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arXiv:cond-mat/0301012 · Physical Review A

Band structure, elementary excitations, and stability of a Bose-Einstein condensate in a periodic potential

Mette Machholm, C. J. Pethick, H. Smith

Published May 28, 2003• 203 citations
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Abstract

We investigate the band structure of a Bose-Einstein condensate in a one-dimensional periodic potential by calculating stationary solutions of the Gross-Pitaevskii equation, which have the form of Bloch waves. We demonstrate that loops (``swallow tails'') in the band structure occur both at the Brillouin zone boundary and at the center of the zone, and they are therefore a generic feature. A physical interpretation of the swallow tails in terms of periodic solitons is given. The linear stability of the solutions is investigated as a function of the strength of the mean-field interaction, the magnitude of the periodic potential, and the wave vector of the condensate. The regions of energetic and dynamical stability are identified by considering the behavior of the Gross-Pitaevskii energy functional for small deviations of the condensate wave function from a stationary state. It is also shown how for long-wavelength disturbances the stability criteria may be obtained within a hydrodynamic approach.

DOI
10.1103/physreva.67.053613

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We investigate the band structure of a Bose-Einstein condensate in a one-dimensional periodic potential by calculating stationary solutions of the Gross-Pitaevskii equation, which have the form of Bloch waves. We demonstrate that loops (``swallow tails'') in the band structure occur both at the Brillouin zone boundary and at the center of the zone, and they are therefore a generic feature. A physical interpretation of the swallow tails in terms of periodic solitons is given. The linear stability of the solutions is investigated as a function of the strength of the mean-field interaction, the magnitude of the periodic potential, and the wave vector of the condensate. The regions of energetic and dynamical stability are identified by considering the behavior of the Gross-Pitaevskii energy functional for small deviations of the condensate wave function from a stationary state. It is also shown how for long-wavelength disturbances the stability criteria may be obtained within a hydrodynamic approach.
Energy band structure and stability of Bose-Einstein condensate with both two- and three-body interactions in one-dimensional periodic optical lattices are discussed. It is demonstrated that the lowest energy band can show a loop structure at the boundary of the first Brillouin zone only when the number of atoms is larger than a critical value. When the loop structure is excited, an analytical expression of the loop width is obtained. The energetical and dynamical stabilities of the system are determined by considering the behavior of the Gross-Pitaevskii energy functional when the condensate wave function diverge away from a stationary state upon small perturbation, and its dependence on the number of atoms, the magnitude of the periodic potential, especially, the strength of the three-body force are studied. In addition, a criteria of the stability for excitations with long wavelengths is given by applying the hydrodynamic approach. The influence of the three-body interaction on the band properties and the stabilities are discussed in detail. It is shown that the critical density for developing the loop structure, the loop width, and the instability criteria are modified dramatically by the three-body interaction.
We study the stability of Bose-Einstein condensates with superfluid currents in a one-dimensional periodic potential. By using the Kronig-Penney model, the condensate and Bogoliubov bands are analytically calculated and the stability of condensates in a periodic potential is discussed. The Landau and dynamical instabilities occur in a Kronig-Penney potential when the quasimomentum of the condensate exceeds certain critical values as in a sinusoidal potential. It is found that the onsets of the Landau and dynamical instabilities coincide with the point where the perfect transmission of low energy excitations through each potential barrier is forbidden. The Landau instability is caused by the excitations with small qqq and the dynamical instability is caused by the excitations with $q=\ensuremath{\pi}∕a$ at their onsets, where qqq is the quasimomentum of excitation and aaa is the lattice constant. A swallow-tail energy loop appears at the edge of the first condensate band when the mean-field energy is sufficiently larger than the strength of the periodic potential. We find that the upper portion of the swallow-tail is always dynamically unstable, but the second Bogoliubov band has a phonon spectrum reflecting the positive effective mass.
All Bloch states of the mean field of a Bose-Einstein condensate in the presence of a one-dimensional lattice are presented in closed analytic form. The band structure is investigated by analyzing the stationary states of the nonlinear Schr\"odinger, or Gross-Pitaevskii, equation for both repulsive and attractive condensates. The appearance of swallowtails in the bands is examined and interpreted in terms of the condensates superfluid properties. The nonlinear stability properties of the Bloch states are described and the stable regions of the bands and swallowtails are mapped out. We find that the Kronig-Penney potential has the same properties as a sinusoidal potential. Bose-Einstein condensates are trapped in sinusoidal optical lattices. The Kronig-Penney potential has the advantage of being analytically tractable, unlike the sinusoidal potential, and, therefore, serves as a good model for understanding experimental phenomena.
All Bloch states of the mean field of a Bose-Einstein condensate in the presence of a one dimensional lattice of impurities are presented in closed analytic form. The band structure is investigated by analyzing the stationary states of the nonlinear Schr\odinger, or Gross-Pitaevskii, equation for both repulsive and attractive condensates. The appearance of swallowtails in the bands is examined and interpreted in terms of the condensates superfluid properties. The nonlinear stability properties of the Bloch states are described and the stable regions of the bands and swallowtails are mapped out. We find that the Kronig-Penney potential has the same properties as a sinusoidal potential; Bose-Einstein condensates are trapped in sinusoidal optical lattices. The Kronig-Penney potential has the advantage of being analytically tractable, unlike the sinusoidal potential, and, therefore, serves as a good model for experimental phenomena.
We consider stationary Bloch waves in a Bose–Einstein condensate placed in a periodic potential for varying strengths of inter‐atomic interactions. Bifurcations of the stationary states are known to occur in this context. These bifurcations generate loops in the energy bands of the Bloch waves near the ends and the center of the Brillouin zone. Using the method of Lyapunov–Schmidt reductions, we show that these bifurcations are of the supercritical pitchfork type. We also characterize the change in stability of the stationary states across the bifurcation point. Analytical results are illustrated by numerical computations.
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.