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

TL;DR

This paper derives an analytic condition for the existence of complex modes in a trapped Bose-Einstein condensate with highly quantized vortices, confirming their presence through comparison with numerical results.

Contribution

It provides the first analytic expression for the condition of complex mode existence in such BECs using a two-mode approximation.

Findings

Complex modes always exist in highly quantized vortex BECs.

Analytic formula matches numerical results for winding numbers 2, 3, and 4.

Identifies quantum numbers associated with complex modes.

Abstract

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.