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A transformational property of the Husimi function and its relation to the wigner function and symplectic tomograms

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Abstract

We consider the Husimi Q-functions, which are quantum quasiprobability distributions in the phase space, and investigate their transformation properties under a scale transformation (q, p) → (λq, λp). We prove a theorem that under this transformation, the Husimi function of a physical state is transformed into a function that is also a Husimi function of some physical state. Therefore, the scale transformation defines a positive map of density operators. We investigate the relation of Husimi functions to Wigner functions and symplectic tomograms and establish how they transform under the scale transformation. As an example, we consider the harmonic oscillator and show how its states transform under the scale transformation.

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References

  1. E. Wigner, Phys. Rev., 40, 749–759 (1932).

    Article  ADS  MATH  Google Scholar 

  2. V. I. Tatarskii, Sov. Phys. Usp., 26, 311–327 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  3. V. V. Dodonov and V. I. Man’ko, Invariants and Evolution of Nonstationary Quantum Systems [in Russian] (Proc. Lebedev Phys. Inst., Vol. 183, M. A. Markov, ed.), Nauka, Moscow (1987); English transl., Nova Science, New York (1989).

    Google Scholar 

  4. K. Husimi, Proc. Phys.-Math. Soc. Japan, 22, 264–314 (1940).

    MATH  Google Scholar 

  5. Y. J. Kano, J. Math. Phys., 6, 1913–1915 (1965).

    Article  ADS  MathSciNet  Google Scholar 

  6. R. J. Glauber, Phys. Rev. Lett., 10, 84–86 (1963).

    Article  ADS  MathSciNet  Google Scholar 

  7. E. C. G. Sudarshan, Phys. Rev. Lett., 10, No. 7, 277–279 (1963).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. R. L. Stratonovich, Sov. Phys. JETP, 4, 891–898 (1957); Sov. Phys. Dokl., 109, 72–75 (1956).

    MathSciNet  Google Scholar 

  9. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory [in Russian], IKI, Moscow (2003); English transl. prev. ed. (North-Holland Ser. Statist. Probab., Vol. 1), North-Holland, Amsterdam (1982).

    Google Scholar 

  10. N. N. Bogolyubov Jr. and A. S. Shumovskii, Proc. Steklov Inst. Math., 191, 87–97 (1992).

    Google Scholar 

  11. W. Pauli, in: Encyclopedia of Physics, Vol. 5, (S. Flügge, ed.), Springer, Berlin (1958), p. 17.

    Google Scholar 

  12. S. Mancini, V. I. Man’ko, and P. Tombesi, Quantum Semiclass. Opt., 7, 615–623 (1995); Phys. Lett. A, 213, 1–6 (1996); arXiv:quant-ph/9603002 (1996); Found. Phys., 27, 801–824 (1997); arXiv:quant-ph/9609026 (1996).

    Article  ADS  Google Scholar 

  13. A. Ibort, V. I. Man’ko, G. Marmo, A. Simoni, and F. Ventriglia, Phys. Scr., 79, 065013 (2009); arXiv:0904.4439 (2009).

    Article  ADS  Google Scholar 

  14. D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett., 70, 1244–1247 (1993).

    Article  ADS  Google Scholar 

  15. P. A. Belousov and R. S. Ismagilov, Theor. Math. Phys., 157, 1365–1369 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  16. R. F. O’Connell and E. P. Wigner, Phys. Lett. A, 83, 145–148 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  17. M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep., 106, 121–167 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  18. K. Takahashi and N. Saitō, Phys. Rev. Lett., 55, 645–648 (1985).

    Article  ADS  MathSciNet  Google Scholar 

  19. D. Biswas and S. Sinha, Phys. Rev. E, 60, 408–415 (1999); arXiv:chao-dyn/9904047 (1999).

    Article  ADS  MathSciNet  Google Scholar 

  20. F. J. Arranz, F. Borondo, and R. M. Benito, Phys. Rev. E, 54, 2458–2464 (1996); J. Chem. Phys., 120, 6516–6523 (2004).

    Article  ADS  Google Scholar 

  21. H. J. Korsch, C. Müller, and H. Wiescher, J. Phys. A, 30, L677–L684 (1997).

    Article  ADS  MATH  Google Scholar 

  22. D. M. Appleby, J. Modern Opt., 46, 825–841 (1999); arXiv:quant-ph/9805054 (1998).

    ADS  MathSciNet  MATH  Google Scholar 

  23. P. Roy, Optics Comm., 221, 145–152 (2003).

    Article  ADS  Google Scholar 

  24. É. A. Akhundova, V. V. Dodonov, and V. I. Man’ko, Theor. Math. Phys., 60, 907–913 (1984).

    Article  MathSciNet  Google Scholar 

  25. H.-Yi. Fan and Q. Guo, Phys. Lett. A, 358, 203–210 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. H.-Yi. Fan and Q. Guo, Ann. Physics, 232, 1502–1528 (2008).

    Article  ADS  Google Scholar 

  27. L. Shunlong, J. Stat. Phys., 102, 1417–1428 (2001).

    Article  MATH  Google Scholar 

  28. F. Olivares, F. Pennini, G. L. Ferri, and A. Plastino, Braz. J. Phys., 39, 503–506 (2009).

    Article  Google Scholar 

  29. D. M. Davidović, D. Lalović, and A. R. Tančić, J. Phys. A, 27, 8247–8250 (1994).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. G. S. Agarwal and K. Tara, Phys. Rev. A, 47, 3160–3166 (1993).

    Article  ADS  Google Scholar 

  31. D. M. Davidović and D. Lalović, J. Phys. A, 26, 5099–5105 (1993).

    Article  ADS  MathSciNet  Google Scholar 

  32. K. E. Cahill and R. J. Glauber, Phys. Rev., 177, 1857–1881, 1882–1902 (1969).

    Article  ADS  Google Scholar 

  33. G. Agarwal and E. Wolf, Phys. Rev. D, 2, 2161–2186 (1970).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. S. S. Mizrahi, Phys. A, 127, 241–264 (1984); 135, 237–250 (1986); 150, 541–554 (1988).

    Article  MathSciNet  Google Scholar 

  35. S. Mancini, O. V. Man’ko, V. I. Man’ko, and P. Tombesi, J. Phys. A, 34, 3461–3476 (2001); arXiv:quant-ph/0005058 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York (1953).

    Google Scholar 

  37. E. C. G. Sudarshan, P. M. Mathews, and J. Rau, Phys. Rev., 121, 920–924 (1961).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Lebedev Physical Institute, RAS, Moscow, Russia

    V. A. Andreev, V. I. Man’ko & M. A. Man’ko

  2. Vinca Institute of Nuclear Sciences, Belgrade, Serbia

    D. M. Davidović

  3. Institute of Physics, Belgrade, Serbia

    L. D. Davidović

  4. Faculty of Civil Engineering, Belgrade University, Belgrade, Serbia

    M. D. Davidović

Authors
  1. V. A. Andreev
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  2. D. M. Davidović
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  3. L. D. Davidović
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  4. M. D. Davidović
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  5. V. I. Man’ko
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  6. M. A. Man’ko
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Correspondence to V. A. Andreev.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 166, No. 3, pp. 410–424, March, 2011.

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Andreev, V.A., Davidović, D.M., Davidović, L.D. et al. A transformational property of the Husimi function and its relation to the wigner function and symplectic tomograms. Theor Math Phys 166, 356–368 (2011). https://doi.org/10.1007/s11232-011-0028-8

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  • DOI: https://doi.org/10.1007/s11232-011-0028-8

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