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arXiv:cond-mat/0504217 · Physical Review A

Period doubling, two-color lattices, and the growth of swallowtails in Bose-Einstein condensates

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

Published Sep 6, 2005• 41 citations
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Abstract

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.

DOI
10.1103/physreva.72.033602

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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.

arXiv:cond-mat/0703045•Physical Review A•Jul 16, 2007•1 citation

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

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In response to Danshita and Tsuchiya's comment on our work on nonlinear band theory, we show that the size of the region of the swallowtail with a negative effective mass is inversely proportional to the interaction strength, i.e., for large interaction strengths, the region becomes negligibly small. We explain why the appearance of swallowtails is not related to superfluidity, but instead to a more universal nonlinear feature valid for both signs of the underlying atomic interactions: period doubling.

Physical Review A•Jul 16, 2007•1 citation

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arXiv:1605.02984•Physical Review A•Dec 23, 2016•22 citations
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.
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.
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.
In response to Danshita and Tsuchiya's comment on our work on nonlinear band theory, we show that the size of the region of the swallowtail with a negative effective mass is inversely proportional to the interaction strength, i.e., for large interaction strengths, the region becomes negligibly small. We explain why the appearance of swallowtails is not related to superfluidity, but instead to a more universal nonlinear feature valid for both signs of the underlying atomic interactions: period doubling.
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 study experimentally the stability of excited, interacting states of bosons in a double-well optical lattice in regimes where the nonlinear interactions are expected to induce ``swallowtail'' looped band structure. A dynamic homogeneous Gross-Pitaevskii calculation indicates that the double-well lattice both stabilizes the looped band structure and allows for dynamic preparation of different initial states, including states within the loop structure. The homogeneous calculation predicts that the loop states, unlike the ground states, should be dynamically stable. An inhomogeneous mean-field calculation including the trap potential, however, implies that the decay is dominated by inhomogeneous effects and that there is little variation in the decay rate among the states prepared within the loop structure. By experimentally preparing different initial coherent states and observing their subsequent decay, we observe distinct decay rates in regimes where multivalued looped band structure is expected, although not the stability predicted by the homogeneous calculation.