Quasiparticle projection method for dynamically unstable Bose-Einstein condensates

TL;DR

This paper introduces a generalized quasiparticle projection method for analyzing the time evolution of dynamically unstable Bose-Einstein condensates, extending previous techniques to complex spectra and unstable regimes.

Contribution

It develops a biorthogonal basis for time-dependent Bogoliubov analysis, enabling accurate mode decomposition in unstable condensates, which was not possible with traditional methods.

Findings

Successfully applied to a 1D attractive condensate

Revealed evolution into localized structures from small perturbations

Extended Bogoliubov analysis to unstable regimes

Abstract

We present a general formalism for performing a time-dependent Bogoliubov analysis of a dynamically unstable Bose-Einstein condensate, which extends the quasiparticle projection method of Morgan et al. [Phys. Rev. A 57, 3818 (1998)] to cases with a complex spectrum. By introducing the proper left eigenvectors associated with each regime, we construct a biorthogonal basis. While the usual Bogoliubov normalization $\langle u | u \rangle - \langle v | v \rangle = 1$ may not hold in this basis, it still allows for a complete mode decomposition and an accurate reconstruction of arbitrary perturbations over time. This approach extends the applicability of the Bogoliubov framework beyond the stable regime, providing a consistent analysis of the time evolution of unstable condensates. As a proof of concept, we apply the method to a one-dimensional condensate with attractive interactions, which is dynamically unstable and evolves into nonstationary localized structures seeded by small perturbations.