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

Elena A. Ostrovskaya and Yuri S. Kivshar

Open Access Open Access

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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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.

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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).

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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.

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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.

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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.

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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).

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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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