Nonlinear Dirac equation in Bose-Einstein condensates: Preparation and stability of relativistic vortices

Abstract

We propose a detailed experimental procedure for preparing relativistic vortices, governed by the nonlinear Dirac equation, in a two-dimensional Bose-Einstein condensate (BEC) in a honeycomb optical lattice. Our setup contains Dirac points, in direct analogy to graphene. We determine a range of practical values for all relevant physical parameters needed to realize relativistic vortices in a BEC of 87Rb atoms. Seven distinct vortex types, including Anderson-Toulouse and Mermin-Ho skyrmion textures and half-quantum vortices, are obtained, and their discrete spectra and stability properties are calculated in a weak harmonic trap. We predict that most vortices are stable, with a lifetime between 1 and 10 s.

Stability of Solitary Waves and Vortices in a 2D Nonlinear Dirac Model

“…The presence of a mass in our case allows not only a direct comparison with NLS (when ω → m ≡ 1), but also generates fundamental differences between our results and those of Refs. [25,26,42], as discussed below as well.…”

“…Thirdly, and perhaps most importantly, NLD starts emerging in physical systems that arise in a diverse set of contexts of considerable interest. These contexts include, in particular, bosonic evolution in honeycomb lattices [25,26] and a growing class of atomically thin 2D Dirac materials [27], such as graphene, silicene, germanene, and transition metal dichalcogenides [28] (notice that in this Letter, we use nD to refer to n spatial dimensions). Recently, the physical aspects of nonlinear optics, such as light propagation in honeycomb photorefractive lattices (the socalled photonic graphene) [29,30], have prompted the consideration of intriguing dynamical features, e.g., conical diffraction in 2D honeycomb lattices [31].…”

“…For this reason, we have confirmed our stability conclusions by comparison with those emerging from the model for square binary waveguides [40], which leads to a massive nonlinear Dirac equation with the same nonlinearity as in Refs. [25,26].…”

“…Both solitary waves and vortex solutions exist in the frequency interval ω ∈ ð0; m ¼ 1Þ, a feature critically distinguishing our models from those of Refs. [25,26]. An intriguing feature of the relevant waveforms is that both the radial profile of the solitary waves and that of the vortices possess a maximum that shifts from r ¼ 0 to a larger r when ω approaches zero (see Fig.…”

“…In that connection, we highlight that although our models of choice may bear a particular nonlinearity, our results suggest that under different nonlinearities including the more physically relevant ones of, e.g., Refs. [25,26] and massive models [40], they still bear stable solitary waves for a suitable wide parametric range of frequencies. Thus, the conclusion of higher-dimensional stability is more general than the specifics of our particular nonlinearity and hence of broad interest.…”