Abstract

The theory of generalized coherent states associated with the SU(2) and SU(1,1) Lie algebras is discussed and applied in an investigation of possible reductions of fluctuations. In the framework of a system of N two-level atoms the squeezing of angular-momentum [SU(2)] fluctuations is exhibited for optical coherent transients involving the photon echo. The SU(1,1) fluctuations are discussed and established for general two-photon processes involving dynamical variables different from the creation and annihilation operators.

© 1985 Optical Society of America

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  1. D. Stoler, Phys. Rev. D 1, 3217 (1970);Phys. Rev. D 4,1925 (1971).
    [Crossref]
  2. J. M. Radcliffe, J. Phys. (Paris) A 4, 313 (1971).
  3. F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972).
    [Crossref]
  4. A. M. Perelomov, Commun. Math. Phys. 26, 222 (1972).
    [Crossref]
  5. For the review article, see A. M. Perelomov, Usp. Fiz. Nauk 123, 23 (1977) [Sov. Phys. Usp. 20, 703 (1977)].
    [Crossref]
  6. A. O. Barut and L. Girardello, Commun. Math. Phys. 21, 41 (1971);D. Bhaumik, T. Nag, and B. Dutta-Roy, J. Phys. (Paris) A 8, 1868 (1975).
    [Crossref]
  7. M. M. Nieto and L. M. Simmons, Phys. Rev. Lett. 41, 207 (1978);Phys. Rev. D 20, 1321, 1332, 1342 (1979);Phys. Rev. A 19, 438 (1979).
    [Crossref]
  8. C. Aragone, G. Guerri, S. Salamo, and J. L. Tani, J. Phys. (Paris) A 15, L149 (1974);C. Aragone, E. Chalbaud, and S. Salamo, J. Math. Phys. 17, 1963 (1976).
    [Crossref]
  9. M. A. Rashid, J. Math. Phys. 19, 1391, 1397 (1978).
    [Crossref]
  10. I. Bialynicki-Birula, Ann. Phys. (NY) 67, 252 (1971);J. Mostowski, Lett. Math. Phys. 2, 1 (1977).
    [Crossref]
  11. J. R. Klauder, Phys. Rev. D 19, 2349 (1979);H. Kuratsuji and T. Suzuki, J. Math. Phys. 21, 472 (1980);C. C. Gerry and S. Silverman, J. Math. Phys. 23, 1995 (1982).
    [Crossref]
  12. F. A. Berezin, Commun. Math. Phys. 40, 153 (1975).
    [Crossref]
  13. E. Y. C. Lu, Nuovo Cimento Lett. 2, 1241 (1971);Nuovo Cimento Lett. 3, 585 (1972).
    [Crossref]
  14. H. P. Yuen, Phys. Lett. 51A, 1 (1975),Phys. Rev. A 13, 2226 (1976).
  15. H. N. Hollenhort, Phys. Rev. D 19, 1669 (1979).
    [Crossref]
  16. H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
    [Crossref]
  17. See for example, C. M. Caves, Phys. Rev. D 23, 1693 (1981);Phys. Rev. D 26, 1817 (1982).
    [Crossref]
  18. W. Becker, M. O. Scully, and M. S. Zubairy, Phys. Rev. Lett. 48, 475 (1982);L. A. Lugiato and G. Stinni, Opt. Commun. 41, 67 (1982);P. Meystre and M. S. Zubairy, Phys. Lett. 89A, 390 (1982).
    [Crossref]
  19. L. Mandel, Phys. Rev. Lett. 49, 136 (1982).
    [Crossref]
  20. D. F. Walls and P. Zoller, Phys. Rev. Lett. 47, 709 (1981).
    [Crossref]
  21. For all the required information about representation of the SU(2) and SU(1,1) Lie algebras see, for example, A. O. Barut and R. Ra̧czka, Theory of Group Representation and Applications (Polish Scientific Publishers, Warsaw, 1977).
  22. K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), Chap. 24.
  23. See, for example, L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).
  24. R. H. Dicke, Phys. Rev. 93, 99 (1954).
    [Crossref]
  25. K. Wódkiewicz, Opt. Commun. 51, 198 (1984).
    [Crossref]
  26. See Ref. 23, Chap. 9, or A. L. Bloom, Phys. Rev.98, 1105 (1955).
    [Crossref]
  27. D. Stoler, Phys. Rev. Lett. 23, 1397 (1974);G. Milburn and D. F. Walls, Opt. Commun. 39, 401 (1981);K. Wódkiewicz and M. S. Zubairy, Phys. Rev. A 27, 2003 (1983).
    [Crossref]
  28. G. Milburn, J. Phys. (Paris) A 17, 737 (1984).
  29. See for example, W. Witschel, J. Phys. (Paris) A 7, 1847 (1974);M. A. M. Santiago and A. N. Vaidya, J. Phys. (Paris A 9, 897 (1976).

1984 (2)

K. Wódkiewicz, Opt. Commun. 51, 198 (1984).
[Crossref]

G. Milburn, J. Phys. (Paris) A 17, 737 (1984).

1982 (2)

W. Becker, M. O. Scully, and M. S. Zubairy, Phys. Rev. Lett. 48, 475 (1982);L. A. Lugiato and G. Stinni, Opt. Commun. 41, 67 (1982);P. Meystre and M. S. Zubairy, Phys. Lett. 89A, 390 (1982).
[Crossref]

L. Mandel, Phys. Rev. Lett. 49, 136 (1982).
[Crossref]

1981 (2)

D. F. Walls and P. Zoller, Phys. Rev. Lett. 47, 709 (1981).
[Crossref]

See for example, C. M. Caves, Phys. Rev. D 23, 1693 (1981);Phys. Rev. D 26, 1817 (1982).
[Crossref]

1980 (1)

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[Crossref]

1979 (2)

H. N. Hollenhort, Phys. Rev. D 19, 1669 (1979).
[Crossref]

J. R. Klauder, Phys. Rev. D 19, 2349 (1979);H. Kuratsuji and T. Suzuki, J. Math. Phys. 21, 472 (1980);C. C. Gerry and S. Silverman, J. Math. Phys. 23, 1995 (1982).
[Crossref]

1978 (2)

M. A. Rashid, J. Math. Phys. 19, 1391, 1397 (1978).
[Crossref]

M. M. Nieto and L. M. Simmons, Phys. Rev. Lett. 41, 207 (1978);Phys. Rev. D 20, 1321, 1332, 1342 (1979);Phys. Rev. A 19, 438 (1979).
[Crossref]

1977 (1)

For the review article, see A. M. Perelomov, Usp. Fiz. Nauk 123, 23 (1977) [Sov. Phys. Usp. 20, 703 (1977)].
[Crossref]

1975 (2)

F. A. Berezin, Commun. Math. Phys. 40, 153 (1975).
[Crossref]

H. P. Yuen, Phys. Lett. 51A, 1 (1975),Phys. Rev. A 13, 2226 (1976).

1974 (3)

C. Aragone, G. Guerri, S. Salamo, and J. L. Tani, J. Phys. (Paris) A 15, L149 (1974);C. Aragone, E. Chalbaud, and S. Salamo, J. Math. Phys. 17, 1963 (1976).
[Crossref]

See for example, W. Witschel, J. Phys. (Paris) A 7, 1847 (1974);M. A. M. Santiago and A. N. Vaidya, J. Phys. (Paris A 9, 897 (1976).

D. Stoler, Phys. Rev. Lett. 23, 1397 (1974);G. Milburn and D. F. Walls, Opt. Commun. 39, 401 (1981);K. Wódkiewicz and M. S. Zubairy, Phys. Rev. A 27, 2003 (1983).
[Crossref]

1972 (2)

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972).
[Crossref]

A. M. Perelomov, Commun. Math. Phys. 26, 222 (1972).
[Crossref]

1971 (4)

J. M. Radcliffe, J. Phys. (Paris) A 4, 313 (1971).

A. O. Barut and L. Girardello, Commun. Math. Phys. 21, 41 (1971);D. Bhaumik, T. Nag, and B. Dutta-Roy, J. Phys. (Paris) A 8, 1868 (1975).
[Crossref]

E. Y. C. Lu, Nuovo Cimento Lett. 2, 1241 (1971);Nuovo Cimento Lett. 3, 585 (1972).
[Crossref]

I. Bialynicki-Birula, Ann. Phys. (NY) 67, 252 (1971);J. Mostowski, Lett. Math. Phys. 2, 1 (1977).
[Crossref]

1970 (1)

D. Stoler, Phys. Rev. D 1, 3217 (1970);Phys. Rev. D 4,1925 (1971).
[Crossref]

1954 (1)

R. H. Dicke, Phys. Rev. 93, 99 (1954).
[Crossref]

Allen, L.

See, for example, L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Aragone, C.

C. Aragone, G. Guerri, S. Salamo, and J. L. Tani, J. Phys. (Paris) A 15, L149 (1974);C. Aragone, E. Chalbaud, and S. Salamo, J. Math. Phys. 17, 1963 (1976).
[Crossref]

Arecchi, F. T.

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972).
[Crossref]

Barut, A. O.

A. O. Barut and L. Girardello, Commun. Math. Phys. 21, 41 (1971);D. Bhaumik, T. Nag, and B. Dutta-Roy, J. Phys. (Paris) A 8, 1868 (1975).
[Crossref]

For all the required information about representation of the SU(2) and SU(1,1) Lie algebras see, for example, A. O. Barut and R. Ra̧czka, Theory of Group Representation and Applications (Polish Scientific Publishers, Warsaw, 1977).

Becker, W.

W. Becker, M. O. Scully, and M. S. Zubairy, Phys. Rev. Lett. 48, 475 (1982);L. A. Lugiato and G. Stinni, Opt. Commun. 41, 67 (1982);P. Meystre and M. S. Zubairy, Phys. Lett. 89A, 390 (1982).
[Crossref]

Berezin, F. A.

F. A. Berezin, Commun. Math. Phys. 40, 153 (1975).
[Crossref]

Bialynicki-Birula, I.

I. Bialynicki-Birula, Ann. Phys. (NY) 67, 252 (1971);J. Mostowski, Lett. Math. Phys. 2, 1 (1977).
[Crossref]

Bloom, A. L.

See Ref. 23, Chap. 9, or A. L. Bloom, Phys. Rev.98, 1105 (1955).
[Crossref]

Caves, C. M.

See for example, C. M. Caves, Phys. Rev. D 23, 1693 (1981);Phys. Rev. D 26, 1817 (1982).
[Crossref]

Courtens, E.

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972).
[Crossref]

Dicke, R. H.

R. H. Dicke, Phys. Rev. 93, 99 (1954).
[Crossref]

Eberly, J. H.

See, for example, L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

Gilmore, R.

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972).
[Crossref]

Girardello, L.

A. O. Barut and L. Girardello, Commun. Math. Phys. 21, 41 (1971);D. Bhaumik, T. Nag, and B. Dutta-Roy, J. Phys. (Paris) A 8, 1868 (1975).
[Crossref]

Gottfried, K.

K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), Chap. 24.

Guerri, G.

C. Aragone, G. Guerri, S. Salamo, and J. L. Tani, J. Phys. (Paris) A 15, L149 (1974);C. Aragone, E. Chalbaud, and S. Salamo, J. Math. Phys. 17, 1963 (1976).
[Crossref]

Hollenhort, H. N.

H. N. Hollenhort, Phys. Rev. D 19, 1669 (1979).
[Crossref]

Klauder, J. R.

J. R. Klauder, Phys. Rev. D 19, 2349 (1979);H. Kuratsuji and T. Suzuki, J. Math. Phys. 21, 472 (1980);C. C. Gerry and S. Silverman, J. Math. Phys. 23, 1995 (1982).
[Crossref]

Lu, E. Y. C.

E. Y. C. Lu, Nuovo Cimento Lett. 2, 1241 (1971);Nuovo Cimento Lett. 3, 585 (1972).
[Crossref]

Mandel, L.

L. Mandel, Phys. Rev. Lett. 49, 136 (1982).
[Crossref]

Milburn, G.

G. Milburn, J. Phys. (Paris) A 17, 737 (1984).

Nieto, M. M.

M. M. Nieto and L. M. Simmons, Phys. Rev. Lett. 41, 207 (1978);Phys. Rev. D 20, 1321, 1332, 1342 (1979);Phys. Rev. A 19, 438 (1979).
[Crossref]

Perelomov, A. M.

For the review article, see A. M. Perelomov, Usp. Fiz. Nauk 123, 23 (1977) [Sov. Phys. Usp. 20, 703 (1977)].
[Crossref]

A. M. Perelomov, Commun. Math. Phys. 26, 222 (1972).
[Crossref]

Ra¸czka, R.

For all the required information about representation of the SU(2) and SU(1,1) Lie algebras see, for example, A. O. Barut and R. Ra̧czka, Theory of Group Representation and Applications (Polish Scientific Publishers, Warsaw, 1977).

Radcliffe, J. M.

J. M. Radcliffe, J. Phys. (Paris) A 4, 313 (1971).

Rashid, M. A.

M. A. Rashid, J. Math. Phys. 19, 1391, 1397 (1978).
[Crossref]

Salamo, S.

C. Aragone, G. Guerri, S. Salamo, and J. L. Tani, J. Phys. (Paris) A 15, L149 (1974);C. Aragone, E. Chalbaud, and S. Salamo, J. Math. Phys. 17, 1963 (1976).
[Crossref]

Scully, M. O.

W. Becker, M. O. Scully, and M. S. Zubairy, Phys. Rev. Lett. 48, 475 (1982);L. A. Lugiato and G. Stinni, Opt. Commun. 41, 67 (1982);P. Meystre and M. S. Zubairy, Phys. Lett. 89A, 390 (1982).
[Crossref]

Shapiro, J. H.

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[Crossref]

Simmons, L. M.

M. M. Nieto and L. M. Simmons, Phys. Rev. Lett. 41, 207 (1978);Phys. Rev. D 20, 1321, 1332, 1342 (1979);Phys. Rev. A 19, 438 (1979).
[Crossref]

Stoler, D.

D. Stoler, Phys. Rev. Lett. 23, 1397 (1974);G. Milburn and D. F. Walls, Opt. Commun. 39, 401 (1981);K. Wódkiewicz and M. S. Zubairy, Phys. Rev. A 27, 2003 (1983).
[Crossref]

D. Stoler, Phys. Rev. D 1, 3217 (1970);Phys. Rev. D 4,1925 (1971).
[Crossref]

Tani, J. L.

C. Aragone, G. Guerri, S. Salamo, and J. L. Tani, J. Phys. (Paris) A 15, L149 (1974);C. Aragone, E. Chalbaud, and S. Salamo, J. Math. Phys. 17, 1963 (1976).
[Crossref]

Thomas, H.

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972).
[Crossref]

Walls, D. F.

D. F. Walls and P. Zoller, Phys. Rev. Lett. 47, 709 (1981).
[Crossref]

Witschel, W.

See for example, W. Witschel, J. Phys. (Paris) A 7, 1847 (1974);M. A. M. Santiago and A. N. Vaidya, J. Phys. (Paris A 9, 897 (1976).

Wódkiewicz, K.

K. Wódkiewicz, Opt. Commun. 51, 198 (1984).
[Crossref]

Yuen, H. P.

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[Crossref]

H. P. Yuen, Phys. Lett. 51A, 1 (1975),Phys. Rev. A 13, 2226 (1976).

Zoller, P.

D. F. Walls and P. Zoller, Phys. Rev. Lett. 47, 709 (1981).
[Crossref]

Zubairy, M. S.

W. Becker, M. O. Scully, and M. S. Zubairy, Phys. Rev. Lett. 48, 475 (1982);L. A. Lugiato and G. Stinni, Opt. Commun. 41, 67 (1982);P. Meystre and M. S. Zubairy, Phys. Lett. 89A, 390 (1982).
[Crossref]

Ann. Phys. (NY) (1)

I. Bialynicki-Birula, Ann. Phys. (NY) 67, 252 (1971);J. Mostowski, Lett. Math. Phys. 2, 1 (1977).
[Crossref]

Commun. Math. Phys. (3)

F. A. Berezin, Commun. Math. Phys. 40, 153 (1975).
[Crossref]

A. M. Perelomov, Commun. Math. Phys. 26, 222 (1972).
[Crossref]

A. O. Barut and L. Girardello, Commun. Math. Phys. 21, 41 (1971);D. Bhaumik, T. Nag, and B. Dutta-Roy, J. Phys. (Paris) A 8, 1868 (1975).
[Crossref]

IEEE Trans. Inf. Theory (1)

H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-26, 78 (1980).
[Crossref]

J. Math. Phys. (1)

M. A. Rashid, J. Math. Phys. 19, 1391, 1397 (1978).
[Crossref]

J. Phys. (Paris) A (4)

G. Milburn, J. Phys. (Paris) A 17, 737 (1984).

See for example, W. Witschel, J. Phys. (Paris) A 7, 1847 (1974);M. A. M. Santiago and A. N. Vaidya, J. Phys. (Paris A 9, 897 (1976).

C. Aragone, G. Guerri, S. Salamo, and J. L. Tani, J. Phys. (Paris) A 15, L149 (1974);C. Aragone, E. Chalbaud, and S. Salamo, J. Math. Phys. 17, 1963 (1976).
[Crossref]

J. M. Radcliffe, J. Phys. (Paris) A 4, 313 (1971).

Nuovo Cimento Lett. (1)

E. Y. C. Lu, Nuovo Cimento Lett. 2, 1241 (1971);Nuovo Cimento Lett. 3, 585 (1972).
[Crossref]

Opt. Commun. (1)

K. Wódkiewicz, Opt. Commun. 51, 198 (1984).
[Crossref]

Phys. Lett. (1)

H. P. Yuen, Phys. Lett. 51A, 1 (1975),Phys. Rev. A 13, 2226 (1976).

Phys. Rev. (1)

R. H. Dicke, Phys. Rev. 93, 99 (1954).
[Crossref]

Phys. Rev. A (1)

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, Phys. Rev. A 6, 2211 (1972).
[Crossref]

Phys. Rev. D (4)

D. Stoler, Phys. Rev. D 1, 3217 (1970);Phys. Rev. D 4,1925 (1971).
[Crossref]

See for example, C. M. Caves, Phys. Rev. D 23, 1693 (1981);Phys. Rev. D 26, 1817 (1982).
[Crossref]

J. R. Klauder, Phys. Rev. D 19, 2349 (1979);H. Kuratsuji and T. Suzuki, J. Math. Phys. 21, 472 (1980);C. C. Gerry and S. Silverman, J. Math. Phys. 23, 1995 (1982).
[Crossref]

H. N. Hollenhort, Phys. Rev. D 19, 1669 (1979).
[Crossref]

Phys. Rev. Lett. (5)

D. Stoler, Phys. Rev. Lett. 23, 1397 (1974);G. Milburn and D. F. Walls, Opt. Commun. 39, 401 (1981);K. Wódkiewicz and M. S. Zubairy, Phys. Rev. A 27, 2003 (1983).
[Crossref]

W. Becker, M. O. Scully, and M. S. Zubairy, Phys. Rev. Lett. 48, 475 (1982);L. A. Lugiato and G. Stinni, Opt. Commun. 41, 67 (1982);P. Meystre and M. S. Zubairy, Phys. Lett. 89A, 390 (1982).
[Crossref]

L. Mandel, Phys. Rev. Lett. 49, 136 (1982).
[Crossref]

D. F. Walls and P. Zoller, Phys. Rev. Lett. 47, 709 (1981).
[Crossref]

M. M. Nieto and L. M. Simmons, Phys. Rev. Lett. 41, 207 (1978);Phys. Rev. D 20, 1321, 1332, 1342 (1979);Phys. Rev. A 19, 438 (1979).
[Crossref]

Usp. Fiz. Nauk (1)

For the review article, see A. M. Perelomov, Usp. Fiz. Nauk 123, 23 (1977) [Sov. Phys. Usp. 20, 703 (1977)].
[Crossref]

Other (4)

For all the required information about representation of the SU(2) and SU(1,1) Lie algebras see, for example, A. O. Barut and R. Ra̧czka, Theory of Group Representation and Applications (Polish Scientific Publishers, Warsaw, 1977).

K. Gottfried, Quantum Mechanics (Benjamin, New York, 1966), Chap. 24.

See, for example, L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

See Ref. 23, Chap. 9, or A. L. Bloom, Phys. Rev.98, 1105 (1955).
[Crossref]

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Equations (95)

Equations on this page are rendered with MathJax. Learn more.

[ X ̂ i , X ̂ j ] = C i j k X ̂ k ,
T g | 0 for g G
Δ X i Δ X j ½ | C i j k X ̂ k | .
( X ̂ i i λ X ̂ j ) | ψ = β | ψ ,
[ Ĵ i , Ĵ j ] = i i j k Ĵ k , i , j , k = 1 , 2 , 3 ,
| θ , ϕ = exp ( α Ĵ + α * Ĵ ) | J , J ,
| τ = ( 1 + | τ | 2 ) J exp ( τ Ĵ + ) | J , J ,
| τ = ( 1 + | τ | 2 ) J M = J J ( 2 J J + M ) 1 / 2 τ J + M | J , M .
τ 1 | τ 2 = ( 1 + τ 1 * τ 2 ) 2 J ( 1 + | τ 1 | 2 ) J ( 1 + | τ 2 | 2 ) J
d μ ( τ ) | τ τ = I ,
d μ ( τ ) = 2 J + 1 π d ( Re τ ) d ( Im τ ) ( 1 + | τ | 2 ) 2 .
Δ J 1 2 = J 2 ( 1 + | τ | 2 ) 2 ( 1 + | τ | 4 τ * 2 τ 2 ) = J 2 ( 1 sin 2 θ cos 2 ϕ ) ,
Δ J 2 2 = J 2 ( 1 + | τ | 2 ) 2 ( 1 + | τ | 4 + τ * 2 + τ 2 ) = J 2 ( 1 sin 2 θ cos 2 ϕ ) ,
| Ĵ 3 | = J | 1 | τ | 2 1 + | τ | 2 | = J | cos θ | .
Δ J 1 Δ J 2 ½ | Ĵ 3 | ,
Ĵ 1 i λ Ĵ 2 | ψ = β | ψ .
| ψ = N exp ( ϕ 0 Ĵ 3 ) exp ( i π 2 Ĵ 2 ) | J , M ,
[ K ̂ 1 , K ̂ 2 ] = i K ̂ 3 ,
[ K ̂ 2 , K ̂ 3 ] = i K ̂ 1 ,
[ K ̂ 3 , K ̂ 1 ] = i K ̂ 2 .
K ̂ 3 | k , n = ( k + n ) | k , n , n = 0 , 1 , k > 0 .
| θ , ϕ = exp ( β K ̂ + β * K ̂ ) | k , 0 ,
| ζ = ( 1 | ζ | 2 ) k exp ( 3 K ̂ + ) | k , 0 ,
| ζ = ( 1 | ζ | 2 ) k n = 0 ( Γ ( n + 2 k ) n ! Γ ( 2 k ) ) 1 / 2 ζ n | n , k .
ζ 1 | ζ 2 = ( 1 | ζ 1 | 2 ) k ( 1 | ζ 2 | 2 ) k ( 1 ζ 1 * ζ 2 ) 2 k ,
d μ ( ζ ) | ζ ζ | = I
d μ ( ζ ) = { 2 k 1 π d ( Re ) d ( Im ζ ) ( 1 | ζ | 2 ) 2 for k ½ 1 π d ( Re ) d ( Im ζ ) ( 1 | ζ | 2 ) 2 for k = ½ .
Δ K 1 2 = k 2 ( 1 | ζ | 2 ) 2 ( ζ * 2 + ζ 2 + | ζ | 4 + 1 ) = k 2 ( 1 + sinh 2 θ cos 2 ϕ ) ,
Δ K 2 2 = k 2 ( 1 | ζ | 2 ) 2 ( 1 + | ζ | 4 ζ * 2 ζ 2 ) = k 2 ( 1 + sinh 2 θ sin 2 ϕ ) ,
| K ̂ 3 | = k 1 + | ζ | 2 1 | ζ | 2 = k cosh θ .
Δ K 1 2 Δ K 2 2 ½ | K ̂ 3 | .
( K ̂ 1 i λ K ̂ 2 ) | ψ = β | ψ ,
| ψ = N exp ( ϕ 0 K ̂ 3 ) exp ( π 2 K ̂ 2 ) | k , n ,
Δ X i 2 ½ | C i j k X ̂ k | or Δ X j 2 ½ | C i j k X ̂ k |
Δ a 1 2 ¼ or Δ a 2 2 ¼ .
Δ J 1 2 ½ | Ĵ 3 | or Δ J 2 2 ½ | Ĵ 3 | .
1 sin 2 θ cos 2 ϕ | cos θ | .
Δ K 1 2 ½ | K ̂ 3 | or Δ K 2 2 ½ | K ̂ 3 | .
( 1 + sinh 2 θ cos 2 ϕ ) cosh θ .
cos 2 ϕ 1 / ( cosh θ + 1 ) ,
H = Δ J 3 + i λ 2 J + i λ 2 Ĵ ,
Ĵ 3 | J , J = J | J , J .
| ψ = exp ( i Δ t 21 Ĵ 3 ) exp ( i Δ t 1 Ĵ 3 + λ t 1 2 Ĵ + λ t 1 2 Ĵ | ) J , J ,
θ = Ω t 1
ϕ = Δ t 21 .
1 sin 2 Ω t 1 cos 2 Δ t 21 | cos Ω t 1 | .
Δ J 1 2 = J 2 ( 1 sin 2 Ω t 1 )
g ( Δ ) = T 2 * π exp [ ( Δ Δ ) 2 T 2 * 2 π ] ,
Δ J 1 2 = ¼ J + 2 + J 2 + J + J + J J + ¯ ¼ J + + J ¯ 2 = 1 4 ( 1 + | τ | 2 [ 2 J ( 2 J 1 ) ( τ * 2 + τ 2 ¯ ) 8 J 2 | τ | 2 + 2 J | τ | 4 + 2 J 4 J 2 ( τ ¯ * + τ ¯ ) 2 ] ,
f ( τ ) ¯ = d Δ g ( Δ ) f [ τ ( Δ ) ] .
J sin 2 θ [ cos 2 Δ t 21 exp ( t 21 2 π T 2 * 2 ) + 1 2 cos 2 Δ t 21 exp ( t 21 2 π 2 T 2 * 2 ) ] + 1 2 + 1 2 cos 2 θ 1 2 cos 2 Δ t 21 × sin 2 θ exp ( t 21 2 π T 2 * 2 ) | cos θ | .
( J ½ ) sin 2 θ + 1 | cos θ | .
| echo ; Δ = exp ( i Δ t 43 Ĵ 3 ) exp ( i Δ t 32 Ĵ 3 + π 2 Ĵ + π 2 Ĵ ) × exp ( i Δ t 21 Ĵ 3 ) exp ( i Δ t 1 Ĵ 3 + θ 1 Ĵ 3 θ 1 Ĵ ) | J , J .
θ = arcos [ cos θ 1 2 Δ Ω sin Δ t 21 sin θ 1 + O ( Δ 2 Ω 2 ) ] ,
ϕ = Δ ( t 43 t 21 ) = Δ T ,
J 1 ¯ J sin θ cos Δ T exp ( T 2 π T 2 * 2 )
J sin 2 θ 1 [ cos 2 Δ T exp ( T 2 π T 2 * 2 ) + 1 2 cos 2 Δ T exp ( T 2 π 2 T 2 * 2 ) ] + 1 2 + 1 2 cos 2 θ 1 1 2 cos 2 Δ T sin 2 θ 1 exp ( T 2 π T 2 * 2 ) | cos θ 1 | .
cos 2 θ 1 | cos θ 1 | ,
Δ J 1 2 = J 2 sin 2 θ 1 ( cos 2 ϕ ¯ cos ϕ 2 ¯ ) + J 2 ( 1 sin 2 θ cos 2 ϕ ¯ ) ,
Δ J 2 2 = J sin 2 θ 1 ( sin 2 ϕ ¯ sin ϕ 2 ¯ ) + J 2 ( 1 sin 2 θ sin 2 ϕ ¯ ) ,
( Δ J 1 2 ) echo = J 2 ( 1 sin 2 θ 1 ) , ( Δ J 2 2 ) echo = J 2 , J 3 | = | J 3 echo | = J 2 | cos θ 1 | ,
( Δ J 1 ) echo ( Δ J 2 ) echo = ½ | J 3 |
Ĥ = ( i λ / 2 ) ( â 2 â + 2 ) ,
K ̂ + = ½ â + 2 , K ̂ = ½ â + 2 , K ̂ 3 = ¼ ( â + â + â â + ) .
K ̂ 3 = ¼ â â + | 0 = ¼ | 0 ,
Ĥ = i λ K ̂ i λ K ̂ + .
| ψ = exp ( λ t K ̂ + + λ t K ̂ ) | 0 .
ξ = tanh ( λ | | t ) .
( 1 + sinh 2 2 Ω t ) cosh 2 Ω t or 1 cosh 2 Ω t .
Ĥ = 2 Ω K ̂ 2
K ̂ 3 = ½ ( N ̂ + ½ ) .
Δ H 2 Ω 2 N ̂ + ½ ,
Δ K 2 2 = : K ̂ 2 2 : + ½ K ̂ 3 .
: K ̂ 2 2 : 0 or : H 2 : 0 .
Δ a 1 2 = K ̂ 3 + ½ ( K ̂ + + K ̂ ) = ¼ ( 1 + ξ ) 2 1 ξ 2 = ¼ e Ω t ,
Δ a 2 2 = K ̂ 3 ½ ( K ̂ + + K ̂ ) = ¼ ( 1 ξ ) 2 1 ξ 2 = ¼ e Ω t ,
( â 1 i λ â 2 ) | ψ .
tanh r = 1 + λ 1 λ ,
â S ̂ | ψ = α S ̂ | ψ ,
S ̂ = exp [ r / 2 ( â â + 2 ) ] .
S ̂ | ψ = | α ,
| ψ = S ̂ + | α ,
Ĥ = i λ ( â + â â + + â + ) ,
K ̂ + = â + + â + , K ̂ = â + â , K ̂ 3 = â + + â + + â + â + 1 .
k = n + n .
exp ( λ + Ĵ + + λ Ĵ + λ 3 Ĵ 3 ) = exp ( Λ + Ĵ + ) exp ( ln Λ 3 Ĵ 3 ) exp ( Λ Ĵ ) ,
exp ( γ + K ̂ + + γ K ̂ + γ 3 K ̂ 3 ) = exp ( Γ + K ̂ + ) exp ( ln Γ 3 K ̂ 3 ) exp ( Γ K ̂ ) ,
Λ 3 = ( cosh α λ 3 2 α sinh α ) 2 ,
Λ ± = 2 λ ± sinh α 2 α cosh α λ 3 sinh α ,
Γ 3 = ( cosh β γ 3 2 β sinh β ) 2 ,
Γ ± = 2 γ ± sinh β 2 β cosh β γ 3 sinh β ,
α 2 = ¼ λ 3 2 + λ + λ , β 2 = ¼ γ 3 2 γ + γ .
τ 1 | exp ( c 1 Ĵ + ) exp ( c 2 Ĵ ) | τ 1 = [ ( 1 + τ 1 * c 1 ) ( 1 + c 2 τ 2 ) + τ 1 * τ 2 ] 2 J ( 1 + | τ 1 | 2 ) J ( 1 + | τ 2 | 2 ) J ,
ξ 1 | exp ( c 1 K ̂ + ) exp ( c 2 K ̂ ) | ζ 2 = ( 1 | ζ 1 | 2 ) k ( 1 | ζ 2 | 2 ) k [ ( 1 ζ 1 * c 1 ) ( 1 ζ 2 c 2 ) ζ 1 * ζ 2 ] 2 k .
ζ | K ̂ + K ̂ | ζ = 2 c 1 c 2 ζ 1 | exp ( c 1 K ̂ + ) exp ( c 2 K ̂ ) | ζ 2 | c 1 = c 2 = 0 ζ 1 = ζ 2 = 4 k 2 | ζ | 2 + 2 k | ζ | 4 ( 1 | ζ | 2 ) 2 .

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