Abstract

We overview our recent theoretical studies on nonlinear atom optics of the Bose-Einstein condensates (BECs) loaded into optical lattices. In particular, we describe the band-gap spectrum and nonlinear localization of BECs in one- and two-dimensional optical lattices. We discuss the structure and stability properties of spatially localized states (matter-wave solitons) in 1D lattices, as well as trivial and vortex-like bound states of 2D gap solitons. To highlight similarities between the behavior of coherent light and matter waves in periodic potentials, we draw useful parallels with the physics of coherent light waves in nonlinear photonic crystals and optically-induced photonic lattices.

© 2004 Optical Society of America

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  1. J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
  2. S.F. Mingaleev and Yu.S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474 (2001).
    [CrossRef] [PubMed]
  3. R. Slusher and B. Eggleton, eds., Nonlinear Photonic Crystals (Springer-Verlag, Berlin, 2003).
  4. Yu.S. Kivshar and G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
  5. J.W. Fleischer, T. Carmon, and M. Segev, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
    [CrossRef] [PubMed]
  6. D. Neshev, E.A. Ostrovskaya, Yu.S. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. 28, 710 (2003).
    [CrossRef] [PubMed]
  7. J.W. Fleischer, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147 (2003).
    [CrossRef] [PubMed]
  8. J.H. Denschlag, J.E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S.L. Rolston, and W.D. Phillips, “A Bose-Einstein condensate in an optical lattice,” J. Phys. B 35, 3095 (2002).
    [CrossRef]
  9. S. Peil, J.V. Porto, B. Laburthe Tolra, J.M. Obrecht, B.E. King, M. Subbotin, S.L. Rolston, and W.D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A 67, 051603 (2003).
    [CrossRef]
  10. M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener tunneling in a periodic potential,” Phys. Rev. Lett.,  91, 230406 (2003).
    [CrossRef] [PubMed]
  11. E.A. Ostrovskaya and Yu.S. Kivshar, “Matter-wave gap solitons in atomic band-gap structures,” Phys. Rev. Lett. 90, 160407 (2003).
    [CrossRef] [PubMed]
  12. O. Zobay, S. Pötting, P. Meystre, and E.M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A 59, 643 (1999).
    [CrossRef]
  13. V.V. Konotop and M. Salerno, “Modulational instability in Bose-Einstein condensates in optical lattices,” Phys. Rev. A 65, 021602 (2002).
    [CrossRef]
  14. P.J. Louis, E.A. Ostrovskaya, C.M. Savage, and Yu.S. Kivshar, “Bose-Einstein condensates in optical lattices: band-gap structure and solitons,” Phys. Rev. A 67, 013602 (2003).
    [CrossRef]
  15. N.K. Efremidis and D.N. Christodoulides, “Lattice solitons in Bose-Einstein condensates,” Phys. Rev. A 67063608 (2003).
    [CrossRef]
  16. E.A. Ostrovskaya and Yu. S. Kivshar, “Localization of two-component Bose-Einstein condensates in optical lattices,” arXiv: http://xxx.arxiv.org/abs/cond-mat/0309127
  17. B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion management for atomic matter waves,” Phys. Rev. Lett. 91, 060402 (2003).
    [CrossRef] [PubMed]
  18. L. Fallani, F. S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada, and M. Inguscio, “Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice,” Phys. Rev. Lett. 91, 240405 (2003).
    [CrossRef] [PubMed]
  19. B. Eiermann, Th. Anker, M. Albeiz, M. Taglieber, and M.K. Oberthaler, “Bright atomic solitons for repulsive interaction”, In: Proceedings of the 16-th International Conference on Laser Spectroscopy (ICOLS’03) (13–18 July 2003, Palm Cove, Australia).
  20. C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149 (1988).
    [CrossRef] [PubMed]
  21. H. Pu, L.O. Baksmaty, W. Zhang, N.P. Bigelow, and P. Meystre, “Effective-mass analysis of Bose-Einstein condensates in optical lattices: Stabilization and levitation,” Phys. Rev. A 67, 43605 (2003).
    [CrossRef]
  22. D. E. Pelinovsky, A.A. Sukhorukov, and Yu.S. Kivshar, “Bifurcations of gap solitons in periodic potentials,” in preparation.
  23. A.A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31 (2003).
    [CrossRef]
  24. A.A. Sukhorukov and Yu. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E 65, 036609 (2002).
    [CrossRef]
  25. N. Aközbek and S. John, “Optical solitary waves in two-and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287 (1998).
    [CrossRef]
  26. J.J. Garcìa-Ripoll and V.M. Pèrez-Garcìa, “Optimizing Schrödinger functionals using Sobolev gradients: Applications to quantum mechanics and nonlinear optics,” SIAM J. Sci. Comput. 23, 1316 (2001).
    [CrossRef]
  27. B. B. Baizakov, V.V. Konotop, and M. Salerno, “Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability,” J. Phys. B 35, 5105 (2002).
    [CrossRef]
  28. N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
    [CrossRef]
  29. E.A. Ostrovskaya, T.J. Alexander, and Yu.S. Kivshar, “Matter-wave gap vortices in two-dimensional optical lattices,” in preparation.

2003 (12)

J.W. Fleischer, T. Carmon, and M. Segev, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

D. Neshev, E.A. Ostrovskaya, Yu.S. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. 28, 710 (2003).
[CrossRef] [PubMed]

J.W. Fleischer, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147 (2003).
[CrossRef] [PubMed]

S. Peil, J.V. Porto, B. Laburthe Tolra, J.M. Obrecht, B.E. King, M. Subbotin, S.L. Rolston, and W.D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A 67, 051603 (2003).
[CrossRef]

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener tunneling in a periodic potential,” Phys. Rev. Lett.,  91, 230406 (2003).
[CrossRef] [PubMed]

E.A. Ostrovskaya and Yu.S. Kivshar, “Matter-wave gap solitons in atomic band-gap structures,” Phys. Rev. Lett. 90, 160407 (2003).
[CrossRef] [PubMed]

P.J. Louis, E.A. Ostrovskaya, C.M. Savage, and Yu.S. Kivshar, “Bose-Einstein condensates in optical lattices: band-gap structure and solitons,” Phys. Rev. A 67, 013602 (2003).
[CrossRef]

N.K. Efremidis and D.N. Christodoulides, “Lattice solitons in Bose-Einstein condensates,” Phys. Rev. A 67063608 (2003).
[CrossRef]

B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion management for atomic matter waves,” Phys. Rev. Lett. 91, 060402 (2003).
[CrossRef] [PubMed]

L. Fallani, F. S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada, and M. Inguscio, “Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice,” Phys. Rev. Lett. 91, 240405 (2003).
[CrossRef] [PubMed]

H. Pu, L.O. Baksmaty, W. Zhang, N.P. Bigelow, and P. Meystre, “Effective-mass analysis of Bose-Einstein condensates in optical lattices: Stabilization and levitation,” Phys. Rev. A 67, 43605 (2003).
[CrossRef]

A.A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

2002 (5)

A.A. Sukhorukov and Yu. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E 65, 036609 (2002).
[CrossRef]

J.H. Denschlag, J.E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S.L. Rolston, and W.D. Phillips, “A Bose-Einstein condensate in an optical lattice,” J. Phys. B 35, 3095 (2002).
[CrossRef]

V.V. Konotop and M. Salerno, “Modulational instability in Bose-Einstein condensates in optical lattices,” Phys. Rev. A 65, 021602 (2002).
[CrossRef]

B. B. Baizakov, V.V. Konotop, and M. Salerno, “Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability,” J. Phys. B 35, 5105 (2002).
[CrossRef]

N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

2001 (2)

S.F. Mingaleev and Yu.S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474 (2001).
[CrossRef] [PubMed]

J.J. Garcìa-Ripoll and V.M. Pèrez-Garcìa, “Optimizing Schrödinger functionals using Sobolev gradients: Applications to quantum mechanics and nonlinear optics,” SIAM J. Sci. Comput. 23, 1316 (2001).
[CrossRef]

1999 (1)

O. Zobay, S. Pötting, P. Meystre, and E.M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A 59, 643 (1999).
[CrossRef]

1998 (1)

N. Aközbek and S. John, “Optical solitary waves in two-and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287 (1998).
[CrossRef]

1988 (1)

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149 (1988).
[CrossRef] [PubMed]

Agrawal, G.P.

Yu.S. Kivshar and G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

Aközbek, N.

N. Aközbek and S. John, “Optical solitary waves in two-and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287 (1998).
[CrossRef]

Albeiz, M.

B. Eiermann, Th. Anker, M. Albeiz, M. Taglieber, and M.K. Oberthaler, “Bright atomic solitons for repulsive interaction”, In: Proceedings of the 16-th International Conference on Laser Spectroscopy (ICOLS’03) (13–18 July 2003, Palm Cove, Australia).

Albiez, M.

B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion management for atomic matter waves,” Phys. Rev. Lett. 91, 060402 (2003).
[CrossRef] [PubMed]

Alexander, T.J.

E.A. Ostrovskaya, T.J. Alexander, and Yu.S. Kivshar, “Matter-wave gap vortices in two-dimensional optical lattices,” in preparation.

Anderlini, M.

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener tunneling in a periodic potential,” Phys. Rev. Lett.,  91, 230406 (2003).
[CrossRef] [PubMed]

Anker, Th.

B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion management for atomic matter waves,” Phys. Rev. Lett. 91, 060402 (2003).
[CrossRef] [PubMed]

B. Eiermann, Th. Anker, M. Albeiz, M. Taglieber, and M.K. Oberthaler, “Bright atomic solitons for repulsive interaction”, In: Proceedings of the 16-th International Conference on Laser Spectroscopy (ICOLS’03) (13–18 July 2003, Palm Cove, Australia).

Arimondo, E.

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener tunneling in a periodic potential,” Phys. Rev. Lett.,  91, 230406 (2003).
[CrossRef] [PubMed]

Baizakov, B. B.

B. B. Baizakov, V.V. Konotop, and M. Salerno, “Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability,” J. Phys. B 35, 5105 (2002).
[CrossRef]

Baksmaty, L.O.

H. Pu, L.O. Baksmaty, W. Zhang, N.P. Bigelow, and P. Meystre, “Effective-mass analysis of Bose-Einstein condensates in optical lattices: Stabilization and levitation,” Phys. Rev. A 67, 43605 (2003).
[CrossRef]

Bigelow, N.P.

H. Pu, L.O. Baksmaty, W. Zhang, N.P. Bigelow, and P. Meystre, “Effective-mass analysis of Bose-Einstein condensates in optical lattices: Stabilization and levitation,” Phys. Rev. A 67, 43605 (2003).
[CrossRef]

Browaeys, A.

J.H. Denschlag, J.E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S.L. Rolston, and W.D. Phillips, “A Bose-Einstein condensate in an optical lattice,” J. Phys. B 35, 3095 (2002).
[CrossRef]

Carmon, T.

J.W. Fleischer, T. Carmon, and M. Segev, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

Cataliotti, F. S.

L. Fallani, F. S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada, and M. Inguscio, “Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice,” Phys. Rev. Lett. 91, 240405 (2003).
[CrossRef] [PubMed]

Catani, J.

L. Fallani, F. S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada, and M. Inguscio, “Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice,” Phys. Rev. Lett. 91, 240405 (2003).
[CrossRef] [PubMed]

Cho, D.

J.H. Denschlag, J.E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S.L. Rolston, and W.D. Phillips, “A Bose-Einstein condensate in an optical lattice,” J. Phys. B 35, 3095 (2002).
[CrossRef]

Christodoulides, D.N.

J.W. Fleischer, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147 (2003).
[CrossRef] [PubMed]

N.K. Efremidis and D.N. Christodoulides, “Lattice solitons in Bose-Einstein condensates,” Phys. Rev. A 67063608 (2003).
[CrossRef]

N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Courtade, E.

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener tunneling in a periodic potential,” Phys. Rev. Lett.,  91, 230406 (2003).
[CrossRef] [PubMed]

Cristiani, M.

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener tunneling in a periodic potential,” Phys. Rev. Lett.,  91, 230406 (2003).
[CrossRef] [PubMed]

de Sterke, C. M.

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149 (1988).
[CrossRef] [PubMed]

Denschlag, J.H.

J.H. Denschlag, J.E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S.L. Rolston, and W.D. Phillips, “A Bose-Einstein condensate in an optical lattice,” J. Phys. B 35, 3095 (2002).
[CrossRef]

Efremidis, N.K.

J.W. Fleischer, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147 (2003).
[CrossRef] [PubMed]

N.K. Efremidis and D.N. Christodoulides, “Lattice solitons in Bose-Einstein condensates,” Phys. Rev. A 67063608 (2003).
[CrossRef]

N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Eiermann, B.

B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion management for atomic matter waves,” Phys. Rev. Lett. 91, 060402 (2003).
[CrossRef] [PubMed]

B. Eiermann, Th. Anker, M. Albeiz, M. Taglieber, and M.K. Oberthaler, “Bright atomic solitons for repulsive interaction”, In: Proceedings of the 16-th International Conference on Laser Spectroscopy (ICOLS’03) (13–18 July 2003, Palm Cove, Australia).

Eisenberg, H. S.

A.A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

Fallani, L.

L. Fallani, F. S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada, and M. Inguscio, “Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice,” Phys. Rev. Lett. 91, 240405 (2003).
[CrossRef] [PubMed]

Fleischer, J.W.

J.W. Fleischer, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147 (2003).
[CrossRef] [PubMed]

J.W. Fleischer, T. Carmon, and M. Segev, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Fort, C.

L. Fallani, F. S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada, and M. Inguscio, “Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice,” Phys. Rev. Lett. 91, 240405 (2003).
[CrossRef] [PubMed]

Garcìa-Ripoll, J.J.

J.J. Garcìa-Ripoll and V.M. Pèrez-Garcìa, “Optimizing Schrödinger functionals using Sobolev gradients: Applications to quantum mechanics and nonlinear optics,” SIAM J. Sci. Comput. 23, 1316 (2001).
[CrossRef]

Haffner, H.

J.H. Denschlag, J.E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S.L. Rolston, and W.D. Phillips, “A Bose-Einstein condensate in an optical lattice,” J. Phys. B 35, 3095 (2002).
[CrossRef]

Helmerson, K.

J.H. Denschlag, J.E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S.L. Rolston, and W.D. Phillips, “A Bose-Einstein condensate in an optical lattice,” J. Phys. B 35, 3095 (2002).
[CrossRef]

Inguscio, M.

L. Fallani, F. S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada, and M. Inguscio, “Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice,” Phys. Rev. Lett. 91, 240405 (2003).
[CrossRef] [PubMed]

Joannopoulos, J.D.

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

John, S.

N. Aközbek and S. John, “Optical solitary waves in two-and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287 (1998).
[CrossRef]

Jona-Lasinio, M.

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener tunneling in a periodic potential,” Phys. Rev. Lett.,  91, 230406 (2003).
[CrossRef] [PubMed]

King, B.E.

S. Peil, J.V. Porto, B. Laburthe Tolra, J.M. Obrecht, B.E. King, M. Subbotin, S.L. Rolston, and W.D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A 67, 051603 (2003).
[CrossRef]

Kivshar, Yu. S.

A.A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

A.A. Sukhorukov and Yu. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E 65, 036609 (2002).
[CrossRef]

E.A. Ostrovskaya and Yu. S. Kivshar, “Localization of two-component Bose-Einstein condensates in optical lattices,” arXiv: http://xxx.arxiv.org/abs/cond-mat/0309127

Kivshar, Yu.S.

P.J. Louis, E.A. Ostrovskaya, C.M. Savage, and Yu.S. Kivshar, “Bose-Einstein condensates in optical lattices: band-gap structure and solitons,” Phys. Rev. A 67, 013602 (2003).
[CrossRef]

E.A. Ostrovskaya and Yu.S. Kivshar, “Matter-wave gap solitons in atomic band-gap structures,” Phys. Rev. Lett. 90, 160407 (2003).
[CrossRef] [PubMed]

D. Neshev, E.A. Ostrovskaya, Yu.S. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. 28, 710 (2003).
[CrossRef] [PubMed]

S.F. Mingaleev and Yu.S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474 (2001).
[CrossRef] [PubMed]

Yu.S. Kivshar and G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

E.A. Ostrovskaya, T.J. Alexander, and Yu.S. Kivshar, “Matter-wave gap vortices in two-dimensional optical lattices,” in preparation.

D. E. Pelinovsky, A.A. Sukhorukov, and Yu.S. Kivshar, “Bifurcations of gap solitons in periodic potentials,” in preparation.

Konotop, V.V.

B. B. Baizakov, V.V. Konotop, and M. Salerno, “Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability,” J. Phys. B 35, 5105 (2002).
[CrossRef]

V.V. Konotop and M. Salerno, “Modulational instability in Bose-Einstein condensates in optical lattices,” Phys. Rev. A 65, 021602 (2002).
[CrossRef]

Krolikowski, W.

Laburthe Tolra, B.

S. Peil, J.V. Porto, B. Laburthe Tolra, J.M. Obrecht, B.E. King, M. Subbotin, S.L. Rolston, and W.D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A 67, 051603 (2003).
[CrossRef]

Louis, P.J.

P.J. Louis, E.A. Ostrovskaya, C.M. Savage, and Yu.S. Kivshar, “Bose-Einstein condensates in optical lattices: band-gap structure and solitons,” Phys. Rev. A 67, 013602 (2003).
[CrossRef]

Malossi, N.

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener tunneling in a periodic potential,” Phys. Rev. Lett.,  91, 230406 (2003).
[CrossRef] [PubMed]

Marzlin, K.-P.

B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion management for atomic matter waves,” Phys. Rev. Lett. 91, 060402 (2003).
[CrossRef] [PubMed]

McKenzie, C.

J.H. Denschlag, J.E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S.L. Rolston, and W.D. Phillips, “A Bose-Einstein condensate in an optical lattice,” J. Phys. B 35, 3095 (2002).
[CrossRef]

Meade, R.D.

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

Meystre, P.

H. Pu, L.O. Baksmaty, W. Zhang, N.P. Bigelow, and P. Meystre, “Effective-mass analysis of Bose-Einstein condensates in optical lattices: Stabilization and levitation,” Phys. Rev. A 67, 43605 (2003).
[CrossRef]

O. Zobay, S. Pötting, P. Meystre, and E.M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A 59, 643 (1999).
[CrossRef]

Mingaleev, S.F.

S.F. Mingaleev and Yu.S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474 (2001).
[CrossRef] [PubMed]

Modugno, M.

L. Fallani, F. S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada, and M. Inguscio, “Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice,” Phys. Rev. Lett. 91, 240405 (2003).
[CrossRef] [PubMed]

Morsch, O.

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener tunneling in a periodic potential,” Phys. Rev. Lett.,  91, 230406 (2003).
[CrossRef] [PubMed]

Müller, J.H.

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener tunneling in a periodic potential,” Phys. Rev. Lett.,  91, 230406 (2003).
[CrossRef] [PubMed]

Neshev, D.

Oberthaler, M.K.

B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion management for atomic matter waves,” Phys. Rev. Lett. 91, 060402 (2003).
[CrossRef] [PubMed]

B. Eiermann, Th. Anker, M. Albeiz, M. Taglieber, and M.K. Oberthaler, “Bright atomic solitons for repulsive interaction”, In: Proceedings of the 16-th International Conference on Laser Spectroscopy (ICOLS’03) (13–18 July 2003, Palm Cove, Australia).

Obrecht, J.M.

S. Peil, J.V. Porto, B. Laburthe Tolra, J.M. Obrecht, B.E. King, M. Subbotin, S.L. Rolston, and W.D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A 67, 051603 (2003).
[CrossRef]

Ostrovskaya, E.A.

E.A. Ostrovskaya and Yu.S. Kivshar, “Matter-wave gap solitons in atomic band-gap structures,” Phys. Rev. Lett. 90, 160407 (2003).
[CrossRef] [PubMed]

P.J. Louis, E.A. Ostrovskaya, C.M. Savage, and Yu.S. Kivshar, “Bose-Einstein condensates in optical lattices: band-gap structure and solitons,” Phys. Rev. A 67, 013602 (2003).
[CrossRef]

D. Neshev, E.A. Ostrovskaya, Yu.S. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. 28, 710 (2003).
[CrossRef] [PubMed]

E.A. Ostrovskaya, T.J. Alexander, and Yu.S. Kivshar, “Matter-wave gap vortices in two-dimensional optical lattices,” in preparation.

E.A. Ostrovskaya and Yu. S. Kivshar, “Localization of two-component Bose-Einstein condensates in optical lattices,” arXiv: http://xxx.arxiv.org/abs/cond-mat/0309127

Peil, S.

S. Peil, J.V. Porto, B. Laburthe Tolra, J.M. Obrecht, B.E. King, M. Subbotin, S.L. Rolston, and W.D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A 67, 051603 (2003).
[CrossRef]

Pelinovsky, D. E.

D. E. Pelinovsky, A.A. Sukhorukov, and Yu.S. Kivshar, “Bifurcations of gap solitons in periodic potentials,” in preparation.

Pèrez-Garcìa, V.M.

J.J. Garcìa-Ripoll and V.M. Pèrez-Garcìa, “Optimizing Schrödinger functionals using Sobolev gradients: Applications to quantum mechanics and nonlinear optics,” SIAM J. Sci. Comput. 23, 1316 (2001).
[CrossRef]

Phillips, W.D.

S. Peil, J.V. Porto, B. Laburthe Tolra, J.M. Obrecht, B.E. King, M. Subbotin, S.L. Rolston, and W.D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A 67, 051603 (2003).
[CrossRef]

J.H. Denschlag, J.E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S.L. Rolston, and W.D. Phillips, “A Bose-Einstein condensate in an optical lattice,” J. Phys. B 35, 3095 (2002).
[CrossRef]

Porto, J.V.

S. Peil, J.V. Porto, B. Laburthe Tolra, J.M. Obrecht, B.E. King, M. Subbotin, S.L. Rolston, and W.D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A 67, 051603 (2003).
[CrossRef]

Pötting, S.

O. Zobay, S. Pötting, P. Meystre, and E.M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A 59, 643 (1999).
[CrossRef]

Pu, H.

H. Pu, L.O. Baksmaty, W. Zhang, N.P. Bigelow, and P. Meystre, “Effective-mass analysis of Bose-Einstein condensates in optical lattices: Stabilization and levitation,” Phys. Rev. A 67, 43605 (2003).
[CrossRef]

Rolston, S.L.

S. Peil, J.V. Porto, B. Laburthe Tolra, J.M. Obrecht, B.E. King, M. Subbotin, S.L. Rolston, and W.D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A 67, 051603 (2003).
[CrossRef]

J.H. Denschlag, J.E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S.L. Rolston, and W.D. Phillips, “A Bose-Einstein condensate in an optical lattice,” J. Phys. B 35, 3095 (2002).
[CrossRef]

Salerno, M.

V.V. Konotop and M. Salerno, “Modulational instability in Bose-Einstein condensates in optical lattices,” Phys. Rev. A 65, 021602 (2002).
[CrossRef]

B. B. Baizakov, V.V. Konotop, and M. Salerno, “Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability,” J. Phys. B 35, 5105 (2002).
[CrossRef]

Savage, C.M.

P.J. Louis, E.A. Ostrovskaya, C.M. Savage, and Yu.S. Kivshar, “Bose-Einstein condensates in optical lattices: band-gap structure and solitons,” Phys. Rev. A 67, 013602 (2003).
[CrossRef]

Sears, S.

N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Segev, M.

J.W. Fleischer, T. Carmon, and M. Segev, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

J.W. Fleischer, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147 (2003).
[CrossRef] [PubMed]

N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

Silberberg, Y.

A.A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

Simsarian, J.E.

J.H. Denschlag, J.E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S.L. Rolston, and W.D. Phillips, “A Bose-Einstein condensate in an optical lattice,” J. Phys. B 35, 3095 (2002).
[CrossRef]

Sipe, J. E.

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149 (1988).
[CrossRef] [PubMed]

Subbotin, M.

S. Peil, J.V. Porto, B. Laburthe Tolra, J.M. Obrecht, B.E. King, M. Subbotin, S.L. Rolston, and W.D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A 67, 051603 (2003).
[CrossRef]

Sukhorukov, A.A.

A.A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

A.A. Sukhorukov and Yu. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E 65, 036609 (2002).
[CrossRef]

D. E. Pelinovsky, A.A. Sukhorukov, and Yu.S. Kivshar, “Bifurcations of gap solitons in periodic potentials,” in preparation.

Taglieber, M.

B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion management for atomic matter waves,” Phys. Rev. Lett. 91, 060402 (2003).
[CrossRef] [PubMed]

B. Eiermann, Th. Anker, M. Albeiz, M. Taglieber, and M.K. Oberthaler, “Bright atomic solitons for repulsive interaction”, In: Proceedings of the 16-th International Conference on Laser Spectroscopy (ICOLS’03) (13–18 July 2003, Palm Cove, Australia).

Treutlein, P.

B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion management for atomic matter waves,” Phys. Rev. Lett. 91, 060402 (2003).
[CrossRef] [PubMed]

Winn, J.N.

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

Wright, E.M.

O. Zobay, S. Pötting, P. Meystre, and E.M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A 59, 643 (1999).
[CrossRef]

Zawada, M.

L. Fallani, F. S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada, and M. Inguscio, “Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice,” Phys. Rev. Lett. 91, 240405 (2003).
[CrossRef] [PubMed]

Zhang, W.

H. Pu, L.O. Baksmaty, W. Zhang, N.P. Bigelow, and P. Meystre, “Effective-mass analysis of Bose-Einstein condensates in optical lattices: Stabilization and levitation,” Phys. Rev. A 67, 43605 (2003).
[CrossRef]

Zobay, O.

O. Zobay, S. Pötting, P. Meystre, and E.M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A 59, 643 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

A.A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. 39, 31 (2003).
[CrossRef]

J. Phys. B (2)

B. B. Baizakov, V.V. Konotop, and M. Salerno, “Regular spatial structures in arrays of Bose-Einstein condensates induced by modulational instability,” J. Phys. B 35, 5105 (2002).
[CrossRef]

J.H. Denschlag, J.E. Simsarian, H. Haffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S.L. Rolston, and W.D. Phillips, “A Bose-Einstein condensate in an optical lattice,” J. Phys. B 35, 3095 (2002).
[CrossRef]

Nature (1)

J.W. Fleischer, M. Segev, N.K. Efremidis, and D.N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147 (2003).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. A (7)

O. Zobay, S. Pötting, P. Meystre, and E.M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A 59, 643 (1999).
[CrossRef]

V.V. Konotop and M. Salerno, “Modulational instability in Bose-Einstein condensates in optical lattices,” Phys. Rev. A 65, 021602 (2002).
[CrossRef]

P.J. Louis, E.A. Ostrovskaya, C.M. Savage, and Yu.S. Kivshar, “Bose-Einstein condensates in optical lattices: band-gap structure and solitons,” Phys. Rev. A 67, 013602 (2003).
[CrossRef]

N.K. Efremidis and D.N. Christodoulides, “Lattice solitons in Bose-Einstein condensates,” Phys. Rev. A 67063608 (2003).
[CrossRef]

S. Peil, J.V. Porto, B. Laburthe Tolra, J.M. Obrecht, B.E. King, M. Subbotin, S.L. Rolston, and W.D. Phillips, “Patterned loading of a Bose-Einstein condensate into an optical lattice,” Phys. Rev. A 67, 051603 (2003).
[CrossRef]

C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodic structures,” Phys. Rev. A 38, 5149 (1988).
[CrossRef] [PubMed]

H. Pu, L.O. Baksmaty, W. Zhang, N.P. Bigelow, and P. Meystre, “Effective-mass analysis of Bose-Einstein condensates in optical lattices: Stabilization and levitation,” Phys. Rev. A 67, 43605 (2003).
[CrossRef]

Phys. Rev. E (3)

N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E 66, 046602 (2002).
[CrossRef]

A.A. Sukhorukov and Yu. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E 65, 036609 (2002).
[CrossRef]

N. Aközbek and S. John, “Optical solitary waves in two-and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287 (1998).
[CrossRef]

Phys. Rev. Lett. (6)

M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Müller, E. Courtade, M. Anderlini, and E. Arimondo, “Asymmetric Landau-Zener tunneling in a periodic potential,” Phys. Rev. Lett.,  91, 230406 (2003).
[CrossRef] [PubMed]

E.A. Ostrovskaya and Yu.S. Kivshar, “Matter-wave gap solitons in atomic band-gap structures,” Phys. Rev. Lett. 90, 160407 (2003).
[CrossRef] [PubMed]

S.F. Mingaleev and Yu.S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474 (2001).
[CrossRef] [PubMed]

J.W. Fleischer, T. Carmon, and M. Segev, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90, 023902 (2003).
[CrossRef] [PubMed]

B. Eiermann, P. Treutlein, Th. Anker, M. Albiez, M. Taglieber, K.-P. Marzlin, and M.K. Oberthaler, “Dispersion management for atomic matter waves,” Phys. Rev. Lett. 91, 060402 (2003).
[CrossRef] [PubMed]

L. Fallani, F. S. Cataliotti, J. Catani, C. Fort, M. Modugno, M. Zawada, and M. Inguscio, “Optically induced lensing effect on a Bose-Einstein condensate expanding in a moving lattice,” Phys. Rev. Lett. 91, 240405 (2003).
[CrossRef] [PubMed]

SIAM J. Sci. Comput. (1)

J.J. Garcìa-Ripoll and V.M. Pèrez-Garcìa, “Optimizing Schrödinger functionals using Sobolev gradients: Applications to quantum mechanics and nonlinear optics,” SIAM J. Sci. Comput. 23, 1316 (2001).
[CrossRef]

Other (7)

E.A. Ostrovskaya, T.J. Alexander, and Yu.S. Kivshar, “Matter-wave gap vortices in two-dimensional optical lattices,” in preparation.

D. E. Pelinovsky, A.A. Sukhorukov, and Yu.S. Kivshar, “Bifurcations of gap solitons in periodic potentials,” in preparation.

B. Eiermann, Th. Anker, M. Albeiz, M. Taglieber, and M.K. Oberthaler, “Bright atomic solitons for repulsive interaction”, In: Proceedings of the 16-th International Conference on Laser Spectroscopy (ICOLS’03) (13–18 July 2003, Palm Cove, Australia).

E.A. Ostrovskaya and Yu. S. Kivshar, “Localization of two-component Bose-Einstein condensates in optical lattices,” arXiv: http://xxx.arxiv.org/abs/cond-mat/0309127

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

R. Slusher and B. Eggleton, eds., Nonlinear Photonic Crystals (Springer-Verlag, Berlin, 2003).

Yu.S. Kivshar and G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

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Figures (6)

Fig. 1.
Fig. 1.

Group velocity and effective diffraction coefficient for Bloch matter waves in an optical lattice shown in the context of the bandgap spectrum with the Bloch bands (shaded) and gaps (open); V 0=2.0.

Fig. 2.
Fig. 2.

Band-gap spectrum of matter waves in an optical lattice shown as the Bloch bands (shaded) and gaps (open), combined with the families of bright gap solitons in (left) repulsive and (right) attractive condensates (V 0=5).

Fig. 3.
Fig. 3.

Examples of a weakly unstable and stable soliton dynamics. Shown is peak density (a) of the repulsive BEC off-site soliton [shown in Fig. 2 (left, b)] in the first gap (µ=3.7), and (b) of the attractive BEC on-site soliton [shown in Fig. 2 (right, a)] in the semi-infinite gap (µ=1.0). In (a) the initial state given by the exact (numerical) stationary solution of Eq. (5) is perturbed by a symmetric excitation at 5% of the soliton peak density. In (b) the antisymmetric internal mode is excited by an initial perturbation at 5% of the initial soliton peak density.

Fig. 4.
Fig. 4.

Left: Dispersion diagram for a 2D square lattice (V 0=1.5); dotted - the line µ=V 0; shaded - spectral bands, open - the lowest, semi-infinite, and the first complete gaps. Below: lattice potential in the cartesian and reciprocal spaces. Right: Spatial structure of the 2D Bloch waves at the extreme high-symmetry points of the first irreducible Brillouine zone.

Fig. 5.
Fig. 5.

Top: Family of bright atomic gap solitons of repulsive BEC in a 2D optical lattice (V 0=1.5). Bottom: spatial structure of the BEC wavefunctions at the marked points of the existence curve inside the gap.

Fig. 6.
Fig. 6.

Spatial structures of the BEC wavefunctions corresponding to higher-order gap solitons (V 0=1.5): (a) dipole (µ=2.0, P=9.02), (b) quadrupole (µ=2.0, P=9.8), (c) charge-one gap vortex (µ=2.0, P=16.2). (d) Phase structure of the gap vortex shown in (c) (color-bar refers to the phase plot only).

Equations (10)

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

i h ¯ Ψ t = { h ¯ 2 2 m 2 + V ( r ) + g 3 D Ψ 2 } Ψ
V ( r ) = 1 2 m ( ω x x 2 + ω y y 2 + ω z z 2 ) + V L ,
V L ( x ) = V 0 sin 2 ( K x ) ,
V L ( x , y ) = V 0 [ sin 2 ( K x ) + sin 2 ( K y ) ] .
i ψ t = { 1 2 2 x 2 + V L ( x ) + σ ψ ( x , t ) 2 } ψ
ϕ ( x ) = b 1 ϕ 1 ( x ) e i k x + b 2 ϕ 2 ( x ) e i k x ,
ψ ( x , t ) = ϕ ( x ) e i μ t + ε [ ( u + i w ) e λ t + ( u * + i w * ) e λ * t ] e i μ t ,
L w = λ u , L + u = λ w ,
i ψ t = { 1 2 2 + V L ( x , y ) + ψ 2 } ψ .
[ 1 2 ( i + k ) 2 + V L ( r ) ] u n , k = μ n ( k ) u n , k .

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