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Nonindependent splittings and Gibbs states

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Abstract

We discuss a nonindependent (beam) splitting for which the related thinning leaves the class of equilibrium states for a one mode electromagnetic field invariant. The thinning affects only the parameters of the state, showing a nonlinear loss of energy. After the splitting, the energy values of both split parts are independent. This independence is a characteristic property of the geometric distribution, the distribution of energy values in the equilibrium state. Also, we observe that the class of states where the full states of the split parts are independent is formed by the so-called phase states.

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References

  1. L. D. Landau and E. M. Lifschitz,Lehrbuch der Theoretischen Physik. Bd. V. Statistische Physik Teil 1, 2nd edition, Akademie-Verlag, Berlin (1984).

    Google Scholar 

  2. R. Lenk,Einführung in die Statistische Physik, VEB Deutscher Verlag der Wissenschaften, Berlin (1978).

    Google Scholar 

  3. K.-H. Fichtner, “Charakterisierung Poissonscher zufälliger Punktfolgen und infinitesimale Verdünnungsschemata,”Math. Nachr. 68, 93–104 (1975).

    MATH  MathSciNet  Google Scholar 

  4. R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) symmetry and photon statistics,”Phys. Rev. A (3),40, No. 3, 1371–1384 (1989).

    Google Scholar 

  5. J. R. Jeffers, N. Imoto, and R. Loudon, “Quantum optics of travelling-wave attenuators and amplifiers,”Phys. Rev. A (3),47, No. 4, 3346–3359 (1993).

    Google Scholar 

  6. K.-H. Fichtner, W. Freudenberg, and V. Liebscher,Beam Splittings and Time Evolutions of Boson Systems, Preprint Math/Inf/96/39, Forschungsergebnisse der Fakultät für Mathematik und Informatik, Jena (1996).

    Google Scholar 

  7. W. Freudenberg, “On a class of quantum Markov chains on the Fock space,” in:Quantum Probability & Related Topics (L. Accardi, editor), Vol. XI, World Sci. Publishing, Singapore (1994), pp. 215–237.

    Google Scholar 

  8. K.-H. Fichtner and W. Freudenberg, “Point processes and the position distribution of infinite boson systems,”J. Statist. Phys.,47, 959–978 (1987).

    Article  MathSciNet  Google Scholar 

  9. A. Vourdas, “Phase States: An analytical approach in the unit disc,”Phys. Scripta,48, 84–86 (1993).

    Google Scholar 

  10. A. M. Perelomov,Generalized Coherent States and Their Applications, Texts Monographs Phys., Springer, Berlin-Heidelberg-New York (1986).

    Google Scholar 

  11. G. G. Emch,Mathematical and Conceptual Foundations of 20th Century Physics, Vol. 100, North-Holland Math. Stud, North-Holland, Amsterdam-New York-Oxford (1984).

    Google Scholar 

  12. K. R. Parthasarathy,An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel-Boston-Berlin (1992).

    Google Scholar 

  13. K.-H. Fichtner and W. Freudenberg, “Characterization of states of infinite boson systems. I. On the construction of states of boson systems,”Comm. Math. Phys.,137, 315–357 (1991).

    MathSciNet  Google Scholar 

  14. T. S. Ferguson, “A characterization of the geometric distribution,”Amer. Math. Monthly,72, 256–260 (1965).

    MATH  MathSciNet  Google Scholar 

  15. K.-H. Fichtner, W. Freudenberg, and V. Liebscher, “Characterization of coherent and mixed coherent states by beam splittings,” (to appear).

  16. V. Liebscher,Characterization of Coherent and Mixed Coherent States by Beam Splittings, Preprint Math/Inf/96/16, Forschungsergebnisse der Fakultät für Mathematik und Informatik, Jena (1996).

    Google Scholar 

  17. U. M. Titulaer and R. J. Glauber, “Density operators for coherent fields,”Phys. Rev.,145, No. 4, 1041–1050 (1966).

    Google Scholar 

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Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 598–605, October, 1998.

This research was partially supported by the International Science Foundation under grant No. 96-0698

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Fichtner, K.H., Freudenberg, W. & Liebscher, V. Nonindependent splittings and Gibbs states. Math Notes 64, 518–523 (1998). https://doi.org/10.1007/BF02314634

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