We consider a trapped Bose--Einstein condensate (BEC) with a highly quantizedvortex. For the BEC with a doubly, triply or quadruply quantized vortex, thenumerical calculations have shown that the Bogoliubov--de Gennes equations,which describe the fluctuation of the condensate, have complex eigenvalues. Inthis paper, we obtain the analytic expression of the condition for theexistence of complex modes, using the method developed by Rossignoli andKowalski [R. Rossignoli and A. M. Kowalski, Phys. Rev. A 72, 032101 (2005)] forthe small coupling constant. To derive it, we make the two-mode approximation.With the derived analytic formula, we can identify the quantum number of thecomplex modes for each winding number of the vortex. Our result is consistentwith those obtained by the numerical calculation in the case that the windingnumber is two, three or four. We prove that the complex modes always exist whenthe condensate has a highly quantized vortex.
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机译：我们考虑了一个具有高度量化涡旋的玻色-爱因斯坦凝聚物（BEC）。对于具有双重，三重或四重量化涡流的BEC，数值计算表明，描述冷凝液波动的Bogoliubov-de Gennes方程具有复杂的特征值。在本文中，我们使用Rossignoli和Kowalski提出的方法获得了存在复杂模式的条件的解析表达式。 Rossignoli和A.M. Kowalski，物理学Rev. A 72，032101（2005）]。为了得到它，我们进行了双模近似。利用推导的解析公式，我们可以识别出每个旋涡数的复模的量子数。在绕组数为二，三或四的情况下，我们的结果与通过数值计算获得的结果一致。我们证明了当冷凝液具有高度量化的涡旋时，复杂的模态始终存在。
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