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Physical Review Letters

Self-Trapped Nonlinear Matter Waves in Periodic Potentials

Tristram J. Alexander, Elena A. Ostrovskaya, Yuri S. Kivshar

Published Jan 30, 2006• 122 citations
Publisher

Abstract

We demonstrate that the recent observation of nonlinear self-trapping of matter waves in one-dimensional optical lattices [Th. Anker, Phys. Rev. Lett. 94, 020403 (2005)10.1103/PhysRevLett.94.020403] can be associated with a novel type of broad nonlinear state existing in the gaps of the matter-wave band-gap spectrum. We find these self-trapped localized modes in one-, two-, and three-dimensional periodic potentials, and demonstrate that such novel gap states can be generated experimentally in any dimension.

DOI
10.1103/physrevlett.96.040401

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We analyze numerically the process of dynamical generation of spatially localized vortices in Bose-Einstein condensates (BEC's) with repulsive atomic interactions confined by two-dimensional optical lattices. Akin to bright solitons in a repulsive condensate, these nonlinear localized states exist only within the gaps of the matter-wave band-gap spectrum imposed by the periodicity of the lattice potential. We discuss the complex structure of matter-wave phase singularities associated with different types of stationary gap vortices and suggest two different excitation methods. In one method, the condensate is adiabatically driven to the edge of the Brillouin zone where a vortex phase is subsequently imprinted onto the condensate wave packet. Alternatively, a vortex is created in a condensate confined in a harmonic trap and then is nonadiabatically released into the lattice potential. We find that only the latter method leads to robust and reliable generation of vortices within the gap of the matter-wave band-gap spectrum. Moreover, the nonadiabatic excitation can lead to the formation of broad gap vortices from the initial BEC wave packets with a large number of atoms. These broad vortices are intimately connected to self-trapped nonlinear states of the BEC recently demonstrated in experiments with a one-dimensional optical lattice. Our numerical simulations also confirm the feasibility of a homodyne interferometric detection of broad gap vortices.
Optical lattices have been used as a versatile toolbox to control Bose-Einstein condensates (BECs) in recent years, and a wealth of emergent nonlinear phenomena have been found, including bright gap solitons and dark ones, among which the former has been realized in experiments. The latter, however, has only theoretical results and its fundamental properties are still not well understood. Here we theoretically and numerically explore an open issue of creating stable matter-wave dark gap solitons in a one-dimensional optical lattice, onto which the BECs with self-defocusing quintic nonlinearity are loaded. Using linear-stability analysis and direct simulations, the formation, structures, and properties of dark gap solitons in quintic nonlinearity have been compared to those upheld by cubic Kerr nonlinearity. In particular, we uncover that the dark gap solitons and soliton clusters are robustly stable in the first finite band gap of the underlying linear spectrum, and are hard to be stabilized in the second gap. The predicted dark gap solitons are observable in current experiments on dense ultracold atoms, using an optical lattice technique, and in the optics domain for nonlinear light propagation in periodic optical media with quintic nonlinearity.
We suggest a method to experimentally obtain two-dimensional matter-wave discrete solitons with a self-repulsive Bose–Einstein condensate in optical lattices. At the edge of the Brillouin zone, a wavepacket effective mass is negative, which could be treated as an inversion of the nonlinearity sign. Above critical nonlinearity this makes the wavepackets collapse partially into localized modes with a chemical potential located in the gap between the first and the second bands. This critical nonlinearity is also associated with the smallest nonlinearity for which the discrete solitons are possible in the gap. Extensive numerical simulations for square and asymmetric honeycomb lattices in the continuous model illustrate every stage of the process.
We analyze the existence and stability of spatially extended (Bloch-type) and localized states of a Bose-Einstein condensate loaded into an optical lattice. In the framework of the Gross-Pitaevskii equation with a periodic potential, we study the band-gap structure of the matter-wave spectrum in both the linear and nonlinear regimes. We demonstrate the existence of families of spatially localized matter-wave gap solitons, and analyze their stability in different band gaps, for both repulsive and attractive atomic interactions.