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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References

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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, andW. 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 67 063608 (2003).
    [CrossRef]
  16. E.A. Ostrovskaya and Yu. S. Kivshar, �??Localization of two-component Bose-Einstein condensates in optical lattices,�?? arXiv: <a href="http://xxx.arxiv.org/abs/cond-mat/0309127">http://xxx.arxiv.org/abs/cond-mat/0309127</a>.
  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.

ICOLS???03 (1)

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).

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)

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]

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]

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)

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]

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 67 063608 (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)

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]

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]

Phys. Rev. Lett. (6)

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]

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]

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]

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 (6)

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

E.A. Ostrovskaya and Yu. S. Kivshar, �??Localization of two-component Bose-Einstein condensates in optical lattices,�?? arXiv: <a href="http://xxx.arxiv.org/abs/cond-mat/0309127">http://xxx.arxiv.org/abs/cond-mat/0309127</a>.

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).

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

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

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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)

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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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