Condition for the Existence of Complex Modes in a Trapped Bose--Einstein Condensate with a Highly Quantized Vortex

We consider a trapped Bose--Einstein condensate (BEC) with a highly quantized vortex. For the BEC with a doubly, triply or quadruply quantized vortex, the numerical calculations have shown that the Bogoliubov--de Gennes equations, which describe the fluctuation of the condensate, have complex eigenvalues. In this paper, we obtain the analytic expression of the condition for the existence of complex modes, using the method developed by Rossignoli and Kowalski [R. Rossignoli and A. M. Kowalski, Phys. Rev. A 72, 032101 (2005)] for the small coupling constant. To derive it, we make the two-mode approximation. With the derived analytic formula, we can identify the quantum number of the complex modes for each winding number of the vortex. Our result is consistent with those obtained by the numerical calculation in the case that the winding number is two, three or four. We prove that the complex modes always exist when the condensate has a highly quantized vortex

Condition for emergence of complex eigenvalues in the Bogoliubov-de Gennes equations

The condition for the appearance of dynamical instability of the Bose-condensed system, characterized by the emergence of complex eigenvalues in the Bogoliubov-de Gennes equations, is studied analytically. We perturbatively expand both the Gross-Pitaevskii and Bogoliubov-de Gennes equations with respect to the coupling constant. It is concluded that the degeneracy between a positive-norm eigenmode and a negative-norm one is essential for the emergence of complex modes. Based on the conclusion, we justify the two-mode approximation applied in our previous work [E. Fukuyama \textit{et al}., Phys. Rev. A {\bf 76}, 043608 (2007)], in which we analytically studied the condition for the existence of complex modes when the condensate has a highly quantized vortex.Comment: 7page

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