Photonic crystals for matter waves: Bose-Einsteincondensates in optical lattices

Elena A. Ostrovskaya and Yuri S. KivsharNonlinear Physics Group and ARC Centre for Quantum-Atom Optics,

Research School of Physical Sciences and Engineering, Australian National University,Canberra ACT 0200, Australia

ysk124@rsphysse.anu.edu.au

http://wwwrsphysse.anu.edu.au/nonlinear/

Abstract: We overview our recent theoretical studies on nonlinear atomoptics of the Bose-Einstein condensates (BECs) loaded into optical lattices.In particular, we describe the band-gap spectrum and nonlinear localizationof BECs in one- and two-dimensional optical lattices. We discuss thestructure and stability properties of spatially localized states (matter-wavesolitons) in 1D lattices, as well as trivial and vortex-like bound states of2D gap solitons. To highlight similarities between the behavior of coherentlight and matter waves in periodic potentials, we draw useful parallelswith the physics of coherent light waves in nonlinear photonic crystals andoptically-induced photonic lattices.

© 2004 Optical Society of America

OCIS codes: (020.0020) Atomic and molecular physics; (190.0190) Nonlinear optics

References and links1. J.D. Joannopoulos, R.D. Meade, and J.N. Winn,Photonic Crystals: Molding the Flow of Light (Princeton Uni-

versity 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).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 waveg-

uide arrays,” Phys. Rev. Lett.90, 023902 (2003).6. D. Neshev, E.A. Ostrovskaya, Yu.S. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,”

Opt. Lett.28, 710 (2003).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,” Nature422, 147 (2003).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. B35, 3095 (2002).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. A67, 051603 (2003).10. M. Jona-Lasinio, O. Morsch, M. Cristiani, N. Malossi, J.H. Muller, E. Courtade, M. Anderlini, and E. Arimondo,

“Asymmetric Landau-Zener tunneling in a periodic potential,” Phys. Rev. Lett.,91, 230406 (2003).11. E.A. Ostrovskaya and Yu.S. Kivshar, “Matter-wave gap solitons in atomic band-gap structures,” Phys. Rev. Lett.

90, 160407 (2003).12. O. Zobay, S. Potting, P. Meystre, and E.M. Wright, “Creation of gap solitons in Bose-Einstein condensates,”

Phys. Rev. A59, 643 (1999).13. V.V. Konotop and M. Salerno, “Modulational instability in Bose-Einstein condensates in optical lattices,” Phys.

Rev. A65, 021602 (2002).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. A67, 013602 (2003).

(C) 2004 OSA 12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 19#3388 - $15.00 US Received 19 November 2003; revised 31 December 2003; accepted 1 January 2004

15. N.K. Efremidis and D.N. Christodoulides, “Lattice solitons in Bose-Einstein condensates,” Phys. Rev. A67063608 (2003).

16. E.A. Ostrovskaya and Yu. S. Kivshar, “Localization of two-component Bose-Einstein condensates in opticallattices,” 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, “Dispersionmanagement for atomic matter waves,” Phys. Rev. Lett.91, 060402 (2003).

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

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

20. C. M. de Sterke and J. E. Sipe, “Envelope-function approach for the electrodynamics of nonlinear periodicstructures,” Phys. Rev. A38, 5149 (1988).

21. H. Pu, L.O. Baksmaty, W. Zhang, N.P. Bigelow, and P. Meystre, “Effective-mass analysis of Bose-Einstein con-densates in optical lattices: Stabilization and levitation,” Phys. Rev. A67, 43605 (2003).

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

23. A.A. Sukhorukov, Yu. S. Kivshar, H. S. Eisenberg, and Y. Silberberg, “Spatial optical solitons in waveguidearrays,” IEEE J. Quantum Electron.39, 31 (2003).

24. A.A. Sukhorukov and Yu. S. Kivshar, “Spatial optical solitons in nonlinear photonic crystals,” Phys. Rev. E65,036609 (2002).

25. N. Akozbek and S. John, “Optical solitary waves in two-and three-dimensional nonlinear photonic band-gapstructures,” Phys. Rev. E57, 2287 (1998).

26. J.J. Garcıa-Ripoll and V.M. Perez-Garcıa, “Optimizing Schrodinger functionals using Sobolev gradients: Appli-cations to quantum mechanics and nonlinear optics,” SIAM J. Sci. Comput.23, 1316 (2001).

27. B. B. Baizakov, V.V. Konotop, and M. Salerno, “Regular spatial structures in arrays of Bose-Einstein condensatesinduced by modulational instability,” J. Phys. B35, 5105 (2002).

28. N.K. Efremidis, S. Sears, D.N. Christodoulides, J.W. Fleischer, and M. Segev, “Discrete solitons in photorefrac-tive optically induced photonic lattices,” Phys. Rev. E66, 046602 (2002).

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

1. Introduction

Photonic band-gap materials [1] – artificial periodic structures fabricated in a dielectric mediumwith a high refractive index contrast – offer new possibilities for the control and manipulationof coherent light waves. Diffraction management, localization, and controlled steering of lightin periodic band-gap structures, such asfiber Bragg gratings andphotonic crystals, has rev-olutionized modern photonics and laid the foundation for the development of novel types ofintegrated photonic devices. The study ofnonlinear photonic crystals made of a Kerr nonlinearmaterial [2, 3], has revealed that such structures can support self-trapped localized modes of theelectromagnetic field in the form ofoptical gap solitons [4] with the energies inside the pho-tonic gaps of a periodic structure. Recent demonstration of the light scattering in dynamicallyreconfigurable photonic structures – optically-induced refractive index gratings in nonlinearmaterials – has opened up novel ways to control light propagation and localization [5, 6, 7].

On the other hand, many recent experimental studies of Bose-Einstein condensates (BECs)in periodic potentials of optical lattices [8, 9, 10] demonstrate an unprecedented level of controland manipulation ofcoherent matter waves in the reconfigurable crystal-like structures createdby light. Due to the inherent nonlinearity of coherent matter waves which is introduced by theinteractions between atoms, BEC in a lattice potential forms a periodic nonlinear system whichis expected to display rich and complex dynamics.

The modelling of both nonlinear optical and nonlinear matter-wave dynamics is often basedon the nonlinear Schrodinger equation, which is used to describe both the electromagneticfield envelope and the BEC macroscopic wavefunction (mean-field). This model reflects sim-ilarities between the physics of coherent light and matter waves, which aid in understandingand predicting the nonlinear behavior of BEC in an optical lattice. Similar to periodic photonic

(C) 2004 OSA 12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 20#3388 - $15.00 US Received 19 November 2003; revised 31 December 2003; accepted 1 January 2004

structures for coherent light waves, optical lattices form band-gap structures for coherent matterwaves [11] which modify the diffraction properties of atomic wavepackets. Different diffractionregimes were predicted to lead to nonlinear localization of condensates with both attractive andrepulsive interactions in the spectral gaps of the atomic band-gap structure [12, 13, 14, 15, 16].Recent experiments on nonlinear dynamics of BEC in a lattice have demonstrated both diffrac-tion management [17, 18] and nonlinear localization of the atomic wavepackets in the form ofthe matter-wave gap solitons [19].

In this paper we describe the structure and stability properties of nonlinear localized states ofBEC in one- and two-dimensional optical lattices, and make links to parallel studies in nonlinearoptics of periodic photonic structures such as nonlinear photonic crystals and optically-inducedphotonic lattices.

2. Model

The dynamics of a Bose-Einstein condensate loaded into an optical lattice can be de-scribed in the mean-field approximation, by the nonlinear Gross-Pitaevskii (GP), or nonlinearSchrodinger, equation for the macroscopic condensate wavefunctionΨ(r, t),

ih∂Ψ∂ t

={− h2

2m∇ 2 +V (r)+g3D|Ψ|2

}Ψ (1)

whereV (r) is the time-independent trapping potential, andg3D = 4πh2as/m characterizes thetwo-body interactions for a condensate with atoms of massm and s-wave scattering lengthas. The scattering lengthas is positive for repulsive interactions and negative for attractiveinteractions.

We consider a trapping potentialV (r) of the form

V (r) =12

m(ωxx2 +ωyy2 +ωzz2)+VL, (2)

where the first term describes an anisotropic parabolic potential due to a magnetic trap. Theeffective periodic potential,VL, is formed by an optical lattice, which we consider to be eitherone-dimensional (1D),

VL(x) = V0sin2(Kx), (3)

or two-dimensional (2D):

VL(x,y) = V0[sin2(Kx)+sin2(Ky)

]. (4)

The 1D (2D) lattice is created by a pair (two pairs) of counter-propagating laser beams with thewavelengthλ = 2π/K, and the lattice depth,V0, is proportional to the intensity of the standinglight wave.

Equation (1) can be made dimensionless using the characteristic lengthaL = 1/K, energyErec = h2/ma2

L, and timeω−1L = h/Erec scales of the lattice. In dimensionless units, the two-

body interaction coefficient is given byg3D = 4π(as/aL), and the lattice depth is measured inunits of the lattice recoil energy,Erec.

3. One-dimensional atomic band-gap structures

We now assume a cigar-shaped condensate weakly trapped inx-direction (ωx � ωy,z ≡ ω⊥). Inthe directions of tight confinement, the condensate wavefunction can be described by the groundstate of a two-dimensional radially symmetric quantum harmonic oscillator potential, with thenormalization

∫ ∞−∞ |Φ|2dr = 1. The 3D wavefunction then separates asΨ(r, t) = Φ(y,z)ψ(x, t),

(C) 2004 OSA 12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 21#3388 - $15.00 US Received 19 November 2003; revised 31 December 2003; accepted 1 January 2004

1 1.5 2 2.5 3µ

-5

0

5

vg

D

Fig. 1. Group velocity and effective diffraction coefficient for Bloch matter waves in anoptical lattice shown in the context of the bandgap spectrum with the Bloch bands (shaded)and gaps (open);V0 = 2.0.

and the transverse dimensions can be intergrated out in the dimensionless Eq. (1), yielding the1D GP equation:

i∂ψ∂ t

={−1

2∂ 2

∂x2 +VL(x)+σ |ψ(x, t)|2}

ψ (5)

where the wavefunction is rescaled asψ → ψ√g1D, g1D = 4(as/aL)(ω⊥/ωL)2, σ = sign(g1D),

and the external potential is approximated by the quasi-1D periodic potential of the opticallattice,VL(x) = sin2(x), by neglecting the contribution of a weak magnetic confinement.

The stationary states of a condensate in a quasi-1D infinite periodic potential are describedby solutions of Eq. (5) of the form:ψ(x, t) = φ(x)exp(−iµt), whereµ is the correspondingchemical potential. The case of a noninteracting condensate formally corresponds tog1D = 0,and the condensate wave function can be represented as a superposition of Bloch waves,

φ(x) = b1φ1(x)eikx +b2φ2(x)e−ikx, (6)

whereφ1,2(x) have periodicity of the lattice potential,b1,2 are constants, andk is the Floquetexponent. The linear matter-wave spectrum consists of bands of eigenvaluesµn,k in whichk(µ)is a (real) wavenumber of amplitude-bounded Bloch waves [14]. The bands are separated byspectral gaps in whichIm(k) �= 0. The solutions at the band edges are stationary Bloch statescorresponding to the condensate density which is strongly and periodically modulated by thelattice.

For an interacting condensate (g1D �= 0), bright solitons can exist for chemical potentialscorresponding to the gaps of the linear matter-wave spectrum. The mechanism for nonlinearlocalization in the gaps was first described for light waves in nonlinear periodic photonic struc-tures by using the Bloch-wave envelope approximation near the band edge [20]. It has beenpointed out that, near the band edges, the characteristic diffraction coefficient for a coherentwave is proportional to the band curvature at that edge, which can be both negative and posi-tive. For a matter wave in an optical lattice, the group velocity,vg = ∂ µ/∂k, and the effectivediffraction coefficient,D = ∂ 2µ/∂k2 (analogous to the inverse effective mass of an electron incrystalline solids [21]), are shown in Fig. 1 for the Bloch waves in the first lowest-order bandsof a relatively shallow lattice. The balance of the repulsive (attractive) interactions and anoma-lous,D < 0, (normal,D > 0) diffraction near the top (bottom) of the bands (see Fig. 1), can lead

(C) 2004 OSA 12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 22#3388 - $15.00 US Received 19 November 2003; revised 31 December 2003; accepted 1 January 2004

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) repul-sive and (right) attractive condensates (V0 = 5).

to formation of localized wavepackets -matter wave solitons - with zero group velocity. Thechemical potentials corresponding to such localized waves lie in the gaps of the Bloch-wavespectrum for the matter waves of the noninteracting condensate (open areas in Fig. 1).

Employing the multi-scale perturbation series expansion for the chemical potential and thegap soliton envelope, it is possible to show that, in general, two types of gap solitons bifurcatefrom each band edge [22]. These are the bright solitons centered on the maxium (off-site)and minimum (on-site) of the lattice potential, respectively. Several families of the lowest-order gap modes are presented in Fig. 2, for the case of both repulsive (left) and attractive(right) condensates. The families are characterized by the norm of the condensate wavefunction,P =

∫φ2(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 firstband [14], which is analogous to the total internal reflection gap of the photonic structures,and where the existence of the localized states is due to conventional self-focusing. Near thebottom (top) gap edge, gap solitons of the repulsive (attractive) BEC are well approximatedby asech-like envelope of the corresponding Bloch wave, centered either on- or off-site. Nearthe 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]. Fora sufficiently wide gap (i.e., in the relatively tight binding regime), the soliton deep inside thegap may be strongly localized around a single well of the lattice potential. The bifurcations ofthe two types of gap solitons from the band edges of the opposite ”polarity” (or sign of theeffective diffraction) are a general feature of the coherent waves in periodic systems, such asspatial optical solitons in discrete waveguide arrays [23].

Stability of extended and localized modes in nonlinear systems is an important issue, sinceonly dynamically stable modes are likely to be generated and observed in experiments. To

(C) 2004 OSA 12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 23#3388 - $15.00 US Received 19 November 2003; revised 31 December 2003; accepted 1 January 2004

0 1000 2000 3000 4000 5000 6000 70000

0.5

1

1.5

2

2.5

3

3.5

4

Time units

Peak

den

sity

0 2000 4000 6000 8000 100000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time units

Peak

den

sity

(a) (b)

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-infinitegap (µ = 1.0). In (a) the initial state given by the exact (numerical) stationary solution ofEq. (5) is perturbed by a symmetric excitation at 5% of the soliton peak density. In (b) theantisymmetric internal mode is excited by an initial perturbation at 5% of the initial solitonpeak density.

determine the stability properties of the localized modes, we consider small perturbations to astationary gap soliton of the GP equation,φ(x), in the form

ψ(x, t) = φ(x)e−iµt + ε[(u+ iw)eλ t +(u∗ + iw∗)eλ ∗t ]e−iµt , (7)

whereε � 1, andu(x), w(x) are spatially-dependent perturbation modes. We linearize the GPequation (5) around the localized solution and obtain, to the first order inε, a linear eigenvalueproblem for the perturbation modes

L−w = λ u, L+u = −λ w, (8)

whereL± = −(1/2)(∂ 2/∂x2)+VL(x)+2σφ2±σφ2−µ. The modes describing the develop-ment of instability have either purely real or complex eigenvaluesλ ; in this latter case, theinstability is calledan oscillatory instability.

We solve the eigenvalue problem for the perturbation modes numerically, as a matrix eigen-value problem with Fourier interpolants for the differential operators. The results reveal that, inagreement with the generalized Vakhitov-Kolokolov criterion (see, e.g., [4]), nodeless localizedstates of the attractive BEC (such as the fundamental solitons in a semi-infinite gap [Fig. 2(a),right]) are linearly stable as long as−∂P/∂ µ > 0. Note that the criterion is “inverted” for ourchoice ofµ. The fundamental gap solitons of repulsive BEC [Fig. 2 (left, a)] are also linearlystable.

The linear stability analysis shows that gap solitons of both attractive and repulsive BEC areassociated with a number of internal modes (Imλ �= 0 and Reλ = 0). The excitation of thesemodes usually leads to a persisting dynamics (e.g., amplitude oscillations) of the localized statethat does not lead to its decay. However, higher-order gap solitons of both attractive and re-pulsive condensates can experience oscillatory instability (Imλ �= 0 and Reλ �= 0) initiated byresonances of internal modes with the bands of the inverted spectrum [22]. This mechanismfor weak spectral instability was first identified in studies of spatial optical solitons in one-dimensional nonlinear photonic crystals [24]. In Fig. 3 we compare temporal evolution of theweakly unstable gap soliton of the repulsive condensate in the first gap (a) with that of a lin-early stable (attractive) BEC soliton in the semi-infinite spectral gap (b). The evolution of thesoliton peak density is shown in Fig. 3 for the cases when initial solitons are perturbed by a5% of the peak density. These results indicate that the oscillatory instability is weak and, on

(C) 2004 OSA 12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 24#3388 - $15.00 US Received 19 November 2003; revised 31 December 2003; accepted 1 January 2004

realistic experimental time scales, can not provide a strong mechanism for the soliton decayinto spatially extended lattice states (which would otherwise prevent observation of localizedstates in experiments).

4. Two-dimensional band-gap structures

The theory of the nonlinear localized matter waves in optical lattices can be extended to thecase of two-dimensional lattices [11]. Nonlinear localization of BEC in higher-dimensionallattices is qualitatively different because both the symmetry and dimensionality of the latticestart to play an important role in the formation and properties of the band-gap structure andthe corresponding nonlinear localized modes. In particular, the problem of the existence andstability of 2D matter-wave gap solitons of BEC with repulsive interatomic interactions loadedinto optical lattices, is analogous to the study of localized states of light waves in nonlinearphotonic crystals [2].

To simplify our analysis of the model (1), we assume that the weak magnetic confinementcharacterized by trap frequenciesωx,y has little effect on the stationary states of the condensatein the 2D lattice formed in the(x,y) plane of the condensate cloud. Under this assumption, thetrap component of the confining potential in the lattice plane can be neglected, and the modelcan be reduced to a two-dimensional GP equation by the dimensionality reduction proceduredescribed above for the 1D case. The condensate wavefunction in the 2D lattice potential isthen described by the following equation:

i∂ψ∂ t

={−1

2∇ 2⊥ +VL(x,y)+ |ψ|2

}ψ. (9)

where∇ ⊥ = ∂ 2/∂x2 + ∂ 2/∂y2, VL(x,y) is the periodic potential of the optical lattice, and thewavefunction is rescaled asψ →ψ√

g2D, with g2D = g3D/√

2. The equation (9) is made dimen-sionless 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 writtenasVL = V0

[sin2(x)+sin2(y)

], whereV0 is the amplitude of the optical lattice.

Stationary (time-independent) states of the condensate in an infinite periodic potential of a 2Doptical lattice are described by solutions of Eq. (9) of the form:ψ(r, t) = φ(r)exp(−iµt), whereµ is the chemical potential. The case of a noninteracting condensate formally corresponds toEq. (9) being linear inψ. According to the Bloch theorem, the stationary wavefunction can thenbe sought in the formφ(r) = uk(r)exp(ikr), where the wave vectork belongs to a Brillouinzone of the square optical lattice, anduk(r) = uk(r + d) is a periodic (Bloch) function withthe periodicity of the lattice. For the values ofk within ann-th Brillouin zone, the dispersionrelation for the 2D Bloch waves,µn,k, is found by solving a linear eigenvalue problem:

[12(−i∇ ⊥ +k)2 +VL(r)

]un,k = µn(k)un,k. (10)

The band-gap structure of the spectrumµ(k) of the atomic Bloch waves in the 2D optical lat-tice is shown in Fig. 4, in the reduced zone representation usually assumed in the theory ofcrystalline solids and photonic crystals. Due to separability of the lattice potential, the Blochwavesu(x,y) in the first band, at the high symmetry points(Γ → X → M) of the first irre-ducible Brillouine zone (see Fig 4), can be found asu(x,y;k) = u(1d)(x;kx)u(1d)(y;ky), whereu(1d) are the corresponding 1D Bloch states. Different structure of the Bloch states in the firstband, at the values ofk corresponding to the three symmetry points, are shown in Fig. 4 (rightcolumn). By applying an envelope theory near the band edges, it can be deduced that each ofthe Bloch states is associated with a (partially) localized state in theΓ, X , or M-gap. Similarly

(C) 2004 OSA 12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 25#3388 - $15.00 US Received 19 November 2003; revised 31 December 2003; accepted 1 January 2004

Γ X M Γ0

1

2

3

4

5

µ(k)

−2

0

2

−2

0

2

0

0.05

0.1

xy

−2

0

2

−2

0

2

−0.1

0

0.1

xy

−2

0

2

−2

0

2

−0.1

0

0.1

xy

Γ

M

X

Fig. 4. Left: Dispersion diagram for a 2D square lattice (V0 = 1.5); dotted - the lineµ = V0;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 2DBloch waves at the extreme high-symmetry points of the first irreducible Brillouine zone.

to optical gap solitons in indirect gaps of higher-dimensional photonic crystals [25], matter-wave gap solitons can be spatially localized in all directions only in the complete gap locatedbetween theM and X-edges of the first and the second band, respectively (see Fig. 4). Thelocalization near the lowerM-edge is possible due to the negative components of the effec-tive diffraction tensor. We find spatially localized stationary solutions of Eq. (9) numerically.Our numerical procedure involves minimization of the normN =

∫f † f dr of the functional

f (φ) = [(1/2)∇ 2⊥−µ −V (r)−|φ|2]φ by following a descent technique with Sobolev precon-

ditioning [26]. The minimization procedure yields a stationary state whenN (φ) → 0.The family of fundamental bright matter-wave solitons in the first complete gap is presented

in Fig. 5 . For the repulsive condensate, such gap solitons exist in all band gaps excluding thesemi-infinite gap below the first band. The spatial structure of the lowest-order gap solitonsnear the lower gap edge [Fig. 5(a)] has a characteristic structure of the symmetric envelopesuperimposed onto the corresponding Bloch state [see Fig. 4]. The threshold value of the num-ber of atoms (and thereforePc) needed for the soliton localization is determined by the theoryof lattice-free self-focusing in the 2D geometry. These weakly localized, low density modesare similar to the near-band-edge modes described above for the case of one-dimensional gapsolitons. 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 barrierheights and positive (normal) effective diffraction. By direct numerical simulations we haveconfirmed that the strongly localized two-dimensional fundamental modes are dynamically sta-

(C) 2004 OSA 12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 26#3388 - $15.00 US Received 19 November 2003; revised 31 December 2003; accepted 1 January 2004

−20 −15 −10 −5 0 5 10 15 20−20

−15

−10

−5

0

5

10

15

20

x

y

−20 −15 −10 −5 0 5 10 15 20−20

−15

−10

−5

0

5

10

15

20

x

y

−20 −15 −10 −5 0 5 10 15 20−20

−15

−10

−5

0

5

10

15

20

xy

1.6 1.8 2 2.2 2.4 2.6 2.8 3µ

0

1

2

3

4

5

6

7

8

9

P

a

b

c

(a) (b) (c)

Fig. 5. Top: Family of bright atomic gap solitons of repulsive BEC in a 2D optical lattice(V0 = 1.5). Bottom: spatial structure of the BEC wavefunctions at the marked points of theexistence curve inside the gap.

ble. The possibility of the formation of periodic trains of such 2D localized structures, triggeredby the modulational instability of the nonlinear Bloch states, has been suggested in Ref. [27].

We have also found different families of higher-order localized gap modes that can be identi-fied as in-phase and out-of-phase bound states of the fundamental gap solitons. These states canbe centered at the lattice potential minima or maxima, similarly to higher-order states in one-dimensional lattices [14], and they can exhibit symmetry-breaking instabilities. In particular,Figs. 6(a-c) showsthree different examples of higher-order gap solitons. The simplest structureof this kind is described by a pair of two fundamental gap solitons which are out-of-phase andform a dipole [see Fig. 6(a)]. The similar structure of four solitons can exist asa quadrupolestate where the neighboring solitons areπ out-of-phase [see Fig. 6(b)]. However, the most in-teresting structure, which is shown in Fig. 6(c), possesses a vortex-like phase dislocation, withthe phase winding by 2π around the low-density centre [as seen in Fig. 6(d)]. Such a boundstate can be identified asa gap vortex [29].

5. Optically-induced photonic lattices

Although many similarities (including the structure of localized modes and their stability prop-erties) exist between nonlinear atom optics in optical lattices and nonlinear optics in fabricatedphotonic structures, the closest analog of the matter-wave band-gap structures can be found inthe new and rapidly developing field of optically-induced photonic lattices [5, 6, 7].

(C) 2004 OSA 12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 27#3388 - $15.00 US Received 19 November 2003; revised 31 December 2003; accepted 1 January 2004

−20 −15 −10 −5 0 5 10 15 20−20

−15

−10

−5

0

5

10

15

20

x

y

−20 −15 −10 −5 0 5 10 15 20−20

−15

−10

−5

0

5

10

15

20

x

y

−20 −10 0 10 20−20

−15

−10

−5

0

5

10

15

20

x

y1

2

3

4

5

6

0

2π

π

−20 −10 0 10 20−20

−15

−10

−5

0

5

10

15

20

x

y

(c) (d)

(a) (b)

Fig. 6. Spatial structures of the BEC wavefunctions corresponding to higher-order gap soli-tons (V0 = 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).

In the existing experimental realizations of optically-induced photonic lattices, a periodicmodulation of the refractive index is induced by a plane wave interference pattern illuminat-ing a photorefractive crystal with a strong electro-optic anisotropy [28]. The spatially periodicinterference pattern modulates the space-charge field in the crystal, which relates to the re-fractive index via electro-optic coefficients. The latter are substantially different for differentpolarizations. As a result, the material nonlinearity experienced by waves polarized in the maindirection of the crystal is up to two orders of magnitude larger than that experienced by theorthogonally polarized ones. When the lattice-forming waves are polarized orthogonally to themain axis, their nonlinear self-action as well as any cross-action from the co-propagating probebeam (polarized along the main axis) can be neglected. Thus, the periodic interference pat-tern propagates in the diffraction-freelinear regime and creates a stationary refractive-indexgrating [5, 6]. The effective induced grating exhibited by the probe light isalmost identical tothe effective periodic potential experienced by matter waves in an optical lattice. Optically in-duced lattices open up an exciting possibility for creating dynamically reconfigurable photonicstructures in bulk nonlinear media, with a degree of control over the parameters of the periodicstructure approaching that of BECs in optical lattices. Nonlinear localization of coherent lightwaves has been observed in both 1D and 2D optically-induced photonic lattices [5, 6, 7], in theself-focusing regime (which is analogous to the attractive matter-wave localization regime).

(C) 2004 OSA 12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 28#3388 - $15.00 US Received 19 November 2003; revised 31 December 2003; accepted 1 January 2004

6. Conclusions

We have studied the spectrum of matter waves and localization of Bose-Einstein condensates inone- and two-dimensional optical lattices and demonstrated that the interaction of the conden-sate with a periodic lattice potential can be compared to the propagation of coherent light wavesin a nonlinear periodic photonic structure. Using the analogy with the physics of photonic band-gap structures, we have described some basic properties of matter-wave band-gap structures anddemonstrated numerically the existence of one- and two-dimensional matter-wave gap solitons,spatially localized states of the condensate existing in the gaps of the matter-wave linear spec-trum. We believe that the analogy between the physics of BEC in optical lattices and light wavesin nonlinear photonic crystals, as well as other types of periodic photonic structures, is usefulto reveal many novel features of the matter-wave dynamics in reconfigurable optically-inducedstructures.

Acknowledgments

We would like to thank A. Sukhorukov, D. Neshev, and D. Pelinovsky for useful discussionsand collaboration on related photonics projects, and P. Louis and C. Savage for collaborationon the BEC projects. This work was supported by the Australian Research Council (ARC)through the Centre of Excellence program. The authors are members of the ARC Centre forQuantum-Atom Optics.

(C) 2004 OSA 12 January 2004 / Vol. 12, No. 1 / OPTICS EXPRESS 29#3388 - $15.00 US Received 19 November 2003; revised 31 December 2003; accepted 1 January 2004