Abstract

We present explicit analytical expressions of all the eigenenergies and eigenstates for two different kinds of four-wave-mixing model in terms of a parameter λ without the so-called Bethe ansatz assumption. The parameter λ is shown to be determined by the roots of a simple polynomial. We also obtain the exact explicit analytical expressions of infinite eigenstates and energies without any unknown parameter.

© 2004 Optical Society of America

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  1. V. A. Andreev and Q. A. Ivanova, J. Phys. A 35, 8587 (2001).
    [CrossRef]
  2. H. Q. Zhou, J. Links, R. H. McKenzie, and X. W. Guan, J. Phys. A 36, L113 (2003).
    [CrossRef]
  3. M. Nielsen and I. Chuang, Quantum Computation and Quantum Communication (Cambridge U. Press, Cambridge, 2000).
  4. Y. Wu and X. Yang, Opt. Lett. 28, 1793 (2003).
    [CrossRef] [PubMed]
  5. Y. Wu and X.-X. Yang, J. Opt. B 5, 414 (2003).
    [CrossRef]
  6. Y. Wu and X. Yang, Phys. Rev. A 68, 013608 (2003).
    [CrossRef]
  7. Y. Wu, X. Yang, and Y. Xiao, Phys. Rev. Lett. 86, 2200 (2001).
    [CrossRef] [PubMed]
  8. Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, Phys. Rev. A 61, 043604 (2000).
    [CrossRef]
  9. P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, Phys. Rev. A 64, 023606 (2001).
    [CrossRef]
  10. J. Heurich, H. Pu, M. G. Moore, and P. Meystre, Phys. Rev. A 63, 033605 (2001).
    [CrossRef]
  11. L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
    [CrossRef]
  12. Y. Wu, J. Saldana, and Y. Zhu, Phys. Rev. A 67, 013811 (2003).
    [CrossRef]
  13. Y. Wu, L. Wen, and Y. Zhu, Opt. Lett. 28, 631 (2003).
    [CrossRef] [PubMed]

2003 (6)

Y. Wu and X.-X. Yang, J. Opt. B 5, 414 (2003).
[CrossRef]

Y. Wu and X. Yang, Phys. Rev. A 68, 013608 (2003).
[CrossRef]

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

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 and X. Yang, Opt. Lett. 28, 1793 (2003).
[CrossRef] [PubMed]

2002 (1)

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

2001 (4)

V. A. Andreev and Q. A. Ivanova, J. Phys. A 35, 8587 (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]

Andreev, V. A.

V. A. Andreev and Q. A. Ivanova, J. Phys. A 35, 8587 (2001).
[CrossRef]

Chuang, I.

M. Nielsen and I. Chuang, Quantum Computation and Quantum Communication (Cambridge U. Press, Cambridge, 2000).

Deng, L.

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

Heurich, J.

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

Ivanova, Q. A.

V. A. Andreev and Q. A. Ivanova, J. Phys. A 35, 8587 (2001).
[CrossRef]

Kozuma, M.

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

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]

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]

Nielsen, M.

M. Nielsen and I. Chuang, Quantum Computation and Quantum Communication (Cambridge U. Press, Cambridge, 2000).

Ohberg, P.

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

Payne, M. G.

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

Pu, H.

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

Sun, C. P.

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

Wu, Y.

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

Y. Wu and X. Yang, Phys. Rev. A 68, 013608 (2003).
[CrossRef]

Y. Wu and X. Yang, Opt. Lett. 28, 1793 (2003).
[CrossRef] [PubMed]

Y. Wu and X.-X. Yang, J. Opt. B 5, 414 (2003).
[CrossRef]

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

Xiao, Y.

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

Yang, X.

Y. Wu and X. Yang, Phys. Rev. A 68, 013608 (2003).
[CrossRef]

Y. Wu and X. Yang, Opt. Lett. 28, 1793 (2003).
[CrossRef] [PubMed]

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]

Yang, X.-X.

Y. Wu and X.-X. Yang, J. Opt. B 5, 414 (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, J. Saldana, and Y. Zhu, Phys. Rev. A 67, 013811 (2003).
[CrossRef]

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

J. Opt. B (1)

Y. Wu and X.-X. Yang, J. Opt. B 5, 414 (2003).
[CrossRef]

J. Phys. A (2)

V. A. Andreev and Q. A. Ivanova, J. Phys. A 35, 8587 (2001).
[CrossRef]

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

Opt. Lett. (2)

Phys. Rev. A (5)

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

Y. Wu and X. Yang, Phys. Rev. A 68, 013608 (2003).
[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]

Phys. Rev. Lett. (2)

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

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

Other (1)

M. Nielsen and I. Chuang, Quantum Computation and Quantum Communication (Cambridge U. Press, Cambridge, 2000).

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Equations (20)

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

Hˆ1=aˆ1aˆ2aˆ3aˆ4+aˆ4aˆ3aˆ2aˆ1,  ω1=ω2+ω3+ω4,
Hˆ1=aˆ1aˆ2aˆ3aˆ4+aˆ4aˆ3aˆ2aˆ1,  ω1+ω3=ω2+ω4,
EM,N,K;λ=Mω2+Nω3+Kω4+gλ,
Hˆ1|ΨM,N,K;λ=λ|ΨM,N,K;λ,
|ΨM,N,K;λ=Sˆaˆ1,aˆ2,aˆ3,aˆ4|vac,Sx1,x2,x3,x4=j=0qαjj!x1jx2M-jx3N-jx4K-j,
x13x2x3x4+x2x3x4x1S=λS.
αj+1=λαj-pj-1αj-1,  0jq,
αj=α0 det Ajλ,  j=0,1,2,,q,
det Aq+1λ=0,  q=minM,N,K,
det Anλ=λn+k=1intn/2-1kβknλn-2k,  n2,βkn=j1=2k-2n-2im=2k-mim-1-2ik=0ik-1-2pi1pimpik,
EM,N,K;λ=ω2M+ω3N+ω4K+gλ,
|ΨM,N,K;λ=j=0minM,N,Kcn|j,M-j,N-j,K-j,cn=c0M-j!N-j!K-j!j!M!N!K!1/2×det Ajλ,
E=ω2M+ω3N+ω4K,
|Ψ=n=0q/2-1nηn|2n,M-2n,N-2n,K-2n,ηn=C2n!M-2n!N-2n!K-2n!1/2M-2n!!N-2n!!K-2n!!2n!!,
x1x32x2x4+x2x42x1x3S=λS,
q=0λ=0, q=1λ=±QR,
q=2λ=0,±1+2Q-12R-11/2,
q=3λ=±3+5Q-1R-1±3u1/2,uR+Q-22+4Q-1R-13+122-14,q=4λ=0,±52Q-32R-3+21±3v21/2,v32R+Q-32+2Q-32R-3+12,
q=5λ=±35Q-2R-2+56+4fsX,Z31/2,
q=6λ=0,±142Q-52R-5+126+4sX,Z31/2,

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