Abstract

We solve exactly a set of fully quantized coupled equations that describe the quantum dynamics, including the evolution of quantum fluctuations, entanglements, and correlations, of four-wave mixing (FWM) with matter waves. The analytical solution reveals Rabi oscillations, collapses and revivals, and agrees with FWM experiments. It also applies to optical FWM and the Rayleigh superradiance in Bose–Einstein condensates.

© 2005 Optical Society of America

Full Article  |  PDF Article

Corrections

Ying Wu and Xiaoxue Yang, "Fully quantized theory of four-wave mixing with bosonic matter waves: erratum," Opt. Lett. 30, 1070-1070 (2005)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-30-9-1070

References

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  1. P. Meystre, Atom Optics (Springer-Verlag, New York, 2001).
    [CrossRef]
  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]
  3. M. Trippenbach, Y. B. Band, and P. S. Julienne, Opt. Express 3, 530 (1998), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  4. E. V. Goldstein and P. Meystre, Phys. Rev. A 59, 1509 (1999).
    [CrossRef]
  5. Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, Phys. Rev. A 61, 043604 (2000).
    [CrossRef]
  6. M. Trippenbach, Y. B. Band, and P. S. Julienne, Phys. Rev. A 62, 023608 (2000).
    [CrossRef]
  7. P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, Phys. Rev. A 64, 023606 (2001).
    [CrossRef]
  8. J. Heurich, H. Pu, M. G. Moore, and P. Meystre, Phys. Rev. A 63, 033605 (2001).
    [CrossRef]
  9. J. M. Vogels, K. Xu, and W. Ketterle, Phys. Rev. Lett. 89, 020401 (2002).
    [CrossRef]
  10. V. A. Yurovsky, Phys. Rev. A 65, 033605 (2002).
    [CrossRef]
  11. 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]
  12. A. Vardi and M. G. Moore, “Abrikosov vortex lattice nonlinear dynamics and its stability in the presence of weak defects,” arXiv.org e-Print archive, , January31, 2002.
  13. M. G. Moore and P. Meystre, Phys. Rev. Lett. 86, 4199 (2001).
    [CrossRef] [PubMed]
  14. W. Ketterle and S. Inouye, Phys. Rev. Lett. 86, 4203 (2001).
    [CrossRef] [PubMed]
  15. H. Christ, C. P. Search, and P. Meystre, Phys. Rev. A 67, 063611 (2003).
    [CrossRef]
  16. M. G. Moore, O. Zobay, and P. Meystre, Phys. Rev. A 60, 1491 (1999).
    [CrossRef]
  17. N. Piovella, M. Cola, and R. Bonifacio, Phys. Rev. A 67, 013817 (2003).
    [CrossRef]
  18. Ö. E. Müstrecapho?lu and L. You, Phys. Rev. A 62, 063615 (2000).
    [CrossRef]
  19. D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003).
    [CrossRef] [PubMed]

2003 (4)

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]

H. Christ, C. P. Search, and P. Meystre, Phys. Rev. A 67, 063611 (2003).
[CrossRef]

N. Piovella, M. Cola, and R. Bonifacio, Phys. Rev. A 67, 013817 (2003).
[CrossRef]

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003).
[CrossRef] [PubMed]

2002 (2)

J. M. Vogels, K. Xu, and W. Ketterle, Phys. Rev. Lett. 89, 020401 (2002).
[CrossRef]

V. A. Yurovsky, Phys. Rev. A 65, 033605 (2002).
[CrossRef]

2001 (4)

M. G. Moore and P. Meystre, Phys. Rev. Lett. 86, 4199 (2001).
[CrossRef] [PubMed]

W. Ketterle and S. Inouye, Phys. Rev. Lett. 86, 4203 (2001).
[CrossRef] [PubMed]

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]

2000 (3)

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

M. Trippenbach, Y. B. Band, and P. S. Julienne, Phys. Rev. A 62, 023608 (2000).
[CrossRef]

Ö. E. Müstrecapho?lu and L. You, Phys. Rev. A 62, 063615 (2000).
[CrossRef]

1999 (3)

M. G. Moore, O. Zobay, and P. Meystre, Phys. Rev. A 60, 1491 (1999).
[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]

E. V. Goldstein and P. Meystre, Phys. Rev. A 59, 1509 (1999).
[CrossRef]

1998 (1)

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]

Band, Y. B.

Bonifacio, R.

N. Piovella, M. Cola, and R. Bonifacio, Phys. Rev. A 67, 013817 (2003).
[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]

Boyd, M.

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003).
[CrossRef] [PubMed]

Christ, H.

H. Christ, C. P. Search, and P. Meystre, Phys. Rev. A 67, 063611 (2003).
[CrossRef]

Cola, M.

N. Piovella, M. Cola, and R. Bonifacio, Phys. Rev. A 67, 013817 (2003).
[CrossRef]

Deng, 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]

Goldstein, E. V.

E. V. Goldstein and P. Meystre, Phys. Rev. A 59, 1509 (1999).
[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]

Hagley, E. W.

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]

Inouye, S.

W. Ketterle and S. Inouye, Phys. Rev. Lett. 86, 4203 (2001).
[CrossRef] [PubMed]

Julienne, P. S.

M. Trippenbach, Y. B. Band, and P. S. Julienne, Phys. Rev. A 62, 023608 (2000).
[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]

M. Trippenbach, Y. B. Band, and P. S. Julienne, Opt. Express 3, 530 (1998), http://www.opticsexpress.org .
[CrossRef] [PubMed]

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]

Ketterle, W.

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003).
[CrossRef] [PubMed]

J. M. Vogels, K. Xu, and W. Ketterle, Phys. Rev. Lett. 89, 020401 (2002).
[CrossRef]

W. Ketterle and S. Inouye, Phys. Rev. Lett. 86, 4203 (2001).
[CrossRef] [PubMed]

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]

Lewenstein, M.

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

Meystre, P.

H. Christ, C. P. Search, and P. Meystre, Phys. Rev. A 67, 063611 (2003).
[CrossRef]

M. G. Moore and P. Meystre, Phys. Rev. Lett. 86, 4199 (2001).
[CrossRef] [PubMed]

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

M. G. Moore, O. Zobay, and P. Meystre, Phys. Rev. A 60, 1491 (1999).
[CrossRef]

E. V. Goldstein and P. Meystre, Phys. Rev. A 59, 1509 (1999).
[CrossRef]

P. Meystre, Atom Optics (Springer-Verlag, New York, 2001).
[CrossRef]

Moore, M. G.

M. G. Moore and P. Meystre, Phys. Rev. Lett. 86, 4199 (2001).
[CrossRef] [PubMed]

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

M. G. Moore, O. Zobay, and P. Meystre, Phys. Rev. A 60, 1491 (1999).
[CrossRef]

A. Vardi and M. G. Moore, “Abrikosov vortex lattice nonlinear dynamics and its stability in the presence of weak defects,” arXiv.org e-Print archive, , January31, 2002.

Müstrecaphoglu, Ö. E.

Ö. E. Müstrecapho?lu and L. You, Phys. Rev. A 62, 063615 (2000).
[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]

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]

Piovella, N.

N. Piovella, M. Cola, and R. Bonifacio, Phys. Rev. A 67, 013817 (2003).
[CrossRef]

Pritchard, D. E.

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003).
[CrossRef] [PubMed]

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]

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]

Schneble, D.

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003).
[CrossRef] [PubMed]

Search, C. P.

H. Christ, C. P. Search, and P. Meystre, Phys. Rev. A 67, 063611 (2003).
[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]

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]

Streed, E. W.

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003).
[CrossRef] [PubMed]

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]

Torii, Y.

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003).
[CrossRef] [PubMed]

Trippenbach, M.

M. Trippenbach, Y. B. Band, and P. S. Julienne, Phys. Rev. A 62, 023608 (2000).
[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]

M. Trippenbach, Y. B. Band, and P. S. Julienne, Opt. Express 3, 530 (1998), http://www.opticsexpress.org .
[CrossRef] [PubMed]

Vardi, A.

A. Vardi and M. G. Moore, “Abrikosov vortex lattice nonlinear dynamics and its stability in the presence of weak defects,” arXiv.org e-Print archive, , January31, 2002.

Villain, P.

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

Vogels, J. M.

J. M. Vogels, K. Xu, and W. Ketterle, Phys. Rev. Lett. 89, 020401 (2002).
[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]

Wu, Y.

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

Xu, K.

J. M. Vogels, K. Xu, and W. Ketterle, Phys. Rev. Lett. 89, 020401 (2002).
[CrossRef]

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

You, L.

Ö. E. Müstrecapho?lu and L. You, Phys. Rev. A 62, 063615 (2000).
[CrossRef]

Yurovsky, V. A.

V. A. Yurovsky, Phys. Rev. A 65, 033605 (2002).
[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]

Zobay, O.

M. G. Moore, O. Zobay, and P. Meystre, Phys. Rev. A 60, 1491 (1999).
[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. Express (1)

Phys. Rev. A (11)

H. Christ, C. P. Search, and P. Meystre, Phys. Rev. A 67, 063611 (2003).
[CrossRef]

M. G. Moore, O. Zobay, and P. Meystre, Phys. Rev. A 60, 1491 (1999).
[CrossRef]

N. Piovella, M. Cola, and R. Bonifacio, Phys. Rev. A 67, 013817 (2003).
[CrossRef]

Ö. E. Müstrecapho?lu and L. You, Phys. Rev. A 62, 063615 (2000).
[CrossRef]

E. V. Goldstein and P. Meystre, Phys. Rev. A 59, 1509 (1999).
[CrossRef]

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

M. Trippenbach, Y. B. Band, and P. S. Julienne, Phys. Rev. A 62, 023608 (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]

V. A. Yurovsky, Phys. Rev. A 65, 033605 (2002).
[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]

Phys. Rev. Lett. (3)

M. G. Moore and P. Meystre, Phys. Rev. Lett. 86, 4199 (2001).
[CrossRef] [PubMed]

W. Ketterle and S. Inouye, Phys. Rev. Lett. 86, 4203 (2001).
[CrossRef] [PubMed]

J. M. Vogels, K. Xu, and W. Ketterle, Phys. Rev. Lett. 89, 020401 (2002).
[CrossRef]

Science (1)

D. Schneble, Y. Torii, M. Boyd, E. W. Streed, D. E. Pritchard, and W. Ketterle, Science 300, 475 (2003).
[CrossRef] [PubMed]

Other (2)

P. Meystre, Atom Optics (Springer-Verlag, New York, 2001).
[CrossRef]

A. Vardi and M. G. Moore, “Abrikosov vortex lattice nonlinear dynamics and its stability in the presence of weak defects,” arXiv.org e-Print archive, , January31, 2002.

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Figures (1)

Fig. 1
Fig. 1

Numbers of atoms n2τ=200-n1τ (left) and n2τ=100-n1τ (right) versus time τ=gt for the initial values of n2=n3=0 and n1=n4=200 (left) and n1=n4=100 (right). Both insets show long-term behavior.

Equations (20)

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

Hˆ=d3rΨˆr,thˆ+2π2σmΨˆr,tΨˆr,t×Ψˆr,t,
iΦkt=ΩjΦj=hˆ+2π2σmn=14Φn2Φj.
Hˆ=gnˆ1nˆ2+nˆ3nˆ4+aˆ1aˆ2aˆ3aˆ4+aˆ2aˆ1aˆ4aˆ3+cˆ,
idaˆ1dτ=aˆ1aˆ2+aˆ4aˆ3aˆ2Jˆ+aˆ2,
idaˆ2dτ=aˆ2aˆ1+aˆ3aˆ4aˆ1Jˆ-aˆ1,
idaˆ3dτ=aˆ2aˆ1+aˆ3aˆ4aˆ4Jˆ-aˆ4,
idaˆ4dτ=aˆ1aˆ2+aˆ4aˆ3aˆ3Jˆ+aˆ3.
Jˆ1-=Jˆ1+=aˆ1aˆ2,    Jˆ2-=Jˆ2+=aˆ4aˆ3,
Jˆ1z=nˆ2-nˆ12,  Jˆ12=nˆ2+nˆ12nˆ2+nˆ12+1,
Jˆ2z=nˆ3-nˆ42,    Jˆ22=nˆ3+nˆ42nˆ3+nˆ42+1,
idαˆdτ=Kˆαˆαˆτ=exp-iKˆταˆ0,
Kˆ=2Jˆz0-1J+0J-00=WˆAˆ+00Aˆ-Wˆ-1,
Wˆ=Aˆ+Aˆ-Jˆ-0Jˆ-0,    W-1=12Ωˆ1-Bˆ--1Bˆ+,
aˆ1τ=expiτωˆ-Cˆτaˆ1-iSˆτωˆ-aˆ1-Jˆ+aˆ2,
aˆ3τ=exp-iτωˆ+Cˆτaˆ3+iSˆτωˆ+aˆ3-Jˆ-aˆ4,
aˆ3τ=exp-iτωˆ+Cˆτaˆ3+iSˆτωˆ+aˆ3-Jˆ-aˆ4,
aˆ4τ=expiτωˆ-Cˆτaˆ4-iSˆτωˆ-aˆ4-Jˆ+aˆ3,
nˆ1;4τ-nˆ1;4=nˆ2;3-nˆ2;3τ=aˆ1;4ωˆ-2-Ωˆ2Sˆ2τaˆ1;4+aˆ2;3Ωˆ2-ωˆ+2Sˆ2τaˆ2;3+iaˆ1;4Cˆτ+iSˆτωˆ-aˆ2;3-H.c.,
j1;2=n1;4+n2;32,    m1;2=n2;3-n1;42.
nˆ1;4τ-n1;4=n2;3-nˆ2;3τ=j=jminj1+j2-1/2C1;2j1-m2Ωj2sin2Ωjτ,

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