Hydrodynamic Description of Spin-1 Bose-Einstein Condensates

We establish a complete set of hydrodynamic equations for a spin-1 Bose-Einstein condensate (BEC), which are equivalent to the multi-component Gross-Pitaevskii equations and expressed in terms of only observable physical quantities: the spin density and the nematic (or quadrupolar) tensor in addition to the density and the mass current that appear in the hydrodynamic description of a scalar BEC. The obtained hydrodynamic equations involve a generalized Mermin-Ho relation that is valid regardless of the spatiotemporal dependence of the spin polarization. Low-lying collec- tive modes for phonons and magnons are reproduced by linearizing the hydrodynamic equations. We also apply the single-mode approximation to the hydrodynamic equations and find a complete set of analytic solutions.