Abstract

We present exact analytical expressions of all the energy eigenstates and eigenvalues in terms of an unknown parameter λ for the model describing interactions among three bosonic modes without using the assumption of the Bethe anzatz. We also derive the explicit expression of a polynomial whose roots give the concrete values of the parameter λ.

© 2004 Optical Society of America

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  8. E. V. Goldstein and P. Meystre, “Phase conjugation of multicomponent Bose–Einstein condensates,” Phys. Rev. A 59, 1509–1513 (1999).
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  9. Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, “Theory of four-wave mixing with matter waves without the undepleted pump approximation,” Phys. Rev. A 61, 043604/1–6 (2000).
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  21. A. Podlipensky, J. Lange, G. Seifert, H. Graener, and I. Cravetchi, “Second-harmonic generation from ellipsoidal silver nanoparticles embedded in silica glass,” Opt. Lett. 28, 716–718 (2003).
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  22. X. X. Yang and Y. Wu, “Coherent superposition states of atoms and molecules in a Bose–Einstein condensate with exactly balanced photoassociations and photodissociations,” Chin. Phys. Lett. 20, 189–191 (2003).
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  23. Y. Wu and X. Yang, “Exact eigenstates for a class of models describing two-mode multiphoton processes,” Opt. Lett. 28, 1793–1795 (2003).
    [CrossRef] [PubMed]
  24. V. A. Andreev and O. A. Ivanova, “Symmetries and reduced systems of equations for three-boson and four-boson interactions,” J. Phys. A 35, 8587–8602 (2002).
    [CrossRef]
  25. G. Alvarez and R. F. Alvarez-Estrada, “Third harmonic generation as a third-order quasi-exactly solvable system,” J. Phys. A 34, 10045–10056 (2001).
    [CrossRef]
  26. Y. Wu, X. Yang, and P. T. Leung, “Theory of microcavity-enhanced Raman gain,” Opt. Lett. 24, 345–347 (1999).
    [CrossRef]
  27. Y. Wu, L. Wen, and Y. Zhu, “Efficient hyper-Raman scattering in resonant coherent media,” Opt. Lett. 28, 631–633 (2003).
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  28. V. P. Karassiov, “G-invariant polynomial extensions of Lie algebras in quantum many-body physics,” J. Phys. A 27, 153–165 (1994).
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  29. Y. Wu, X. Yang, and Y. Xiao, “Analytical method for yrast line states in interacting Bose–Einstein condensates,” Phys. Rev. Lett. 86, 2200–2203 (2001).
    [CrossRef] [PubMed]
  30. Y. Wu and X. Yang, “Analytical results for energy spectrum and eigenstates of Bose–Einstein condensate in Mott insulator state,” Phys. Rev. A 68, 013608/1–7 (2003).
    [CrossRef]
  31. M. Alexanian and S. K. Bose, “Unitary transformation and the dynamics of a three-level atom interacting with two quantized field modes,” Phys. Rev. A 52, 2218–2224 (1995).
    [CrossRef] [PubMed]
  32. Y. Wu, “Effective Raman theory for a three-level atom in the Λ configuration,” Phys. Rev. A 54, 1586–1592 (1996).
    [CrossRef] [PubMed]
  33. Y. Wu and X. Yang, “Effective two-level model for a three-level atom in the Ξ configuration,” Phys. Rev. A 56, 2443–2446 (1997).
    [CrossRef]
  34. M. Alexanian and S. K. Bose, “Comment on ‘Generation of phase states by two-photon absorption’,” Phys. Rev. Lett. 85, 1136 (2000).
    [CrossRef]
  35. M. Alexanian, S. K. Bose, and L. Chow, “Trapping and photon number states in a two-photon micromaser,” J. Lumin. 76–77, 677–680 (1998).
    [CrossRef]
  36. M. Alexanian and S. K. Bose, “Generation of photonic superposition states by two-photon absorption,” J. Lumin. 94, 815–819 (2001).
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    [CrossRef]
  39. H. Pu and P. Meystre, “Creating macroscopic atomic Einstein–Podolsky–Rosen states from Bose–Einstein condensates,” Phys. Rev. Lett. 85, 3987–3990 (2000).
    [CrossRef] [PubMed]
  40. L.-M. Duan, A. Sorensen, J. I. Cirac, and P. Zoller, “Squeezing and entanglement of atomic beams,” Phys. Rev. Lett. 85, 3991–3994 (2000).
    [CrossRef] [PubMed]
  41. M. Alexanian, “Cavity coherent-state cloning via Raman scattering,” Phys. Rev. A 67, 033809/1–6 (2003).
    [CrossRef]
  42. C. W. Gardiner, “Particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross–Pitaevskii equation for a highly condensed Bose gas,” Phys. Rev. A 56, 1414–1423 (1997).
    [CrossRef]
  43. M. D. Girardeau, “Comment on ‘Particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross–Pitaevskii equation for a highly condensed Bose gas’,” Phys. Rev. A 58, 775–778 (1998).
    [CrossRef]
  44. Y. Castin and R. Dum, “Low-temperature Bose–Einstein condensates in time-dependent traps: Beyond the U(1) symmetry-breaking approach,” Phys. Rev. A 57, 3008–3021 (1998).
    [CrossRef]
  45. Liwei Wang, R. R. Puri, and J. H. Eberly, “Coupled-channel cavity QED model and exact solutions,” Phys. Rev. A 46, 7192–7209 (1992).
    [CrossRef] [PubMed]
  46. Y. Wu, “Simple algebraic method to solve a coupled-channel cavity QED model,” Phys. Rev. A 54, 4534–4543 (1996).
    [CrossRef] [PubMed]
  47. X. Yang, Y. Wu, and Y. Li, “A unified and standardized procedure to solve various nonlinear Jaynes–Cummings models,” Phys. Rev. A 55, 4545–4551 (1997).
    [CrossRef]
  48. I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, and J. Timonen, “Algebraic approach to the Tavis–Cummings problem,” Phys. Rev. A 67, 053808/1–12 (2003).
    [CrossRef]
  49. C. K. Law, H. Pu, and N. P. Bigelow, “Quantum spins in spinor Bose–Einstein condensates,” Phys. Rev. Lett. 81, 5257–5261 (1998).
    [CrossRef]
  50. H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow, “Spin-mixing dynamics of a spinor Bose–Einstein condensate,” Phys. Rev. A 60, 1463–1470 (1999).
    [CrossRef]
  51. M. Koashi and M. Ueda, “Exact eigenstates and magnetic response of spin-1 and spin-2 Bose–Einstein condensates,” Phys. Rev. Lett. 84, 1066–1069 (2000).
    [CrossRef] [PubMed]
  52. S. Raghavan, H. Pu, C. K. Law, J. H. Eberly, and N. P. Bigelow, “Properties of spinor Bose condensates,” J. Low Temp. Phys. 119, 437–460 (2000).
    [CrossRef]
  53. O. E. Müstecaplioǧlu, M. Zhang, and L. You, “Spin squeezing and entanglement in spinor condensates,” Phys. Rev. A 66, 033611/1–9 (2002).
    [CrossRef]
  54. Y. Wu, X. Yang, and C. Sun, “Systematical method to study general structure of Bose–Einstein condensates with arbitrary spin,” Phys. Rev. A 62, 063603/1–4 (2000).
    [CrossRef]
  55. Y. Wu and X. Yang, “Algebraic method for solving a class of coupled-channel cavity QED models,” Phys. Rev. A 63, 043816/1–5 (2001).
    [CrossRef]
  56. T. Papenbrock, “Universal solutions for interacting bosons in one-dimensional harmonic traps,” Phys. Rev. A 65, 063606/1–5 (2002).
    [CrossRef]
  57. R. Franzosi and V. Penna, “Spectral properties of coupled Bose–Einstein condensates,” Phys. Rev. A 63, 043609/1–8 (2001).
    [CrossRef]

2003

Y. Wu, J. Saldana, and Y. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811/1–5 (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, “Nonlinear phase mismatch and optimal input combination in atomic four-wave mixing in Bose–Einstein condensates,” Phys. Rev. A 67, 013603/1–7 (2003).
[CrossRef]

C. Anceau, S. Brasselet, J. Zyss, and P. Gadenne, “Local second-harmonic generation enhancement on gold nanostructures probed by two-photon microscopy,” Opt. Lett. 28, 713–715 (2003).
[CrossRef] [PubMed]

A. Podlipensky, J. Lange, G. Seifert, H. Graener, and I. Cravetchi, “Second-harmonic generation from ellipsoidal silver nanoparticles embedded in silica glass,” Opt. Lett. 28, 716–718 (2003).
[CrossRef] [PubMed]

X. X. Yang and Y. Wu, “Coherent superposition states of atoms and molecules in a Bose–Einstein condensate with exactly balanced photoassociations and photodissociations,” Chin. Phys. Lett. 20, 189–191 (2003).
[CrossRef]

Y. Wu and X. Yang, “Exact eigenstates for a class of models describing two-mode multiphoton processes,” Opt. Lett. 28, 1793–1795 (2003).
[CrossRef] [PubMed]

Y. Wu, L. Wen, and Y. Zhu, “Efficient hyper-Raman scattering in resonant coherent media,” Opt. Lett. 28, 631–633 (2003).
[CrossRef] [PubMed]

Y. Wu and X. Yang, “Analytical results for energy spectrum and eigenstates of Bose–Einstein condensate in Mott insulator state,” Phys. Rev. A 68, 013608/1–7 (2003).
[CrossRef]

L. You, “Creating maximally entangled atomic states in a Bose–Einstein condensate,” Phys. Rev. Lett. 90, 030402/1–4 (2003).
[CrossRef]

M. Alexanian, “Cavity coherent-state cloning via Raman scattering,” Phys. Rev. A 67, 033809/1–6 (2003).
[CrossRef]

I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, and J. Timonen, “Algebraic approach to the Tavis–Cummings problem,” Phys. Rev. A 67, 053808/1–12 (2003).
[CrossRef]

2002

O. E. Müstecaplioǧlu, M. Zhang, and L. You, “Spin squeezing and entanglement in spinor condensates,” Phys. Rev. A 66, 033611/1–9 (2002).
[CrossRef]

T. Papenbrock, “Universal solutions for interacting bosons in one-dimensional harmonic traps,” Phys. Rev. A 65, 063606/1–5 (2002).
[CrossRef]

S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giogis, “Second-harmonic generation in amorphous silicon nitride microcavities,” Appl. Phys. Lett. 81, 4706–4708 (2002).
[CrossRef]

V. A. Andreev and O. A. Ivanova, “Symmetries and reduced systems of equations for three-boson and four-boson interactions,” J. Phys. A 35, 8587–8602 (2002).
[CrossRef]

J. M. Vogels, K. Xu, and W. Ketterle, “Generation of macroscopic pair-correlated atomic beams by four-wave mixing in Bose–Einstein condensates,” Phys. Rev. Lett. 89, 020401/1–4 (2002).
[CrossRef]

L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, “Opening optical four-wave mixing channels with giant enhancement using ultraslow pump waves,” Phys. Rev. Lett. 88, 143902/1–4 (2002).
[CrossRef] [PubMed]

2001

P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, “Four-wave mixing in degenerate atomic gases,” Phys. Rev. A 64, 023606/1–5 (2001).
[CrossRef]

J. Heurich, H. Pu, M. G. Moore, and P. Meystre, “Instabilities and self-oscillations in atomic four-wave mixing,” Phys. Rev. A 63, 033605/1–7 (2001).
[CrossRef]

M. G. Moore and P. Meystre, “Atomic four-wave mixing: fermions versus bosons,” Phys. Rev. Lett. 86, 4199–4202 (2001).
[CrossRef] [PubMed]

W. Ketterle and S. Inouye, “Does matter wave amplification work for fermions?,” Phys. Rev. Lett. 86, 4203–4206 (2001).
[CrossRef] [PubMed]

Y. Wu and X. Yang, “Quantum theory for micro-cavity enhancement of second-harmonic generation,” J. Phys. B 34, 2281–2288 (2001).
[CrossRef]

G. Alvarez and R. F. Alvarez-Estrada, “Third harmonic generation as a third-order quasi-exactly solvable system,” J. Phys. A 34, 10045–10056 (2001).
[CrossRef]

M. K. Olsen, V. I. Kruglov, and M. J. Collett, “Effects of χ(3) nonlinearities in second-harmonic generation,” Phys. Rev. A 63, 033801/1–7 (2001).
[CrossRef]

A. B. Klimov, L. L. Sánchez-Soto, and J. Delgado, “Mimicking a Kerrlike medium in the dispersive regime of second-harmonic generation,” Opt. Commun. 191, 419–426 (2001).
[CrossRef]

Y. Wu, X. Yang, and Y. Xiao, “Analytical method for yrast line states in interacting Bose–Einstein condensates,” Phys. Rev. Lett. 86, 2200–2203 (2001).
[CrossRef] [PubMed]

M. Alexanian and S. K. Bose, “Generation of photonic superposition states by two-photon absorption,” J. Lumin. 94, 815–819 (2001).
[CrossRef]

R. Franzosi and V. Penna, “Spectral properties of coupled Bose–Einstein condensates,” Phys. Rev. A 63, 043609/1–8 (2001).
[CrossRef]

Y. Wu and X. Yang, “Algebraic method for solving a class of coupled-channel cavity QED models,” Phys. Rev. A 63, 043816/1–5 (2001).
[CrossRef]

2000

Y. Wu, X. Yang, and C. Sun, “Systematical method to study general structure of Bose–Einstein condensates with arbitrary spin,” Phys. Rev. A 62, 063603/1–4 (2000).
[CrossRef]

M. Koashi and M. Ueda, “Exact eigenstates and magnetic response of spin-1 and spin-2 Bose–Einstein condensates,” Phys. Rev. Lett. 84, 1066–1069 (2000).
[CrossRef] [PubMed]

S. Raghavan, H. Pu, C. K. Law, J. H. Eberly, and N. P. Bigelow, “Properties of spinor Bose condensates,” J. Low Temp. Phys. 119, 437–460 (2000).
[CrossRef]

M. Alexanian and S. K. Bose, “Comment on ‘Generation of phase states by two-photon absorption’,” Phys. Rev. Lett. 85, 1136 (2000).
[CrossRef]

H. Pu and P. Meystre, “Creating macroscopic atomic Einstein–Podolsky–Rosen states from Bose–Einstein condensates,” Phys. Rev. Lett. 85, 3987–3990 (2000).
[CrossRef] [PubMed]

L.-M. Duan, A. Sorensen, J. I. Cirac, and P. Zoller, “Squeezing and entanglement of atomic beams,” Phys. Rev. Lett. 85, 3991–3994 (2000).
[CrossRef] [PubMed]

Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, “Theory of four-wave mixing with matter waves without the undepleted pump approximation,” Phys. Rev. A 61, 043604/1–6 (2000).
[CrossRef]

1999

Y. Wu, X. Yang, and P. T. Leung, “Theory of microcavity-enhanced Raman gain,” Opt. Lett. 24, 345–347 (1999).
[CrossRef]

E. V. Goldstein and P. Meystre, “Phase conjugation of multicomponent Bose–Einstein condensates,” Phys. Rev. A 59, 1509–1513 (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, “Four-wave mixing with matter waves,” Nature 398, 218–220 (1999).
[CrossRef]

H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow, “Spin-mixing dynamics of a spinor Bose–Einstein condensate,” Phys. Rev. A 60, 1463–1470 (1999).
[CrossRef]

1998

C. K. Law, H. Pu, and N. P. Bigelow, “Quantum spins in spinor Bose–Einstein condensates,” Phys. Rev. Lett. 81, 5257–5261 (1998).
[CrossRef]

M. D. Girardeau, “Comment on ‘Particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross–Pitaevskii equation for a highly condensed Bose gas’,” Phys. Rev. A 58, 775–778 (1998).
[CrossRef]

Y. Castin and R. Dum, “Low-temperature Bose–Einstein condensates in time-dependent traps: Beyond the U(1) symmetry-breaking approach,” Phys. Rev. A 57, 3008–3021 (1998).
[CrossRef]

M. Trippenbach, Y. B. Band, and P. S. Julienne, “Four-wave mixing in the scattering of Bose–Einstein condensates,” Opt. Express 3, 530–537 (1998).
[CrossRef] [PubMed]

M. Alexanian, S. K. Bose, and L. Chow, “Trapping and photon number states in a two-photon micromaser,” J. Lumin. 76–77, 677–680 (1998).
[CrossRef]

1997

Y. Wu and X. Yang, “Effective two-level model for a three-level atom in the Ξ configuration,” Phys. Rev. A 56, 2443–2446 (1997).
[CrossRef]

C. W. Gardiner, “Particle-number-conserving Bogoliubov method which demonstrates the validity of the time-dependent Gross–Pitaevskii equation for a highly condensed Bose gas,” Phys. Rev. A 56, 1414–1423 (1997).
[CrossRef]

X. Yang, Y. Wu, and Y. Li, “A unified and standardized procedure to solve various nonlinear Jaynes–Cummings models,” Phys. Rev. A 55, 4545–4551 (1997).
[CrossRef]

1996

Y. Wu, “Simple algebraic method to solve a coupled-channel cavity QED model,” Phys. Rev. A 54, 4534–4543 (1996).
[CrossRef] [PubMed]

Y. Wu, “Effective Raman theory for a three-level atom in the Λ configuration,” Phys. Rev. A 54, 1586–1592 (1996).
[CrossRef] [PubMed]

1995

M. Alexanian and S. K. Bose, “Unitary transformation and the dynamics of a three-level atom interacting with two quantized field modes,” Phys. Rev. A 52, 2218–2224 (1995).
[CrossRef] [PubMed]

1994

V. P. Karassiov, “G-invariant polynomial extensions of Lie algebras in quantum many-body physics,” J. Phys. A 27, 153–165 (1994).
[CrossRef]

1992

Liwei Wang, R. R. Puri, and J. H. Eberly, “Coupled-channel cavity QED model and exact solutions,” Phys. Rev. A 46, 7192–7209 (1992).
[CrossRef] [PubMed]

Alexanian, M.

M. Alexanian, “Cavity coherent-state cloning via Raman scattering,” Phys. Rev. A 67, 033809/1–6 (2003).
[CrossRef]

M. Alexanian and S. K. Bose, “Generation of photonic superposition states by two-photon absorption,” J. Lumin. 94, 815–819 (2001).
[CrossRef]

M. Alexanian and S. K. Bose, “Comment on ‘Generation of phase states by two-photon absorption’,” Phys. Rev. Lett. 85, 1136 (2000).
[CrossRef]

M. Alexanian, S. K. Bose, and L. Chow, “Trapping and photon number states in a two-photon micromaser,” J. Lumin. 76–77, 677–680 (1998).
[CrossRef]

M. Alexanian and S. K. Bose, “Unitary transformation and the dynamics of a three-level atom interacting with two quantized field modes,” Phys. Rev. A 52, 2218–2224 (1995).
[CrossRef] [PubMed]

Alvarez, G.

G. Alvarez and R. F. Alvarez-Estrada, “Third harmonic generation as a third-order quasi-exactly solvable system,” J. Phys. A 34, 10045–10056 (2001).
[CrossRef]

Alvarez-Estrada, R. F.

G. Alvarez and R. F. Alvarez-Estrada, “Third harmonic generation as a third-order quasi-exactly solvable system,” J. Phys. A 34, 10045–10056 (2001).
[CrossRef]

Anceau, C.

Andreev, V. A.

V. A. Andreev and O. A. Ivanova, “Symmetries and reduced systems of equations for three-boson and four-boson interactions,” J. Phys. A 35, 8587–8602 (2002).
[CrossRef]

Ballarini, V.

S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giogis, “Second-harmonic generation in amorphous silicon nitride microcavities,” Appl. Phys. Lett. 81, 4706–4708 (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, “Four-wave mixing with matter waves,” Nature 398, 218–220 (1999).
[CrossRef]

Band, Y. B.

Bigelow, N. P.

S. Raghavan, H. Pu, C. K. Law, J. H. Eberly, and N. P. Bigelow, “Properties of spinor Bose condensates,” J. Low Temp. Phys. 119, 437–460 (2000).
[CrossRef]

H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow, “Spin-mixing dynamics of a spinor Bose–Einstein condensate,” Phys. Rev. A 60, 1463–1470 (1999).
[CrossRef]

C. K. Law, H. Pu, and N. P. Bigelow, “Quantum spins in spinor Bose–Einstein condensates,” Phys. Rev. Lett. 81, 5257–5261 (1998).
[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, “Nonlinear phase mismatch and optimal input combination in atomic four-wave mixing in Bose–Einstein condensates,” Phys. Rev. A 67, 013603/1–7 (2003).
[CrossRef]

Bose, S. K.

M. Alexanian and S. K. Bose, “Generation of photonic superposition states by two-photon absorption,” J. Lumin. 94, 815–819 (2001).
[CrossRef]

M. Alexanian and S. K. Bose, “Comment on ‘Generation of phase states by two-photon absorption’,” Phys. Rev. Lett. 85, 1136 (2000).
[CrossRef]

M. Alexanian, S. K. Bose, and L. Chow, “Trapping and photon number states in a two-photon micromaser,” J. Lumin. 76–77, 677–680 (1998).
[CrossRef]

M. Alexanian and S. K. Bose, “Unitary transformation and the dynamics of a three-level atom interacting with two quantized field modes,” Phys. Rev. A 52, 2218–2224 (1995).
[CrossRef] [PubMed]

Brasselet, S.

Castin, Y.

Y. Castin and R. Dum, “Low-temperature Bose–Einstein condensates in time-dependent traps: Beyond the U(1) symmetry-breaking approach,” Phys. Rev. A 57, 3008–3021 (1998).
[CrossRef]

Chow, L.

M. Alexanian, S. K. Bose, and L. Chow, “Trapping and photon number states in a two-photon micromaser,” J. Lumin. 76–77, 677–680 (1998).
[CrossRef]

Cirac, J. I.

L.-M. Duan, A. Sorensen, J. I. Cirac, and P. Zoller, “Squeezing and entanglement of atomic beams,” Phys. Rev. Lett. 85, 3991–3994 (2000).
[CrossRef] [PubMed]

Collett, M. J.

M. K. Olsen, V. I. Kruglov, and M. J. Collett, “Effects of χ(3) nonlinearities in second-harmonic generation,” Phys. Rev. A 63, 033801/1–7 (2001).
[CrossRef]

Cravetchi, I.

Delgado, J.

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L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, “Opening optical four-wave mixing channels with giant enhancement using ultraslow pump waves,” Phys. Rev. Lett. 88, 143902/1–4 (2002).
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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, “Four-wave mixing with matter waves,” Nature 398, 218–220 (1999).
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H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow, “Spin-mixing dynamics of a spinor Bose–Einstein condensate,” Phys. Rev. A 60, 1463–1470 (1999).
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Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, “Nonlinear phase mismatch and optimal input combination in atomic four-wave mixing in Bose–Einstein condensates,” Phys. Rev. A 67, 013603/1–7 (2003).
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L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, “Opening optical four-wave mixing channels with giant enhancement using ultraslow pump waves,” Phys. Rev. Lett. 88, 143902/1–4 (2002).
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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, “Four-wave mixing with matter waves,” Nature 398, 218–220 (1999).
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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, “Four-wave mixing with matter waves,” Nature 398, 218–220 (1999).
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J. Heurich, H. Pu, M. G. Moore, and P. Meystre, “Instabilities and self-oscillations in atomic four-wave mixing,” Phys. Rev. A 63, 033605/1–7 (2001).
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M. Trippenbach, Y. B. Band, and P. S. Julienne, “Four-wave mixing in the scattering of Bose–Einstein condensates,” Opt. Express 3, 530–537 (1998).
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Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, “Nonlinear phase mismatch and optimal input combination in atomic four-wave mixing in Bose–Einstein condensates,” Phys. Rev. A 67, 013603/1–7 (2003).
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Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, “Nonlinear phase mismatch and optimal input combination in atomic four-wave mixing in Bose–Einstein condensates,” Phys. Rev. A 67, 013603/1–7 (2003).
[CrossRef]

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A. B. Klimov, L. L. Sánchez-Soto, and J. Delgado, “Mimicking a Kerrlike medium in the dispersive regime of second-harmonic generation,” Opt. Commun. 191, 419–426 (2001).
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M. Koashi and M. Ueda, “Exact eigenstates and magnetic response of spin-1 and spin-2 Bose–Einstein condensates,” Phys. Rev. Lett. 84, 1066–1069 (2000).
[CrossRef] [PubMed]

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L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, “Opening optical four-wave mixing channels with giant enhancement using ultraslow pump waves,” Phys. Rev. Lett. 88, 143902/1–4 (2002).
[CrossRef] [PubMed]

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S. Raghavan, H. Pu, C. K. Law, J. H. Eberly, and N. P. Bigelow, “Properties of spinor Bose condensates,” J. Low Temp. Phys. 119, 437–460 (2000).
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H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow, “Spin-mixing dynamics of a spinor Bose–Einstein condensate,” Phys. Rev. A 60, 1463–1470 (1999).
[CrossRef]

C. K. Law, H. Pu, and N. P. Bigelow, “Quantum spins in spinor Bose–Einstein condensates,” Phys. Rev. Lett. 81, 5257–5261 (1998).
[CrossRef]

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S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giogis, “Second-harmonic generation in amorphous silicon nitride microcavities,” Appl. Phys. Lett. 81, 4706–4708 (2002).
[CrossRef]

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Lewenstein, M.

P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, “Four-wave mixing in degenerate atomic gases,” Phys. Rev. A 64, 023606/1–5 (2001).
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X. Yang, Y. Wu, and Y. Li, “A unified and standardized procedure to solve various nonlinear Jaynes–Cummings models,” Phys. Rev. A 55, 4545–4551 (1997).
[CrossRef]

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S. Lettieri, S. D. Finizio, P. Maddalena, V. Ballarini, and F. Giogis, “Second-harmonic generation in amorphous silicon nitride microcavities,” Appl. Phys. Lett. 81, 4706–4708 (2002).
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M. G. Moore and P. Meystre, “Atomic four-wave mixing: fermions versus bosons,” Phys. Rev. Lett. 86, 4199–4202 (2001).
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J. Heurich, H. Pu, M. G. Moore, and P. Meystre, “Instabilities and self-oscillations in atomic four-wave mixing,” Phys. Rev. A 63, 033605/1–7 (2001).
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H. Pu and P. Meystre, “Creating macroscopic atomic Einstein–Podolsky–Rosen states from Bose–Einstein condensates,” Phys. Rev. Lett. 85, 3987–3990 (2000).
[CrossRef] [PubMed]

E. V. Goldstein and P. Meystre, “Phase conjugation of multicomponent Bose–Einstein condensates,” Phys. Rev. A 59, 1509–1513 (1999).
[CrossRef]

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I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, and J. Timonen, “Algebraic approach to the Tavis–Cummings problem,” Phys. Rev. A 67, 053808/1–12 (2003).
[CrossRef]

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J. Heurich, H. Pu, M. G. Moore, and P. Meystre, “Instabilities and self-oscillations in atomic four-wave mixing,” Phys. Rev. A 63, 033605/1–7 (2001).
[CrossRef]

M. G. Moore and P. Meystre, “Atomic four-wave mixing: fermions versus bosons,” Phys. Rev. Lett. 86, 4199–4202 (2001).
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O. E. Müstecaplioǧlu, M. Zhang, and L. You, “Spin squeezing and entanglement in spinor condensates,” Phys. Rev. A 66, 033611/1–9 (2002).
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Q. Yang, J. T. Seo, S. Greekmore, D. A. Temple, P. Ye, C. Bonner, M. Namkung, S. S. Jung, and J. H. Kim, “Nonlinear phase mismatch and optimal input combination in atomic four-wave mixing in Bose–Einstein condensates,” Phys. Rev. A 67, 013603/1–7 (2003).
[CrossRef]

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P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, “Four-wave mixing in degenerate atomic gases,” Phys. Rev. A 64, 023606/1–5 (2001).
[CrossRef]

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M. K. Olsen, V. I. Kruglov, and M. J. Collett, “Effects of χ(3) nonlinearities in second-harmonic generation,” Phys. Rev. A 63, 033801/1–7 (2001).
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L. Deng, M. Kozuma, E. W. Hagley, and M. G. Payne, “Opening optical four-wave mixing channels with giant enhancement using ultraslow pump waves,” Phys. Rev. Lett. 88, 143902/1–4 (2002).
[CrossRef] [PubMed]

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R. Franzosi and V. Penna, “Spectral properties of coupled Bose–Einstein condensates,” Phys. Rev. A 63, 043609/1–8 (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, “Four-wave mixing with matter waves,” Nature 398, 218–220 (1999).
[CrossRef]

Podlipensky, A.

Pu, H.

J. Heurich, H. Pu, M. G. Moore, and P. Meystre, “Instabilities and self-oscillations in atomic four-wave mixing,” Phys. Rev. A 63, 033605/1–7 (2001).
[CrossRef]

S. Raghavan, H. Pu, C. K. Law, J. H. Eberly, and N. P. Bigelow, “Properties of spinor Bose condensates,” J. Low Temp. Phys. 119, 437–460 (2000).
[CrossRef]

H. Pu and P. Meystre, “Creating macroscopic atomic Einstein–Podolsky–Rosen states from Bose–Einstein condensates,” Phys. Rev. Lett. 85, 3987–3990 (2000).
[CrossRef] [PubMed]

H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow, “Spin-mixing dynamics of a spinor Bose–Einstein condensate,” Phys. Rev. A 60, 1463–1470 (1999).
[CrossRef]

C. K. Law, H. Pu, and N. P. Bigelow, “Quantum spins in spinor Bose–Einstein condensates,” Phys. Rev. Lett. 81, 5257–5261 (1998).
[CrossRef]

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Liwei Wang, R. R. Puri, and J. H. Eberly, “Coupled-channel cavity QED model and exact solutions,” Phys. Rev. A 46, 7192–7209 (1992).
[CrossRef] [PubMed]

Raghavan, S.

S. Raghavan, H. Pu, C. K. Law, J. H. Eberly, and N. P. Bigelow, “Properties of spinor Bose condensates,” J. Low Temp. Phys. 119, 437–460 (2000).
[CrossRef]

H. Pu, C. K. Law, S. Raghavan, J. H. Eberly, and N. P. Bigelow, “Spin-mixing dynamics of a spinor Bose–Einstein condensate,” Phys. Rev. A 60, 1463–1470 (1999).
[CrossRef]

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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, “Four-wave mixing with matter waves,” Nature 398, 218–220 (1999).
[CrossRef]

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I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, and J. Timonen, “Algebraic approach to the Tavis–Cummings problem,” Phys. Rev. A 67, 053808/1–12 (2003).
[CrossRef]

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Y. Wu, J. Saldana, and Y. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811/1–5 (2003).
[CrossRef]

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A. B. Klimov, L. L. Sánchez-Soto, and J. Delgado, “Mimicking a Kerrlike medium in the dispersive regime of second-harmonic generation,” Opt. Commun. 191, 419–426 (2001).
[CrossRef]

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P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, “Four-wave mixing in degenerate atomic gases,” Phys. Rev. A 64, 023606/1–5 (2001).
[CrossRef]

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P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, “Four-wave mixing in degenerate atomic gases,” Phys. Rev. A 64, 023606/1–5 (2001).
[CrossRef]

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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, “Nonlinear phase mismatch and optimal input combination in atomic four-wave mixing in Bose–Einstein condensates,” Phys. Rev. A 67, 013603/1–7 (2003).
[CrossRef]

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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, “Four-wave mixing with matter waves,” Nature 398, 218–220 (1999).
[CrossRef]

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L.-M. Duan, A. Sorensen, J. I. Cirac, and P. Zoller, “Squeezing and entanglement of atomic beams,” Phys. Rev. Lett. 85, 3991–3994 (2000).
[CrossRef] [PubMed]

Sun, C.

Y. Wu, X. Yang, and C. Sun, “Systematical method to study general structure of Bose–Einstein condensates with arbitrary spin,” Phys. Rev. A 62, 063603/1–4 (2000).
[CrossRef]

Sun, C. P.

Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, “Theory of four-wave mixing with matter waves without the undepleted pump approximation,” Phys. Rev. A 61, 043604/1–6 (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, “Nonlinear phase mismatch and optimal input combination in atomic four-wave mixing in Bose–Einstein condensates,” Phys. Rev. A 67, 013603/1–7 (2003).
[CrossRef]

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I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, and J. Timonen, “Algebraic approach to the Tavis–Cummings problem,” Phys. Rev. A 67, 053808/1–12 (2003).
[CrossRef]

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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, “Four-wave mixing with matter waves,” Nature 398, 218–220 (1999).
[CrossRef]

M. Trippenbach, Y. B. Band, and P. S. Julienne, “Four-wave mixing in the scattering of Bose–Einstein condensates,” Opt. Express 3, 530–537 (1998).
[CrossRef] [PubMed]

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M. Koashi and M. Ueda, “Exact eigenstates and magnetic response of spin-1 and spin-2 Bose–Einstein condensates,” Phys. Rev. Lett. 84, 1066–1069 (2000).
[CrossRef] [PubMed]

Vadeiko, I. P.

I. P. Vadeiko, G. P. Miroshnichenko, A. V. Rybin, and J. Timonen, “Algebraic approach to the Tavis–Cummings problem,” Phys. Rev. A 67, 053808/1–12 (2003).
[CrossRef]

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P. Villain, P. Ohberg, L. Santos, A. Sanpera, and M. Lewenstein, “Four-wave mixing in degenerate atomic gases,” Phys. Rev. A 64, 023606/1–5 (2001).
[CrossRef]

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J. M. Vogels, K. Xu, and W. Ketterle, “Generation of macroscopic pair-correlated atomic beams by four-wave mixing in Bose–Einstein condensates,” Phys. Rev. Lett. 89, 020401/1–4 (2002).
[CrossRef]

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Liwei Wang, R. R. Puri, and J. H. Eberly, “Coupled-channel cavity QED model and exact solutions,” Phys. Rev. A 46, 7192–7209 (1992).
[CrossRef] [PubMed]

Wang, Y. Q.

Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, “Theory of four-wave mixing with matter waves without the undepleted pump approximation,” Phys. Rev. A 61, 043604/1–6 (2000).
[CrossRef]

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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, “Four-wave mixing with matter waves,” Nature 398, 218–220 (1999).
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Wu, Y.

Y. Wu, L. Wen, and Y. Zhu, “Efficient hyper-Raman scattering in resonant coherent media,” Opt. Lett. 28, 631–633 (2003).
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X. X. Yang and Y. Wu, “Coherent superposition states of atoms and molecules in a Bose–Einstein condensate with exactly balanced photoassociations and photodissociations,” Chin. Phys. Lett. 20, 189–191 (2003).
[CrossRef]

Y. Wu and X. Yang, “Exact eigenstates for a class of models describing two-mode multiphoton processes,” Opt. Lett. 28, 1793–1795 (2003).
[CrossRef] [PubMed]

Y. Wu and X. Yang, “Analytical results for energy spectrum and eigenstates of Bose–Einstein condensate in Mott insulator state,” Phys. Rev. A 68, 013608/1–7 (2003).
[CrossRef]

Y. Wu, J. Saldana, and Y. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67, 013811/1–5 (2003).
[CrossRef]

Y. Wu and X. Yang, “Quantum theory for micro-cavity enhancement of second-harmonic generation,” J. Phys. B 34, 2281–2288 (2001).
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Y. Wu, X. Yang, and Y. Xiao, “Analytical method for yrast line states in interacting Bose–Einstein condensates,” Phys. Rev. Lett. 86, 2200–2203 (2001).
[CrossRef] [PubMed]

Y. Wu and X. Yang, “Algebraic method for solving a class of coupled-channel cavity QED models,” Phys. Rev. A 63, 043816/1–5 (2001).
[CrossRef]

Y. Wu, X. Yang, and C. Sun, “Systematical method to study general structure of Bose–Einstein condensates with arbitrary spin,” Phys. Rev. A 62, 063603/1–4 (2000).
[CrossRef]

Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, “Theory of four-wave mixing with matter waves without the undepleted pump approximation,” Phys. Rev. A 61, 043604/1–6 (2000).
[CrossRef]

Y. Wu, X. Yang, and P. T. Leung, “Theory of microcavity-enhanced Raman gain,” Opt. Lett. 24, 345–347 (1999).
[CrossRef]

Y. Wu and X. Yang, “Effective two-level model for a three-level atom in the Ξ configuration,” Phys. Rev. A 56, 2443–2446 (1997).
[CrossRef]

X. Yang, Y. Wu, and Y. Li, “A unified and standardized procedure to solve various nonlinear Jaynes–Cummings models,” Phys. Rev. A 55, 4545–4551 (1997).
[CrossRef]

Y. Wu, “Simple algebraic method to solve a coupled-channel cavity QED model,” Phys. Rev. A 54, 4534–4543 (1996).
[CrossRef] [PubMed]

Y. Wu, “Effective Raman theory for a three-level atom in the Λ configuration,” Phys. Rev. A 54, 1586–1592 (1996).
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Y. Wu, X. Yang, and Y. Xiao, “Analytical method for yrast line states in interacting Bose–Einstein condensates,” Phys. Rev. Lett. 86, 2200–2203 (2001).
[CrossRef] [PubMed]

Xu, K.

J. M. Vogels, K. Xu, and W. Ketterle, “Generation of macroscopic pair-correlated atomic beams by four-wave mixing in Bose–Einstein condensates,” Phys. Rev. Lett. 89, 020401/1–4 (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, “Nonlinear phase mismatch and optimal input combination in atomic four-wave mixing in Bose–Einstein condensates,” Phys. Rev. A 67, 013603/1–7 (2003).
[CrossRef]

Yang, X.

Y. Wu and X. Yang, “Analytical results for energy spectrum and eigenstates of Bose–Einstein condensate in Mott insulator state,” Phys. Rev. A 68, 013608/1–7 (2003).
[CrossRef]

Y. Wu and X. Yang, “Exact eigenstates for a class of models describing two-mode multiphoton processes,” Opt. Lett. 28, 1793–1795 (2003).
[CrossRef] [PubMed]

Y. Wu, X. Yang, and Y. Xiao, “Analytical method for yrast line states in interacting Bose–Einstein condensates,” Phys. Rev. Lett. 86, 2200–2203 (2001).
[CrossRef] [PubMed]

Y. Wu and X. Yang, “Quantum theory for micro-cavity enhancement of second-harmonic generation,” J. Phys. B 34, 2281–2288 (2001).
[CrossRef]

Y. Wu and X. Yang, “Algebraic method for solving a class of coupled-channel cavity QED models,” Phys. Rev. A 63, 043816/1–5 (2001).
[CrossRef]

Y. Wu, X. Yang, and C. Sun, “Systematical method to study general structure of Bose–Einstein condensates with arbitrary spin,” Phys. Rev. A 62, 063603/1–4 (2000).
[CrossRef]

Y. Wu, X. Yang, C. P. Sun, X. J. Zhou, and Y. Q. Wang, “Theory of four-wave mixing with matter waves without the undepleted pump approximation,” Phys. Rev. A 61, 043604/1–6 (2000).
[CrossRef]

Y. Wu, X. Yang, and P. T. Leung, “Theory of microcavity-enhanced Raman gain,” Opt. Lett. 24, 345–347 (1999).
[CrossRef]

Y. Wu and X. Yang, “Effective two-level model for a three-level atom in the Ξ configuration,” Phys. Rev. A 56, 2443–2446 (1997).
[CrossRef]

X. Yang, Y. Wu, and Y. Li, “A unified and standardized procedure to solve various nonlinear Jaynes–Cummings models,” Phys. Rev. A 55, 4545–4551 (1997).
[CrossRef]

Yang, X. X.

X. X. Yang and Y. Wu, “Coherent superposition states of atoms and molecules in a Bose–Einstein condensate with exactly balanced photoassociations and photodissociations,” Chin. Phys. Lett. 20, 189–191 (2003).
[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, “Nonlinear phase mismatch and optimal input combination in atomic four-wave mixing in Bose–Einstein condensates,” Phys. Rev. A 67, 013603/1–7 (2003).
[CrossRef]

You, L.

L. You, “Creating maximally entangled atomic states in a Bose–Einstein condensate,” Phys. Rev. Lett. 90, 030402/1–4 (2003).
[CrossRef]

O. E. Müstecaplioǧlu, M. Zhang, and L. You, “Spin squeezing and entanglement in spinor condensates,” Phys. Rev. A 66, 033611/1–9 (2002).
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Equations (36)

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Hˆ=Hˆ0+gHˆ1=j=13ωjaˆjaˆj+g(aˆ1aˆ2aˆ3+aˆ3aˆ2aˆ1),
Mˆ=aˆ1aˆ1+aˆ2aˆ2,Nˆ=aˆ1aˆ1+aˆ3aˆ3,
H=ω2Mˆ+ω3Nˆ+g(aˆ1aˆ2aˆ3+aˆ3aˆ2aˆ1).
E=ω2M+ω3N+gλ,
|ΨM,N,λ=j=0qcj|j, M-j, N-j,
|ΨM,N,λ=S(aˆ1, aˆ2, aˆ3)|vac,
x1 2x2x3+x2x3 x1S(x1, x2, x3)=λS,
S(x1, x2, x3)=j=0qαjx1jx2M-jx3N-j/j!x2Mx3Nf(z),
z3f+[1-(M+N-1)z2]f+MNzf=λf,
f(z)=CNzNF-N2, 1-N2+M2, 12z2+CMzMF-M2, 1-M2+N2, 12z2,
F(a, b, x)F1(a, b, x)=k=0 (a)k(b)k xkk!,
f(z)=Ck=0q/2(-1)k z2k(M-2k)!!(N-2k)!!(2k)!!,
|ΨM,N,λ=0=Ck=0q/2(-1)k [(2k)!(M-2k)!(N-2k)!]1/2(2k)!!(N-2k)!!(M-2k)!!×|2k, M-2k, N-2k,
αj+1=λαj-pj-1αj-10jq,
det A(q+1)(λ)=0q=min(M, N),
αj=α0 det A(j)(λ)j=0, 1, 2,,q,
det A(n+1)=λ det A(n)-pn-1 det A(n-1)n2
E=ω2M+ω3N+gλM,N=0,1,,
|ΨM,N,λ=k=0min(M,N)ck|k, M-k, N-k,
ck=c0(M-k)!(N-k)!k!M!N!1/2 det A(k)(λ),
det A(n)(λ)=λn+k=1int(n/2)(-1)kβk(n)λn-2kn2,
βk(n)=i1=2k-2n-2i2=2k-4i1-2im=2(k-m)im-1-2ik=0ik-1-2pi1pi2pik,
λ=0  q=0,
λ=±Q1/2q=1,
λ=0,±(4Q-2)1/2q=2,
λ=±{5(Q-1)±[16(Q-1)2+9]1/2}1/2
q=3,
λ=0,±{10Q±3[(2Q-3)2+8]1/2-15}1/2
q=4,
λ=±13 35(Q-2)+4[112Q(Q-4)+691]1/2×cos(2sπ+θ)31/2,
θ=cos-120(Q-2)(32Q2-128Q+371)[112Q(Q-4)+691]3/2,
s=0, 1, 2q=5,
λ=0,±13 28(2Q-5)+4R cos(2sπ+ϕ)31/2,
s=0, 1, 2,
ϕ=cos-14Q[43Q(2Q-15)+9615]-23950)R3,
R=[196Q(Q-5)+2035]1/2q=6.

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