Skip to main content
Log in

Probability representation of the quantum evolution and energy-level equations for optical tomograms

  • Published:
Journal of Russian Laser Research Aims and scope

Abstract

The von Neumann evolution equation for the density matrix and the Moyal equation for the Wigner function are mapped onto the evolution equation for the optical tomogram of the quantum state. The connection with the known evolution equation for the symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for optical tomograms. The classical Liouville equation for optical tomogram is obtained. An example of the parametric oscillator is considered in detail.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Mancini, V. I. Man’ko and P. Tombesi, Phys. Lett. A, 213, 1 (1996).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. S. Mancini, V. I. Man’ko, and P. Tombesi, Found. Phys., 27, 801 (1997).

    Article  ADS  MathSciNet  Google Scholar 

  3. A. Ibort, V. I. Man’ko, G. Marmo, et al., Phys. Scr., 79, 065013 (2009).

    Article  ADS  Google Scholar 

  4. M. A. Man’ko and V. I. Man’ko, Found. Phys., doi:10.1007/s10701-009-9403-9 (2009).

  5. J. Radon, Ber. Verh. Sachs. Akad., 69, 262 (1917).

    Google Scholar 

  6. J. Bertrand and P. Bertrand, Found. Phys., 17, 397 (1987).

    Article  ADS  MathSciNet  Google Scholar 

  7. K. Vogel and H. Risken, Phys. Rev. A, 40, 2847 (1989).

    Article  ADS  Google Scholar 

  8. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett., 70, 1244 (1993).

    Article  ADS  Google Scholar 

  9. A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys., 81, 299 (2009).

    Article  ADS  Google Scholar 

  10. S. Mancini, V. I. Man’ko, and P. Tombesi, Quantum Semiclass. Opt., 7, 615 (1995).

    Article  ADS  Google Scholar 

  11. G. M. D’Ariano, S. Mancini, V. I. Man’ko, and P. Tombesi, J. Opt. B: Quantum Semiclass. Opt., 8, 1017 (1996).

    Article  ADS  Google Scholar 

  12. O. V. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 18, 407 (1997).

    Article  Google Scholar 

  13. V. I. Man’ko and R. V. Mendes, Physica D, 145, 330 (2000).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  14. S. Mancini, O. V. Man’ko, V. I. Man’ko, and P. Tombesi, J. Phys. A: Math. Gen., 34, 3461 (2001).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin (1932).

    MATH  Google Scholar 

  16. J. E. Moyal, Proc. Cambridge Philos. Soc., 45, 99 (1949).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  17. E. Wigner, Phys. Rev., 40, 749, (1932).

    Article  MATH  ADS  Google Scholar 

  18. V. N. Chernega and V. I. Man’ko, J. Russ. Laser Res., 29, 43 (2008).

    Article  Google Scholar 

  19. A. S. Arkhipov and V. I. Man’ko, J. Russ. Laser Res., 25, 468 (2004).

    Article  Google Scholar 

  20. V. V. Dodonov, M. A. Marchiolli, Ya. A. Korennoy, et al., Phys. Scr., 58, 4087 (1998).

    Article  Google Scholar 

  21. A. Erdélyi (ed.), Bateman Manuscript Project: Higher Transcendental Functions, McGraw-Hill, New York (1953).

    Google Scholar 

  22. G. Szegö, Orthogonal Polynomials, American Mathematical Society, Providence, RI (1959).

    MATH  Google Scholar 

  23. I. A. Malkin and V. I. Man’ko, Phys. Lett. A, 31, 243 (1970).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yakov A. Korennoy.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Korennoy, Y.A., Man’ko, V.I. Probability representation of the quantum evolution and energy-level equations for optical tomograms. J Russ Laser Res 32, 74–85 (2011). https://doi.org/10.1007/s10946-011-9191-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10946-011-9191-5

Keywords

Navigation