Exact nonlinear dynamics of Spinor BECs applied to nematic quenches

In this thesis we study the nonlinear dynamics of spin-1 and spin-2 Bose-Einstein condensates, with particular application to antiferromagnetic systems exhibiting nematic (beyond magnetic) order. Firstly, we give a derivation of the spinor energy functionals with a focus on the connections between the nonlinear terms. We derive a hierarchy of nonlinear irreducible multipole observables sensitive to different levels of nematic order, and explore the various nematic states in terms of their multipolar order, representations of their symmetries, and topological defects. We then develop an exact solution to the nonlinear dynamics of spinor Bose-Einstein condensates. We use this solution to construct efficient and accurate numerical algorithms to evolve the spinor Gross-Pitaevskii equation in time. We demonstrate the advantages of our algorithms with several 1D numerical test problems, comparing with existing methods in the literature. We apply our numerical methods to simulating quenches of the condensate between various antiferromagnetic phases for spin-1 and spin-2. For spin-1, we carry out quenches for a theoretical uniform system in 2D, and then specialize to the parameters used in a recent harmonically trapped experiment in 3D. We connect the long-time coarsening growth law of the relevant order parameter to the decay of half-quantum vortices, which are the relevant topological defects of the ground state. For the spin-2 system, we investigate a novel quench from two different quadrupolar-nematic phases to an octupolar-nematic “cyclic” phase which supports 1/3 fractional vortices. We develop appropriate order parameter observables which couple to the spin and superfluid currents generated by these defects, and show that a new growth law appears with exponent 1/3