## Photonic crystals for matter waves: Bose-Einstein condensates in optical lattices

Optics Express, Vol. 12, Issue 1, pp. 19-29 (2004)

http://dx.doi.org/10.1364/OPEX.12.000019

Acrobat PDF (1775 KB)

### 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

## 1. Introduction

*fiber Bragg gratings and photonic crystals*, has revolutionized modern photonics and laid the foundation for the development of novel types of integrated photonic devices. The study of

*nonlinear photonic crystals*made of a Kerr nonlinear material [2

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]

*optical gap solitons*[4] with the energies inside the photonic gaps of a periodic structure. Recent demonstration of the light scattering in dynamically reconfigurable photonic structures – optically-induced refractive index gratings in nonlinear materials – has opened up novel ways to control light propagation and localization [5

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]

*coherent matter waves*in the reconfigurable crystal-like structures created by light. Due to the inherent nonlinearity of coherent matter waves which is introduced by the interactions between atoms, BEC in a lattice potential forms a periodic nonlinear system which is expected to display rich and complex dynamics.

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: 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]

*the matter-wave gap solitons*[19].

## 2. Model

**Ψ**(

**r**,

*t*),

*V*(

**r**) is the time-independent trapping potential, and

*g3D*=4

*πh̄*

^{2}

*a*

_{s}

*/m*characterizes the two-body interactions for a condensate with atoms of mass

*m*and s-wave scattering length

*a*

_{s}. The scattering length

*a*

_{s}is positive for repulsive interactions and negative for attractive interactions.

*V*(

**r**) of the form

*V*

_{L}, is formed by an optical lattice, which we consider to be either one-dimensional (1D),

*π/K*, and the lattice depth,

*V*

_{0}, is proportional to the intensity of the standing light wave.

*a*

_{L}=1/

*K*, energy

*E*

_{rec}=

*h̅*

^{2}/

*h̄*/

*E*

_{rec}scales of the lattice. In dimensionless units, the two-body interaction coefficient is given by

*g3D*=4

*π(a*

_{s}

*/a*

_{L}

*)*, and the lattice depth is measured in units of the lattice recoil energy,

*E*

_{rec}.

## 3. One-dimensional atomic band-gap structures

*x*-direction (

*ω*

_{x}≪

*ω*

_{y,z}≡ω

_{⊥}). In the directions of tight confinement, the condensate wavefunction can be described by the ground state of a two-dimensional radially symmetric quantum harmonic oscillator potential, with the normalization ∫

^{∞}

_{-∞}|Φ|

^{2}

*d*

**r**=1. The 3D wavefunction then separates as Ψ(

**r**,

*t*)=Φ(

*y, z*)

*ψ*(

*x, t*), and the transverse dimensions can be intergrated out in the dimensionless Eq. (1), yielding the 1D GP equation:

*g*1

*D*=4(

*a*

_{s}/

*a*

_{l})(ω

_{L}/

*ω*

_{L})

^{2}, σ=sign(

*g*1

*D*), and the external potential is approximated by the quasi-1D periodic potential of the optical lattice,

*V*

_{L}(

*x*)=sin

^{2}(

*x*), by neglecting the contribution of a weak magnetic confinement.

*ψ*(

*x, t*)=

*ϕ*(

*x*)exp(-

*iµt*), where

*µ*is the corresponding chemical potential. The case of a noninteracting condensate formally corresponds to

*g*1

*D*=0, and the condensate wave function can be represented as a superposition of Bloch waves,

*ϕ*

_{1,2}(

*x*) have periodicity of the lattice potential,

*b*

_{1,2}are constants, and

*k*is the Floquet exponent. The linear matter-wave spectrum consists of bands of eigenvalues

*µ*

_{n,k}in which

*k*(

*µ*) is a (real) wavenumber of amplitude-bounded Bloch waves [14

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]

*Im*(

*k*)≠0. The solutions at the band edges are stationary Bloch states corresponding to the condensate density which is strongly and periodically modulated by the lattice.

*g*1

*D*≠0), bright

*solitons*can exist for chemical potentials corresponding to the gaps of the linear matter-wave spectrum. The mechanism for nonlinear localization in the gaps was first described for light waves in nonlinear periodic photonic structures by using the Bloch-wave envelope approximation near the band edge [20

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]

*v*

_{g}=

*∂µ*/

*∂k*, and the effective diffraction coefficient,

*D*=

*∂*

^{2}

*µ*/

*∂k*

^{2}(analogous to the inverse effective mass of an electron in crystalline solids [21

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]

*D*<0, (normal,

*D*>0) diffraction near the top (bottom) of the bands (see Fig. 1), can lead to formation of localized wavepackets -

*matter wave solitons*- with zero group velocity. The chemical potentials corresponding to such localized waves lie in the gaps of the Bloch-wave spectrum for the matter waves of the noninteracting condensate (open areas in Fig. 1).

*P*=∫

*ϕ*

^{2}(

*x*)

*dx*, which is proportional to the number of atoms in a localized state. Such localized states exist in all band gaps, including the semi-infinite gap of the spectrum below the first band [14

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]

*sech*-like envelope of the corresponding Bloch wave, centered either on- or off-site. Near the opposite edge of the gap, i.e. approaching the spectral band, the solitons develop extended “tails” with the spatial structure of the Bloch wave at the corresponding band edge [14

**67**, 013602 (2003). [CrossRef]

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]

*ϕ*(

*x*), in the form

*ε*≪1, and

*u*(

*x*),

*w*(

*x*) are spatially-dependent perturbation modes. We linearize the GP equation (5) around the localized solution and obtain, to the first order in

*ε*, a linear eigenvalue problem for the perturbation modes

*L*±=-(1/2)(

*∂*

^{2}/

*∂x*

^{2})+

*V*

_{L}(

*x*)+2

*σϕ*

^{2}±

*σϕ*

^{2}-

*µ*. The modes describing the development of instability have either purely real or complex eigenvalues λ ; in this latter case, the instability is called

*an oscillatory instability*.

*∂P/∂µ*>0. Note that the criterion is “inverted” for our choice of

*µ*. The fundamental gap solitons of repulsive BEC [Fig. 2 (left, a)] are also linearly stable.

*λ*≠0 and Re

*λ*=0). The excitation of these modes usually leads to a persisting dynamics (e.g., amplitude oscillations) of the localized state that does not lead to its decay. However, higher-order gap solitons of both attractive and repulsive condensates can experience oscillatory instability (Im

*λ*≠0 and Re

*λ*≠0) initiated by resonances of internal modes with the bands of the inverted spectrum [22]. This mechanism for weak spectral instability was first identified in studies of spatial optical solitons in one-dimensional nonlinear photonic crystals [24

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

## 4. Two-dimensional band-gap structures

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]

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]

*ω*

_{x,y}has little effect on the stationary states of the condensate in the 2D lattice formed in the (

*x,y*) plane of the condensate cloud. Under this assumption, the trap component of the confining potential in the lattice plane can be neglected, and the model can be reduced to a two-dimensional GP equation by the dimensionality reduction procedure described above for the 1D case. The condensate wavefunction in the 2D lattice potential is then described by the following equation:

_{⊥}=

*∂*

^{2}/

*∂x*

^{2}+

*∂*

^{2}/

*∂y*

^{2},

*V*

_{L}(

*x,y*) is the periodic potential of the optical lattice, and the wavefunction is rescaled as

*g*

_{2D}=

*g*

_{3D}/√2. The equation (9) is made dimensionless by using the “natural” lattice units of energy, length, and frequency, described above. In the simplest case of the square optical lattice, the potential of the optical lattice can be written as

*V*

_{L}=

*V*

_{0}[sin

^{2}(

*x*)+sin

^{2}(

*y*)], where

*V*

_{0}is the amplitude of the optical lattice.

*ψ*(

**r**,

*t*)=

*ϕ*(

**r**)exp(-

*iµt*), where

*µ*is the chemical potential. The case of a noninteracting condensate formally corresponds to Eq. (9) being linear in

*ψ*. According to the Bloch theorem, the stationary wavefunction can then be sought in the form

*ϕ*(

**r**)=

*u*

_{k}(

**r**)exp(

*i*

**kr**), where the wave vector

**k**belongs to a Brillouin zone of the square optical lattice, and

*u*

_{k}(

**r**)=

*u*

_{k}(

**r**+

**d**) is a periodic (Bloch) function with the periodicity of the lattice. For the values of

**k**within an

*n*-th Brillouin zone, the dispersion relation for the 2D Bloch waves,

*µ*

_{n,k}, is found by solving a linear eigenvalue problem:

*µ*(

**k**) of the atomic Bloch waves in the 2D optical lattice is shown in Fig. 4, in the reduced zone representation usually assumed in the theory of crystalline solids and photonic crystals. Due to separability of the lattice potential, the Bloch waves

*u*(

*x,y*) in the first band, at the high symmetry points (Γ→

*X*→

*M*) of the first irreducible Brillouine zone (see Fig 4), can be found as

*u*(

*x,y*;

**k**)=

*u*

^{(1d)}(

*x; k*

_{x})

*u*

^{(1d)}(

*y; k*

_{y}), where

*u*

^{(1d)}are the corresponding 1D Bloch states. Different structure of the Bloch states in the first band, at the values of

**k**corresponding to the three symmetry points, are shown in Fig. 4 (right column). By applying an envelope theory near the band edges, it can be deduced that each of the Bloch states is associated with a (partially) localized state in the Γ,

*X*, or

*M*-gap. Similarly to optical gap solitons in indirect gaps of higher-dimensional photonic crystals [25

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]

*M*and

*X*-edges of the first and the second band, respectively (see Fig. 4). The localization near the lower

*M*-edge is possible due to the negative components of the effective diffraction tensor. We find spatially localized stationary solutions of Eq. (9) numerically. Our numerical procedure involves minimization of the norm

*𝓝*=∫

*f*†

*f*dr of the functional

*f*(

*ϕ*)=[(1/2)

*µ*-

*V*(

**r**)-|ϕ|

^{2}]

*ϕ*by following a descent technique with Sobolev preconditioning [26

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]

*𝓝*(

*ϕ*)→0.

*P*

_{c}) needed for the soliton localization is determined by the theory of lattice-free self-focusing in the 2D geometry. These weakly localized, low density modes are similar to the near-band-edge modes described above for the case of one-dimensional gap solitons. Deeper inside the gap, the matter-wave soliton becomes strongly localized [Fig. 5(b)], before delocalizing again near the upper gap edge. Near that edge, extended tails [see Fig. 5(c)] are developed in the directions along which the tunneling is assisted by lower inter-well barrier heights and positive (normal) effective diffraction. By direct numerical simulations we have confirmed that the strongly localized two-dimensional fundamental modes are dynamically stable. The possibility of the formation of periodic trains of such 2D localized structures, triggered by the modulational instability of the nonlinear Bloch states, has been suggested in Ref. [27

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]

**67**, 013602 (2003). [CrossRef]

*three different examples*of higher-order gap solitons. The simplest structure of this kind is described by a pair of two fundamental gap solitons which are out-of-phase and form a

*dipole*[see Fig. 6(a)]. The similar structure of four solitons can exist as

*a quadrupole state*where the neighboring solitons are

*π*out-of-phase [see Fig. 6(b)]. However, the most interesting structure, which is shown in Fig. 6(c), possesses a vortex-like phase dislocation, with the phase winding by 2

*π*around the low-density centre [as seen in Fig. 6(d)]. Such a bound state can be identified as a

*gap vortex*[29].

## 5. Optically-induced photonic lattices

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]

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]

*linear*regime and creates a stationary refractive-index grating [5

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]

*almost identical to the effective periodic potential experienced by matter waves in an optical lattice*. Optically induced lattices open up an exciting possibility for creating dynamically reconfigurable photonic structures in bulk nonlinear media, with a degree of control over the parameters of the periodic structure approaching that of BECs in optical lattices. Nonlinear localization of coherent light waves has been observed in both 1D and 2D optically-induced photonic lattices [5

**90**, 023902 (2003). [CrossRef] [PubMed]

**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]

## 6. Conclusions

## Acknowledgments

## References and links

1. | J.D. Joannopoulos, R.D. Meade, and J.N. Winn, |

2. | S.F. Mingaleev and Yu.S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. |

3. | R. Slusher and B. Eggleton, eds., |

4. | Yu.S. Kivshar and G.P. Agrawal, |

5. | J.W. Fleischer, T. Carmon, and M. Segev, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. |

6. | D. Neshev, E.A. Ostrovskaya, Yu.S. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. |

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 |

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 |

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 |

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. , |

11. | E.A. Ostrovskaya and Yu.S. Kivshar, “Matter-wave gap solitons in atomic band-gap structures,” Phys. Rev. Lett. |

12. | O. Zobay, S. Pötting, P. Meystre, and E.M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A |

13. | V.V. Konotop and M. Salerno, “Modulational instability in Bose-Einstein condensates in optical lattices,” Phys. Rev. A |

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 |

15. | N.K. Efremidis and D.N. Christodoulides, “Lattice solitons in Bose-Einstein condensates,” Phys. Rev. A |

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. |

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. |

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 |

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 |

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. |

24. | A.A. Sukhorukov and Yu. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E |

25. | N. Aközbek and S. John, “Optical solitary waves in two-and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E |

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. |

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 |

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 |

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

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(190.0190) Nonlinear optics : Nonlinear optics

**ToC Category:**

Focus Issue: Cold atomic gases in optical lattices

**History**

Original Manuscript: November 19, 2003

Revised Manuscript: December 31, 2003

Published: January 12, 2004

**Citation**

Elena Ostrovskaya and Yuri Kivshar, "Photonic crystals for matter waves: Bose-Einstein condensates in optical lattices," Opt. Express **12**, 19-29 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-19

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### References

- J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
- 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]
- 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).
- 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, andW. Krolikowski, �??Spatial solitons in optically induced gratings,�?? Opt. Lett. 28, 710 (2003). [CrossRef] [PubMed]
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