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Nonlinear Band Structure in Bose Einstein Condensates: The Nonlinear Schr\

B. T. Seaman, Lincoln D. Carr, Murray Holland

Published Oct 13, 2004• 0 citations
OpenAlex

Abstract

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.

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Comment on “Nonlinear band structure in Bose-Einstein condensates: Nonlinear Schrödinger equation with a Kronig-Penney potential”

Ippei Danshita, Shunji Tsuchiya

In their recent paper [Phys. Rev. A 71, 033622 (2005)], Seaman et al. studied Bloch states of the condensate wave function in a Kronig-Penney potential and calculated the band structure. They argued that the effective mass is always positive when a swallowtail energy loop is present in the band structure. In this Comment, we reexamine their argument by actually calculating the effective mass. It is found that there exists a region where the effective mass is negative even when a swallowtail is present. Based on this fact, we discuss the interpretation of swallowtails in terms of superfluidity.

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arXiv:cond-mat/0301012•Physical Review A•May 28, 2003•203 citations
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.
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.
The band structure of a Bose-Einstein condensate is studied for lattice traps of sinusoidal, Jacobi elliptic, and Kronig-Penney form, all in the context of the nonlinear Schr\"odinger equation. It is demonstrated that the physical properties of the system are independent of the type of lattice. The Kronig-Penney potential, which admits a full exact solution in closed analytical form, is then used to understand the swallowtails, or loops, that form in the band structure. The appearance of swallowtails is explained by adiabatically tuning a second lattice with half the period. Such a two-color lattice, which can be easily realized in experiments, has intriguing physical properties. For instance, swallowtails appear even for weak nonlinearity, which is the experimental regime. We determine the stability properties of this system and relate them to current experiments.
In their recent paper [Phys. Rev. A 71, 033622 (2005)], Seaman et al. studied Bloch states of the condensate wave function in a Kronig-Penney potential and calculated the band structure. They argued that the effective mass is always positive when a swallowtail energy loop is present in the band structure. In this Comment, we reexamine their argument by actually calculating the effective mass. It is found that there exists a region where the effective mass is negative even when a swallowtail is present. Based on this fact, we discuss the interpretation of swallowtails in terms of superfluidity.
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.
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.
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.