Abstract

We present an efficient approach to studying the spectra and eigenstates for nonlinear models describing multiphoton processes. We obtain the exact explicit analytical expressions of all the energy spectra and eigenstates in terms of a parameter for a class of models describing two-mode multiphoton processes. The parameter is shown to be determined by the roots of a simple polynomial.

© 2003 Optical Society of America

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  1. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1985).
  2. Y. Wu and X. Yang, J. Phys. B 34, 2281 (2001).
    [CrossRef]
  3. S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giogis, Appl. Phys. Lett. 81, 4706 (2002).
    [CrossRef]
  4. M. K. Olsen, V. I. Kruglov, and M. J. Collett, Phys. Rev. A 63, 033801 (2001).
    [CrossRef]
  5. H. Q. Zhou, J. Links, R. H. McKenzie, and X. W. Guan, J. Phys. A 36, L113 (2003).
    [CrossRef]
  6. V. P. Karassiov and V. P. Klimov, Phys. Lett. A 189, 43 (1994).
    [CrossRef]
  7. V. A. Andreev and O. A. Ivanova, J. Phys. A 35, 8587 (2002).
    [CrossRef]
  8. G. Alvarez and R. F. Alvarez-Estrada, J. Phys. A 34, 10045 (2001).
    [CrossRef]
  9. Y. Wu, X. Yang, and P. T. Leung, Opt. Lett. 24, 345 (1999).
    [CrossRef]
  10. Y. Wu, L. Wen, and Y. Zhu, Opt. Lett. 28, 631 (2003).
    [CrossRef] [PubMed]
  11. G. S. Agarwal and S. P. Tewari, Phys. Rev. Lett. 70, 1417 (1993).
    [CrossRef] [PubMed]
  12. L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
    [CrossRef]
  13. Y. Wu, J. Saldana, and Y. Zhu, Phys. Rev. A 67, 013811 (2003).
    [CrossRef]
  14. L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
    [CrossRef]
  15. Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, Phys. Rev. A 61, 043604 (2000).
    [CrossRef]
  16. P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, Phys. Rev. A 64, 023606 (2001).
    [CrossRef]
  17. J. Heurich, H. Pu, M. G. Moore, and P. Meystre, Phys. Rev. A 63, 033605 (2001).
    [CrossRef]
  18. Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, Phys. Rev. A 67, 013603 (2003).
    [CrossRef]
  19. C. W. Gardiner, Phys. Rev. A 56, 1414 (1997).
    [CrossRef]
  20. M. D. Girardeau, Phys. Rev. A 58, 775 (1998).
    [CrossRef]
  21. Y. Castin and R. Dum, Phys. Rev. A 57, 3008 (1998).
    [CrossRef]
  22. Y. Wu, X. Yang, and Y. Xiao, Phys. Rev. Lett. 86, 2200 (2001).
    [CrossRef] [PubMed]

2003 (4)

H. Q. Zhou, J. Links, R. H. McKenzie, and X. W. Guan, J. Phys. A 36, L113 (2003).
[CrossRef]

Y. Wu, L. Wen, and Y. Zhu, Opt. Lett. 28, 631 (2003).
[CrossRef] [PubMed]

Y. Wu, J. Saldana, and Y. Zhu, Phys. Rev. A 67, 013811 (2003).
[CrossRef]

Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, Phys. Rev. A 67, 013603 (2003).
[CrossRef]

2002 (3)

L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
[CrossRef]

V. A. Andreev and O. A. Ivanova, J. Phys. A 35, 8587 (2002).
[CrossRef]

S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giogis, Appl. Phys. Lett. 81, 4706 (2002).
[CrossRef]

2001 (6)

M. K. Olsen, V. I. Kruglov, and M. J. Collett, Phys. Rev. A 63, 033801 (2001).
[CrossRef]

Y. Wu and X. Yang, J. Phys. B 34, 2281 (2001).
[CrossRef]

G. Alvarez and R. F. Alvarez-Estrada, J. Phys. A 34, 10045 (2001).
[CrossRef]

P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, Phys. Rev. A 64, 023606 (2001).
[CrossRef]

J. Heurich, H. Pu, M. G. Moore, and P. Meystre, Phys. Rev. A 63, 033605 (2001).
[CrossRef]

Y. Wu, X. Yang, and Y. Xiao, Phys. Rev. Lett. 86, 2200 (2001).
[CrossRef] [PubMed]

2000 (1)

Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, Phys. Rev. A 61, 043604 (2000).
[CrossRef]

1999 (2)

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
[CrossRef]

Y. Wu, X. Yang, and P. T. Leung, Opt. Lett. 24, 345 (1999).
[CrossRef]

1998 (2)

M. D. Girardeau, Phys. Rev. A 58, 775 (1998).
[CrossRef]

Y. Castin and R. Dum, Phys. Rev. A 57, 3008 (1998).
[CrossRef]

1997 (1)

C. W. Gardiner, Phys. Rev. A 56, 1414 (1997).
[CrossRef]

1994 (1)

V. P. Karassiov and V. P. Klimov, Phys. Lett. A 189, 43 (1994).
[CrossRef]

1993 (1)

G. S. Agarwal and S. P. Tewari, Phys. Rev. Lett. 70, 1417 (1993).
[CrossRef] [PubMed]

1985 (1)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1985).

Agarwal, G. S.

G. S. Agarwal and S. P. Tewari, Phys. Rev. Lett. 70, 1417 (1993).
[CrossRef] [PubMed]

Alvarez, G.

G. Alvarez and R. F. Alvarez-Estrada, J. Phys. A 34, 10045 (2001).
[CrossRef]

Alvarez-Estrada, R. F.

G. Alvarez and R. F. Alvarez-Estrada, J. Phys. A 34, 10045 (2001).
[CrossRef]

Andreev, V. A.

V. A. Andreev and O. A. Ivanova, J. Phys. A 35, 8587 (2002).
[CrossRef]

Ballarini, V.

S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giogis, Appl. Phys. Lett. 81, 4706 (2002).
[CrossRef]

Band, Y.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
[CrossRef]

Bonner, C.

Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, Phys. Rev. A 67, 013603 (2003).
[CrossRef]

Castin, Y.

Y. Castin and R. Dum, Phys. Rev. A 57, 3008 (1998).
[CrossRef]

Collett, M. J.

M. K. Olsen, V. I. Kruglov, and M. J. Collett, Phys. Rev. A 63, 033801 (2001).
[CrossRef]

Deng, L.

L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
[CrossRef]

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
[CrossRef]

Dum, R.

Y. Castin and R. Dum, Phys. Rev. A 57, 3008 (1998).
[CrossRef]

Finizio, S. D.

S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giogis, Appl. Phys. Lett. 81, 4706 (2002).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner, Phys. Rev. A 56, 1414 (1997).
[CrossRef]

Giogis, F.

S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giogis, Appl. Phys. Lett. 81, 4706 (2002).
[CrossRef]

Girardeau, M. D.

M. D. Girardeau, Phys. Rev. A 58, 775 (1998).
[CrossRef]

Greekmore, S.

Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, Phys. Rev. A 67, 013603 (2003).
[CrossRef]

Guan, X. W.

H. Q. Zhou, J. Links, R. H. McKenzie, and X. W. Guan, J. Phys. A 36, L113 (2003).
[CrossRef]

Hagley, E. W.

L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
[CrossRef]

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
[CrossRef]

Helmerson, K.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
[CrossRef]

Heurich, J.

J. Heurich, H. Pu, M. G. Moore, and P. Meystre, Phys. Rev. A 63, 033605 (2001).
[CrossRef]

Ivanova, O. A.

V. A. Andreev and O. A. Ivanova, J. Phys. A 35, 8587 (2002).
[CrossRef]

Julienne, P. S.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
[CrossRef]

Jung, S. S.

Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, Phys. Rev. A 67, 013603 (2003).
[CrossRef]

Karassiov, V. P.

V. P. Karassiov and V. P. Klimov, Phys. Lett. A 189, 43 (1994).
[CrossRef]

Kim, J. H.

Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, Phys. Rev. A 67, 013603 (2003).
[CrossRef]

Klimov, V. P.

V. P. Karassiov and V. P. Klimov, Phys. Lett. A 189, 43 (1994).
[CrossRef]

Kozuma, M.

L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
[CrossRef]

Kruglov, V. I.

M. K. Olsen, V. I. Kruglov, and M. J. Collett, Phys. Rev. A 63, 033801 (2001).
[CrossRef]

Lettieri, S.

S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giogis, Appl. Phys. Lett. 81, 4706 (2002).
[CrossRef]

Leung, P. T.

Lewenstein, M.

P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, Phys. Rev. A 64, 023606 (2001).
[CrossRef]

Links, J.

H. Q. Zhou, J. Links, R. H. McKenzie, and X. W. Guan, J. Phys. A 36, L113 (2003).
[CrossRef]

Maddalena, P.

S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giogis, Appl. Phys. Lett. 81, 4706 (2002).
[CrossRef]

McKenzie, R. H.

H. Q. Zhou, J. Links, R. H. McKenzie, and X. W. Guan, J. Phys. A 36, L113 (2003).
[CrossRef]

Meystre, P.

J. Heurich, H. Pu, M. G. Moore, and P. Meystre, Phys. Rev. A 63, 033605 (2001).
[CrossRef]

Moore, M. G.

J. Heurich, H. Pu, M. G. Moore, and P. Meystre, Phys. Rev. A 63, 033605 (2001).
[CrossRef]

Namkung, M.

Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, Phys. Rev. A 67, 013603 (2003).
[CrossRef]

Ohberg, P.

P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, Phys. Rev. A 64, 023606 (2001).
[CrossRef]

Olsen, M. K.

M. K. Olsen, V. I. Kruglov, and M. J. Collett, Phys. Rev. A 63, 033801 (2001).
[CrossRef]

Payne, M. G.

L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
[CrossRef]

Phillips, W. D.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
[CrossRef]

Pu, H.

J. Heurich, H. Pu, M. G. Moore, and P. Meystre, Phys. Rev. A 63, 033605 (2001).
[CrossRef]

Rolston, S. L.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
[CrossRef]

Saldana, J.

Y. Wu, J. Saldana, and Y. Zhu, Phys. Rev. A 67, 013811 (2003).
[CrossRef]

Sanpera, A.

P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, Phys. Rev. A 64, 023606 (2001).
[CrossRef]

Santos, L.

P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, Phys. Rev. A 64, 023606 (2001).
[CrossRef]

Seo, J. T.

Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, Phys. Rev. A 67, 013603 (2003).
[CrossRef]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1985).

Simsarian, J. E.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
[CrossRef]

Sun, C. P.

Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, Phys. Rev. A 61, 043604 (2000).
[CrossRef]

Temple, D. A.

Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, Phys. Rev. A 67, 013603 (2003).
[CrossRef]

Tewari, S. P.

G. S. Agarwal and S. P. Tewari, Phys. Rev. Lett. 70, 1417 (1993).
[CrossRef] [PubMed]

Trippenbach, M.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
[CrossRef]

Villain, P.

P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, Phys. Rev. A 64, 023606 (2001).
[CrossRef]

Wang, Y. Q.

Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, Phys. Rev. A 61, 043604 (2000).
[CrossRef]

Wen, J.

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
[CrossRef]

Wen, L.

Wu, Y.

Y. Wu, L. Wen, and Y. Zhu, Opt. Lett. 28, 631 (2003).
[CrossRef] [PubMed]

Y. Wu, J. Saldana, and Y. Zhu, Phys. Rev. A 67, 013811 (2003).
[CrossRef]

Y. Wu and X. Yang, J. Phys. B 34, 2281 (2001).
[CrossRef]

Y. Wu, X. Yang, and Y. Xiao, Phys. Rev. Lett. 86, 2200 (2001).
[CrossRef] [PubMed]

Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, Phys. Rev. A 61, 043604 (2000).
[CrossRef]

Y. Wu, X. Yang, and P. T. Leung, Opt. Lett. 24, 345 (1999).
[CrossRef]

Xiao, Y.

Y. Wu, X. Yang, and Y. Xiao, Phys. Rev. Lett. 86, 2200 (2001).
[CrossRef] [PubMed]

Yang, Q.

Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, Phys. Rev. A 67, 013603 (2003).
[CrossRef]

Yang, X.

Y. Wu, X. Yang, and Y. Xiao, Phys. Rev. Lett. 86, 2200 (2001).
[CrossRef] [PubMed]

Y. Wu and X. Yang, J. Phys. B 34, 2281 (2001).
[CrossRef]

Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, Phys. Rev. A 61, 043604 (2000).
[CrossRef]

Y. Wu, X. Yang, and P. T. Leung, Opt. Lett. 24, 345 (1999).
[CrossRef]

Ye, P.

Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, Phys. Rev. A 67, 013603 (2003).
[CrossRef]

Zhou, H. Q.

H. Q. Zhou, J. Links, R. H. McKenzie, and X. W. Guan, J. Phys. A 36, L113 (2003).
[CrossRef]

Zhou, X. J.

Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, Phys. Rev. A 61, 043604 (2000).
[CrossRef]

Zhu, Y.

Y. Wu, L. Wen, and Y. Zhu, Opt. Lett. 28, 631 (2003).
[CrossRef] [PubMed]

Y. Wu, J. Saldana, and Y. Zhu, Phys. Rev. A 67, 013811 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giogis, Appl. Phys. Lett. 81, 4706 (2002).
[CrossRef]

J. Phys. B (1)

Y. Wu and X. Yang, J. Phys. B 34, 2281 (2001).
[CrossRef]

J. Phys. A (3)

V. A. Andreev and O. A. Ivanova, J. Phys. A 35, 8587 (2002).
[CrossRef]

G. Alvarez and R. F. Alvarez-Estrada, J. Phys. A 34, 10045 (2001).
[CrossRef]

H. Q. Zhou, J. Links, R. H. McKenzie, and X. W. Guan, J. Phys. A 36, L113 (2003).
[CrossRef]

Nature (1)

L. Deng, E. W. Hagley, J. Wen, M. Trippenbach, Y. Band, P. S. Julienne, J. E. Simsarian, K. Helmerson, S. L. Rolston, and W. D. Phillips, Nature 398, 218 (1999).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

V. P. Karassiov and V. P. Klimov, Phys. Lett. A 189, 43 (1994).
[CrossRef]

Phys. Rev. A (9)

Y. Wu, J. Saldana, and Y. Zhu, Phys. Rev. A 67, 013811 (2003).
[CrossRef]

M. K. Olsen, V. I. Kruglov, and M. J. Collett, Phys. Rev. A 63, 033801 (2001).
[CrossRef]

Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, Phys. Rev. A 61, 043604 (2000).
[CrossRef]

P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, Phys. Rev. A 64, 023606 (2001).
[CrossRef]

J. Heurich, H. Pu, M. G. Moore, and P. Meystre, Phys. Rev. A 63, 033605 (2001).
[CrossRef]

Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, Phys. Rev. A 67, 013603 (2003).
[CrossRef]

C. W. Gardiner, Phys. Rev. A 56, 1414 (1997).
[CrossRef]

M. D. Girardeau, Phys. Rev. A 58, 775 (1998).
[CrossRef]

Y. Castin and R. Dum, Phys. Rev. A 57, 3008 (1998).
[CrossRef]

Phys. Rev. Lett. (3)

Y. Wu, X. Yang, and Y. Xiao, Phys. Rev. Lett. 86, 2200 (2001).
[CrossRef] [PubMed]

G. S. Agarwal and S. P. Tewari, Phys. Rev. Lett. 70, 1417 (1993).
[CrossRef] [PubMed]

L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
[CrossRef]

Other (1)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1985).

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Equations (16)

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

Hˆω1aˆaˆ+ω2bˆbˆ+gaˆsbˆr+bˆraˆs,
HˆωNˆ+gHˆ1,    Hˆ1aˆsbˆr+bˆraˆs.
Hˆ1|Ψλ,M;na0,nb0=λ|Ψλ,M;na0,nb0,
|Ψ=Sˆaˆ,bˆ|vac,Sˆaˆ,bˆ=n=0Mαnaˆns+na0bˆM-nr+nb0ns+na0!,
xsrSyr+yrsSxs=λS.
αn+1=λαn-pn-1αn-1,    0nM,
pn=ns+s+na0!ns+na0!Mr-nr+nb0!Mr-nr-r+nb0!.
αn=α0 det Anλ,    n=1,2,,M,
det AM+1λ=0,
det An=λn+k=1intn/2-1kλn-2kj=1kij=2k-jij-1-2pij,
E=rsM+rna0+snb0ω+gλ,|Ψλ,M;na0,nb0=α0n=0Mdet Anλ|ns+na0,Mr-nr+nb0ns+na0!3/2M-nr+nb0!1/2,
E=rsM+rna0+snb0ω,    M=0,2,4,6,,|Ψλ=0,M;na0,nb0=m=0intM/2-1mcm×|2ms+na0,Mr-2mr+nb0,cmα0j=1mp2m-2j2ms+na0!3/2Mr-2mr+nb0!1/2,
M=0λ=0,
M=1λ=±s+na0!r+nb0!na0!nb0!1/2,M=2λ=0,±n=01ns+s+na0!2r-nr+nb0!ns+na0!r-nr+nb0!1/2,
M=3λ=±u±u2-v,
M=4λ=0,±u1±u12-v1,

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