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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 12 — Dec. 1, 2000
  • pp: 2315–2318

Quasi-probability distributions for the simplest dynamical groups

A. B. Klimov and S. M. Chumakov  »View Author Affiliations

JOSA A, Vol. 17, Issue 12, pp. 2315-2318 (2000)

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We prove that the Wigner–Stratonovich–Agarwal operator that defines the quasi-probability distribution on the sphere [for the SU(2) dynamical group] can be written as an integral of the SU(2) (irreducible unitary) representation element with respect to a single variable that labels the orbits in the coadjoint representation. This allows us to consider contractions of the SU(2) quasi-probability distribution to the cases of the Heisenberg–Weyl group and the two-dimensional Euclidean group.

© 2000 Optical Society of America

OCIS Codes
(000.1600) General : Classical and quantum physics

A. B. Klimov and S. M. Chumakov, "Quasi-probability distributions for the simplest dynamical groups," J. Opt. Soc. Am. A 17, 2315-2318 (2000)

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  1. J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
  2. A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).
  3. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
  4. A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
  5. A. Royer, “Wigner function as the expectation value of the parity operator,” Phys. Rev. A 15, 449–450 (1977).
  6. R. L. Stratonovich, “On distribution in representation space,” Sov. Phys. JETP 4, 891–898 (1957) [ J. Exp. Theor. Phys. 31, 1012–1020 (1956)].
  7. G. S. Agarwal, “Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions,” Phys. Rev. A 24, 2889–2896 (1981).
  8. J. C. Várilly and J. M. Garcia-Bondía, “The Moyal representation for spin,” Ann. Phys. (Paris) 190, 107–148 (1989); C. Brif and A. Mann, “Phase space formulation of quantum mechanics and quantum state reconstruction for physical systems with Lie-group symmetries,” Phys. Rev. A 59, 971–987 (1999).
  9. J. P. Dowling, G. S. Agarwal, and W. P. Schleich, “Wigner distribution of a general angular momentum state: application to a collection of two-level atoms,” Phys. Rev. A 49, 4101–4109 (1994).
  10. K. B. Wolf, “Wigner distribution function for paraxial polychromatic optics,” Opt. Commun. 132, 343–352 (1996).
  11. N. M. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998); S. M. Chumakov, A. Frank, and K. B. Wolf, “Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1822 (1999).
  12. S. T. Ali, N. A. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. (to be published).
  13. S. M. Chumakov, A. B. Klimov, and K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101–1–034101–3 (2000).
  14. L. M. Nieto, N. A. Atakishiyev, S. M. Chumakov, and K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
  15. J.-P. Amiet and S. Weigert, “Contracting the Wigner-kernel of a spin to the Wigner-kernel of a particle,” Phys. Rev. A (to be published).
  16. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskǐ, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
  17. F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
  18. A. Frank and P. Vansacker, Algebraic Methods in Molecular and Nuclear Structure (Wiley Interscience, New York, 1994).
  19. E. Inönü and E. P. Wigner, “On the contraction of groups and their representations,” Proc. Natl. Acad. Sci. USA 39, 510–524 (1953).
  20. N. Ya. Vilenkin and A. U. Klimyk, Representations of Lie Groups and Special Functions (Kluwer Academic, Dordrecht, The Netherlands, 1991), Vol. 1.
  21. K. B. Wolf, M. A. Alonso, and G. W. Forbes, “Wigner function for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).

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