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Orthogonal Operators and Phase Space Distributions in Quantum Optics

  • S. Billis
  • E. A. Mishkin
Conference paper

Abstract

Electromagnetic fields at optical frequencies are excited by indeterministic sources. Their statistical description is usually given by means of the density operator ρ̂ or a phase-space distribution which is, in general, a quasi-probability distribution. Expansions of the density matrix ρ̂ in terms of a given complete set of orthogonal operators establishes a one-to-one correspondence between the expanded operator ρ̂ and the quasi-probability phase-space distribution that appears in said expansion. The time evolution of the field’s statistics are obtained then as a solution of the equation of motion of the density operator ρ̂ or the differential equation of the corresponding phase-space distribution. The latter method has the advantage of being unburdened by problems of commutativity associated with the ρ̂ operator. With the statistical information vested in the phase-space distribution, in lieu of the density matrix ρ̂ the expected value of an observable F̂ is given by an integral of the phase-space distribution multiplied by a weighting function which is representative of the observable F̂. This is essentially the method first introduced by Wigner[l] and Moyal[2].

Keywords

Coherent State Density Operator Reduce Density Matrix Interaction Picture Phase Space Distribution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    E. P. Wigner, Phys. Rev. 40, 749 (1932).ADSMATHCrossRefGoogle Scholar
  2. 2.
    J. E. Moyal, Proc. Camb. Phil. Soc. 45, 99 (1948) and 45, 545 (1949).MathSciNetMATHGoogle Scholar
  3. 3.
    H. Weyl, The Theory of Groups and Quantum Mechanics ( Dover, New York, 1931 ), pp. 274–277.MATHGoogle Scholar
  4. 4.
    J. C. T. Pool, J. Math. Phys. 7, 66 (1966).MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    K. E. Cahill and R. J. Glauber, Phys. Rev. 177, 1857 (1969); 179, 1882 (1969).ADSCrossRefGoogle Scholar
  6. 6.
    I.N. Herstein, Topics in Algebra ( Blaisdell Publishing Co. Massachusetts, 1964 ), pp. 48–50.MATHGoogle Scholar
  7. 7.
    R. J. Glauber, Phys. Rev. Letters 10, 84 (1963); Phys. Rev. 131, 2766 (1963); J. R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics ( Benjamin, New York, 1968 ).MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    J. R. Klauder, J. Math. Phys. 4, 1055 (1963); 5, 177 (1964).MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    W. H. Louisell, Radiation and Noise in Quantum Electronics, (McGraw-Hill Book Co., 1964), Chap. 3.Google Scholar
  10. 10.
    Contents of this section are based on the work of P. P. Betrand, Ph.D. thesis, Polytechnic Institute of Brooklyn, 1969, and K. Moy, M.Sc. thesis, Polytechnic Institute of Brooklyn, 1968. See also Ref. 11, 12.Google Scholar
  11. 11.
    P. P. Betrand and E. A. Mishkin, Bull. Am. Phys. Soc. 15, 89 Jan (1970).Google Scholar
  12. 12.
    P. P. Betrand, K. Moy and E. A. Mishkin, Phys. Rev. 4, 1909 (1971).ADSGoogle Scholar
  13. 13.
    A. E. Glassgold and D. Holliday, Phys. Rev. 139, A1717 (1965).ADSCrossRefGoogle Scholar
  14. 14.
    P. P. Betrand and E. A. Mishkin, Phys. Letters 25A, 204 (1967).ADSCrossRefGoogle Scholar
  15. 15.
    M. M. Miller and E. A. Mishkin, Phys. Letters 24A, 188 (1967).Google Scholar
  16. 16.
    For an analysis of these constraints see Ref. 5.Google Scholar
  17. 17.
    R. J. Glauber, Phys. Rev. 131, 2766 (1963).MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    P. Kelly and W. H. Kluner, Phys. Rev. 136, 316 (1964).MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    R. J. Glauber, Phys. Rev. 130, 2529 (1963).MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    R. J. Glauber, Quantum Optics and Electronics (Les Houches, 1964 ) edited by C. de Witt, et al., p. 63, ( Gordon and Breach, New York, 1965 ).Google Scholar
  21. 21.
    E. C. B. Sudarshan, Phys. Rev. Letters 10, 277 (1963).MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    R. Bonifacio, L. M. Narducci, and E. Montaldi, Phys. Rev. Letters 16, 1125 (1966); Nuovo Cimento 47, 890 (1967).CrossRefGoogle Scholar
  23. 23.
    M. M. Miller and E. A. Mishkin, Phys. Rev. 164, 1610 (1967).MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    W. H. Louisell, Quantum Optics, (Proceedings of the Inter-national School of Physics “Enrico Fermi”, 1967 ), edited by R. J. Glauber ( Academic Press, New York, 1969 ).Google Scholar

Copyright information

© Plenum Press, New York 1973

Authors and Affiliations

  • S. Billis
    • 1
  • E. A. Mishkin
    • 1
  1. 1.Polytechnic Institute of BrooklynBrooklynUSA

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