L. H. Haddad

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

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 Schrodinger 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.

The nonlinear Dirac equation in Bose–Einstein condensates: I. Relativistic solitons in armchair nanoribbon optical lattices

We present a thorough analysis of soliton solutions to the quasi-one-dimensional (quasi-1D) nonlinear Dirac equation (NLDE) for a Bose?Einstein condensate in a honeycomb lattice with armchair geometry. Our NLDE corresponds to a quasi-1D reduction of the honeycomb lattice along the zigzag direction, in direct analogy to graphene nanoribbons. Excitations in the remaining large direction of the lattice correspond to the linear subbands in the armchair nanoribbon spectrum. Analytical as well as numerical soliton Dirac spinor solutions are obtained. We analyze the solution space of the quasi-1D NLDE by finding fixed points, delineating the various regions in solution space, and through an invariance relation which we obtain as a first integral of the NLDE. We obtain spatially oscillating multi-soliton solutions as well as asymptotically flat single soliton solutions using five different methods: by direct integration; an invariance relation; parametric transformation; a series expansion; and by numerical shooting. By tuning the ratio of the chemical potential to the nonlinearity for a fixed value of the energy?momentum tensor, we can obtain both bright and dark solitons over a nonzero density background.

The nonlinear Dirac equation in Bose-Einstein condensates: Vortex solutions and spectra in a weak harmonic trap

We analyze the vortex solution space of the

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

(2 +1)

The nonlinear Dirac equation in Bose–Einstein condensates: II. Relativistic soliton stability analysis

-dimensional nonlinear Dirac equation for bosons in a honeycomb optical lattice at length scales much larger than the lattice spacing. Dirac point relativistic covariance combined with s-wave scattering for bosons leads to a large number of vortex solutions characterized by different functional forms for the internal spin and overall phase of the order parameter. We present a detailed derivation of these solutions which include skyrmions, half-quantum vortices, Mermin-Ho and Anderson-Toulouse vortices for vortex winding

Lightspeed at a Snail's Pace: Relativity Meets Ultracold Physics

\ell = 1

The nonlinear Dirac equation in Bose–Einstein condensates: Foundation and symmetries

. For

The nonlinear Dirac equation in Bose–Einstein condensates: superfluid fluctuations and emergent theories from relativistic linear stability equations

\ell \ge 2

The nonlinear Dirac equation in Bose-Einstein condensates: Relativistic solitons in armchair and zigzag geometries

we obtain topological as well as non-topological solutions defined by the asymptotic radial dependence. For arbitrary values of

The nonlinear Dirac equation: Relativistic vortices and experimental realization in Bose-Einstein condensates

\ell