.

    S. J. Freedman and J. S. Clauser, Phys. Rev. Lett. 28, 938 (1972).

    ADS  Google Scholar 

  8. 8.

    D. M. Greenberger, M. A. Horne, A. Shimony, et al., Am. J. Phys. 58, 1131 (1990); D. M. Greenberger, M. A. Horne, and A. Zeilinger, Phys. Today 46 (8), 22 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  9. 9.

    D. Bouwmeester, J.-W. Pan, M. Daniell, et al., Phys. Rev. Lett. 82, 1345 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  10. 10.

    J.-W. Pan, M. Daniell, S. Gasparoni, et al., Phys. Rev. Lett. 86, 4435 (2001).

    Article  ADS  Google Scholar 

  11. 11.

    M. Eibl, S. Gaertner, M. Bourennane, et al., Phys. Rev. Lett. 90, 200403 (2003).

  12. 12.

    N. Korolkova and G. Leuchs, in Coherence and Statistics of Photons and Atoms, Ed. by J. Perina (Wiley, New York, 2001), p. 111.

    Google Scholar 

  13. 13.

    E. Yu. Morozov and A. S. Chirkin, J. Opt. A: Pure Appl. Opt. 5, 233 (2003).

    Article  Google Scholar 

  14. 14.

    A. S. Chirkin, V. V. Volkov, G. D. Laptev, et al., Kvantovaya Élektron. (Moscow) 30, 847 (2000).

    Article  Google Scholar 

  15. 15.

    N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (Nauka, Moscow, 1965; American Mathematical Society, Providence, R.I., 1968).

    Google Scholar 

  16. 16.

    J. Wei and E. Norman, J. Math. Phys. 4, 575 (1963).

    MathSciNet  Google Scholar 

  17. 17.

    M. J. Collett, Phys. Rev. A 38, 2233 (1988).

    ADS  MathSciNet  Google Scholar 

  18. 18.

    I. V. Bargatin, B. A. Grishanin, and V. N. Zadkov, Usp. Fiz. Nauk 171, 625 (2001) [Phys. Usp. 44, 597 (2001)].

    Google Scholar 

  19. 19.

    A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, et al., Phys. Rev. A 60, R4209 (1999); L. A. Krivitskii, S. P. Kulik, A. N. Penin, et al., Zh. Éksp. Teor. Fiz. 124, 943 (2003) [JETP 97, 846 (2003)].

  20. 20.

    S. A. Akhmanov, Yu. E. D’yakov, and A. S. Chirkin, Introduction to Statistical Radio Physics and Optics (Nauka, Moscow, 1981).

    Google Scholar 

Download references

Author information



Additional information


Translated from Pis’ma v Zhurnal Éksperimental’no\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) i Teoretichesko\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} \) Fiziki, Vol. 79, No. 6, 2004, pp. 311–314.

Original Russian Text Copyright © 2004 by Rodionov, Chirkin.

An erratum to this article is available at http://dx.doi.org/10.1134/1.1787108.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rodionov, A.V., Chirkin, A.S. Entangled photon states in consecutive nonlinear optical interactions. Jetp Lett. 79, 253–256 (2004). https://doi.org/10.1134/1.1759404

Download citation

PACS numbers

  • 42.50.Dv
  • 42.65.Ky
  • 03.67.Mn
  • 03.65.Ud