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Laser Physics

Instabilities and sound speed of a Bose-Einstein condensate in the Kronig-Penney potential

Xi Dong, Biao Wu

Published Feb 1, 2007• 11 citations
Publisher

Abstract

We analyze the full set of Bloch wave stationary solutions for a Bose-Einstein condensate in the Kronig-Penney potential. We investigate the Landau instability and dynamical instability of the Bloch states in the lowest Bloch band, including the loop if it appears. The stability phase diagrams are shown to be similar as in the case of the sinusoidal optical lattice potential. We also compute the speed of sound as a function of the potential strength. The trend is shown to be similar to the sinusoidal case, reflecting our general conclusion that, in any one-dimensional periodic potential, the sound speed always falls monotonically with increasing potential strength, no matter whether the atomic interaction is weak or strong. The Kronig-Penney potential, being analytically tractable and hence more advantageous than the sinusoidal potential, therefore serves as a good model for understanding the phenomena of Bose-Einstein condensation.

DOI
10.1134/s1054660x07020247

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We analyze the full set of Bloch wave stationary solutions for a Bose-Einstein condensate in the Kronig-Penney potential. We investigate the Landau instability and dynamical instability of the Bloch states in the lowest Bloch band, including the loop if it appears. The stability phase diagrams are shown to be similar as in the case of the sinusoidal optical lattice potential. We also compute the speed of sound as a function of the potential strength. The trend is shown to be similar to the sinusoidal case, reflecting our general conclusion that, in any one-dimensional periodic potential, the sound speed always falls monotonically with increasing potential strength, no matter whether the atomic interaction is weak or strong. The Kronig-Penney potential, being analytically tractable and hence more advantageous than the sinusoidal potential, therefore serves as a good model for understanding the phenomena of Bose-Einstein condensation.
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.
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 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.
A set of exact closed-form Bloch-state solutions to the stationary Gross-Pitaevskii equation are obtained for a Bose-Einstein condensate in a one-dimensional periodic array of quantum wells, i.e. a square-well periodic potential. We use these exact solutions to comprehensively study the Bloch band, the compressibility, effective mass and the speed of sound as functions of both the potential depth and interatomic interaction. According to our study, a periodic array of quantum wells is more analytically tractable than the sinusoidal potential and allows an easier experimental realization than the Krönig-Penney potential, therefore providing a useful theoretical model for understanding Bose-Einstein condensates in a periodic potential.
The speed of sound of a Bose-Einstein condensate in an optical lattice is studied both analytically and numerically in all spatial three dimensions. Our investigation shows that the sound speed depends strongly on the strength of the lattice. In the one-dimensional case, the speed of sound falls monotonically with increasing lattice strength. The dependence on lattice strength becomes much richer in two and three dimensions. In the two-dimensional case, when the interaction is weak, the sound speed first increases and then decreases as the lattice strength increases. For the three-dimensional lattice, the sound speed can even oscillate with the lattice strength. These rich behaviors can be understood in terms of compressibility and effective mass. Our analytical results in the limit of weak lattices also offer an interesting perspective to help with our understanding: they show that the lattice component perpendicular to the sound propagation increases the sound speed while the lattice component parallel to the propagation decreases the sound speed. The various dependences of the sound speed on the lattice strength are the result of this competition.