A weakly interacting, spin-orbit-coupled, ultracold, dilute Bose gas on a two-dimensional square lattice with an external Zeeman field is studied. We explore the plane- and stripe-wave phases of the system involving nonzero condensate momenta, which occur when the Zeeman field is below a critical value. Their excitation spectra are found using Bogoliubov theory and by two different routes. The validity of each method to obtain the excitation spectrum is discussed, and it is found that projection on the lowest single-particle band is an excellent approximation in the plane-wave phase, while it is a poor approximation in the stripe-wave phase. While the plane-wave phase has a phonon minimum at its single condensate momentum, revealing a nonzero sound velocity of the excitations, the stripe-wave phase has quadratic minima at its two condensate momenta showing zero sound velocity of the excitations. We discuss how the presence of more than one condensate momentum is essential for these differences between the two phases. Additionally, it is emphasized that the zero sound velocity in the stripe-wave phase is a lattice effect, since continuum studies of the same phase have shown nonzero sound velocity.

中文翻译：

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具有塞曼场的光学晶格中自旋轨道耦合的玻色-爱因斯坦凝聚物的平面波和条纹波相位

研究了具有外部塞曼场的二维方格上的弱相互作用，自旋轨道耦合，超冷稀Bose气体。我们探索了包含非零凝结动量的系统的平面波和条纹波相位，当塞曼场低于临界值时会发生这种情况。它们的激发光谱是使用Bogoliubov理论并通过两种不同的途径找到的。讨论了获得激发光谱的每种方法的有效性，发现在最低单粒子带上的投影在平面波相位上是极好的近似，而在条纹波相位上则是较差的近似。虽然平面波相位在其单个凝结动量处具有最小的声子，但揭示了激发的非零声速，条纹波相位在其两个凝结矩处具有二次极小值，表明激发的声速为零。我们讨论了不止一个凝结动量的存在对于两个阶段之间的这些差异至关重要。另外，需要强调的是，条纹波相位中的零声速是晶格效应，因为对相同相位的连续研究表明非零声速。