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AIP conference proceedings

Elementary Excitations of Condensates in a Kronig-Penney Potential

Ippei Danshita, Shinichiro Kurihara, S. Tsuchiya

Published Jan 1, 2006• 0 citations
Publisher

Abstract

We investigate the excitation spectrum of a Bose‐Einstein condensate in a Kronig‐Penney potential. We solve the Bogoliubov equations analytically and obtain the band structure of the excitation spectrum. We find that the excitation spectrum is gapless and linear at low energies. This property is found to be attributed to the anomalous tunneling behavior of low energy excitations, which has been predicted by Kagan et al.

DOI
10.1063/1.2354597

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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
We investigate the excitation spectrum of a Bose‐Einstein condensate in a Kronig‐Penney potential. We solve the Bogoliubov equations analytically and obtain the band structure of the excitation spectrum. We find that the excitation spectrum is gapless and linear at low energies. This property is found to be attributed to the anomalous tunneling behavior of low energy excitations, which has been predicted by Kagan et al.
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.
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.