Optical, symplectic and fresnel tomographies of quantum states

  • Margarita A. Man’ko
  • Sergio De Nicola
  • Renato Fedele
  • Vladimir I. Man’ko
Article

Abstract

The description of photon quantum states by means of probability-distribution functions (tomograms) of three different kinds (optical, symplectic and Fresnel ones) is presented. Mutual relations between the optical, symplectic and Fresnel tomograms are established. Evolution equation for states of Bose-Einstein condensate (Gross-Pitaevskii nonlinear equation) is given in the tomographic-probability representation. Entropy of solitons related to the Shannon entropy of the tomographic-probability representation is considered.

Keywords

quantum tomography symplectic tomography Fresnel tomography nonlinear Schrödinger equation solitons Gross-Pitaevskii equation Bose-Einstein condensates 

PACS

03.65.Wj 03.75.Lm 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Mancini, V.I. Man’ko and P. Tombesi, Quantum Semiclass. Opt. 7 (1995) 615.CrossRefADSGoogle Scholar
  2. 2.
    V.I. Man’ko and R.V. Mendes, Phys. Lett. A263 (1999) 53.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    M.A. Man’ko, J. Russ. Laser Res. 20 (1999) 225; ibid. 21 (2000) 411.Google Scholar
  4. 4.
    J. von Neumann, Mathematische Grundlagen der Quantummechanik, Springer, Berlin, 1932.Google Scholar
  5. 5.
    E.P. Wigner, Phys. Rev. 40 (1932) 749.CrossRefMATHADSGoogle Scholar
  6. 6.
    J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Berichte Sachsische Akademie der Wissenschaften, Leipzig, Math.-Physik. Klasse 69 (1917) 262.Google Scholar
  7. 7.
    J. Bertrand and P. Bertrand, Found. Phys. 17 (1987) 397.CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    K. Vogel and H. Risken, Phys. Rev. A40 (1989) 2847.CrossRefADSGoogle Scholar
  9. 9.
    S. Mancini, V.I. Man’ko and P. Tombesi, Phys. Lett. A213 (1996) 1; Found. Phys. 27 (1997) 801.CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    S. De Nicola, R. Fedele, M.A. Man’ko and V.I. Man’ko, J. Opt. B: Quantum Semiclass. Opt. 5 (2003) 95; Wigner picture and tomographic representation of envelope solitons, in Proc. Int. Workshop ‘Nonlinear Physics: Theory and Experiment. II’, eds. M.J. Ablowitz, M. Boiti and F. Pempinelli, Gallipoli, Lecce, Italy, June–July 2002, World Scientific, Singapore, 2003, p. 372; M.A. Man’ko, Soliton signals in tomographic representation, in Proc. Int. Conf. on Squeezed States and Uncertainty Relations, eds. H. Moya-Cessa, R. Jàuregui, S. Hacyan and O. Castaños, Puebla, Mexico, June 2003, Rinton Press, Princeton, NJ, 2003, p. 246.CrossRefADSGoogle Scholar
  11. 11.
    E.P. Gross, Nuovo Cim. 20 (1961) 454; L.P. Pitaevskii, Zh. Eksp. Teor. Fiz. 40 (1961) 646 [Sov. Phys. JETP 13 (1961) 451].CrossRefMATHGoogle Scholar
  12. 12.
    S. De Nicola, R. Fedele, M.A. Man’ko and V.I. Man’ko, Eur. Phys. J. B36 (2003) 385.ADSGoogle Scholar
  13. 13.
    S. De Nicola, R. Fedele, M.A. Man’ko and V.I. Man’ko, Tomographic analysis of envelope solitons: Concepts and applications, in Abstracts of the Int. Workshop ‘Optics in Computing’. International Optical Congress ‘Optics XXI Century’ (St. Petersburg, Russia, November 2002); Fresnel tomography: a novel approach to wave function reconstruction based on the Fresnel representation of tomograms, submitted to Theor. Math. Phys. (2004).Google Scholar
  14. 14.
    P. Lougovski, E. Solano, Z.M. Zhang, H. Walter, H. Mack and W.P. Schleich, Phys. Rev. Lett. 91 (2003) 010401–1; O. Crasser, H. Mack and W.P. Schleich, Is Fresnel optics quantum mechanics in phase space? quant-ph/0402115.CrossRefADSGoogle Scholar
  15. 15.
    V.I. Man’ko, L. Rosa and P. Vitale, Phys. Rev. A58 (1989) 3291.Google Scholar
  16. 16.
    M.G. Raymer, Contemp. Phys. 38 (1997) 343.CrossRefADSGoogle Scholar
  17. 17.
    S. Schiller, G. Breitenbach, S.F. Pereira, T. Mikker and J. Mlynek, Phys. Rev. Lett. 77 (1996) 2933.CrossRefADSGoogle Scholar
  18. 18.
    D.T. Smithey, M. Beck, M.G. Raymer and A. Faridani, Phys. Rev. Lett. 70 (1993) 1244.CrossRefADSGoogle Scholar
  19. 19.
    M.G. Raymer, M. Beck and D.F. McAlister, Phys. Rev. Lett. 72 (1994) 1137.CrossRefMATHADSMathSciNetGoogle Scholar
  20. 20.
    Z.P. Karkuszewski, K. Sacha and J. Zakrzewski, Phys. Rev. A 63 (2001) 061601(R).ADSGoogle Scholar
  21. 21.
    C.E. Shannon, Bell Tech. J. 27 (1848) 379.MathSciNetGoogle Scholar
  22. 22.
    O.V. Man’ko and V.I. Man’ko, J. Russ. Laser Res. 25 (2004) 115.CrossRefGoogle Scholar
  23. 23.
    M.A. Man’ko, J. Russ. Laser Res. 22 (2001) 168.CrossRefGoogle Scholar
  24. 24.
    S. De Nicola, R. Fedele, M.A. Man’ko and V.I. Man’ko, J. Russ. Laser Res. 25 (2004) 95.Google Scholar

Copyright information

© Akadémiai Kiadó 2004

Authors and Affiliations

  • Margarita A. Man’ko
    • 1
  • Sergio De Nicola
    • 2
  • Renato Fedele
    • 3
    • 4
  • Vladimir I. Man’ko
    • 1
  1. 1.P.N. Lebedev Physical InstituteMoscowRussia
  2. 2.Istituto di Cibernetica “Eduardo Caianiello” del CNR Comprensorio “A. Olivetti” Fabbr. 70Pozzuoli (NA)Italy
  3. 3.Dipartimento di Scienze FisicheUniversità “Federico II” di NapoliNapoliItaly
  4. 4.Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, ComplessoUniversitario di Monte Sant AngeloNapoliItaly

Personalised recommendations