Abstract

Using the counterpart of the Gross-Pitaevskii equation, we study a system of atomic and molecular condensates in equilibrium in the presence of photoassociating light. All equilibria except a special case with only molecules are prone to an analog of the modulational instability in second-harmonic generation. The nature of the instability is such that the atoms and molecules aggregate in dense clumps.

© 1999 Optical Society of America

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References

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  1. P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates, ” Phys. Rev. A 58, R797–800 (1998).
    [Crossref]
  2. J. Javanainen and M. Mackie, “Probability of photoassociation from a quasicontinuum approach, ” Phys. Rev. A 58, R789–92 (1998).
    [Crossref]
  3. M. Mackie and J. Javanainen, “Quasicontinuum modeling of photoassociation, ” Phys. Rev. A 60, 3174–87 (1999).
    [Crossref]
  4. J. Javanainen and M. Mackie, “Coherent photoassociation of a Bose-Einstein condensate, ” Phys. Rev. A 59, R3186–9 (1999).
    [Crossref]
  5. P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates, ” Phys. Rev. Lett. 81, 3055–8 (1998).
    [Crossref]
  6. S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation, ” Optics Lett. 20, 438–40 (1995).
    [Crossref]
  7. H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity, ” Opt. Commun. 123, 394–402 (1996).
    [Crossref]
  8. M. Koštrunet al., unpublished.
  9. P. Tommasini, E. Timmermans, M. Hussein, and A. Kerman, “Feshbach Resonance and Hybrid Atomic/Molecular BEC-Systems, ” preprint http://xxx.lanl.gov/abs/cond-mat/9804015.
  10. H. He, M. J. Werner, and P. D. Drummond, “Simultaneous solitary-wave solutions in a nonlinear parametric waveguide, ” Phys. Rev. E 54, 896–911 (1996).
    [Crossref]
  11. P. A. Ruprecht, M. Edwards, K. Burnett, and C. W. Clark, “Probing the linear and nonlinear excitations of Bose-condensed neutral atoms in a trap, ” Phys. Rev. A 54, 4178–87 (1996).
    [Crossref] [PubMed]
  12. E. V. Goldstein and P. Meystre, “Quasiparticle instabilities in multicomponent atomic condensates, ” Phys. Rev. A 55, 2935–40 (1997).
    [Crossref]
  13. B. D. Esry and C. H. Greene, “Low-lying excitations of double Bose-Einstein condensates, ” Phys. Rev. A 57, 1265–71 (1998).
    [Crossref]
  14. H. Pu and N. P. Bigelow, “Collective excitations, metastability, and nonlinear response of a trapped two-species Bose-Einstein condensate, ” Phys. Rev. Lett. 80, 1134–7 (1998).
    [Crossref]
  15. A. L. Fetter, “Nonuniform states of an imperfect Bose gas, ” Ann. Phys. (N.Y.) 70, 67–101 (1972).
    [Crossref]
  16. P. Öhberg and S. Stenholm, “Hartree-Fock treatment of the two-component Bose-Einstein condensate, ” Phys. Rev. A 57, 1272–9 (1998).
    [Crossref]
  17. D. Gordon and G. M. Savage, “Excitation spectrum and instability of a two-species Bose-Einstein condensate, ” Phys. Rev. A 58, 1440–4 (1998).
    [Crossref]
  18. B. D. Esry and C. H. Greene, “Spontaneous spatial symmetry breaking in two-component Bose-Einstein condensates, ” Phys. Rev. A 59, 1457–60 (1999).
    [Crossref]
  19. A. Sinatra, P. O. Fedichev, Y. Castin, J. Dalibard, and G. V. Shlyapnikov, “Dynamics of two interacting Bose-Einstein condensates, ” Phys. Rev. Lett. 82, 251–4 (1999).
    [Crossref]
  20. 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, 1463–70 (1999).
    [Crossref]
  21. D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media, ” Opt. Commun. 148, 79–84 (1998).
    [Crossref]

1999 (5)

M. Mackie and J. Javanainen, “Quasicontinuum modeling of photoassociation, ” Phys. Rev. A 60, 3174–87 (1999).
[Crossref]

J. Javanainen and M. Mackie, “Coherent photoassociation of a Bose-Einstein condensate, ” Phys. Rev. A 59, R3186–9 (1999).
[Crossref]

B. D. Esry and C. H. Greene, “Spontaneous spatial symmetry breaking in two-component Bose-Einstein condensates, ” Phys. Rev. A 59, 1457–60 (1999).
[Crossref]

A. Sinatra, P. O. Fedichev, Y. Castin, J. Dalibard, and G. V. Shlyapnikov, “Dynamics of two interacting Bose-Einstein condensates, ” Phys. Rev. Lett. 82, 251–4 (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, 1463–70 (1999).
[Crossref]

1998 (8)

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media, ” Opt. Commun. 148, 79–84 (1998).
[Crossref]

B. D. Esry and C. H. Greene, “Low-lying excitations of double Bose-Einstein condensates, ” Phys. Rev. A 57, 1265–71 (1998).
[Crossref]

H. Pu and N. P. Bigelow, “Collective excitations, metastability, and nonlinear response of a trapped two-species Bose-Einstein condensate, ” Phys. Rev. Lett. 80, 1134–7 (1998).
[Crossref]

P. Öhberg and S. Stenholm, “Hartree-Fock treatment of the two-component Bose-Einstein condensate, ” Phys. Rev. A 57, 1272–9 (1998).
[Crossref]

D. Gordon and G. M. Savage, “Excitation spectrum and instability of a two-species Bose-Einstein condensate, ” Phys. Rev. A 58, 1440–4 (1998).
[Crossref]

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates, ” Phys. Rev. Lett. 81, 3055–8 (1998).
[Crossref]

P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates, ” Phys. Rev. A 58, R797–800 (1998).
[Crossref]

J. Javanainen and M. Mackie, “Probability of photoassociation from a quasicontinuum approach, ” Phys. Rev. A 58, R789–92 (1998).
[Crossref]

1997 (1)

E. V. Goldstein and P. Meystre, “Quasiparticle instabilities in multicomponent atomic condensates, ” Phys. Rev. A 55, 2935–40 (1997).
[Crossref]

1996 (3)

H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity, ” Opt. Commun. 123, 394–402 (1996).
[Crossref]

H. He, M. J. Werner, and P. D. Drummond, “Simultaneous solitary-wave solutions in a nonlinear parametric waveguide, ” Phys. Rev. E 54, 896–911 (1996).
[Crossref]

P. A. Ruprecht, M. Edwards, K. Burnett, and C. W. Clark, “Probing the linear and nonlinear excitations of Bose-condensed neutral atoms in a trap, ” Phys. Rev. A 54, 4178–87 (1996).
[Crossref] [PubMed]

1995 (1)

S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation, ” Optics Lett. 20, 438–40 (1995).
[Crossref]

1972 (1)

A. L. Fetter, “Nonuniform states of an imperfect Bose gas, ” Ann. Phys. (N.Y.) 70, 67–101 (1972).
[Crossref]

Band, Y. B.

P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates, ” Phys. Rev. A 58, R797–800 (1998).
[Crossref]

Bigelow, N. P.

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, 1463–70 (1999).
[Crossref]

H. Pu and N. P. Bigelow, “Collective excitations, metastability, and nonlinear response of a trapped two-species Bose-Einstein condensate, ” Phys. Rev. Lett. 80, 1134–7 (1998).
[Crossref]

Burnett, K.

P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates, ” Phys. Rev. A 58, R797–800 (1998).
[Crossref]

P. A. Ruprecht, M. Edwards, K. Burnett, and C. W. Clark, “Probing the linear and nonlinear excitations of Bose-condensed neutral atoms in a trap, ” Phys. Rev. A 54, 4178–87 (1996).
[Crossref] [PubMed]

Castin, Y.

A. Sinatra, P. O. Fedichev, Y. Castin, J. Dalibard, and G. V. Shlyapnikov, “Dynamics of two interacting Bose-Einstein condensates, ” Phys. Rev. Lett. 82, 251–4 (1999).
[Crossref]

Clark, C. W.

P. A. Ruprecht, M. Edwards, K. Burnett, and C. W. Clark, “Probing the linear and nonlinear excitations of Bose-condensed neutral atoms in a trap, ” Phys. Rev. A 54, 4178–87 (1996).
[Crossref] [PubMed]

Dalibard, J.

A. Sinatra, P. O. Fedichev, Y. Castin, J. Dalibard, and G. V. Shlyapnikov, “Dynamics of two interacting Bose-Einstein condensates, ” Phys. Rev. Lett. 82, 251–4 (1999).
[Crossref]

Drummond, P. D.

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates, ” Phys. Rev. Lett. 81, 3055–8 (1998).
[Crossref]

H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity, ” Opt. Commun. 123, 394–402 (1996).
[Crossref]

H. He, M. J. Werner, and P. D. Drummond, “Simultaneous solitary-wave solutions in a nonlinear parametric waveguide, ” Phys. Rev. E 54, 896–911 (1996).
[Crossref]

Eberly, J. H.

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, 1463–70 (1999).
[Crossref]

Edwards, M.

P. A. Ruprecht, M. Edwards, K. Burnett, and C. W. Clark, “Probing the linear and nonlinear excitations of Bose-condensed neutral atoms in a trap, ” Phys. Rev. A 54, 4178–87 (1996).
[Crossref] [PubMed]

Esry, B. D.

B. D. Esry and C. H. Greene, “Spontaneous spatial symmetry breaking in two-component Bose-Einstein condensates, ” Phys. Rev. A 59, 1457–60 (1999).
[Crossref]

B. D. Esry and C. H. Greene, “Low-lying excitations of double Bose-Einstein condensates, ” Phys. Rev. A 57, 1265–71 (1998).
[Crossref]

Fedichev, P. O.

A. Sinatra, P. O. Fedichev, Y. Castin, J. Dalibard, and G. V. Shlyapnikov, “Dynamics of two interacting Bose-Einstein condensates, ” Phys. Rev. Lett. 82, 251–4 (1999).
[Crossref]

Ferro, P.

S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation, ” Optics Lett. 20, 438–40 (1995).
[Crossref]

Fetter, A. L.

A. L. Fetter, “Nonuniform states of an imperfect Bose gas, ” Ann. Phys. (N.Y.) 70, 67–101 (1972).
[Crossref]

Firth, W. J.

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media, ” Opt. Commun. 148, 79–84 (1998).
[Crossref]

Goldstein, E. V.

E. V. Goldstein and P. Meystre, “Quasiparticle instabilities in multicomponent atomic condensates, ” Phys. Rev. A 55, 2935–40 (1997).
[Crossref]

Gordon, D.

D. Gordon and G. M. Savage, “Excitation spectrum and instability of a two-species Bose-Einstein condensate, ” Phys. Rev. A 58, 1440–4 (1998).
[Crossref]

Greene, C. H.

B. D. Esry and C. H. Greene, “Spontaneous spatial symmetry breaking in two-component Bose-Einstein condensates, ” Phys. Rev. A 59, 1457–60 (1999).
[Crossref]

B. D. Esry and C. H. Greene, “Low-lying excitations of double Bose-Einstein condensates, ” Phys. Rev. A 57, 1265–71 (1998).
[Crossref]

He, H.

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates, ” Phys. Rev. Lett. 81, 3055–8 (1998).
[Crossref]

H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity, ” Opt. Commun. 123, 394–402 (1996).
[Crossref]

H. He, M. J. Werner, and P. D. Drummond, “Simultaneous solitary-wave solutions in a nonlinear parametric waveguide, ” Phys. Rev. E 54, 896–911 (1996).
[Crossref]

Hussein, M.

P. Tommasini, E. Timmermans, M. Hussein, and A. Kerman, “Feshbach Resonance and Hybrid Atomic/Molecular BEC-Systems, ” preprint http://xxx.lanl.gov/abs/cond-mat/9804015.

Javanainen, J.

M. Mackie and J. Javanainen, “Quasicontinuum modeling of photoassociation, ” Phys. Rev. A 60, 3174–87 (1999).
[Crossref]

J. Javanainen and M. Mackie, “Coherent photoassociation of a Bose-Einstein condensate, ” Phys. Rev. A 59, R3186–9 (1999).
[Crossref]

J. Javanainen and M. Mackie, “Probability of photoassociation from a quasicontinuum approach, ” Phys. Rev. A 58, R789–92 (1998).
[Crossref]

Julienne, P. S.

P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates, ” Phys. Rev. A 58, R797–800 (1998).
[Crossref]

Kerman, A.

P. Tommasini, E. Timmermans, M. Hussein, and A. Kerman, “Feshbach Resonance and Hybrid Atomic/Molecular BEC-Systems, ” preprint http://xxx.lanl.gov/abs/cond-mat/9804015.

Kheruntsyan, K. V.

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates, ” Phys. Rev. Lett. 81, 3055–8 (1998).
[Crossref]

Koštrun, M.

M. Koštrunet al., unpublished.

Law, C. K.

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, 1463–70 (1999).
[Crossref]

Mackie, M.

M. Mackie and J. Javanainen, “Quasicontinuum modeling of photoassociation, ” Phys. Rev. A 60, 3174–87 (1999).
[Crossref]

J. Javanainen and M. Mackie, “Coherent photoassociation of a Bose-Einstein condensate, ” Phys. Rev. A 59, R3186–9 (1999).
[Crossref]

J. Javanainen and M. Mackie, “Probability of photoassociation from a quasicontinuum approach, ” Phys. Rev. A 58, R789–92 (1998).
[Crossref]

Malomed, B. A.

H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity, ” Opt. Commun. 123, 394–402 (1996).
[Crossref]

Meystre, P.

E. V. Goldstein and P. Meystre, “Quasiparticle instabilities in multicomponent atomic condensates, ” Phys. Rev. A 55, 2935–40 (1997).
[Crossref]

Öhberg, P.

P. Öhberg and S. Stenholm, “Hartree-Fock treatment of the two-component Bose-Einstein condensate, ” Phys. Rev. A 57, 1272–9 (1998).
[Crossref]

Pu, H.

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, 1463–70 (1999).
[Crossref]

H. Pu and N. P. Bigelow, “Collective excitations, metastability, and nonlinear response of a trapped two-species Bose-Einstein condensate, ” Phys. Rev. Lett. 80, 1134–7 (1998).
[Crossref]

Raghavan, S.

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, 1463–70 (1999).
[Crossref]

Ruprecht, P. A.

P. A. Ruprecht, M. Edwards, K. Burnett, and C. W. Clark, “Probing the linear and nonlinear excitations of Bose-condensed neutral atoms in a trap, ” Phys. Rev. A 54, 4178–87 (1996).
[Crossref] [PubMed]

Savage, G. M.

D. Gordon and G. M. Savage, “Excitation spectrum and instability of a two-species Bose-Einstein condensate, ” Phys. Rev. A 58, 1440–4 (1998).
[Crossref]

Shlyapnikov, G. V.

A. Sinatra, P. O. Fedichev, Y. Castin, J. Dalibard, and G. V. Shlyapnikov, “Dynamics of two interacting Bose-Einstein condensates, ” Phys. Rev. Lett. 82, 251–4 (1999).
[Crossref]

Sinatra, A.

A. Sinatra, P. O. Fedichev, Y. Castin, J. Dalibard, and G. V. Shlyapnikov, “Dynamics of two interacting Bose-Einstein condensates, ” Phys. Rev. Lett. 82, 251–4 (1999).
[Crossref]

Skryabin, D. V.

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media, ” Opt. Commun. 148, 79–84 (1998).
[Crossref]

Stenholm, S.

P. Öhberg and S. Stenholm, “Hartree-Fock treatment of the two-component Bose-Einstein condensate, ” Phys. Rev. A 57, 1272–9 (1998).
[Crossref]

Stwalley, W. C.

P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates, ” Phys. Rev. A 58, R797–800 (1998).
[Crossref]

Timmermans, E.

P. Tommasini, E. Timmermans, M. Hussein, and A. Kerman, “Feshbach Resonance and Hybrid Atomic/Molecular BEC-Systems, ” preprint http://xxx.lanl.gov/abs/cond-mat/9804015.

Tommasini, P.

P. Tommasini, E. Timmermans, M. Hussein, and A. Kerman, “Feshbach Resonance and Hybrid Atomic/Molecular BEC-Systems, ” preprint http://xxx.lanl.gov/abs/cond-mat/9804015.

Trillo, S.

S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation, ” Optics Lett. 20, 438–40 (1995).
[Crossref]

Werner, M. J.

H. He, M. J. Werner, and P. D. Drummond, “Simultaneous solitary-wave solutions in a nonlinear parametric waveguide, ” Phys. Rev. E 54, 896–911 (1996).
[Crossref]

Ann. Phys. (N.Y.) (1)

A. L. Fetter, “Nonuniform states of an imperfect Bose gas, ” Ann. Phys. (N.Y.) 70, 67–101 (1972).
[Crossref]

Opt. Commun. (2)

H. He, P. D. Drummond, and B. A. Malomed, “Modulational stability in dispersive optical systems with cascaded nonlinearity, ” Opt. Commun. 123, 394–402 (1996).
[Crossref]

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media, ” Opt. Commun. 148, 79–84 (1998).
[Crossref]

Optics Lett. (1)

S. Trillo and P. Ferro, “Modulational instability in second-harmonic generation, ” Optics Lett. 20, 438–40 (1995).
[Crossref]

Phys. Rev. A (11)

P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, “Stimulated Raman molecule production in Bose-Einstein condensates, ” Phys. Rev. A 58, R797–800 (1998).
[Crossref]

J. Javanainen and M. Mackie, “Probability of photoassociation from a quasicontinuum approach, ” Phys. Rev. A 58, R789–92 (1998).
[Crossref]

M. Mackie and J. Javanainen, “Quasicontinuum modeling of photoassociation, ” Phys. Rev. A 60, 3174–87 (1999).
[Crossref]

J. Javanainen and M. Mackie, “Coherent photoassociation of a Bose-Einstein condensate, ” Phys. Rev. A 59, R3186–9 (1999).
[Crossref]

P. Öhberg and S. Stenholm, “Hartree-Fock treatment of the two-component Bose-Einstein condensate, ” Phys. Rev. A 57, 1272–9 (1998).
[Crossref]

D. Gordon and G. M. Savage, “Excitation spectrum and instability of a two-species Bose-Einstein condensate, ” Phys. Rev. A 58, 1440–4 (1998).
[Crossref]

B. D. Esry and C. H. Greene, “Spontaneous spatial symmetry breaking in two-component Bose-Einstein condensates, ” Phys. Rev. A 59, 1457–60 (1999).
[Crossref]

P. A. Ruprecht, M. Edwards, K. Burnett, and C. W. Clark, “Probing the linear and nonlinear excitations of Bose-condensed neutral atoms in a trap, ” Phys. Rev. A 54, 4178–87 (1996).
[Crossref] [PubMed]

E. V. Goldstein and P. Meystre, “Quasiparticle instabilities in multicomponent atomic condensates, ” Phys. Rev. A 55, 2935–40 (1997).
[Crossref]

B. D. Esry and C. H. Greene, “Low-lying excitations of double Bose-Einstein condensates, ” Phys. Rev. A 57, 1265–71 (1998).
[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, 1463–70 (1999).
[Crossref]

Phys. Rev. E (1)

H. He, M. J. Werner, and P. D. Drummond, “Simultaneous solitary-wave solutions in a nonlinear parametric waveguide, ” Phys. Rev. E 54, 896–911 (1996).
[Crossref]

Phys. Rev. Lett. (3)

P. D. Drummond, K. V. Kheruntsyan, and H. He, “Coherent molecular solitons in Bose-Einstein condensates, ” Phys. Rev. Lett. 81, 3055–8 (1998).
[Crossref]

H. Pu and N. P. Bigelow, “Collective excitations, metastability, and nonlinear response of a trapped two-species Bose-Einstein condensate, ” Phys. Rev. Lett. 80, 1134–7 (1998).
[Crossref]

A. Sinatra, P. O. Fedichev, Y. Castin, J. Dalibard, and G. V. Shlyapnikov, “Dynamics of two interacting Bose-Einstein condensates, ” Phys. Rev. Lett. 82, 251–4 (1999).
[Crossref]

Other (2)

M. Koštrunet al., unpublished.

P. Tommasini, E. Timmermans, M. Hussein, and A. Kerman, “Feshbach Resonance and Hybrid Atomic/Molecular BEC-Systems, ” preprint http://xxx.lanl.gov/abs/cond-mat/9804015.

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Figures (2)

Figure 1.
Figure 1.

The largest imaginary part among the four evolution frequencies of small perturbations of the joint atom-molecule condensate is plotted as a function of the laser detuning δ̄ and the wave number of the plane-wave like excitation p. The excitation and the light are taken to propagate in orthogonal directions.

Figure 2.
Figure 2.

Evolution of the absolute square of the atomic field is plotted as a function of position x and time t. At t=0 the fields start with the equilibrium values, though a small amount of noise is articially added to bring forth the instability. The parameters are δ̄=0, ξ=0.

Equations (15)

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

𝓗 = ψ ( δ 2 4 m ) ψ + φ ( 2 2 m ) φ 1 2 [ E + · ψ d 3 r ' φ ( r + 1 2 r ' ) d ( r ' ) φ ( r 1 2 r ' ) + H . C . ] .
d kk ' = 1 V d 3 r e i ( k k ' ) 2 · r d ( r ) ,
1 2 E + · ψ d 3 r ' φ ( r + 1 2 r ' ) d ( r ' ) φ ( r 1 2 r ' )
1 2 E + · [ d 3 r ' d ( r ' ) ] ψ φ φ = Ω 2 ρ e i q · r ψ φ φ .
Ω = lim υ 0 2 π 2 Γ ( υ ) ρ μ 2 υ .
1 V d 3 r ( φ 2 + 2 ψ 2 ) = 1 .
i φ ˙ = 1 2 2 φ μ φ φ * ψ ,
i ψ ˙ = 1 4 2 ψ + 1 2 i q · ψ ( 2 μ δ ) ψ 1 2 φ 2 .
ψ 0 = 1 2 , φ 0 = 0 , μ = 1 2 δ ̅ .
ψ 0 = μ = 6 + δ 2 δ 6 , φ 0 = 1 2 ψ 0 2 .
φ ( r , t ) = φ 0 + u φ e i ( p · r ω t ) + υ φ * e i ( p · r ω * t ) ,
[ 1 2 p 2 + μ + ω ] u φ + ψ 0 υ φ + φ 0 u ψ = 0 ,
[ 1 2 p 2 + μ + ω ] υ φ + ψ 0 u φ + φ 0 υ ψ = 0 ,
[ 1 4 ( p 2 + 2 p · q ) + 2 μ δ + ω ] u ψ + φ 0 u φ = 0 ,
[ 1 4 ( p 2 + 2 p · q ) + 2 μ δ + ω ] υ ψ + φ 0 υ φ = 0 .

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