Abstract

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

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  1. J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
    [CrossRef]
  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).
    [CrossRef]
  4. A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
    [CrossRef]
  5. A. Royer, “Wigner function as the expectation value of the parity operator,” Phys. Rev. A 15, 449–450 (1977).
    [CrossRef]
  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).
    [CrossRef]
  8. J. C. Várilly, J. M. Garcia-Bondía, “The Moyal representation for spin,” Ann. Phys. (Paris) 190, 107–148 (1989);C. Brif, 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).
    [CrossRef]
  9. J. P. Dowling, G. S. Agarwal, 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).
    [CrossRef] [PubMed]
  10. K. B. Wolf, “Wigner distribution function for paraxial polychromatic optics,” Opt. Commun. 132, 343–352 (1996).
    [CrossRef]
  11. N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998);S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1822 (1999).
    [CrossRef]
  12. S. T. Ali, N. A. Atakishiyev, S. M. Chumakov, 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, K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101-1–034101-3 (2000).
    [CrossRef]
  14. L. M. Nieto, N. A. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
    [CrossRef]
  15. J.-P. Amiet, 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, V. K. Khersonskı̌, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
  17. F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
    [CrossRef]
  18. A. Frank, P. Vansacker, Algebraic Methods in Molecular and Nuclear Structure (Wiley Interscience, New York, 1994).
  19. E. Inönü, E. P. Wigner, “On the contraction of groups and their representations,” Proc. Natl. Acad. Sci. USA 39, 510–524 (1953).
    [CrossRef] [PubMed]
  20. N. Ya. Vilenkin, 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, G. W. Forbes, “Wigner function for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
    [CrossRef]

2000 (1)

S. M. Chumakov, A. B. Klimov, K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101-1–034101-3 (2000).
[CrossRef]

1999 (1)

1998 (2)

N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998);S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1822 (1999).
[CrossRef]

L. M. Nieto, N. A. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
[CrossRef]

1997 (1)

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

1996 (1)

K. B. Wolf, “Wigner distribution function for paraxial polychromatic optics,” Opt. Commun. 132, 343–352 (1996).
[CrossRef]

1994 (1)

J. P. Dowling, G. S. Agarwal, 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).
[CrossRef] [PubMed]

1989 (1)

J. C. Várilly, J. M. Garcia-Bondía, “The Moyal representation for spin,” Ann. Phys. (Paris) 190, 107–148 (1989);C. Brif, 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).
[CrossRef]

1981 (1)

G. S. Agarwal, “Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions,” Phys. Rev. A 24, 2889–2896 (1981).
[CrossRef]

1980 (1)

A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).

1977 (1)

A. Royer, “Wigner function as the expectation value of the parity operator,” Phys. Rev. A 15, 449–450 (1977).
[CrossRef]

1972 (1)

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
[CrossRef]

1957 (1)

R. L. Stratonovich, “On distribution in representation space,” Sov. Phys. JETP 4, 891–898 (1957) [J. Exp. Theor. Phys. 31, 1012–1020 (1956)].

1953 (1)

E. Inönü, E. P. Wigner, “On the contraction of groups and their representations,” Proc. Natl. Acad. Sci. USA 39, 510–524 (1953).
[CrossRef] [PubMed]

1949 (1)

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

1932 (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Agarwal, G. S.

J. P. Dowling, G. S. Agarwal, 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).
[CrossRef] [PubMed]

G. S. Agarwal, “Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions,” Phys. Rev. A 24, 2889–2896 (1981).
[CrossRef]

Ali, S. T.

S. T. Ali, N. A. Atakishiyev, S. M. Chumakov, K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. (to be published).

Alonso, M. A.

Amiet, J.-P.

J.-P. Amiet, S. Weigert, “Contracting the Wigner-kernel of a spin to the Wigner-kernel of a particle,” Phys. Rev. A (to be published).

Arecchi, F. T.

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
[CrossRef]

Atakishiyev, N. A.

L. M. Nieto, N. A. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
[CrossRef]

S. T. Ali, N. A. Atakishiyev, S. M. Chumakov, K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. (to be published).

Atakishiyev, N. M.

N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998);S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1822 (1999).
[CrossRef]

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Chumakov, S. M.

S. M. Chumakov, A. B. Klimov, K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101-1–034101-3 (2000).
[CrossRef]

L. M. Nieto, N. A. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
[CrossRef]

N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998);S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1822 (1999).
[CrossRef]

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

S. T. Ali, N. A. Atakishiyev, S. M. Chumakov, K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. (to be published).

Courtens, E.

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
[CrossRef]

Dowling, J. P.

J. P. Dowling, G. S. Agarwal, 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).
[CrossRef] [PubMed]

Forbes, G. W.

Frank, A.

A. Frank, P. Vansacker, Algebraic Methods in Molecular and Nuclear Structure (Wiley Interscience, New York, 1994).

Garcia-Bondía, J. M.

J. C. Várilly, J. M. Garcia-Bondía, “The Moyal representation for spin,” Ann. Phys. (Paris) 190, 107–148 (1989);C. Brif, 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).
[CrossRef]

Gilmore, R.

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
[CrossRef]

Inönü, E.

E. Inönü, E. P. Wigner, “On the contraction of groups and their representations,” Proc. Natl. Acad. Sci. USA 39, 510–524 (1953).
[CrossRef] [PubMed]

Khersonski?, V. K.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskı̌, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

Klimov, A. B.

S. M. Chumakov, A. B. Klimov, K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101-1–034101-3 (2000).
[CrossRef]

Klimyk, A. U.

N. Ya. Vilenkin, A. U. Klimyk, Representations of Lie Groups and Special Functions (Kluwer Academic, Dordrecht, The Netherlands, 1991), Vol. 1.

Lohmann, A.

A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).

Moskalev, A. N.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskı̌, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

Moyal, J. E.

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

Nieto, L. M.

L. M. Nieto, N. A. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
[CrossRef]

Rivera, A. L.

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Royer, A.

A. Royer, “Wigner function as the expectation value of the parity operator,” Phys. Rev. A 15, 449–450 (1977).
[CrossRef]

Schleich, W. P.

J. P. Dowling, G. S. Agarwal, 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).
[CrossRef] [PubMed]

Stratonovich, R. L.

R. L. Stratonovich, “On distribution in representation space,” Sov. Phys. JETP 4, 891–898 (1957) [J. Exp. Theor. Phys. 31, 1012–1020 (1956)].

Thomas, H.

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
[CrossRef]

Vansacker, P.

A. Frank, P. Vansacker, Algebraic Methods in Molecular and Nuclear Structure (Wiley Interscience, New York, 1994).

Várilly, J. C.

J. C. Várilly, J. M. Garcia-Bondía, “The Moyal representation for spin,” Ann. Phys. (Paris) 190, 107–148 (1989);C. Brif, 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).
[CrossRef]

Varshalovich, D. A.

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskı̌, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

Weigert, S.

J.-P. Amiet, S. Weigert, “Contracting the Wigner-kernel of a spin to the Wigner-kernel of a particle,” Phys. Rev. A (to be published).

Wigner, E. P.

E. Inönü, E. P. Wigner, “On the contraction of groups and their representations,” Proc. Natl. Acad. Sci. USA 39, 510–524 (1953).
[CrossRef] [PubMed]

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, K. B.

S. M. Chumakov, A. B. Klimov, K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101-1–034101-3 (2000).
[CrossRef]

K. B. Wolf, M. A. Alonso, G. W. Forbes, “Wigner function for Helmholtz wave fields,” J. Opt. Soc. Am. A 16, 2476–2487 (1999).
[CrossRef]

L. M. Nieto, N. A. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
[CrossRef]

N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998);S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1822 (1999).
[CrossRef]

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

K. B. Wolf, “Wigner distribution function for paraxial polychromatic optics,” Opt. Commun. 132, 343–352 (1996).
[CrossRef]

S. T. Ali, N. A. Atakishiyev, S. M. Chumakov, K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. (to be published).

Ya. Vilenkin, N.

N. Ya. Vilenkin, A. U. Klimyk, Representations of Lie Groups and Special Functions (Kluwer Academic, Dordrecht, The Netherlands, 1991), Vol. 1.

Ann. Phys. (Paris) (1)

J. C. Várilly, J. M. Garcia-Bondía, “The Moyal representation for spin,” Ann. Phys. (Paris) 190, 107–148 (1989);C. Brif, 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).
[CrossRef]

J. Math. Phys. (1)

N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for finite systems,” J. Math. Phys. 39, 6247–6261 (1998);S. M. Chumakov, A. Frank, K. B. Wolf, “Finite Kerr medium: macroscopic quantum superposition states and Wigner functions on the sphere,” Phys. Rev. A 60, 1817–1822 (1999).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. A (1)

L. M. Nieto, N. A. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Wigner distribution function for Euclidean systems,” J. Phys. A 31, 3875–3895 (1998).
[CrossRef]

Opt. Commun. (2)

K. B. Wolf, “Wigner distribution function for paraxial polychromatic optics,” Opt. Commun. 132, 343–352 (1996).
[CrossRef]

A. Lohmann, “The Wigner function and its optical production,” Opt. Commun. 42, 32–37 (1980).

Phys. Rev. (1)

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. A (6)

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K. B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase spaces,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

A. Royer, “Wigner function as the expectation value of the parity operator,” Phys. Rev. A 15, 449–450 (1977).
[CrossRef]

J. P. Dowling, G. S. Agarwal, 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).
[CrossRef] [PubMed]

G. S. Agarwal, “Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions,” Phys. Rev. A 24, 2889–2896 (1981).
[CrossRef]

S. M. Chumakov, A. B. Klimov, K. B. Wolf, “Connection between two Wigner functions for spin systems,” Phys. Rev. A 61, 034101-1–034101-3 (2000).
[CrossRef]

F. T. Arecchi, E. Courtens, R. Gilmore, H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211–2237 (1972).
[CrossRef]

Proc. Cambridge Philos. Soc. (1)

J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Cambridge Philos. Soc. 45, 99–124 (1949).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

E. Inönü, E. P. Wigner, “On the contraction of groups and their representations,” Proc. Natl. Acad. Sci. USA 39, 510–524 (1953).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

R. L. Stratonovich, “On distribution in representation space,” Sov. Phys. JETP 4, 891–898 (1957) [J. Exp. Theor. Phys. 31, 1012–1020 (1956)].

Other (5)

A. Frank, P. Vansacker, Algebraic Methods in Molecular and Nuclear Structure (Wiley Interscience, New York, 1994).

J.-P. Amiet, S. Weigert, “Contracting the Wigner-kernel of a spin to the Wigner-kernel of a particle,” Phys. Rev. A (to be published).

D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskı̌, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

S. T. Ali, N. A. Atakishiyev, S. M. Chumakov, K. B. Wolf, “The Wigner function for general Lie groups and the wavelet transform,” Ann. Inst. Henri Poincaré Phys. Theor. (to be published).

N. Ya. Vilenkin, A. U. Klimyk, Representations of Lie Groups and Special Functions (Kluwer Academic, Dordrecht, The Netherlands, 1991), Vol. 1.

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Equations (38)

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Wρ(θ, ϕ)=Tr(wˆρ),
wˆ(θ, ϕ)=4π2S+1 L=02SM=-LLYL,M(θ, ϕ)*TˆL,M(S).
Tr wˆ(θ, ϕ)=1,2S+14πS2dΩwˆ(θ, ϕ)=I.
TL,M(S)=2L+12S+1 m,m=-SSCS,m;L,MS,m|S, mS, m|,
2S+14πS2dΩWρ(θ, ϕ)WA(θ, ϕ)=Tr(ρA),
WA(θ, ϕ)=Tr(wˆA).
ρ=2S+14πS2dΩwˆ(θ, ϕ)Wρ(θ, ϕ),
exp(-iy·S)=2π2S+1L=02SM=-LL(-i)LχLS(ω)×YL,M(n)*TˆL,M(S),
χLS(w)=iLMexp(-iMω)CSM,L0SM.
02πdωχLS(ω)χLS(ω)=2πδL,L2S+12L+1.
wˆ(θ, ϕ)=02πdω exp(-iωn·S)f(ω),
f(ω)=12πL=02SiL2L+12S+1χLS(ω).
f(ω)(-1)S[δ(ω-π)-(i/S)δ(ω-π)],
S,
Wρ(θ, ϕ)=2π2S+1L=02SΩL,S×M=-LLYL,M(θ, ϕ)* Tr[TˆL,M(S)ρ]
f(ω)=12πL=02SiL2L+12S+1χLS(ω)ΩL,S.
S+=2Sa,S-=2Sa,Sz=N-S,
[N, a]=a,[N, a]=-a,[a, a]=1-N/S.
N=aa+N2-N2S.
wˆ(θ, ϕ)(-1)S02πdω expiωS2[a exp(-iφ)+a exp(iφ)]sin θ+(N-S)cos θ×[δ(ω-π)-(i/S)δ(ω-π)].
wˆ(α)=2 exp[iπ(a+α*)(a+α)],
Tˆ(φ,ϑ,ψ)=exp(-iφSz)exp(-iϑSx)exp(-iψSz)=cos φ cos ψ-sin φ cos ϑ sin ψ-cos φ sin ψ-sin φ cos ϑ cos ψsin ϑ sin φsin φ cos ψ+cos φ cos ϑ sin ψ-sin φ sin ψ+cos φ cos ϑ cos ψ-sin ϑ cos φsin ϑ sin ψsin ϑ cos ψcos ϑ,
ϑ=r/R,rfinite.
Tˆ(φ, r/R, ψ)xyREˆ(α, r, φ)xyR,
Eˆ(α, r, ϕ)=cos α-sin αrRsin φsin αcos α-rRcos φ001.
Δ=2ϑ2+cot ϑϑ+1sin2 ϑ2φ2R22x2+2y2).
cosω2=cosϑ2cosφ+ψ2,
ωφ+ψ=α.
Ex=Sx/R,Ey=Sy/R,SzSz,
[Sz, Ex]=iEy,[Ey, Sz]=iEx,[Ex, Ey]=0.
Sz|n=n|n,n=-,, .
wˆ(r, φ)=(-1)S exp(-iφSz) exp(-irEx)exp(-iφSz)×exp(-iπSz)=(-1)S exp[-ir(Ex cos φ+Ey sin φ)]×exp(-iπSz).
E±=Ex±iEy,E±|n=k|n±1,
exp(-iπSz)E± exp(-iπSz)=-E±
wˆ(r, φ)=(-1)SD(z/2)exp(-iπSz)D(z/2)=wˆ(r, φ),
D(z)=exp[-i(z*E++zE-)],D(z)=D(-z),
z=(r/2)exp(iφ).
wˆ(r, φ)=exp[-iπSz+(π/2)(zE+-z*E-)].

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