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

TL;DR

This paper analytically derives the condition for dynamical instability in Bose-condensed systems, showing that degeneracy between specific eigenmodes leads to complex eigenvalues, and justifies a two-mode approximation used in prior work.

Contribution

It provides an analytical condition for the emergence of complex eigenvalues in the Bogoliubov-de Gennes equations, emphasizing the role of mode degeneracy and validating a previous two-mode approximation.

Findings

Degeneracy between positive- and negative-norm eigenmodes is essential for complex eigenvalues.

Analytical condition for dynamical instability in Bose condensates.

Justification of the two-mode approximation for vortex states.

Abstract

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.