Relativistic linear stability equations for the nonlinear Dirac equation in Bose-Einstein condensates

L. H. Haddad^a and L. D. Carr^b

Department of Physics, Colorado School of Mines - Golden, CO 80401, USA

^a haddad@mines.edu
^b lcarr@mines.edu

Received: 8 January 2011
Accepted: 19 April 2011

Abstract

We present relativistic linear stability equations (RLSE) for quasi-relativistic cold atoms in a honeycomb optical lattice. These equations are derived from first principles and provide a method for computing stabilities of arbitrary localized solutions of the nonlinear Dirac equation (NLDE), a relativistic generalization of the nonlinear Schrödinger equation. We present a variety of such localized solutions: skyrmions, solitons, vortices, and half-quantum vortices, and study their stabilities via the RLSE. When applied to a uniform background, our calculations reveal an experimentally observable effect in the form of Cherenkov radiation.